Animal Biodiversity and Conservation issue 27.1 (1) (2004) pp 1-296 by Museu Ciències Naturals de Barcelona MCNB

Formerly Miscel·lània Zoològica

2004

and

Animal Biodiversity Conservation 27.1

Dibuix de la coberta de Jordi Domènech Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de redacció / Secretaría de redacción / Editorial Office

Secretària de redacció / Secretaria de redacción / Managing Editor Montserrat Ferrer

Museu de Ciències Naturals de la Ciutadella Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail abc@mail.bcn.es

Consell assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Editors / Editores / Editors Pere Abelló Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Javier Alba–Tercedor Univ. de Granada, Granada, Spain Antonio Barbadilla Univ. Autònoma de Barcelona, Bellaterra, Spain Xavier Bellés Centre d' Investigació i Desenvolupament–CSIC, Barcelona, Spain Juan Carranza Univ. de Extremadura, Cáceres, Spain Luís Mª Carrascal Museo Nacional de Ciencias Naturales–CSIC, Madrid, Spain Michael J. Conroy Univ. of Georgia, Athens, USA Adolfo Cordero Univ. de Vigo, Vigo, Spain Mario Díaz Univ. de Castilla–La Mancha, Toledo, Spain Xavier Domingo–Roura Inst. de Recerca i Tecnologia Agroalimentàries, Cabrils, Spain Gary D. Grossman Univ. of Georgia, Athens, USA Damià Jaume IMEDEA–CSIC, Univ. de les Illes Balears, Spain Jordi Lleonart Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Jorge M. Lobo Museo Nacional de Ciencias Naturales–CSIC, Madrid, Spain Pablo J. López–González Univ de Sevilla, Sevilla, Spain Vicente M. Ortuño Univ. de Alcalá de Henares, Alcalá de Henares, Spain Miquel Palmer IMEDEA–CSIC, Univ. de les Illes Balears, Spain Francisco Palomares Estación Biológica de Doñana, Sevilla, Spain Francesc Piferrer Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Montserrat Ramón Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Ignacio Ribera Nacional de Ciencias Naturales–CSIC, Madrid, Spain Pedro Rincón Museo Nacional de Ciencias Naturales–CSIC, Madrid, Spain Alfredo Salvador Museo Nacional de Ciencias Naturales–CSIC, Madrid, Spain José Luís Tellería Univ. Complutense de Madrid, Madrid, Spain Francesc Uribe Museu de Ciències Naturals de la Ciutadella, Barcelona, Spain Consell Editor / Consejo editor / Editorial Board José A. Barrientos Univ. Autònoma de Barcelona, Bellaterra, Spain Jean C. Beaucournu Univ. de Rennes, Rennes, France David M. Bird McGill Univ., Québec, Canada Mats Björklund Uppsala Univ., Uppsala, Sweden Jean Bouillon Univ. Libre de Bruxelles, Brussels, Belgium Miguel Delibes Estación Biológica de Doñana–CSIC, Sevilla, Spain Dario J. Díaz Cosín Univ. Complutense de Madrid, Madrid, Spain Alain Dubois Museum national d’Histoire naturelle–CNRS, Paris, France John Fa Durrell Wildlife Conservation Trust, Jersey, United Kingdom Marco Festa–Bianchet Univ. de Sherbrooke, Québec, Canada Rosa Flos Univ. Politècnica de Catalunya, Barcelona, Spain Josep Mª Gili Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Edmund Gittenberger Rijksmuseum van Natuurlijke Historie, Leiden, The Netherlands Fernando Hiraldo Estación Biológica de Doñana–CSIC, Sevilla, Spain Patrick Lavelle Inst. Français de recherche scient. pour le develop. en cooperation, Bondy, France Santiago Mas–Coma Univ. de Valencia, Valencia, Spain Joaquín Mateu Estación Experimental de Zonas Áridas–CSIC, Almería, Spain Neil Metcalfe Univ. of Glasgow, Glasgow, United Kingdom Jacint Nadal Univ. de Barcelona, Barcelona, Spain Stewart B. Peck Carleton Univ., Ottawa, Canada Eduard Petitpierre Univ. de les Illes Balears, Palma de Mallorca, Spain Taylor H. Ricketts Stanford Univ., Stanford, USA Joandomènec Ros Univ. de Barcelona, Barcelona, Spain Valentín Sans–Coma Univ. de Málaga, Málaga, Spain Tore Slagsvold Univ. of Oslo, Oslo, Norway Animal Biodiversity and Conservation 27.1, 2004 © 2004 Museu de Ciències Naturals de la Ciutadella, Institut de Cultura, Ajuntament de Barcelona Autoedició: Montserrat Ferrer Fotomecànica i impressió: Sociedad Cooperativa Librería General ISSN: 1578–665X Dipòsit legal: B–16.278–58 The journal is freely available online at: http://www.bcn.cat/ABC

Animal Biodiversity and Conservation 27.1 (2004)

Actes del Congrès Internacional EURING 2003, Alemanya / Actas del Congreso Internacional EURING 2003, Alemania / Proceedings of the Internatioanl Conference EURING 2003, Deutschland

Editors executius / Editores ejecutivos / Executive Editors Joan Carles Senar Museu Ciències Naturals de la Ciutadella, Barcelona, Spain André Dhondt Cornell Univ., Ithaca, U.S.A. Michael J. Conroy Univ. of Georgia, Athens, U.S.A. Secretària de redacció / Secretaria de redacción / Managing Editor Montserrat Ferrer Museu Ciències Naturals de la Ciutadella, Barcelona, Spain Editors / Editores / Editors Evolutionary Biology Session Charles R. Brown Univ. of Tulsa, Tulsa, U.S.A. David L. Thomson Max Planck Inst. For Demographic Research, Rostock, Germany Random Effects Session Kenneth P. Burnham Colorado State Univ., Fort Collins, U.S.A. Multistate Models Session Neil Arnason Univ. of Manitoba, Winnipeg, Canada Emmanuelle Cam Univ. Paul Sabatier, Toulouse, France Methodological Advances Session Jean–Dominique Lebreton CEFE / CNRS, Montpellier, France Kenneth H. Pollock North Carolina State Univ., U.S.A. Computing and Softward Session Jim Hines Patuxent Wildlife Research Center, Laurel, Maryland, U.S.A. Gary C. White Colorado State Univ., Fort Collins, U.S.A. Decision Making Session Michael J. Conroy Univ. of Georgia, Athens, U.S.A. Danny C. Lee Pacific Southwest Res. Station, Arcata, U.S.A. The Analysis of Movement: Migration and Dispersal Franz Bairlein Inst. of Avian Research, Wilhelmshaven, Deutchland Carl J. Schwarz Simon Fraser Univ., Burnaby, Canada The Analysis of Data from Ringing Schemes Stephen Baillie BTO The Nunnery, Norfolk, U.K. Paul Doherty Colorado State Univ., Fort Collins, U.S.A. Population Estimation and Conservation Darryl MacKenzie Proteus Research & Consulting ltd., Dunedin, New Zealand James D. Nichols Patuxent Wildlife Research Center, Laurel, U.S.A. Population Dynamics, Wave and Cycle André Dhondt Cornell Univ., Ithaca, U.S.A. Evan Cooch Cornell Univ., Ithaca, U.S.A.

Poster Session Charles M. Francis Canadian Wildlife Service, Ontario, Canada EURING Short Course Evan Cooch Cornell Univ., Ithaca, U.S.A.

"La tortue greque" Oeuvres du Comte de Lacépède comprenant L'Histoire Naturelle des Quadrupèdes Ovipares, des Serpents, des Poissons et des Cétacés; Nouvelle édition avec planches coloriées dirigée par M. A. G. Desmarest; Bruxelles: Th. Lejeuné, Éditeur des oeuvres de Buffon, 1836. Pl. 7

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Editors / Editores / Editors Antonio Barbadilla Univ. Autònoma de Barcelona, Bellaterra, Spain Xavier Bellés Centre d' Investigació i Desenvolupament CSIC, Barcelona, Spain Juan Carranza Univ. de Extremadura, Cáceres, Spain Luís Mª Carrascal Museo Nacional de Ciencias Naturales CSIC, Madrid, Spain Adolfo Cordero Univ. de Vigo, Vigo, Spain Mario Díaz Univ. de Castilla–La Mancha, Toledo, Spain Xavier Domingo Univ. Pompeu Fabra, Barcelona, Spain Francisco Palomares Estación Biológica de Doñana, Sevilla, Spain Francesc Piferrer Inst. de Ciències del Mar CSIC, Barcelona, Spain Ignacio Ribera The Natural History Museum, London, United Kingdom Alfredo Salvador Museo Nacional de Ciencias Naturales, Madrid, Spain José Luís Tellería Univ. Complutense de Madrid, Madrid, Spain Francesc Uribe Museu de Zoologia de Barcelona, Barcelona, Spain Consell Editor / Consejo editor / Editorial Board José A. Barrientos Univ. Autònoma de Barcelona, Bellaterra, Spain Jean C. Beaucournu Univ. de Rennes, Rennes, France David M. Bird McGill Univ., Québec, Canada Mats Björklund Uppsala Univ., Uppsala, Sweden Jean Bouillon Univ. Libre de Bruxelles, Brussels, Belgium Miguel Delibes Estación Biológica de Doñana CSIC, Sevilla, Spain Dario J. Díaz Cosín Univ. Complutense de Madrid, Madrid, Spain Alain Dubois Museum national d’Histoire naturelle CNRS, Paris, France John Fa Durrell Wildlife Conservation Trust, Trinity, United Kingdom Marco Festa–Bianchet Univ. de Sherbrooke, Québec, Canada Rosa Flos Univ. Politècnica de Catalunya, Barcelona, Spain Josep Mª Gili Inst. de Ciències del Mar CMIMA–CSIC, Barcelona, Spain Edmund Gittenberger Rijksmuseum van Natuurlijke Historie, Leiden, The Netherlands Fernando Hiraldo Estación Biológica de Doñana CSIC, Sevilla, Spain Patrick Lavelle Inst. Français de recherche scient. pour le develop. en cooperation, Bondy, France Santiago Mas–Coma Univ. de Valencia, Valencia, Spain Joaquín Mateu Estación Experimental de Zonas Áridas CSIC, Almería, Spain Neil Metcalfe Univ. of Glasgow, Glasgow, United Kingdom Jacint Nadal Univ. de Barcelona, Barcelona, Spain Stewart B. Peck Carleton Univ., Ottawa, Canada Eduard Petitpierre Univ. de les Illes Balears, Palma de Mallorca, Spain Taylor H. Ricketts Stanford Univ., Stanford, USA Joandomènec Ros Univ. de Barcelona, Barcelona, Spain Valentín Sans–Coma Univ. de Málaga, Málaga, Spain Tore Slagsvold Univ. of Oslo, Oslo, Norway

Animal Biodiversity and Conservation 24.1, 2001 © 2001 Museu de Zoologia, Institut de Cultura, Ajuntament de Barcelona Autoedició: Montserrat Ferrer Fotomecànica i impressió: Sociedad Cooperativa Librería General ISSN: 1578–665X Dipòsit legal: B–16.278–58

III

Animal Biodiversity and Conservation 27.1 (2004)

Índex / Índice / Contents

1–2 Senar, J. C., Dhondt, A. A. & Conroy, M. J. The quantitative study of marked individuals in ecology, evolution and conservation biology: a foreword to the EURING 2003 Conference

Multi–state Models Session

3–19 Honour Speaker Nichols, J. D. Evolution of quantitative methods for the study and management of avian populations: on the importance of individual contributions

97–107 Plenary paper Kendall, W. L. Coping with unobservable and mis–classified states in capture–recapture studies

Evolutionary Biology Session 21–22 Introductory paper Brown, C. R. & Thomson, D. L. Evolutionary biology and life histories 23–34 Plenary paper Brown, C. R. & Brown, M. B. Mark–recapture and behavioral ecology: a case study of Cliff Swallows 35–46 Reed, E. T., Gauthier, G. & Giroux, J.–F. Effects of spring conditions on breeding propensity of Greater Snow Goose females 47–58 Gaillard, J.–M., Viallefont, A., Loison, A. & Festa– Bianchet, M. Assessing senescence patterns in populations of large mammals 59–72 Marshall, M. R., Diefenbach, D. R., Wood, L. A. & Cooper, R. J. Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: study designs that may help

Random Effects Session 73–85 Schaub, M. & Lebreton, J.–M. Testing the additive versus the compensatory hypothesis of mortality from ring recovery data using a random effects model 87–91 Link, W. A. Individual heterogeneity and identifiability in capture–recapture models

93–95 Introductory paper Arnason, A. N. & Cam, E. Multi–state models: metapopulation and life history analyses

109–111 Barbraud, C. & Weimerskirch, H. Modelling the effects of environmental and individual variability when measuring the costs of first reproduction 113–128 Jamieson, L. E. & Brooks, S. P. Density dependence in North American ducks 129–131 Kéry, M. & Gregg, K. B. Demographic estimation methods for plants with dormancy 133–146 Senar, J. C. & Conroy, M. J. Multi–state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): the importance of estimating recapture rates

Methodological Advances Session 147–148 paper Lebreton, J.–D. & Pollock, K. H. Methodological advances

Introductory

149–155 Bonner, S. J. & Schwarz, C. J. Continuous time–dependent Individual covariates and the Cormack–Jolly–Seber model 157–173 Otis, D. L. & White, G. C. Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations

Computing and Software Session

Dispersal and Migration Session

175–176 Introductory paper White, G. C. & Hines, J. E. Computing and software

297–298 Introductory paper Schwarz, C. & Bairlein, F. Dispersal and migration

177–185 Plenary paper Barker, R. J. & White, G. C. Towards the mother–of–all–models: customised construction of the mark–recapture likelihood function

299–317 Plenary paper Siriwardena, G. M., Wernham, C. V. & Baillie, S. R. Quantifying variation in migratory strategies using ring–recoveries

187–205 Rotella, J. J., Dinsmore, S. J. & Shaffer, T. L. Modeling nest–survival data: a comparison of recently developed methods that can be implemented in MARK and SAS 207–215 Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O., Lebreton, J.–D. M–SURGE: new software specifically designed for multistate capture–recapture models 217–228 Efford, M. G., Dawson, D. K. & Robbins, C. S. Density: software for analysing capture–recapture data from passive detector arrays

Decision Analysis Session 229–230 Introductory paper Conroy, M. J. & Lee, D. C. Population dynamics and monitoring applied to decision–making 231–245 Plenary paper Haas, T. C. Ecosystem management via interacting models of political and ecological processes 247–266 Hoyle, S. D. & Maunder, M. N. A Bayesian integrated population dynamics model to analyze data for protected species 267–281 Fonnesbeck, C. J. & Conroy, M. J. Application of integrated Bayesian modeling and Markov chain Monte Carlo methods to the conservation of a harvested species 283–285 Drechsler, M. & Wätzold, F. A decision model for the efficient management of a conservation fund over time 287–296 Moore, C. T. & Kendall, W. L. Costs of detection bias in index–based population monitoring

319–329 Thorup, K. & Rahbek, C. How do geometric constraints influence migration patterns? 331–341 Drake, K. L. & Alisauskas, R. T. Breeding dispersal by Ross’s geese in the Queen Maud Gulf metapopulation 343–354 King, R. & Brooks, S. P. Bayesian analysis of the Hector's Dolphin data 355–356 Sokolov, L., Chernetsov, N.,Kosarev, K., Leoke, D., Markovets, M., Tsvey, Q. & Shapoval, A. Spatial distribution of breeding Pied Flycatchers Ficedula hypoleuca in respect to their natal sites 357–368 Lokki, H. & Saurola, P. Comparing timing and routes of migration based on ring encounters and randomization methods 369– 370 Traylor, J. J., Alisauskas, R. T. & Kehoe, F. P. Multistate modeling of brood amalgamation in White–winged Scoters Melanitta fusca deglandi

Analysis of Ringing Data Session 371–373 Introductory paper Baillie, S. R. & Doherty, P. F. Analysis using large–scale ringing data 375–386 Plenary paper Royle, J. A. Generalized estimators of avian abundance from count survey data 387–402 Doligez, B., Thomson, D. L. & van Noordwijk, A. J. Using large–scale data analysis to assess life history and behavioural traits: the case of the reintroduced White stork Ciconia ciconia population in the Netherlands

Animal Biodiversity and Conservation 27.1 (2004)

403–415 Saurola, P. & Francis, C. Estimating population dynamics and dispersal distances of owls from nationally coordinated ringing data in Finland

471–488 Plenary paper Caswell, H. & Fujiwara, M. Beyond survival estimation: mark–recapture, matrix population models, and population dynamics

417–426 Julliard, R. Estimating the contribution of survival and recruitment to large scale population dynamics

489–502 Francis, C. M. & Saurola, P. Estimating components of variance in demographic parameters of Tawny Owls, Strix aluco

427–435 Yoccoz, N. G. & Ims, R. A. Spatial population dynamics of small mammals: some methodological and practical issues

503–514 Gauthier, G. & Lebreton, J.–D. Population models for Greater Snow Geese: a comparison of different approaches to assess potential impacts of harvest

Abundance Estimation and Conservation Biology Session

515–529 Brooks, S. P., King, R. & Morgan, B. J. T. A Bayesian approach to combining animal abundance and demographic data

437–439 Nichols, J. D. & MacKenzie, D. I. Abundance estimation and Conservation Biology 441–449 Plenary paper Link, W. A. & Barker, R. J. Hierarchial mark–recapture models: a framework for inference about demographic processes 451–460 Alisauskas, R. T., Traylor, J. J., Swoboda, C. J. & Kehoe, F. P. Components of population growth rate for White–winged Scoters in Saskatchewan, Canada 461–467 Mackenzie, D. I. & Nichols, J. D. Occupancy as a surrogate for abundance estimation

Population Dynamics 469–470 Cooch, E. G. & Dhondt, A. A. Population dynamics

Posters Session 531–540 Schreiber, E. A., Doherty, P. F. Jr. & Schenk, G. A. Dispersal and survival rates of adult and juvenile Red–tailed tropicbirds (Phaethon rubricauda) exposed to potential contaminants 541–548 Frederiksen, M., Wanless, S. & Harris, M. P. Estimating true age–dependence in survival when only adults can be observed: an example with Black–legged Kittiwakes 549–560 Lahoreau, G., Gignoux, J. & Julliard, R. Exploring mark–resighting–recovery models to study savannah tree demographics

EURING Short Course 561–572 Gimenez, O., Viallefont, A., Catchpole, E. A., Choquet, R. & Morgan, B. J. T. Methods for investigating parameter redundancy

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

The quantitative study of marked individuals in ecology, evolution and conservation biology: a foreword to the EURING 2003 Conference J. C. Senar, A. A. Dhondt & M. J. Conroy

Senar, J. C., Dhondt, A. A. & Conroy, M. J., 2004. The quantitative study of marked individuals in ecology, evolution and conservation biology: a foreword to the EURING 2003 Conference. Animal Biodiversity and Conservation, 27.1: Few fields in modern ecology have developed as fast as the analysis of marked individuals in the study of wild animal populations (Seber & Schwarz, 2002). This is the topic of EURING Conferences, which from 1986 have been the premier forum for advances in capture–recapture methodology. In this sense, EURING Conferences still maintain the flavour that originally inspired scientific meetings: to disseminate the very last findings, ideas and results on the field. Traditionally, EURING Conferences have been published in the form of Proceedings, which because of their relevant content, become a required reading to anyone interested in the capture–recapture methodology. EURING 2003 was held in Radolfzell (Germany), hosted by the Max Planck Research Centre for Ornithology, and the Proceedings appear as a special issue of Animal Biodiversity and Conservation. The full title of the 2003 meeting was "The quantitative study of marked individuals in ecology, evolution and conservation biology", which stands for one of the main aims of the meeting: to establish the capture-recapture approach as one of the standard methodologies in studies within these fields. One of the shared views is that capture–recapture methodologies have reached a considerable maturity, but the need still exists to spread their use as a "standard" methodology. The nice review paper by Lebreton et al. (1993) in Trends in Ecology and Evolution is still applicable, in that general ecologists and evolutionary biologists still resist their general use. The same applies to conservation biology, where the analysis of marked individuals may also be a key tool in its development. We hope, with the spread of 2003 Proceedings, to help to fill this gap. The Proceedings follow the same general structure as the Conference. We organised the EURING meeting in 10 technical sessions, covering what we considered as fastest growing areas in the field. We appointed for each session, two chairs, which were charged with selecting 4–7 talks on the topic of their session. Each session additionally included a plenary conference intended to summarise or to provide a general but synthetic flavour of the topic. As a novelty in EURING conferences, we asked session chairs to include at least one talk dealing with study species other than birds. This is the result of a heated but fruitful discussion at EURING 2000 in Point Reyes, and fits with the general aim to spread the capture– recapture methodology beyond zoological groups: although EURING as an organization, deals with birds, and conferences have traditionally focused on this group, the capture–recapture approach is becoming a standard way to address biologically relevant questions on populations and individuals (Schwarz, 2002), for any zoological group. This volume, contains several nice examples of taxa other than birds. As far as possible, we selected chairs so that each session was delineated with a good balance between the biological and the statistician emphasis. This balance has in fact characterised EURING conferences, which in addition to the workshop atmosphere always present, has lead to very fruitful exchanges. Session

Juan Carlos Senar, Museu de Ciències Naturals, Psg. Picasso s/n., Parc de la Ciutadella, 08003 Barcelona, Spain. E–mail: jcsenar@mail.bcn.es André Dhondt, Cornell Univ., 159 Sapsucker Woods Road, Ithaca, NY 14850, U.S.A. E–mail: aad4@cornell.edu Michael J. Conroy, USGS, Georgia Cooperative Fish and Wildlife Research Unit, Univ. of Georgia, Athens, GA, U.S.A. E–mail: conroy@fisher.forestry.uga.edu ISSN: 1578–665X

Senar et al.

chairs were also asked to act as editors for the papers within their session. All the papers were hence subjected to peer review, as in any other issue of Animal Biodiversity and Conservation, and presentation of the paper in the Conference did not assure publication in the Proceedings. This has lead to an even higher quality of the papers presented at the Conference. Editors were additionally asked to write a short summary on their session. Given that these summaries also present the views of the Editors on the different topics presented, we have preferred each introduction to appear as a short paper in the front of each one of the sessions, so that it can be cited as a regular paper. The Proceedings start with the Honour Speaker Talk by James Nichols (Nichols, 2004). This talk is traditionally the last one in the Conference, but we think that it nicely summarises how and why capture– recapture has developed to its current healthy state. The talk is in fact a tribute to David Anderson, to whom, as Nichols says, all of us are more or less in debt. Hence, we have preferred to move the Honour Talk to the front position of the Proceedings, and we would like this to be our humble tribute to David. At the end of the Proceedings appear a few papers which were presented in poster format, and a paper summarising several of the main topics presented at the traditional short course on capture–recapture, this time organized by the unflagging Evan Cooch. We would like to thank all the people who helped in one way or another to the successful completion of the EURING Conference and the Proceedings. We thank to the Session Chairs, their dedication and enthusiasm in organizing the sessions and also in editing the different papers. All their names appear in the front page of the Proceedings as credits. We thank Wolfgang Fiedler for the local organization of the event: a very difficult and exhausting task that is not always properly recognized. Jean Clobert, although unfortunately unable to attend the Conference, supported us with ideas and friendship meanwhile preparing the scientific program. Evan Cooch maintained the always successful web page (which probably will also become a classic in EURING conferences…), and organized the traditional course on capture–recapture. Charles Francis very efficiently organized the poster session and acted as editor for the papers sent for publication. Finally we thank the Ministerio de Ciencia y Tecnología for financial support to the publication of this special issue of Animal Biodiversity and Conservation (B.O.S. 2002–12283–E) and to the Natural History Museum of Barcelona for their support. References Lebreton, J. D., Pradel, R. & Clobert, J., 1993. The statistical analysis of survival in animal populations. Trends in Ecology and Evolution, 8: 91–95. Nichols, J. D., 2004. Evolution of quantitative methods for the study and management of avian populations: on the importance of individual contributions. Animal Biodversity and Conservation, 27.1: 3–19. Schwarz, C. J. 2002. Real and quasi–experiments in capture–recapture studies. Journal of Applied Statistics, 29: 459–473. Seber, G. A. F. & Schwarz, C. J., 2002. Capture–recapture: before and after EURING 2000. Journal of Applied Statistics, 29: 1–4.

Animal Biodiversity and Conservation 27.1 (2004)

Evolution of quantitative methods for the study and management of avian populations: on the importance of individual contributions J. D. Nichols

Nichols, J. D., 2004. Evolution of quantitative methods for the study and management of avian populations: on the importance of individual contributions. Animal Biodversity and Conservation, 27.1: 3–19. Abstract Evolution of quantitative methods for the study and management of avian populations: on the importance of individual contributions.— The EURING meetings and the scientists who have attended them have contributed substantially to the growth of knowledge in the field of estimating parameters of animal populations. The contributions of David R. Anderson to process modeling, parameter estimation and decision analysis are briefly reviewed. Metrics are considered for assessing individual contributions to a field of inquiry, and it is concluded that Anderson’s contributions have been substantial. Important characteristics of Anderson and his career are the ability to identify and focus on important topics, the premium placed on dissemination of new methods to prospective users, the ability to assemble teams of complementary researchers, and the innovation and vision that characterized so much of his work. The paper concludes with a list of interesting current research topics for consideration by EURING participants. Key words: Animal population dynamics, David R. Anderson, EURING meetings, Individual contributions, Management, Science. Resumen Evolución de los métodos cuantitativos para el estudio y gestión de las poblaciones de aves: sobre la importancia de las contribuciones individuales.— Los congresos EURING y los científicos que asistieron a los mismos han contribuido de forma significativa al aumento de conocimientos en el campo de la estimación de parámetros de las poblaciones animales. En el presente estudio se revisan brevemente las aportaciones de David R. Anderson a la modelación de procesos, la estimación de parámetros y el análisis de toma de decisiones. Se consideran los distintos modos en que se puede cuantificar la contribucion realizada por un investigador al desarrollo de un campo de la ciéncia, llegándose a la conclusión de que las contribuciones de Anderson han resultado fundamentales. De entre las destacadas características de Anderson y su carrera cabe mencionar su capacidad para identificar temas clave y centrarse en los mismos, la importancia que le da a la diseminación de los nuevos métodos entre los posibles usuarios, la capacidad para formar equipos de investigadores complementarios, y la innovación y visión estratégica que caracterizan gran parte de su trabajo. El estudio concluye con una interesante lista de temas de investigación actuales para que los participantes de EURING tomen en consideración. Palabras clave: Dinámica poblacional de los animales, David R. Anderson, Reuniones de EURING, Contribuciones individuales, Gestión, Ciencia. James D. Nichols, Patuxent Wildlife Research Center, Laurel, Maryland 20708–4017, U.S.A.

ISSN: 1578–665X

Nichols

Introduction At the first EURING meeting I was able to attend, held in Montpellier in 1992, George M. Jolly (1993) presented the introductory lecture on the topic of "Instinctive Statistics". In this lecture, he reviewed the contributions to biometry in general, and to studies of marked animals in particular, of four "original thinkers", A. Quetelet, C. H. N. Jackson, R. A. Fisher, and P. H. Leslie. In more recent history, and especially over the last two decades, participants in the EURING conferences have included many of the original thinkers and primary contributors to methods for studying population dynamics using marked animals. In fact, these participants, and the EURING conferences themselves, have played an important role in the rapid growth in capture–recapture and band recovery methodologies over the last two decades. This year marked the retirement of one such participant, Dr. David R. Anderson. Although it is extremely unlikely that his contributions will end with his retirement, this event does provide an appropriate motivation for reviewing his contributions to our field. In this paper, I will thus try to trace the modern evolution of studies of band recovery and capture– recapture methods through the contributions of David Anderson, one of the truly original thinkers in our field. As strongly emphasized in several of the presentations in EURING–03, views of estimation for population dynamics are expanding to include integration of marked–animal data with observation–based approaches. Thus, the contributions of Anderson to perhaps the most important observation–based estimation approach, distance sampling, will also be briefly reviewed. Throughout this methodological review, I will emphasize a general approach developed by Anderson to introduce biologists and managers to new classes of methods and to expedite assimilation of such methods into the fields of population ecology and management. I also emphasize the Anderson theme of striving to permit flexibility, as provided by multiple models, and his efforts to confront the resulting problem of model selection. I note that although EURING conferences have focused on estimation issues, it is important to recall that estimation is not a "stand–alone" activity or an inherently useful endeavor and attains value primarily in the context of a larger process, such as science or management. Population modeling can be viewed as a class of methods required for the conduct of science and management, and decision–theoretic and optimization methods are also essential to informed management. Anderson has made important contributions to both of these classes of methods, and these will be briefly reviewed as well. I briefly explore the nature of individual contributions to a field and discuss the personal and research attributes that separate the exceptional contributors from the rest of us. In particular, I propose hypotheses about characteristics of Anderson’s research program that set it

apart from other programs. Finally, I offer some opinions about the current state of our field and note some of the opportunities and possibilities that seem to be especially useful and exciting. Band recovery models The use of data from individually banded birds that have been recovered as shot or found dead have a long history of use for estimation of annual survival and, for hunted species, indices to hunting mortality rate (see reviews in Lebreton, 2001; Nichols & Tautin, in press). Modern approaches to estimation are often traced back to Haldane (1953, 1955), with the first general (permitting time–specific estimates), stochastic models attributed to Seber (1970) and Robson & Youngs (1971). When these latter two papers appeared, Anderson was working with the U.S. Fish and Wildlife Service studying the population dynamics of the mallard Anas platyrynchos in North America. He immediately recognized the utility of these new models and wanted to extend them to multiple age classes. Thus, he developed software for implementing the Seber–Robson–Youngs model (Anderson et al., 1974) and funded D. S. Robson and C. Brownie, at Cornell University, to develop age– specific extensions. Anderson and colleague K. P. Burnham, together with Brownie and Robson, produced an impressive body of methodological work, much of which was incorporated into the monograph, "Statistical Inference from Band Recovery Data – A Handbook" (Brownie et al., 1978). This monograph described a set of models to be fit to band recovery data in order to draw inferences about survival rates and band recovery rates (indices to harvest rates). Estimators and associated variances and covariances were presented for models permitting closed form expressions, and computer programs were described for computing estimates and test statistics for goodness–of–fit and between–model tests. The software corresponded to single–age (ESTIMATE) and 2–age (BROWNIE) models designed for data from a single banding period each year. However, the monograph also described models for 3 age classes and models for designs with two banding periods per year. Test statistics were recommended for drawing inferences about sources of variation in underlying survival and recovery rate parameters, and a chapter was devoted to the design of studies, including necessary sample sizes. The Brownie et al. (1978; 2nd addition in 1985) handbook is simply the most influential publication ever written on band recovery models for drawing inference about avian survival and recovery rates. Beyond that distinction, I believe it to have been a landmark publication for two other reasons. The first reason is the suggestion that multiple models be fit to the same data sets. The second reason is that the monograph provided a model for the introduction of statistical inference methods to biologists and managers.

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With only a few exceptions (e.g., Darroch, 1958), the history of parameter estimation for animal populations prior to the work of Brownie et al. (1978) had been characterized by the development of single, general models. General models were a focus because of the greater likelihood that model assumptions would be met, so such models were viewed as most likely to be useful. The basic, general models in the sets introduced by Brownie et al. (1978) included time–specific survival and recovery rates for 1–age (Seber, 1970; Robson & Youngs, 1971) and 2–age (Brownie & Robson, 1976) data. The model sets also included even more general models in which adult recovery rates the first year after banding differed from adult recovery rates in subsequent years following banding. These models were developed to incorporate the biological realism of high recovery rates following initial banding, a phenomenon encountered by Anderson in his analyses of mallard data sets. The model sets also included models in which parameters (survival and/or recovery rates) were constrained to be constant over time. The consideration of multiple models leads to the question of how to select the "best" model for a given data set. Brownie et al. (1978) presented a clear discussion of the principle of parsimony, recommending use of the simplest model that fits the data adequately. All of the models in their model sets were nested, so Brownie et al. (1978) recommended using sequential hypothesis–testing as a selection procedure. The user began with the most general model and assessed model fit. Conditional on fit of the general model, a likelihood ratio test statistic can be computed between the general model and the next most general model, and this procedure can be repeated for all neighboring pairs of models in the sequence going from most general to most restrictive. If a likelihood ratio test between two models is judged to be "significant", then it is concluded that the extra parameters of the more general model are needed to describe the variation in the data (i.e., the simpler model is not adequate). If the likelihood ratio test is not significant, then this is taken as evidence that the extra parameters of the more general model are not really needed, so the user selects the less general model for estimation, as associated estimates will be more precise than those of the more general model. In addition to introducing multiple models and the concept of model selection to biological readers, Brownie et al. (1978) provided an extremely successful model for the introduction of new statistical inference procedures to biologists and managers. An important component of the model was a monograph written for a biological readership. Although it was detailed and rigorous, Brownie et al. (1978) was not written in the terse manner typical of statistical contributions, but included descriptions designed to be understandable to biologists and wildlife managers. Another component was the presentation of model sets that provided the user with unprecedented flexibility. This flexibility included the ability to fit very

general models motivated by biological realism (the models with different direct and indirect recovery rates), as well as reduced–parameter models designed for parsimonious estimation. Perhaps the component of the Brownie et al. (1978) model most responsible for the rapid assimilation of the methods into biological research and management was the accompanying software. BROWNIE and ESTIMATE were user–oriented, and their output was described in comprehensive worked examples in the monograph itself. All estimates, summary statistics, and test statistics needed for inference were computed by the software, and it was not necessary for the user to perform any secondary computations. A final component of the model for introducing statistical methods to biologists and managers was the conduct of accompanying workshops and short courses. Anderson and Burnham recognized that the monograph contained many new concepts that might require additional explanation for some readers, so they taught a number of 2–5 day workshops for the purpose of introducing potential users to the models and underlying concepts of their methods. The workshops included multiple computer exercises designed to familiarize attendees with software use. As will be noted subsequently, Anderson and colleagues have used this basic model repeatedly for the introduction of new statistical estimation methodologies to biologists. Following the publication of Brownie et al. (1978), Anderson collaborated on papers that clearly demonstrated the superiority of these band recovery models to estimators based on different approaches that had dominated the literature prior to publication of the handbook (Anderson et al., 1981, 1985; Burnham & Anderson, 1979). He also worked on issues involving bias expected to result from failure of underlying model assumptions (Anderson & Burnham, 1980; Nelson et al., 1980). The next major step in the evolution of band recovery models was the development of software facilitating estimation under user–defined models (White, 1983; Conroy & Williams, 1984; Conroy et al., 1989). This step can be viewed as a logical extension of Anderson’s emphasis on modeling flexibility exemplified by the model sets of Brownie et al. (1978). Now, instead of the statistician providing model sets consisting of a relatively small number of models, the practitioner was given the ability to tailor models to specific situations and questions of interest. This ability included the fitting of ultrastructural models in which survival and recovery parameters were themselves modeled as functions of relevant covariates (North & Morgan, 1979). This emphasis on increased flexibility has continued and is now exemplified by program MARK (White & Burnham, 1999). The modeling flexibility made possible by developments in the 1980s has resulted in many useful extensions of band recovery models. For example, recovery rate estimates can be used to estimate harvest rates with the addition of information about band reporting rate. Such information can be ob-

tained from reward band studies (e.g., Henny & Burnham, 1976). Flexible modeling software permitted the development of models that could be used for direct estimation and inference about reporting and harvest rates (e.g., Conroy & Blandin, 1984; Conroy et al., 1989; Nichols et al., 1991, 1995; Pollock et al., 2001). Special models similar to those initially discussed by Brownie et al. (1978) were developed for hunted species that were banded at multiple times per year (Conroy & Williams, 1984; Conroy et al., 1989). Such models have been used to draw inferences about seasonal survival rates (e.g., Blohm et al., 1987), survival rates of young birds between hatch and the time of fledging (e.g., Hestbeck et al., 1989), and harvest mortality rates (Hearn et al., 1998). Band recovery models dealing with movement among banding and recovery strata have been developed (Schwarz et al., 1988, 1993). Important models have been developed for nonharvested species as well, including the special case in which all birds are banded as juveniles (e.g., Seber, 1971; North & Cormack, 1981; Catchpole et al., 1998) and the situation where numbers of banded birds are unknown (Burnham, 1990). This list of advances and extensions is not intended to be complete (see Williams et al., 2002 for other developments) but provides evidence of substantial development built upon the base provided by Brownie et al. (1978). Capture–recapture models Closed–population models David Anderson moved to the Cooperative Fish and Wildlife Research Unit at Utah State University in 1975 and began a project on capture–recapture models for estimating abundance when populations are closed to gains and losses between sampling periods. Collaborators on this project were D. L. Otis, K. P. Burnham and G. C. White, and the group recognized that there was an opportunity to provide a synthetic treatment of the different models and estimators that had been developed since the early work of such investigators as Petersen (1896), Lincoln (1930) and Schnabel (1938). Key efforts to be included in the synthesis were the constant and time–specific models for capture probability developed by Darroch (1958), the model and jackknife estimator for the case of individual heterogeneity in capture probability developed by Burnham & Overton (1978), and the synthetic treatment of sources of variation in capture probability, including trap response, found in K. H. Pollock’s (1974) Ph. D. Thesis. The monograph resulting from this work (Otis et al., 1978) provided an extremely valuable synthesis of models for closed populations. Instead of searching through various papers scattered throughout the literature, the biologist interested in these models could go to a single authoritative source. Otis et al. (1978) provided a conceptual framework for thinking

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of different models in terms of hypotheses about the sources of variation in capture probability that they incorporated. There was a constant capture probability model, models with single sources of variation (time, behavior, heterogeneity) and all possible combinations of sources. Estimation was not possible with all models, but they were all included in the model set, thus retaining Anderson’s emphasis on flexibility. Multiple models again led to the question of model selection, but this question could not be addressed using sequential hypothesis tests between nested models, because the closed population models were not all nested. Otis et al. (1978) thus developed a clever approach to the problem by simulating data under the various models, computing the goodness– of–fit and between–model tests that could be conducted, and using the probabilities associated with these tests to build a discriminant analysis classification function. The various test statistics are then computed for an actual data set and used in the discriminant function to compute a score that is the basis for model selection (Otis et al., 1978). The Otis et al. (1978) monograph adhered to the approach taken by Brownie et al. (1978). Although it provided statistical detail, it was written with a biological readership in mind. Otis et al. (1978) included numerous worked examples, as well as important sections on study design and sample size. The monograph was accompanied by program CAPTURE, user–oriented software that could be used to fit the models and provide estimators under most of the models. As with the band recovery methods, Anderson and colleagues conducted workshops introducing biologists and managers to the models and associated software. Apparently, some biologists found the monograph intimidating despite the authors’ efforts, so the authors wrote an additional monograph (White et al., 1982) explaining the general modeling and design concepts without the statistical detail of Otis et al. (1978). Once again Anderson and his group placed a premium on explaining the methods to a relatively naïve user group. The Otis et al. (1978) monograph represents another landmark publication, and although additional useful developments have followed its publication (Chao & Huggins, in press), these developments always refer back to the monograph framework. Subsequent work has included models for capture probability as functions of covariates (e.g., Pollock et al., 1984; Huggins, 1989, 1991). A great deal of effort has been expended on estimators for dealing with heterogeneity models. These efforts include different moment–based estimators (e.g., Pollock & Otto, 1983; Chao, 1987), estimators based on sample coverage (e.g., Chao et al., 1992; Lee & Chao, 1994), finite mixture models (e.g., Norris & Pollock, 1996; Pledger, 2000), and models based on continuous mixtures (Dorazio & Royle, 2003). Stanley & Burnham (1998) have investigated both model selection procedures and model– averaged estimators. Program CAPTURE has been updated to include some of these estimators

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(Rexstad & Burnham, 1991) and program MARK (White & Burnham, 1999) can be used to fit a variety of user–defined models. Open–population models Although David Anderson had used capture–recapture estimators for data from open populations (e.g., Anderson & Sterling, 1974), his first major effort to contribute to this methodology involved work on experimental data collected for the purpose of drawing inferences about survival. The emphasis was thus not simply on estimation, but on drawing inferences about the effects of specific treatment factors on short and long–term survival. Despite the more narrow focus than the band recovery and closed model monographs, Anderson recognized a need for synthesizing a variety of topics relevant to open–population capture–recapture models. Anderson and colleagues K. P. Burnham, G. C. White, C. Brownie, and K. H. Pollock built on the initial open capture–recapture models developed by Cormack (1964), Jolly (1965) and Seber (1965) and on subsequent work to assess model fit (Pollock et al., 1985) and to introduce age–specificity (Pollock, 1981), capture history dependence (Pollock, 1975; Sandland & Kirkwood, 1981; Brownie & Robson, 1983), and reduced–parameter models (Jolly, 1982; Brownie et al., 1986). Anderson’s vision of the kind of synthesis possible with open capture–recapture models was colored by his recognition that these models could be viewed as generalizations of band recovery models. The resulting monograph (Burnham et al., 1987) contains a nice development of theory underlying open–population capture–recapture models, including data structures, modeling, goodness–of–fit testing, and between–model testing. Important ancillary topics such as quasi–likelihood, variance components, and bias approximation are developed nicely as well, and the monograph is literally filled with useful methods and information for capture– recapture practitioners. The core of the monograph deals with a nested set of models that differ from each other in the number and kinds of parameters (time–specific survival and capture probabilities) that are shared by, as opposed to modeled separately for, two or more groups. Sequential likelihood ratio tests are used to address questions about treatment effects and also to select the most appropriate model for use with the data. This monograph once again followed the model of Brownie et al. (1978) in several important ways. It included statistical development, yet was written for a biological readership. It contained several worked examples and an important chapter on study design. User–oriented software, program RELEASE, accompanied the monograph and computed estimates, between–model test statistics and goodness–of–fit test statistics for the most general model. The experimental situation involved multiple groups (treatments), and modeling flexibility was provided by development of models representing

different groups of shared parameters. Model selection and tests for treatment effects were accomplished by sequential between–model tests. Anderson and colleagues once again held workshops on the use of program RELEASE to implement the models presented in the monograph. While completing the Burnham et al. (1987) monograph, Anderson became aware of flexible software, SURGE, developed by J.–D. Lebreton and colleague J. Clobert (e.g., Clobert & Lebreton, 1985), to implement user–defined open–population capture–recapture models. Early uses of SURGE emphasized the important ability to model survival and capture parameters as functions of time–specific covariates. Because of his emphasis on flexible modeling, Anderson was quite interested in this work to the extent that he and Burnham joined Lebreton and Clobert in a collaborative project on modeling capture–recapture data for open populations. This collaboration resulted in another landmark monograph that followed the Brownie et al. (1978) model in many ways. Lebreton et al. (1992) presented a thorough development of open–population capture–recapture models and data structures, with instructions about how to fit the models in SURGE. This software was not quite as flexible as SURVIV (White, 1983) but was more user friendly. Few limits were placed on the SURGE user, and a wide variety of user– defined models could be fit to the data. Because the models did not have to be nested, the problem of model selection became very important. Possible models were not specified a priori as with the closed models of Otis et al. (1978), so the discriminant function approach to model selection was not appropriate. Burnham and Anderson (1992) had been exploring the use of Akaike’s Information Criterion (AIC; Akaike, 1973) for the purpose of model selection and indeed recommended this approach in Lebreton et al. (1992). AIC treated model selection not as a problem in sequential hypothesis testing but as a direct optimization problem. The optimization criterion involved the magnitude of the likelihood and the number of model parameters, and was based on the principle of parsimony. The Lebreton et al. (1992) monograph was written for biologists and contained several worked examples with actual data sets. All of the authors were involved in subsequent workshops to explain the methods and software to biologist practitioners. The foundation provided by Burnham et al. (1987), Lebreton et al. (1992) and some other key publications (e.g., Pollock et al., 1990) led to rapid development of modeling capture–recapture data from open populations. Anderson and Burnham became very interested in model selection. They used simulation to investigate the properties of open–population estimators based on models selected by AIC and other competitor approaches to model selection (Anderson et al., 1994; Burnham et al., 1994, 1995) and then wrote two books on the topic of model selection (Burnham & Anderson, 1998, 2002). Anderson has also maintained inter-

est in software development, collaborating with White and Burnham on the enormously useful program MARK (White et al., 2001). Other developments in open–population capture– recapture modeling cannot be reviewed here (see Williams et al., 2002), but I will note a few representative developments. Different models of capture history dependence have been presented including the trap response model of Pradel (1993) and the transient model of Pradel et al. (1997). Models and corresponding software for use in estimating parameters as functions of individual animal covariates were developed by Skalski et al. (1993) and Smith et al. (1994). Multistate models were developed in the 1970s by Arnason (1972, 1973), but lay dormant until the 1990s (Brownie et al., 1993; Schwarz et al., 1993) and are now seeing substantial use (Lebreton & Pradel, 2002; Fujiwara & Caswell, 2002a). These models can be implemented in programs MSSURVIV (Hines, 1994), MARK (White & Burnham, 1999) and MSURGE (Choquet et al., 2003). Open models permitting estimation under certain forms of temporary emigration have been recently developed (Fujiwara & Caswell, 2002b; Kendall & Nichols, 2002). Alternative parameterizations of the Jolly–Seber likelihood permit direct inference about recruitment and rate of population change based on open–model data (Pradel, 1996; Schwarz & Arnason, 1996; Nichols et al., 2000). These approaches can be implemented using program MARK (White & Burnham, 1999) and program POPAN (Schwarz & Arnason, 1996). Models for multiple data sources on marked animals Some studies yield different kinds of encounters for marked animals, and it is sometimes useful to tailor models to this situation. To my knowledge, David Anderson was the first person to publish the idea that different encounter types might provide information that could be used to draw inferences about parameters that could not be studied using traditional methods and models. Anderson & Sterling (1974) obtained both band recoveries and recaptures of molting drake pintails (Anas acuta) banded at a study site in Saskatchewan. Anderson recognized that the complements of survival rate estimates based on band recoveries reflected only mortality, whereas the complements of capture–recapture survival estimates included both death and permanent emigration. Anderson & Sterling (1974) presented an ad hoc estimator for the probability of permanent emigration using both capture–recapture and band recovery survival estimates. This basic idea was later formalized by Burnham (1993; also see Barker, 1997), and his model is now widely used to estimate both survival and fidelity. The robust design of Pollock (1982) combines capture–recapture data from open– and closed– population sampling, permitting robust estimation of the usual open model parameters as well as permitting inference about other processes such as

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temporary emigration (Kendall et al., 1997) and potential problems such as uncertain state assignment (Kendall et al., 2003). Barker (1997) considered models in the situation where observations of animals could be recorded between formal sampling periods. His models permit more precise estimation of standard parameters and sometimes provide inferences about movement as well. The combination of band recovery data and recapture data from both closed– and open–population time scales permits separate estimation of true survival and both temporary and permanent emigration (Lindberg et al., 2001). Observational data and distance sampling During work on his M. S. degree at Colorado State University in the 1960s, Anderson and fellow graduate student, R. S. Pospahala, worked on waterfowl production in the San Luis Valley in southern Colorado. They walked line transects looking for duck nests and took perpendicular measurements to each nest to use in estimation. Anderson & Pospahala (1970) provided an intuitive development for a nonparametric approach to nest density estimation using distance sampling. Burnham & Anderson (1976) then formally developed the framework for distance sampling, presenting a general density estimator that provides the basis for both parametric and nonparametric modeling. Building on the early parametric modeling of Hayne (1949) and subsequently Gates et al. (1968) and Eberhardt (1968), and on their own seminal work (Burnham & Anderson, 1976), Burnham and Anderson proceeded to develop various modeling approaches for use in analysis of distance data (e.g., Anderson et al. 1978, 1979a, 1979b; Burnham, 1979; Burnham et al., 1979). They recognized the need for synthesis and produced yet another landmark monograph, Burnham et al. (1980). Burnham et al. (1980) provided a thorough conceptual development for distance sampling, and recommended a robust estimation approach based on Fourier series. The Anderson theme of flexibility was again emphasized, as the user was to decide on the number of terms to be included in the Fourier series based on goodness–of–fit and sequential likelihood ratio tests. Various parametric models were also described and investigated. The model of Brownie et al. (1978) for dissemination of described methods was followed, as Burnham et al. (1980) was again written for biologists. It included many worked examples and substantial discussion of design considerations. The monograph contained a description of a comprehensive computer program, TRANSECT, designed to fit models, compute estimates, compute test statistics, and carry out necessary computations. Anderson has remained heavily involved in research on distance sampling and has been a collaborator on the two major synthetic books written to update the state of distance sampling methods and modeling (Buckland

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et al., 1993, 2001). He has also been associated with the comprehensive software package, DISTANCE, written to replace TRANSECT. In addition to these seminal contributions to traditional distance sampling, Anderson is responsible for another innovation in the use of these methods. He considered application of this approach to trapping data. As there is no natural gradient in capture probability on standard trapping grids, Anderson et al. (1983) recommended a web configuration of traps designed to induce such a gradient and thus permit analysis using distance sampling methods. The trapping web is an ingenious idea that permits direct estimation of density from trapping data and that appears to work well in practice (Parmenter et al., 2003). Methods for the conduct of science and management: estimation in context The focus of the EURING meetings over the past two decades has been on estimation of parameters relevant to population dynamics and management. However, estimation should not be viewed as a stand–alone activity. Absent the context provided by the processes of science or management, estimates are not very useful and are of little intrinsic value. Two other broad classes of methods are useful in the conduct of science and management, those associated with dynamic process modeling and decision analysis (Williams et al., 2002). Here, I note that David Anderson has made seminal contributions to these other methodological components of the processes of science and management as well as to the estimation component. Finally, I argue that the emphasis of Burnham & Anderson (1992, 1998, 2002) on model selection has resulted in renewed interest in the multiple model approach to the conduct of science (e.g., Chamberlin, 1897; Hilborn & Mangel, 1997). Modeling biological processes The most common use of mathematical models in the conduct of science is to deduce consequences of associated hypotheses (e.g., Nichols, 2001; Williams et al., 2002). We have one or multiple hypotheses about a system of interest and develop models for each hypothesis of interest. Each model is used to make a prediction about system response to an observed perturbation or experimental treatment. These predictions are then compared to the estimate of system response (an important role of estimation), and the distance between estimate and predictions is used to either reject a single hypothesis or not, or to modify relative degrees of faith in the different models in a multiple hypothesis context. During his early work on mallard population dynamics, David Anderson was asked to address questions about the effects of hunting on mallard survival and populations dynamics. In another land-

mark monograph, Anderson & Burnham (1976) provided a conceptual framework and developed a model structure for addressing these questions in a formal manner. Define Si as the annual survival rate (probability of surviving all mortality sources) for year i, S 0 as the probability of surviving nonhunting mortality sources in the absence of hunting, and Ki as the probability of dying in year i as a result of hunting in the absence of nonhunting mortality. Then various hypotheses about the effects of hunting on survival can be expressed in the following model: Si = S0(1 – bKi)

(1)

If b = 1, then the model in (1) corresponds to the completely additive mortality hypothesis. Instantaneous risks associated with hunting and nonhunting are additive, and annual survival decreases in a linear manner with increases in hunting mortality. If b = 0, then (1) corresponds to the completely compensatory mortality hypothesis, such that for a range of hunting mortality rates less than some threshold c, K < c, changes in hunting mortality rate bring about no corresponding change in total survival. Intermediate values of b, 0 < b < 1, correspond to intermediate models exhibiting partial compensation (Conroy & Krementz, 1990). As simple as this construction now seems, it represented an important step in the scientific process, the articulation of a model from which testable predictions could be clearly deduced. In their development of methods for estimating survival rates, Anderson and Burnham explored ways of incorporating this process model into the statistical models used for parameter estimation. For example, the model (S., fi) was closely associated with the compensatory mortality hypothesis (Burnham & Anderson, 1984), as annual survival is constant in the face of time–varying hunting mortality (recall that recovery rates, fi, index hunting mortality). When the appropriate numerical methods became available (White, 1983), they also developed ultrastructural models in which annual survival was modeled as a function of scaled recovery rate, permitting direct estimation of b and related parameters (Burnham et al., 1984). Anderson used process models in other aspects of his work on mallard population dynamics. Pospahala et al. (1974) developed autoregressive models of pond numbers in prairie Canada (an important environmental covariate) as a function of rainfall. In a monograph on mallard population dynamics, Anderson (1975a) used a number of models. He modeled mallard reproductive rate as a nonlinear function of mallard breeding density and environmental conditions (pond numbers). He developed a 2–sex deterministic matrix model for the purpose of investigating asymptotic population growth rate and sex ratio. He then developed a stochastic analog that incorporated covariances among the age– and sex–specific survival rates (Anderson, 1975a).

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In summary, Anderson was well aware of the role of models in the conduct of science and developed and used them as needed. The use of mathematical modeling was not widespread in animal population research in the late 1960s and early 1970s, and I believe that Anderson can be viewed as one of our field’s pioneers with respect to model use. With respect to the question of effects of hunting, we are now exploring models (e.g., Johnson et al., 1993; Williams et al., 2002) that are less phenomenological and more mechanistic than the original models of Anderson & Burnham (1976). Nevertheless, as noted by Lebreton (in press) in a recent review of effects of exploitation on animal populations, we have made surprisingly little progress in modeling the exploitation process since the important work of Anderson & Burnham (1976; also Burnham & Anderson, 1984; Burnham et al., 1984). Decision analysis When the process of interest is management or conservation, the class of methods associated with decision analysis is needed. Management problems involve five key elements (Kendall, 2001; Williams et al., 2002): 1. Management requires a clear articulation of objectives, in the form of an objective function; 2. Management requires a set of possible actions that can be taken; 3. Management requires models (or at least a single model) that reflect our understanding of the system and that permit prediction of system response to management actions; 4. In the case of multiple models (structural uncertainty about system dynamics), we require measures of our relative faith in the different models (sometimes referred to as "model weights"); 5. Finally, informed management requires a monitoring program providing estimates of system state for the purpose of making state– dependent decisions. Armed with these elements, at each decision point in the time frame, the manager would like to select the management action that is "best" with respect to achieving objectives. The step in the management process of deciding the best management action is a problem in dynamic optimization. Each decision carries a specified consequence for the objectives and drives the system to a new state. Our decisions must be state– specific and must account for system dynamics. To make the problem more difficult, because of environmental variation and other forms of uncertainty, our system model will typically be stochastic, permitting only probabilistic predictions about system state in the subsequent time step. We thus require a method that will yield optimal decisions for dynamic systems that permit only stochastic predictions (Williams et al., 2002). While working on mallard harvest problems, Anderson recognized the potential utility of stochastic dynamic programming (Bellman, 1957) for solving problems in optimal stochastic control. For his Ph. D. research at the University of Maryland, Anderson

used dynamic programming to explore optimal decision policies for mallard hunting regulations under two contrasting models, completely compensatory mortality and totally additive mortality. The state of the system was characterized by the number of breeding mallards each spring and the number of ponds (environmental state), the decision involved the total mallard harvest (management action), and the objective was to maximize total harvest over a long time horizon. The computing difficulties associated with dynamic programming were substantial in the early 1970s, yet Anderson (1975b) was able to obtain optimal state–specific policies for the two models reflecting different hypotheses about hunting effects. The demonstration of very different optimal policies for these two competing models was important in demonstrating the importance to management of distinguishing between these two alternatives. I view Anderson’s recognition that duck harvest management is a problem in optimal stochastic control as an extremely important development in wildlife management. Most decisions in wildlife management and conservation are made very subjectively in the absence of some of the important components of a management process (e.g., no clear statement of objectives, no specific model(s) predicting system responses to management actions). Anderson not only identified the necessary components of an objective process, but he also found an optimization approach to compute optimal management decisions. Anderson’s (1975b) use of stochastic dynamic programming is one of the earliest uses in natural resource management and the first use, to my knowledge, in wildlife management. Duck hunting regulations for mallards in North America are now established using a formal program of adaptive management (Walters, 1986; Williams et al., 2002), and the process represents a model for this approach. I believe that the articulation of competing hypotheses and models for the effects of hunting on survival (e.g., Anderson & Burnham, 1976), the recognition that the establishment of hunting regulations was a classic problem in decision analysis, and the use of stochastic dynamic programming as a means of obtaining optimal policies (Anderson, 1975b) represent key points in the evolution of the North American program of adaptive harvest management for ducks (Nichols, 2000). Without these contributions, I very much doubt that we would be using adaptive management today. Model selection, science and management Here, I no longer consider the separate methodological components of science and management and move to the overall processes, themselves. Although much of the original motivation of Burnham and Anderson for studying model selection appeared to be to choose estimators with good properties, their work has had substantial influence in the conduct of science. The multiple hypothesis approach to science articulated by Chamberlin (1897) saw little use for

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nearly a century, but has now become fairly popular (e.g., Hilborn & Mangel, 1997; Williams et al., 2002). We admit that truth is unknowable and, even if it were not, that it would be incomprehensible to us and too complex to use for prediction. The task of the scientist is then to develop simplifying hypotheses, together with their corresponding models, and to use them to deduce predictions that are then confronted with data. This confrontation leads to changes in the relative degrees of faith held in the different hypotheses and in the predictions of their associated models. Science then becomes a task of selecting among competing hypotheses and models. A current trend in parameter estimation is to develop model–averaged estimators (e.g., Buckland et al., 1997; Burnham & Anderson, 1998, 2002). The scientific analog occurs when we need to make predictions, as when the scientific process is embedded in a management program. In such cases, we turn to something akin to weighted model averaged predictions (Williams et al., 2002), where the model weights reflect the relative degrees of faith in the different hypotheses. These model weights are themselves a result of the scientific process, and are based on the past predictive abilities of the models as judged against estimates of true system state. The point of this brief development is simply that the model selection philosophy popularized by Burnham and Anderson has extended well beyond estimation to the conduct of science and management. On individual contributions The EURING meetings are very important and contribute substantially to defining the state of the art with respect to methods for studying animal populations. The strength of the meetings is the attendees, who represent the most important contributors to this field. Here, I would like to consider the metrics by which individual contributions can be judged, using David Anderson as an example. First, I consider a metric that is best viewed as an abstraction, hopefully useful in defining what constitutes an important contributor, though probably not useful in actual measurement. Then I consider metrics that can be actually measured and that are hopefully correlated with real contributions. Fisher’s (1930) reproductive value is a metric that has proven useful in the study of demography. It is a function of age–specific rates of survival and fecundity and essentially quantifies the relative contributions of different individuals (in this case differing by age or stage) to future population growth (e.g., see Stearns, 1976). One way to view reproductive value is to consider removal of an individual from the population and to consider the population size at some time period in the distant future relative to the case where a different individual (different age or stage) is removed. The ratio of the future population sizes should reflect the ratio of reproductive values of the two individuals.

Although very much an abstraction in the case of individual contributions, this is the sort of question we would like to ask in order to judge the relative contribution of a single individual to the growth of knowledge. We would like to summon Clarence Odbody, angel second class, from the movie "It’s A Wonderful Life". Just as Clarence showed George Bailey what Bedford Falls, N.Y. would have looked like had George never been born, we could ask Clarence to show us the current state of knowledge in our field had any individual of interest "never been born" or otherwise never contributed. In the case of David Anderson, I would argue that the difference (current knowledge with and without his contributions) would be substantial. With respect to estimation, I would guess that the modeling of band recovery data, capture– recapture data, and distance sampling data would not be nearly as advanced as they now are. The landmark synthetic monographs (Brownie et al., 1978, 1985; Buckland et al., 1993, 2002; Burnham et al., 1980, 1987; Lebreton et al., 1992; Otis et al., 1978; White et al., 1982) have simply been too important in providing a base for further development. I believe it is nearly impossible to overstate the importance of such synthetic points of departure for new work in a field. It is also unlikely that comprehensive software development would be nearly as advanced as it now is. Although there were computer programs available to compute capture–recapture estimators before Anderson’s work, these were stripped down computational programs with little effort devoted to either flexibility or user friendliness. Except perhaps for the software developed by Arnason (e.g., Arnason & Baniuk, 1980), the user–friendly and comprehensive programs developed as part of Anderson’s early work (e.g., BROWNIE, ESTIMATE, TRANSECT, CAPTURE) were unique. These programs can be viewed as the early ancestors of such current software as MARK (White & Burnham, 1999), MSURGE (Cloquet et al., 2003) and POPAN (Arnason & Schwarz, 1999). I doubt that software evolution would have produced anything resembling the current state of development, in the absence of this work during the 1970’s. Certainly, I believe that the number of biologists and managers possessing a working knowledge of modern estimation methods would be much smaller in the absence of Anderson’s contributions. With respect to specific innovations, I suspect that eventually we would have developed joint band recovery and capture–recapture models even in the absence of the suggestions of Anderson & Sterling (1974), but I doubt that the trapping web would be with us had it not been for Anderson et al. (1983). With respect to the larger issues involving the conduct of science and management, I have argued above and elsewhere (Nichols, 2000) that there would probably be no formal program for adaptive waterfowl harvest management had it not been for Anderson’s early work on mallard populations. Specifically, his key contributions were:

1. Development of competing hypotheses and associated models for responses of survival rates to hunting mortality; 2. Incorporation of these survival models into larger population–dynamic models; and 3. Identification and use of stochastic dynamic programming for computing optimal harvest policies. With respect to the conduct of science, I believe that the recent emphasis on model selection by Burnham and Anderson has popularized the multiple–hypothesis approach to science and focused needed attention on the manner in which hypotheses are evaluated (e.g., Johnson, 1999; Franklin et al., 2001). Of course all of the above speculation represents an exercise in a posteriori story–telling, as I have no ability to test any of these stories about what our state of knowledge might be like in the absence of Anderson’s varied contributions. When we search for more tangible metrics that might be correlated with the reproductive value abstraction, we might focus on publications and citations of his work. David Anderson has published about 150 journal articles and book chapters and 15 books and research monographs, and there are approximately 6,500 citations of his work in the scientific literature. Another approach would be to quantifying contributions using empirical data would be to investigate the scientific collaboration network (Newman, 2001) in the field of animal population parameter estimation. We would select a group of interest; for example all scientists who have authored at least one EURING proceedings article during the last five EURING meetings. For each member of this set, or for a randomly selected subset of these individuals, we would compute the path length (e.g., Watts & Strogatz, 1998), or the number of steps required to link each individual, via collaboration and coauthorship, with David Anderson. For example, only one step is required to link me with David as we have written papers together. Juan–Carlos Senar, to my knowledge, has not coauthored a paper with Anderson. Senar has authored papers with Mike Conroy, who has coauthored papers with Anderson, so the path length linking Senar to Anderson would be 2. I speculate that average path length required to link scientists with Anderson would be substantially less than required to link scientists with most other focal individuals. If this speculation is correct, then it would provide yet another metric reflecting the disproportionate influence and contribution of David Anderson to our field. In closing this section on individual contributions, I believe it is wise to examine the characteristics and habits of important contributors, as such knowledge may make better contributors of the rest of us. One attribute of Anderson is his ability to discriminate between important and mediocre topics and to devote his valuable time to the important ones. The ability to discriminate may involve not only knowledge but also intuition, and I am guessing that this is something that cannot necessarily be learned. However, given that the investigator has decided that some topics are more important than others, he/she certainly has the

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ability to focus energy on the important questions. Too frequently, research topics are selected based on such factors as funding sources, interests of collaborators, and projected ease of study. While these factors are of some relevance, they should be viewed as minor relative to our judgments about the potential importance of the contributions. Human life spans are finite, and we should be extremely jealous and protective of our time. Another important characteristic of Anderson, in my opinion, is the premium placed by him on disseminating his completed work to potential users. In most areas of endeavor, the value of a contribution is defined not so much by intrinsic characteristics such as novelty and potential utility, as by the degree to which the work is actually used. As emphasized in the sections on estimation, Anderson’s approach of writing monographs for a biological readership, providing accompanying user–oriented software, and conducting workshops to explain the methods to users has provided our discipline with an immensely successful model for methodological dissemination. Anderson is an excellent verbal communicator with a knack for explaining complex issues in a simple and straightforward manner, and this ability has no doubt proven useful in explaining new methods to biologists and managers. Another idea that comes to mind regarding Anderson’s success is his tendency to surround himself with other good scientists with similar interests and distinct, yet broadly overlapping, skills and abilities. The resulting synergies have proven to be extremely productive and useful to all participants. Many of the best scientists in our field, including many EURING participants, have been collaborators and students of Anderson. Finally, I believe that Anderson has been very innovative with respect to both specific methods (e.g., distance sampling, trapping web) and broader visions of the future (e.g., comprehensive methodological software packages, institutional adoption of decision–theoretic approaches to animal population management). Although I do not claim that innovation can be learned, I do suspect that some of us do not spend adequate time thinking beyond well–defined problems and at least attempting to be more visionary. Estimation methods, present and future The preceding discussion on the evolution of methods has been historical in nature. Such discussions are only useful if past failures and, in this case, successes are informative of promising paths for future work. Here I comment on some of the aspects of our current state that seem especially promising, exciting, and likely to provide bases for useful future work. The preceding text included a short section dealing with models for multiple data sources on marked animals. Such combination models tend to exhibit the advantages of increased precision, increased flexibil-

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ity and ability to obtain separate estimates of parameters that are confounded with single data sources. In addition to some of the combinations discussed in the historical review, work is either in progress or has been recently completed on new combinations. For example, Skvarla et al. (in press) have implemented a robust design in a multistate framework, permitting estimation of temporary emigration from the study system as well as survival and movement among study sites. Bailey et al. (in press) have used a special kind of open robust design based on amphibians captured while entering and leaving breeding ponds to estimate parameters of interest, including probability of temporary emigration, which is equivalent to probability of nonbreeding in this system. Kendall et al. (in review) have extended this modeling to multiple breeding ponds. Barker (in review) and Kendall (pers. comm.) have developed robust design models that also include band recoveries and incidental observations. I continue to believe that there remain many opportunities for combining radio telemetry and capture–recapture data in useful ways (e.g., Powell et al., 2000). Powell (in press) has recently developed multistate models to use the extra information from isotope or genetic signatures that identify the location of a captured animal the previous time period, regardless of whether or not the animal was caught that period. An issue of special concern in studies of marked animals is the appropriate treatment of uncertainty. Lukacs & Burnham (in review) have considered the sampling of low quality sources of DNA (e.g., hair samples, fecal samples) for the purpose of identifying individual animals using microsatellite molecular data. Different kinds of error in identification can result from such data and constitute a source of uncertainty that should be incorporated into the modeling and estimation. Kendall et al. (2003) use the robust design to permit estimation under multistate models in cases where errors can be made in state assignment. Pradel (in review) and Nichols et al. (in press) consider the problem of uncertainty in assignment of sex to individuals. In cases where it is possible to observe at least some behaviors that are definitive of sex, and even some cases where this is not true, estimation is possible within an open population framework. Such modeling is preferred to a typical ad hoc approach, which yields sex–specific survival estimates that are positively biased. Individual heterogeneity is an important topic that was discussed at the previous EURING conference (Link et al., 2002; Cam et al., 2002). This previous work was based on a sampling situation in which capture probability was effectively 1, permitting hierarchical modeling and direct estimation. Link (2003) has incorporated heterogeneous capture probabilities into a standard open model framework using a Bayesian approach, and Pledger et al. (2003) have developed finite mixture models for dealing with heterogeneity under the robust design. Link’s (2003) demonstration of the nonidentifiability of abundance in the presence of heterogeneous

capture probabilities in closed population models is both important and sobering. It would be valuable to be able to identify characteristics of distributions that are most likely to cause serious problems. In the absence of such a classification, this result should place a premium on studies of known populations or subpopulations, as this will become our only means of gaining confidence in particular estimators and models. As the EURING conferences are becoming increasingly broad in subject matter, I note several interesting developments in the analysis of data from observational studies. One involves the decomposition of detection probability estimates (Pollock, pers. comm.) into components associated with: 1. Detection conditional on both presence in the study area and on the organism not being invisible (e.g., submerged manatee, silent songbird); 2. Visibility given presence in the study area; and 3. Presence in the study area. The detection probabilities associated with some estimation methods (e.g., Royle & Nichols, 2003) include all three components, others (e.g., Farnsworth et al., 2002) include components 1 and 2, and still others (Buckland et al., 2001) deal only with component 1. This hierarchy permits estimation of the individual components in situations where this might be useful. The consideration of detectability in observation–based studies brings up a recurring design issue for large–scale monitoring programs. How much effort should be allocated to the estimation of detection probability within sample units, versus better dealing with geographic variation by surveying more sample units (Pollock et al., 2002)? Can double sampling approaches be used such that detection probability is estimated on only a subset of sample units? Large–scale monitoring programs may also benefit from consideration of state variables other than abundance, and the recent work on estimation of occupancy may be useful in this regard (Mackenzie et al., 2002, 2003, 2004; Mackenzie & Bailey, in press; Royle & Nichols, 2003; Royle, 2004; Royle & Link, in review). Royle (pers. comm.) has noted that the large majority of modeling effort in distance sampling has involved detection probabilities, whereas the densities themselves are the quantities of real interest. Royle (pers. comm.) has noted the potential to embed models for both spatial and temporal variation in densities in joint likelihoods for distance sampling data. Link et al. (2003) have recently modeled observational data on whooping cranes, in which observed birds are categorized by state (e.g., young and adult). The deterministic nature of state transitions (knowledge of the number of years spent as young) permits inference about survival. Finally, J. M. Nichols et al. (in press) are exploring the possibility of using attractor–based methods (e.g., Pecora et al., 1995, 1997; Schiff et al., 1996) developed for the analysis of nonlinear time series data to detect coupling (dynamical interdependence) based on time series of abundance estimates from potentially coupled systems. The methods

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should be preferable to the linear cross–correlation typically used in ecology (e.g., Bjornstad et al., 1999; Koenig, 1999) because the attractor–based approaches do not rely on the assumption of linearity and because they can detect asymmetry in the coupling. One of the most exciting areas of research is the combination of data from both observations and marked animals. Observation–based data provide information on abundance, density and rate of change in abundance, whereas marked animal data provide information on vital rates, abundance, and rate of change in abundance. Various combinations of such data permit direct estimation of all important components of population dynamics in a synthetic and consistent manner. Variations on this general idea have been provided by Trenkel et al. (2000), Besbeas et al. (2002), White & Lubow (2002), Gove et al. (2002), as well as in EURING– 03 presentations by Brooks, Caswell, King, Lebreton, Morgan, and Otto. Estimation approaches range from least squares approaches to use of the Kalman filter and hierarchical modeling within a Bayesian framework. Finally, Fonnesbeck & Conroy (in review) have presented a synthetic view of decision theory combining modeling, estimation and optimization in a single Bayesian framework for use in management programs. Summary As noted in the introduction, the EURING meetings and their participants have played an important role in the field of parameter estimation for animal populations. David Anderson is certainly one of the outstanding contributors in this field of endeavor, with seminal publications not only in the EURING specialty area of estimation, but also in the methodological areas of modeling and decision theory, and even in overall approaches to the conduct of science and management. It would be wise for all of us to examine the characteristics of Anderson and his work in an effort to try to insure that our research efforts are maximally useful. Our field is relatively mature now, yet there are many exciting topics worthy of our efforts. Some of the more important of these topics involve efforts to integrate different data sources, methods, and general approaches in an effort to create a synthetic view and treatment of estimation, modeling and decision theory. I fully expect the participants in this and other EURING conferences to be among the leading contributors to these promising developments. Acknowledgements I thank the meeting organizers and volume editors, M. J. Conroy, A. A. Dhondt, and J. C. Senar for extending the invitation to present this paper and for making constructive comments on the initial manuscript.

References Akaike, H., 1973. Information theory and an extension of the maximum likelihood principle. In: Second international symposium on information theory: 267–281 (B. N. Petran & F. Csàaki, Eds.). Akadèemiai Kiadi, Budapest, Hungary. Anderson, D. R., 1975a. Population ecology of the mallard. V. Temporal and geographic estimates of survival, recovery and harvest rates. U.S. Fish & Wildlife Service Resource Publication, 125. – 1975b. Optimal exploitation strategies for an animal population in a Markovian environment: A theory and an example. Ecology, 56: 1281–1297. Anderson, D. R. & Burnham, K. P., 1976. Population ecology of the mallard. VI. The effect of exploitation on survival. U.S. Fish & Wildlife Service Resource Publication, 128. – 1980. Effect of delayed reporting of ban recoveries on survival estimates. Journal of Field Ornithology, 51: 244–247. Anderson, D. R., Burnham, K. P. & Crain, B. R., 1978. A log–linear model approach to estimation of population size using the line transect sampling method. Ecology, 59: 190–193. – 1979b. Line transect estimation of population size; the exponential case with grouped data. Communications in Statistics–Theory and Methods, A8: 487–507. Anderson, D. R., Burnham, K. P. & White, G. C., 1985. Problems in estimating age–specific survival rates from recovery data of birds ringed as young. Journal of Animal Ecology, 54: 89–98. – 1994. AIC model selection in overdispersed capture–recapture data. Ecology, 75: 1780–1793. Anderson, D. R., Burnham, K. P., White, G. C. & Otis, D. L., 1983. Density estimation of small– mammal populations using a trapping web and distance sampling methods. Ecology, 64: 674– 680. Anderson, D. R., Fiehrer, F. R. & Kimball, C. F., 1974. A computer program for estimating survival and recovery rates. Journal of Wildlife Management, 38: 369–370. Anderson, D. R., Laake, J. L., Crain, B. R. & Burnham, K. P., 1979a. Guidelines for line transect sampling of biological populations. Journal of Wildlife Management, 43: 70–78. Anderson, D. R. & Pospahala, R. S., 1970. Correction of bias in belt transect studies of immotile objects. Journal of Wildlife Management, 34: 141– 146. Anderson, D. R. & Sterling, R. T., 1974. Population dynamics of molting pintail drakes banded in south–central Saskatchewan. Journal of Wildlife Management, 38: 266–274. Anderson, D. R., Wywialowski, A. P. & Burnham, K. P., 1981. Tests of the assumptions underlying life table methods for estimating parameters from cohort data. Ecology, 62: 1121–1124. Arnason, A. N., 1972. Parameter estimates from mark–recapture experiments on two populations subject to migration and death. Researches on

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Population Ecology, 13: 97–113. – 1973. The estimation of population size, migration rates, and survival in a stratified population. Researches on Population Ecology, 15:1–8. Arnason, A. N. & Baniuk, L., 1980. A computer system for mark–recapture analysis of open populations. Journal of Wildlife Management, 44: 325–332. Bailey, L. L., Kendall, W. L., Church, D. R. & Wilbur, H. M. (in press). Estimating survival and breeding probability for pond–breeding amphibians: A modified robust design. Ecology, 85. Barker, R. J., 1997. Joint modeling of live–recapture, tag–resight, and tag–recovery data. Biometrics, 53: 666–677. Bellman, R., 1957. Dynamic programming. Princeton University Press, Princeton, New Jersey. Besbeas, P., Freeman, S. N., Morgan, B. J. Y. & Catchpole, E. A., 2002. Integrating mark–recapture–recovery and census data to estimate animal abundance and demographic parameters. Biometrics, 58: 540–547. Bjornstad, O. N., Ims, R. A. & Lambin, X., 1999. Spatial population dynamics: analyzing patterns and processes of population synchrony. Trends in Ecology and Evolution, 14: 427–432. Blohm, R. J., Reynolds, R. E., Bladen, J. P., Nichols, J. D., Hines, J. E., Pollock, K. H. & Eberhardt, R. T., 1987. Mallard mortality rates on key breeding and wintering areas. Transactions of the North American Wildlife and Natural Resources Conference, 52: 246–257. Brownie, C., Anderson, D. R., Burnham, K. P. & Robson, D. S., 1978. Statistical inference from band recovery data – a handbook. U.S. Fish and Wildlife Service Resource Publication, 131. – 1985. Statistical inference from band recovery data – a handbook, 2nd ed. U.S. Fish and Wildlife Service Resource Publication, 156. Brownie, C., Hines, J. E. & Nichols, J. D., 1986. Constant–parameter capture–recapture models. Biometrics, 42: 561–574. Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H. & Hestbeck, J. B., 1993. Capture–recapture studies for multiple strata including non– Markovian transition probabilities. Biometrics, 49: 1173–1187. Brownie, C. & Robson, D. S., 1976. Models allowing for age–dependent survival rates for band– return data. Biometrics, 32: 305–323. – 1983. Estimation of time–specific survival rates from tag–resighting samples: a generalization of the Jolly–Seber model. Biometrics, 39: 437–453. Buckland, S. T., Anderson, D. R., Burnham, K. P. & Laake, J. L., 1993. Distance sampling: Estimating abundance of biological populations. Chapman and Hall, New York. Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L. & Thomas, L., 2001. Introduction to distance sampling: Estimating abundance of biological populations. Oxford, Oxford, UK. Buckland, S. T., Burnham, K. P. & Augustin, N. H.,

1997. Model selection: an integral part of inference. Biometrics, 38: 469–477. Burnham, K. P., 1979. A parametric generalization of the Hayne estimator for line transect sampling. Biometrics, 35: 587–595. – 1990. Survival analysis of recovery data from birds ringed as young: Efficiency of analyses when numbers of ringed are not known. Ring, 13: 115–132. – 1993. A theory for combined analysis of ring recovery and recapture data. In: Marked individuals in the study of bird population: 199–213 (J. –D. Lebreton & P. M. North, Eds.). Birkhauser Verlag, Basel, Switzerland. Burnham, K. P. & Anderson, D. R., 1976. Mathematical models for nonparametric inferences from line transect data. Biometrics, 32: 325–336. – 1979. The composite–dynamic method as evidence for age–specific waterfowl mortality. Journal of Wildlife Management, 43: 356–366. – 1984. Tests of compensatory vs. additive hypotheses of mortality in mallards. Ecology, 65: 105–112. – 1992. Data–based selection of an appropriate biological model: the key to modern data analysis. In: Wildlife 2001:populations: 16–30 (D. R. McCullough & R. H. Barrett, Eds.). Elsevier, New York. – 1998. Model selection and inference: A practical information–theoretic approach. Springer–Verlag, New York. – 2002. Model selection and inference: a practical information–theoretic approach. 2 nd edition. Springer–Verlag, New York. Burnham, K. P., Anderson, D. R. & Laake, J. L., 1979. Robust estimation from line transect data. Journal of Wildlife Management, 43: 992–996. – 1980. Estimation of density from line transect sampling of biological populations. Wildlife Monographs, 72. Burnham, K. P., Anderson, D. R. & White, G. C., 1994. Evaluation of the Kullback–Liebler discrepancy for model selection in open population capture–recapture models. Biometrical Journal, 38: 299–315. Burnham, K. P., Anderson, D. R., White, G. C., Brownie, C. & Pollock, K. H., 1987. Design and analysis methods for fish survival experiments based on release–recapture. American Fisheries Society Monograph, 5. Burnham, K. P. & Overton, W. S., 1978. Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika, 65: 625–633. Burnham, K. P., White, G. C. & Anderson, D. R., 1984. Estimating the effect of hunting on annual survival rates of adult mallards. Journal of Wildlife Management, 48: 350–361. – 1995. Model selection strategy in the analysis of capture–recapture data. Biometrics, 51: 888–898. Cam, E., Link, W. L., Cooch, E. G., Monnat, J.–Y. & Danchin, E., 2002. Individual covariation in life history traits: seeing the trees despite the forest.

American Naturalist, 159: 96–105. Catchpole, E. A., Morgan, B. J. T. & Freeman, S. N., 1998. Estimation in parameter–redundant models. Biometrika, 85: 462–468. Chamberlin, T. C., 1897. The method of multiple working hypotheses. Journal of Geology, 5: 837–848. Chao, A., 1987. Estimating the population size for capture–recapture data with unequal catchability. Biometrics, 43: 783–791. Chao, A., Lee S. M. & Jeng, S. L., 1992. Estimating population size for capture–recapture data when capture probabilities vary by time and individual animal. Biometrics, 48: 201–216. Chao, A. & Huggins, R. M. (in press). Modern closed population models. In: Handbook of capture–recapture methods (B. F. J. Manly, T. McDonald & S. Armstrup, Eds.). Princeton University Press, Princeton, New Jersey. Choquet, R., Reboulet, A. M., Pradel, R., Gimenez, O. & Lebreton, J. D., 2003. User’s manual for M– SURGE 1.01. Mimeographed document, CEFE/ CNRS, Montpellier. ftp://ftp.cefe.cnrs–mop.fr/biom/Soft–CR Clobert, J. & Lebreton, J.–D., 1985. Dependance de facteurs de milieu dans les estimations de taux de survie par capture–recapture. Biometrics, 41: 1031–1037. Conroy, M. J. & Blandin, W. W., 1984. Geographic and temporal differences in band reporting rates for American black ducks. Journal of Wildlife Management, 48: 23–36. Conroy, M. J., Hines, J. E. & Williams, B. K., 1989. Procedures for the analysis of band–recovery data and user instructions for program MULT. U.S. Fish and Wildlife Service Resource Publication, 175. Conroy, M. J. & Krementz, D. G., 1990. A review of the evidence for the effects of hunting on American black duck populations. Transactions of the North American Wildlife and Natural Resources Conference, 55: 511–517. Conroy, M. J. & Williams, B. K., 1984. A general methodology for maximum likelihood inference from band recovery data. Biometrics, 40: 739–748. Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438. Darroch, J. N., 1958. The multiple–recapture census: I. Estimation of a closed population. Biometrika, 45: 343–359. Dorazio, R. M. & Royle, J. A., 2003. Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics, 59: 351–364. Eberhardt, T. L., 1968. A preliminary appraisal of line transects. Journal of Wildlife Management, 32: 82–88. Farnsworth, G. L., Pollock, K. H., Nichols, J. D., Simons, T. R., Hines, J. E. & Sauer, J. R., 2002. A removal model for estimating detection probabilities from point–count surveys. Auk, 119: 414–425. Fisher, R. A., 1930. The genetical theory of natural selection. Clarendon Press, Oxford.

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Franklin, A. B., Shenk, T. M., Anderson, D. R. & Burnham, K. P., 2001. Statistical model selection: an alternative to null hypothesis testing. In: Modeling in natural resource management: 75–90 (T. M. Shenk & A. B. Franklin, Eds.). Island Press, Washington, D.C. Fujiwara, M. & Caswell, H., 2002a. Estimating population projection matrices from multi–stage mark–recapture data. Ecology, 83: 3257–3265. – 2002b. Temporary emigration in mark–recapture analysis. Ecology, 83: 3266–3275. Gates, C. E., Marshall, W. H. & Olson, D. P., 1968. Line transect method of estimating grouse population densities. Biometrics, 36: 155–160. Gove, N. E., Skalski, J. R., Zager, P. & Townsend, R. L., 2002. Statistical models for population reconstruction using age–at–harvest data. Journal of Wildlife Management, 66: 310–320. Haldane, J. B. S., 1953. Some animal life tables. Journal of the Insitute of Actuaries, 79: 83–89. – 1955. The calculation of mortality rates from ringing data. Proceedings of the International Congress of Ornithologists, 9: 454–458. Hayne, D. W., 1949. An examination of the strip census method for estimating animal populations. Journal of Wildlife Management, 13: 145–157. Hearn, W. S., Pollock, K. H. & Brooks, E. N., 1998. Pre– and post–season tagging models: estimation of reporting rate and fishing and natural mortality rates. Canadian Journal of Fisheries and Aquatic Sciences, 55: 199–205. Henny, C. J. & Burnham, K. P., 1976. A reward band study of mallards to estimate reporting rates. Journal of Wildlife Management, 40: 1–14. Hestbeck, J. B., Dzubin, A., Gollop, J. B. & Nichols, J. D., 1989. Mallard survival from local to immature stage in southwestern Saskatchewan. Journal of Wildlife Management, 53: 428–431. Hilborn, R. & Mangel, M., 1997. The ecological detective. Confronting models with data. Princeton University Press, Princeton, New Jersey. Hines, J. E., 1994. MSSURVIV user’s manual. National Biological Survey, Laurel, Maryland. Huggins, R. M., 1989. On the statistical analysis of capture experiments. Biometrika, 76: 133–140. – 1991. Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics, 47: 725–732. Johnson, D. H., 1999. The insignificance of significance testing. Journal of Wildlife Management, 63: 763–772. Johnson, F. A., Williams, B. K., Nichols, J. D., Hines, J. E., Kendall, W. L., Smith, G. W. & Caithamer, D. F., 1993. Developing an adaptive management strategy for harvesting waterfowl in North America. Transactions of the North American Wildlife and Natural Resources Conference, 58: 565–583. Jolly, G. M., 1965. Explicit estimates from capture– recapture data with both death and immigration– stochastic model. Biometrika, 52: 225–247. – 1982. Mark–recapture models with parameters constant in time. Biometrics, 38: 301–321.

Animal Biodiversity and Conservation 27.1 (2004)

– 1993. Instinctive statistics. In: Marked individuals in the study of bird population: 1–5 (J.–D. Lebreton & P. M. North, Eds.). Birhauser Verlag, Basel, Switzerland. Kendall, W. L., 2001. Using models to facilitate complex decisions. In: Modeling in natural resource management: 147–170 (T. M. Shenk & A. B. Franklin, Eds.). Island Press, Washington, D.C. Kendall, W. L., Hines, J. E. & Nichols, J. D., 2003. Adjusting multi–state capture–recapture models for misclassification bias: manatee breeding proportions. Ecology, 84: 1058–1066. Kendall, W. L. & Nichols, J. D., 2002. Estimating state–transition probabilities for unobservable states using capture–recapture/resighting data. Ecology, 83: 3276–3284. Kendall, W. L., Nichols, J. D. & Hines, J. E., 1997. Estimating temporary emigration using capture– recapture data with Pollock’s robust design. Ecology, 78: 563–578. Koenig, W. D., 1999. Spatial autocorrelation of ecological phenomena. Trends in Ecology and Evolution, 14: 22–26. Lebreton, J.–D., 2001. The use of bird rings in the study of survival. Ardea, 89: 85–100. – (in press). Dynamical and statistical models for exploited populations. Australian and New Zealand Journal of Statistics. Lebreton, J. D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modelling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118. Lebreton, J. D. & Pradel, R., 2002. Multistate recapture models: modeling incomplete individual histories. Journal of Applied Statistics: 353–369. Lee, S.–M. & Chao, A., 1994. Estimating population size via sample coverage for closed capture–recapture models. Biometrics, 50: 88–97. Lincoln, F. C., 1930. Calculating waterfowl abundance on the basis of banding returns. U.S. Department of Agriculture Circular Number, 118: 1–4. Lindberg, M. S., Kendall, W. L., Hines, J. E. & Anderson, M. G., 2001. Combining band recovery data and Pollock’s Robust Design to model temporary and permanent emigration. Biometrics, 57: 273–281. Link, W. A., 2003. Nonidentifiability of population size from capture–recapture data with heterogeneous detection probabilities. Biometrics, 59: 1125–1132. Link, W. A., Cooch, E. & Cam, E., 2002. Model– based estimation of individual fitness. Journal of Applied Statistics, 29: 207–224. Link, W. A., Royle, J. R. & Hatfield, J. S., 2003. Demographic analysis from summaries of an age– structured population. Biometrics, 59: 778–785. Mackenzie, D. I. & Bailey, L. L. (in press). Assessing the fit of site occupancy models. Journal of Agricultural, Biological and Environmental Statistics. Mackenzie, D. I., Bailey, L. L. & Nichols, J. D., 2004. Investigating patterns of species co–oc-

currence when species are detected imperfectly. Journal of Animal Ecology, 73: 546–555. Mackenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A. & Langtimm, C. A., 2002. Estimating site occupancy rates when detection probabilities are less than one. Ecology, 83: 2248–2255. Mackenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G. & Franklin, A. B., 2003. Estimating site occupancy, colonization and local extinction when a species is detected imperfectly. Ecology, 84: 2200–2207. Nelson, L. J., Anderson, D. R. & Burnham, K. P., 1980. The effect of band loss on estimates of annual survival. Journal of Field Ornithology, 51: 30–38. Newman, A. E. J., 2001. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences U.S.A., 98: 404–409. Nichols, J. D., 2000. Evolution of harvest management for North American waterfowl: selective pressures and preadaptations for adaptive harvest management. Transactions of the North American Wildlife and Natural Resources Conference, 65: 65–77. – 2001. Using models in the conduct of science and management of natural resources. In: Modeling in natural resource management: 11– 34 (T. M. Shenk & A. B. Franklin, Eds.). Island Press, Washington, D.C. Nichols, J. D., Blohm, R. J., Reynolds, R. E., Trost, R. E., Hines, J. E. & Bladen, J. P., 1991. Band reporting rates for mallards with reward bands of different dollar values. Journal of Wildlife Management, 55: 119–126. Nichols, J. D., Hines, J. E., Lebreton, J.–D. & Pradel, R., 2000. The relative contributions of demographic components to population growth: a direct estimation approach based on reverse– time capture–recapture. Ecology, 81: 3362–3376. Nichols, J. D., Kendall, W. L., Hines, J. E. & Spendelow, J. A. (in press). Estimation of sex– specific survival from capture–recapture data when sex is not always known. Ecology. Nichols, J. M., Moniz, L., Nichols, J. D., Pecora, L. M. & Cooch, E. (in press). Assessing spatial coupling in complex population dynamics using mutual prediction and continuity statistics. Theoretical Population Biology. Nichols, J. D., Reynolds, R. E., Blohm, R. J., Trost, R. E., Hines, J. E. & Bladen, J. P., 1995. Geographic variation in band reporting rates for mallards based on reward banding. Journal of Wildlife Management, 59: 697–708. Nichols, J. D. & Tautin, J. (in press). North American bird banding and population ecology. In: History of North American bird banding (J. A. Jackson, Ed.). Norris, J. L. III & Pollock, K. H., 1996. Nonparametric MLE under two closed capture–recapture models with heterogeneity. Biometrics, 52: 639–649. North, P. M. & Cormack, R. M., 1981. On Seber’s method for estimating age–specific bird survival

rates from ringing recoveries. Biometrics, 37: 103–112. North, P. M. & Morgan, B. J. T., 1979. Modeling heron survival using weather data. Biometrics, 35: 667–681. Otis, D. L., Burnham, K. P., White, G. C. & Anderson, D. R., 1978. Statistical inference from capture data on closed animal populations. Wildlife Monographs, 62. Parmenter, R. R., Yates, T. L., Anderson, D. R., Burnham, K. P., Dunnum, J. L., Franklin, A. B., Friggens, M. T., Lubow, B. C., Miller, M., Olson, G. S., Parmenter, C. A., Pollard, J., Rexstad, E., Shenk, T. M., Stanley, T. R. & White, G. C., 2003. Small–mammal density estimation: a field comparison of grid–based vs. web–based density estimators. Ecological Monographs, 73: 1–26. Pecora, L. M., Carroll, T. L. & Heagy, J. F., 1995. Statistics for mathematical properties of maps between time series embeddings. Physical Review E, 52: 3420–3439. Petersen, C. J. N., 1896. The yearly immigration of young plaice into the Limfjord from the German Sea. Report of the Danish Biological Station, 6: 1–48. Pledger, S., 2000. Unified maximum likelihood estimates for closed capture–recapture models for mixtures. Biometrics, 56: 434–442. Pledger, S., Pollock, K. H. & Norris, J. L., 2003. Open capture–recapture models with heterogeneity: I. Cormack–Jolly–Seber model. Biometrics, 59: 786–794. Pollock, K. H., 1974. The assumption of equal catchability of animals in tag–recapture experiments. Ph. D. Thesis, Cornell University, Ithaca, New York. – 1975. A K–sample tag–recapture model allowing for unequal survival and catchability. Biometrika, 62: 577–583. – 1981. Capture–recapture models allowing for age–dependent survival and capture rates. Biometrics, 37: 521–529. – 1982. A capture–recapture design robust to unequal probability of capture. Journal of Wildlife Management, 46: 752–757. Pollock, K. H., Hines, J. E. & Nichols, J. D., 1984. The use of auxiliary variables in capture–recapture and removal experiments. Biometrics, 40: 329–340. – 1985. Goodness–of–fit tests for open capture– recapture models. Biometrics, 41: 399–410. Pollock, K. H., Hoenig, J. M., Hearn, W. S. & Calingaert, B., 2001. Tag reporting rate estimation: 1. An evaluation of the high–reward tagging method. North American Journal of Fisheries Management, 21: 521–532. Pollock, K. H., Nichols, J. D., Brownie, C. & Hines, J. E., 1990. Statistical inference for capture–recapture experiments. Wildlife Monographs, 107. Pollock, K. H., Nichols, J. D., Simons, T.R., Farnsworth, G. L., Bailey, L. & Sauer, J. R., 2002. Large scale wildlife monitoring studies: statistical methods for design and analysis.

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Environmetrics, 13: 105–119. Pollock, K. H., & Otto, M. C., 1983. Robust estimation of population size in closed animal populations from capture–recapture experiments. Biometrics, 39: 1035–1049. Pospahala, R. S., Anderson, D. R. & Henny, C. J., 1974. Population ecology of the mallard. II. Breeding habitat conditions, size of the breeding populations, and production indices. U.S. Fish and Wildlife Service Resource Publication, 115. Powell, L. A. (in press). A multi–state capture–recapture model using stable isotope data to enhance estimation of movement rates. Condor. Powell, L. A., Conroy, M. J., Hines, J. E., Nichols, J. D. & Krementz, D. G., 2000. Simultaneous use of mark–recapture and radio telemetry to estimate survival, movement, and capture rates. Journal of Wildlife Management, 64: 302– 313. Pradel, R., 1993. Flexibility in survival analysis from recapture data: handling trap–dependence. In: Marked individuals in the study of bird population: 29–37 (J.–D. Lebreton & P. M. North, Eds.). Birkhauser Verlag, Basel, Switzerland. – 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Pradel, R., Hines, J. E., Lebreton, J.–D. & Nichols, J. D., 1997. Capture–recapture survival models taking account of transients. Biometrics, 53: 60–72. Rexstad, E. & Burnham, K. P., 1991. User’s guide for interactive program CAPTURE. Colorado Cooperative Fish and Wildlife Research Unit, Fort Collins, Colorado. Robson, D. S. & Youngs, W. D., 1971. Statistical analysis of reported tag–recaptures in the harvest from an exploited population. Biometrics Unit, Cornell University, Ithaca, New York. BU– 369–M. Royle, J. A., 2004. N–mixture models for estimating population size from spatially replicated counts. Biometrics, 60: 108–115. Royle, J. A. & Nichols, J. D., 2003. Estimating abundance from repeated presence–absence data or point counts. Ecology, 84: 777–790. Sandland, R. L. & Kirkwood, P., 1981. Estimation of survival in marked populations with possibly dependent sighting probabilities. Biometrika, 68: 531–541. Schiff, S. J., So, P., Chang, T., Burke, R. E. & Sauer, T., 1996. Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble. Physical Review E, 54: 6708–6724. Schnabel, Z. E., 1938. The estimation of the total fish population of a lake. American Mathematical. Monthly, 45: 348–352. Schwarz, C. J. & Arnason, A. N., 1996. A general methodology for the analysis of capture–recapture experiments in open populations. Biometrics, 52: 860–873. Schwarz, C. J., Burnham, K. P. & Arnason, A. N.,

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1988. Post–release stratification in band–recovery models. Biometrics, 44: 765–785. Schwarz, C. J., Schweigert, J. F. & Arnason, A. N., 1993. Estimating migration rates using tag recovery data. Biometrics, 49: 177–193. Seber, G. A. F., 1965. A note on the multiple– recapture census. Biometrika, 52: 249–259. – 1970. Estimating time–specific survival and reporting rates for adult birds from band returns. Biometrika, 57: 313–318. – 1971. Estimating age–specific survival rates for birds from bird–band returns when the reporting rate is constant. Biometrika, 58: 491–497. – 1982. The estimation of animal abundance and related parameters. MacMillan, New York. Skalski, J. R., Hoffman, A. & Smith, S. G., 1993. Testing the significance of individual– and cohort–level covariates in animal survival studies. In: Marked individuals in the study of bird population: 39–49 (J.–D. Lebreton & P. M. North, Eds.). Birkhauser Verlag, Basel, Switzerland. Skvarla, J., Nichols, J. D., Hines, J. E. & Waser, P. M. (in press). Modeling interpopulation dispersal by banner–tailed kangaroo rats. Ecology. Smith, S. G., Skalski, J. R., Schlechte, W., Hoffman, A. & Cassen, V., 1994. SURPH.1 Manual. Statistical survival analysis for fish and wildlife tagging studies. Bonneville Power Administration, Portland, Oregon. Stanley, T. R. & Burnham, K. P., 1998. Estimator selection for closed–population capture–recapture. Journal of Agricultural, Biological, and Environmental Statistics, 3: 131–150. Stearns, S. C., 1976. Life history tactics: a review

of ideas. Quarterly Review of Biology, 51: 3–47. Trenkel, V. M., Elston, D. A., & Buckland, S. T., 2000. Fitting population dynamics models to count and cull data using sequential importance sampling. Journal of the American Statistical Association, 95: 363–374. Walters, C. J., 1986. Adaptive management of renewable resources. MacMillan, New York. Watts, D. J. & Strogatz, S. H., 1998. Collective dynamics of "small–world" networks. Nature, 393: 440–442. White, G. C., 1983. Numerical estimation of survival rates from band recovery and biotelemetry data. Journal of Wildlife Management, 47: 716–728. White, G. C., Anderson, D. R., Burnham, K. P. & Otis, D. L., 1982. Capture–recapture and removal methods for sampling closed populations. Los Alamos National Laboratory, LA–8787–NERP. White, G. C. & Burnham, K. P., 1999. Program MARK: survival rate estimation from both live and dead encounters. Bird Study, 46: 120–139. White, G. C., Burnham, K. P. & Anderson, D. R., 2001. Advanced features of program MARK. In: Wildlife, land and people: priorities for the 21st century: 368–377 (R. Field, R. J. Warren, H. Okarma & P. R. Sievert, Eds.). The Wildlife Society, Bethesda, Maryland. White, G. C. & Lubow, B.C., 2002. Fitting population models to multiple sources of observed data. Journal of Wildlife Management, 66: 300–309. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, New York.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

Evolutionary biology and life histories C. R. Brown & D. L. Thomson

Brown, C. R. & Thomson, D. L., 2004. Evolutionary biology and life histories. Animal Biodiversity and Conservation, 27.1: 21–22. The demographic processes that drive the spread of populations through environments and in turn determine the abundance of organisms are the same demographic processes that drive the spread of genes through populations and in turn determine gene frequencies and fitness. Conceptually, marked similarities exist in the dynamic processes underlying population ecology and those underlying evolutionary biology. Central to an understanding of both disciplines is life history and its component demographic rates, such as survival, fecundity, and age of first breeding, and biologists from both fields have a vested interest in good analytical machinery for the estimation and analysis of these demographic rates. In the EURING conferences, we have been striving since the mid 1980s to promote a quantitative understanding of demographic rates through interdisciplinary collaboration between ecologists and statisticians. From the ecological side, the principal impetus has come from population biology, and in particular from wildlife biology, but the importance of good quantitative insights into demographic processes has long been recognized by a number of evolutionary biologists (e.g., Nichols & Kendall, 1995; Clobert, 1995; Cooch et al., 2002). In organizing this session, we have aimed to create a forum for those committed to gaining the best possible understanding of evolutionary processes through the application of modern quantitative methods for the collection and interpretation of data on marked animal populations. Here we present a short overview of the material presented in the session on evolutionary biology and life histories. In a plenary talk, Brown & Brown (2004) explored how mark–recapture methods have allowed a better understanding of the evolution of group–living and alternative reproductive tactics in colonial cliff swallows (Petrochelidon pyrrhonota). By estimating the number of transient birds passing through colonies of different sizes, they showed that the number of ectoparasites at a colony site depends in part on how many transient swallows visit and introduce bugs from outside the group. Brown & Brown (2004) found that annual survival was related to the likelihood of a bird engaging in extra–pair copulation or intraspecific brood parasitism and could thus infer the relative costs and benefits of these reproductive tactics. These authors also showed that first–year survival patterns can explain the observed clutch–size distribution, in which clutches of intermediate size tend to be the most productive in most years even though larger clutches always fledge more offspring. The importance of using mark–recapture for questions in behavioral ecology was emphasized by Brown & Brown’s (2004) study. The likelihood that a sexually mature individual chooses to breed in a given year was explored by Reed et al. (2004) in a study of greater snow geese (Chen caerulescens atlantica). By combining information from mark–recapture, telemetry, and nest surveys, these authors found that adult snow geese were more likely to breed in years with low snow cover and that breeding propensity was positively related to nest density in a given year. Reed et al.’s (2004) results illustrate surprisingly high variation in reproductive effort among geese between years and emphasize the importance of environmental conditions (in this case, the timing of spring snow melt) in determining reproductive success of snow geese and other arctic–nesting birds. The trade–off between survival and reproduction, and to what degree this trade–off reflects a cost of reproduction, was addressed by Tavecchia et al. (2005) for Soay sheep (Ovis aries) and Rivalan et al. (pers. comm.) for leatherback sea turtles (Dermochelys coriacea). In sheep, the cost of reproduction was a quadratic function of a mother’s age, being greatest for the youngest and oldest females, but this cost applied only during Charles R. Brown, Dept. of Biological Sciences, Univ. of Tulsa, Tulsa, Oklahoma 74104, U.S.A. David L. Thomson, Max Planck Inst. for Demographic Research, Konrad–Zuse–Str. 1, D–18057, Rostock Germany. ISSN: 1578–665X

Brown & Thomson

severe environmental conditions (cold and wet winters) that coincided with high population density (Tavecchia et al., 2005). The significant interaction between age and time may have been responsible for maintaining differences among cohorts in demographic parameters such as survival (Tavecchia et al., 2005). In turtles, Rivalan et al. (pers. comm.) found evidence of a typical cost of reproduction in which the number of reproductive seasons in an individual’s lifetime was inversely related to the extent of reproductive investment in a given season. By statistically accounting for loss of tags in these turtles, which can be substantial and thus may potentially bias state–transition probabilities, the authors discovered that fitness was roughly equivalent for all females, regardless of how often they attempted to breed. The studies by Tavecchia et al. (2005) and Rivalan et al. (pers. comm.) illustrate the benefits of using multi–state models in accounting for recapture/re– sighting probabilities. The pattern of senescence in wild populations was a focus of the work by Catchpole et al. (2004) and Gaillard et al. (2004). In the well–studied red deer (Cervus elaphus) of Rum, senility was suggested by a survival probability that declined with age among the oldest age classes in both males and females, although there was little evidence for age–dependent survival among the younger age classes (Catchpole et al., 2004). As in many species, some animals dispersed from the study area, and notably Catchpole et al. (2004) accounted for dispersal in their estimates of age–dependent survival. In a comparative study of roe deer (Capreolus capreolus) and bighorn sheep (Ovis canadensis), Gaillard et al. (2004) combined traditional mark– recapture analysis with the commonly used Gompertz and Weibull models to describe senescence patterns. They found that senescence in these species can be generally described by the Gompertz model, a result likely to be of interest to researchers working on senescence in a variety of taxa. Both studies (Catchpole et al., 2004; Gaillard et al., 2004) represent useful refinements of the typical approach in senescence studies of simply fitting linear or quadratic relationships between survival and age. Two studies addressed important methodological issues. E. Cam et al. (in press) examined the assumption of homogeneous survival probabilities when estimating recruitment to breeding status. Various studies have explored temporal variation in age of first breeding and how this co–varies with social or environmental factors, but most of these studies have ignored the size of the pool of prebreeding individuals, most of whom are not encountered prior to actual recruitment. Methods to date have assumed no difference in pre–breeder survival among the groups of recruiting individuals being compared. Using numerical simulations, Cam et al. (in press) addressed violations of this assumption, finding that recruitment estimates can be both positively or negatively influenced by variation in pre–breeder survival, depending on circumstances, and suggest that consideration of pre–breeder survival should be taken into account in analyses of recruitment. Marshall et al. (2004) address a problem afflicting most mark– recapture studies, that of marked individuals settling beyond the boundaries of a study area and how this may affect estimates of survival. These authors simulated re–sightings of birds in areas surrounding a study "core" and found that, at least for organisms such as songbirds distributed in contiguous habitat, re–sightings in the progressively larger study plots around the core can be used to achieve relatively unbiased survival estimates. The problem addressed by Marshall et al. (2004) remains an important one, and further exploration of this issue is urgently needed. References Brown, C. R. & Brown, M. B., 2004. Mark–recapture and behavioral ecology: a case study of Cliff Swallows. Animal Biodiversity and Conservation, 27.1: 23–34. Cam, E., Cooch, E. G. & Monnat, J.–Y. (in press). Earlier recruitment or earlier death? On the assumption of homogeneous survival rates in capture–recapture models to estimate recruitment. Ecological Monographs. Catchpole, E. A., Fan, Y., Morgan, B. J. T., Clutton–Brock, T. H. & Coulson, T. N., 2004. Modelling senility and dispersal of red deer. Journal of Agricultural, Biological and Environmental Statistics, 9: 1–26. Clobert, J., 1995. Capture–recapture and evolutionary ecology: a difficult wedding? Journal of Applied Statistics, 22: 989–1008. Cooch, E. G., Cam, E. & Link, W., 2002. Occam’s shadow: levels of analysis in evolutionary ecology –where to next? Journal of Applied Statistics, 29: 19–48. Gaillard, J.–M., Viallefont, A., Loison, A. & Festa–Bianchet, M., 2004. Assessing senescence patterns in populations of large mammals. Animal Biodiversity and Conservation, 27.1: 47–58. Marshall, M. R., Diefenbach, D. R., Wood, L. A. & Cooper, R. J., 2004. Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: study designs that may help. Animal Biodiversity and Conservation, 27.1: 59–72. Nichols, J. D. & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. Journal of Applied Statistics, 22: 835–846. Reed, E. T., Gauthier, G. & Giroux, J.–F., 2004. Effects of spring conditions on breeding propensity of greater snow goose females. Animal Biodiversity and Conservation, 27.1: 35–46. Tavecchia, G., Coulson, T. N., Morgan, B. J. T., Pemberton, J. M., Pilkington, J. C., Gurland, F. M. D. & Clutton–Brock, T. H., 2005. Predictors of reproductive cost in female Soay sheep. Journal of Animal Ecology, 74: 201–213.

Animal Biodiversity and Conservation 27.1 (2004)

Mark–recapture and behavioral ecology: a case study of Cliff Swallows C. R. Brown & M. B. Brown

Brown, C. R. & Brown, M. B., 2004. Mark–recapture and behavioral ecology: a case study of Cliff Swallows Animal Biodiversity and Conservation, 27.1: 23–34. Abstract Mark–recapture and behavioral ecology: a case study of Cliff Swallows.— Mark–recapture and the statistical analysis methods associated with it offer great potential for investigating fitness components associated with particular behavioral traits. However, few behavioral ecologists have used these techniques. We illustrate the insights that have come from a long–term mark–recapture study of social behavior in Cliff Swallows (Petrochelidon pyrrhonota). The number of transient swallows passing through a colony per hour increased with colony size and was responsible in part for increased rates of ectoparasite introduction from outside the group into the larger colonies. Annual survival probabilities of males engaging in extra–pair copulation attempts were lower than those of males not seen to commit extra–pair copulations, suggesting that males who engage in this behavior may be inferior individuals and that females do not benefit from copulating with them. Females engaging in intraspecific brood parasitism had higher annual survival probabilities than ones either parasitized by others or not known to be either hosts or parasites. This suggests that parasitic females are high–quality birds and that brood parasitism is an effective reproductive tactic for increasing their fitness. By estimating first–year survival of chicks, we found that a clutch size of 4 eggs is often the most productive, on average, as measured by recruitment of offspring as breeders, although birds laying the more uncommon clutch size of 5 fledge more young on average. This helps to explain the observed clutch–size distribution in which clutch size 4 is the most commonly produced. Key words: Clutch size, Coloniality, Parasitism, Social behavior. Resumen Marcaje–recaptura y ecología del comportamiento: el ejemplo de las golondrinas de frente canela.—El método de marcaje–recaptura y los métodos de análisis estadísticos asociados al mismo brindan un enorme potencial para investigar componentes del estado de salud asociados a determinados rasgos de comportamiento. Sin embargo, son pocos los ecólogos del comportamiento que han empleado dichas técnicas. En este artículo se presentan los resultados de un estudio a largo plazo de las golondrinas de frente canela (Petrochelidon pyrrhonota), en el que se empleó la técnica de marcaje–recaptura para analizar su comportamiento social. Observamos que cuanto mayor era el tamaño de la colonia, más elevado era el número de golondrinas que pasaba por la misma cada hora, siendo esto parcialmente responsable, en las colonias de mayor tamaño, de un aumento en las tasas de introducción de ectoparásitos desde fuera del grupo. Las probabilidades de supervivencia anual de los machos que intentaron llevar a cabo cópulas fuera de la pareja fueron inferiores a las de los machos a los que no se les había visto copular con una hembra distinta a la de su pareja, lo que sugiere que los machos que adoptan este comportamiento pueden ser individuos inferiores, y que las hembras no se benefician de copular con ellos. Las probabilidades de supervivencia anual de las hembras que participaron en parasitismo de nidada intraespecífico fueron más elevadas que las de aquellas que habían sido parasitadas por otras, o que las de aquellas que no eran conocidas como huéspedes ni como parásitas. Ello sugiere que las hembras parásitas son aves de alta calidad y que utilizan el parasitismo de nidada como una táctica reproductora eficaz para aumentar su buen estado de salud. La estimación de supervivencia de los polluelos durante el primer año de vida nos permitió determinar que una puesta de cuatro huevos suele ser, por término medio, la más productiva, medido ISSN: 1578–665X

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según el reclutamiento de polluelos como aves reproductoras, aunque las aves que tienen una puesta poco usual de cinco huevos se desarrollan, por término medio, a una edad más temprana. Esto ayuda a explicar la distribución observada del tamaño de las puestas, siendo la de cuatro huevos la que se produce con mayor frecuencia. Palabras clave: Tamaño de la puesta, Colonialidad, Parasitismo, Comportamiento social. Charles R. Brown & Mary Bomberger Brown, Dept. of Biological Sciences, Univ. of Tulsa, Tulsa, Oklahoma 74104, U.S.A. Corresponding to: C. R. Brown. E–mail: charles-brown@utulsa.edu

Animal Biodiversity and Conservation 27.1 (2004)

Introduction That the behavior of animals is adapted to the environmental or social conditions under which those animals live is a central tenet of behavioral ecology. Studying that adaptation usually requires measuring the fitness associated with a particular behavioral trait under particular ecological conditions. Although fitness is one of the most fundamental concepts in evolutionary biology dating back, in behavioral ecology at least, to W. D. Hamilton’s (1964) work on social insects, there is debate about the best way to measure it in natural populations (Grafen, 1988). Fitness is probably best defined as the intrinsic rate of natural increase of a phenotype within a population, with the three major components of fitness being age at maturity, survival, and fecundity (Roff, 2002). The latter two parameters tend to be ones commonly measured in ornithological field studies, especially fecundity as reflected in annual nesting success. Most of the information on survival probabilities in natural populations has come from applied work done with wildlife management or conservation objectives. Surprisingly few studies in behavioral ecology have directly measured either survival or lifetime reproductive success (a parameter based on survival) in relation to a behavioral trait of interest. For example, of the 117 papers published in 2002 in the field’s premier journal Behavioral Ecology (volume 13), only three (2.6%) incorporated mark–recapture into their study design, and only one of those used modern statistical methods to analyze the data. Although some of these papers were based on laboratory studies for which mark–recapture may not have been appropriate, in many cases the same questions could have been addressed in the field and the behavioral traits under study linked to fitness. For example, mark–recapture methods permit estimation of survival associated with different mating strategies or different levels of sexual ornamentation and thus the estimation of fitness components associated with these different behavior patterns and their evolution. Many behavioral studies have tended instead to relate the expression of a behavioral trait to indirect correlates of fitness such as energetic gain (e.g., from foraging) or observed mating success (e.g., in comparing extra–pair mating strategies). In part, this has been because many studies in behavioral ecology have been short–term or exclusively experimental in focus, and have not followed large numbers of marked animals over multiple– time periods in natural environments. In this paper, we illustrate the insights made possible by a long–term mark–recapture study of colonially nesting Cliff Swallows (Petrochelidon pyrrhonota), and how this work has furthered our understanding of questions in behavioral ecology. Our work focuses broadly on the adaptive significance of coloniality and the consequences of living in groups (colonies) of different sizes. Specifically, here we address how estimating the number of transient birds has been used in an experimental

study of between–group parasite transmission, how annual survival probabilities have provided information on alternative reproductive tactics such as extra–pair copulation and intraspecific brood parasitism, and how annual survival probabilities have been used to investigate tradeoffs associated with different clutch sizes. This paper reviews previously published empirical work, and additional methodological details (as well as other data and analyses) can be found in the original studies (Brown & Brown, 1998, 1999, 2004). We conclude by urging more behavioral ecologists to take advantage of mark– recapture methods and the many statistical analysis tools now available, particularly for studies that require estimation of fitness in relation to behavior. Background Cliff Swallows are highly colonial passerines that breed throughout most of western North America (Brown & Brown, 1995). They build gourd–shaped mud nests and attach them to the vertical faces of cliff walls, rock outcrops, or artificial sites such as the eaves of buildings or bridges. Their nests tend to be stacked closely together, often sharing walls. Cliff Swallows are migratory, wintering in southern South America, and have a relatively short breeding season in North America. They begin to arrive at our study site in late April or early May and depart by late July. They generally raise only one brood. Cliff Swallows are associated with a variety of ectoparasites, endoparasites, and viruses throughout their range (Monath et al., 1980; Scott et al., 1984; Brown & Brown, 1995; Brown et al., 2001). The ectoparasites, in particular the hematophagous swallow bug (Hemiptera, Cimicidae: Oeciacus vicarius), are responsible for much of the nestling mortality and nest failures that occur in our study area (Brown & Brown, 1986, 1996). We have studied ectoparasitism in Cliff Swallows by removing swallow bugs from nests by spraying (fumigating) them with a dilute solution of an insecticide, Dibrom, that is highly effective in killing swallow bugs. Our study site is centered at the Cedar Point Biological Station (41°13' N, 101°39' W) near Ogallala, in Keith County, along the North and South Platte Rivers, and also includes portions of Deuel, Garden, and Lincoln counties, southwestern Nebraska, USA. We have studied Cliff Swallows there since 1982. There are approximately 160 Cliff Swallow colony sites in our 150 × 50 km study area, with about a third of these not used in a given year. Colony size varies widely; in our study area, it ranges from 2 to 3700 nests, with some birds nesting solitarily. Over a 20–year period, mean (! SE) colony size (n = 1363) was 363 (! 16) nests. Each colony site tends to be separated from the next nearest by 1–10 km but in a few cases by m 20 km. In our study area, the birds nest on both natural cliff faces and artificial structures such as bridges, buildings, and highway culverts. The study site is described in detail by Brown & Brown (1996).

Transient birds and parasite transmission A number of field studies on various taxa have shown that parasitism by ectoparasites (or infection by microparasitic pathogens) increases as group size or density increases (e.g., Brown & Brown, 1986; Moore et al., 1988; Rubenstein & Hohmann, 1989; Hieber & Uetz, 1990; Davies et al., 1991; Côté & Poulin, 1995; Hoogland, 1995; Arneberg et al., 1998; Krasnov et al., 2002). Although this seems to be a common pattern, we know little about the factors that cause it. One possibility is that larger groups of hosts represent a larger “target area” for parasites seeking hosts, and consequently more parasites successfully immigrate into areas with large concentrations of hosts. Another possibility is that once introduced into a group, parasites or pathogens are more easily spread by the greater spatial proximity of hosts in large colonies. The relative importance of these mechanisms has not been empirically investigated in any species, although epidemiological theory has recognized that immigration of parasites between host groups can be critical for sustaining epidemics and preventing local extinction of parasite populations (Cliff et al., 1981; Loehle 1995; White et al., 1996; Swinton et al., 1998; Grenfell et al., 2001). Cliff Swallows show the typical increase in levels of ectoparasitism with group size; infestations of swallow bugs per nest increased significantly over an observed range of 1–1,600 nests in colony sizes (Brown & Brown, 1986, 1996). In addition, an encephalitis–related alphavirus (Buggy Creek virus) associated with Cliff Swallows also increased with colony size, as measured by per–nest infection probabilities of swallow bugs that vector the virus (Brown et al., 2001). Larger Cliff Swallow colonies contain more bugs for several possible reasons that include greater introduction of bugs into a colony from the outside, greater transmission of bugs within a colony, and larger colonies being more likely to be re–occupied by birds each year, promoting bug survival (Brown & Brown, 1996). In order to understand some of the mechanisms responsible for increased parasitism in larger groups, in 1999–2002 we experimentally measured the transmission of swallow bugs between colony sites in Cliff Swallows (Brown & Brown, 2004). We quantified transmission by fumigating entire colonies and counting the number of parasites appearing in the weekly interval between fumigations. This experiment showed that the number of bugs introduced into a colony per nest per week increased significantly with colony size (Brown & Brown, 2004), suggesting that at least some of the increased parasitism in larger Cliff Swallow colonies can be attributed to a greater likelihood of bugs being brought into a colony from outside the group. The increased immigration of swallow bugs into larger Cliff Swallow colonies could occur because (1) more transient birds visit large colonies and/or (2) the transient birds visiting large colonies are more likely to be infested with dispersing bugs than

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are those visiting small colonies. Being wingless, bugs disperse only by clinging to the feet and legs of swallows that move from one colony to another; dispersal seems most likely to occur when a nest or entire colony site is not occupied in a given season, yet transient birds briefly visit those sites while investigating old nests. There is a relatively large pool of non–breeding Cliff Swallows in our study area in any given year, and these birds circulate among different colony sites, apparently assessing them for future years (Brown et al., 2000). These individuals are transients at each site in that they tend to be caught once at a colony, then vanish from that site sometimes to be caught again elsewhere. We used mark–recapture to estimate the number of transient birds visiting colonies of different sizes. By setting mist nets across the upwind end of culverts containing nests, we caught birds as they exited the colony site. Days on which birds were captured at the experimental colonies (usually 3– 3.5 hours with a net open per day) ranged from 9 to 33 at a site per season and extended from the period of the birds’ arrival until most had departed from the colonies for the year. Total bird captures at the experimental colonies, in order of ascending colony size, were 529, 264, 613, 680, 2478, 2858, 3825, 4180, 4520, 5710, 3477, and 4149. All birds were banded with U.S. Fish and Wildlife Service bands upon initial capture (further details in Brown & Brown, 2004). Transients are defined as birds not resident at a colony that pass through the site on a temporary basis. Those individuals caught only once at a colony include the transient class, but they also may include some residents who just happened to never be caught again. To estimate the fraction of the one–time captures that consisted of transients, we estimated the daily proportion of transients among those birds captured during each netting session with the method of Pradel et al., (1997). By fitting an age–dependent model to the capture data, the "first–year" age class approximates the transients, who, by virtue of not reappearing at a site, have much lower apparent survival, &, than the residents who tend to be caught multiple times. The estimate of "first–year" survival allows one to calculate $t, the proportion of transients in each time interval (t), as 1 – (f1t / f2t), where f1t is apparent survival probability of the "first–year" age class and f2t is apparent survival probability of the "older" age class (Pradel et al., 1997). The proportion of transients, $t, was multiplied by the number of newly caught birds during each capture session and divided by the number of hours that the net was open that day to produce the number of transients per hour per day. The calculation of $t for each netting session (Pradel et al., 1997) specifically excludes the fraction of one–time captures attributed to residents who were never caught again. Survival models were fit, and parameter estimates produced, by the program MARK (White & Burnham, 1999). Each colony was analyzed separately, as the number of capture occasions, dates of sam-

Animal Biodiversity and Conservation 27.1 (2004)

40 Mean (± SE) number of transient birds per hour per day

pling, and intervals between the occasions were different for each site. For each colony, the best–fitting model (used to estimate $t) was usually one with time–dependent survival probabilities for the "first–year" age class, time–constant survival for the "older" age class, and time–dependent recapture probability the same among both age classes. Any differences in recapture probabilities among the different colonies (these tended to vary each day because of differences in weather conditions, time nets were open, etc…) were accounted for in the estimates of & (and thus $t) calculated by MARK. See Brown & Brown (2004) for more details. More transient Cliff Swallows passed through the larger colonies (fig. 1). Averaged over all days throughout the season, the total number of transient birds per hour per day increased significantly with colony size (fig. 1). This indicates that the increased ectoparasitism in larger colonies is attributable, in part, to more transient birds passing through those colonies. In addition, the weekly change in the average number of transients at a site tended to match the weekly change in the number of bugs introduced, indicating that transient presence is a determinant of bug immigration rates (Brown & Brown, 2004). Although there are other factors that also contribute both to the higher rates of between–group transmission and the increased overall incidence of parasitism in larger Cliff Swallow colonies (Brown & Brown, 2004), this analysis reveals that transient visitation of colonies is not uniform. Often considered a nuisance effect when one is trying to estimate survival of residents, the presence of transients in this case is biologically interesting. Formal mark–recapture models (e.g., Pradel et al., 1997) allow one to estimate the number of transients in a way that accounts for the fraction of resident birds who also were caught only once at a site. Without this approach, it would be impossible to estimate the total number of transient birds based strictly on how many times an individual was caught.

500

1000 Colony size

1500

2000

Fig. 1. Mean (! SE) number of transient Cliff Swallows per hour per day over the entire nesting season in relation to Cliff Swallow colony size. The mean number of transient birds increased significantly with colony size (rs = 0.66, P = 0.02, n = 12 colonies). Total sample sizes (number of birds caught) used to generate the estimates of the number of transients are given in the text (Brown & Brown, 2004). Fig. 1. Promedio (! EE) de golondrinas de frente canela transeuntes, por hora y día, a lo largo de toda la estación de nidificación según el tamaño de la colonia. El promedio de aves transeuntes aumentó significativamente con el tamaño de la colonia (rs = 0,66, P = 0,02, n = 12 colonias). En el texto se detallan los tamaños de muestras totales (número de aves capturadas) empleados para generar las estimaciones del número de aves transeuntes (Brown & Brown, 2004).

Annual survival and alternative reproductive tactics One of the most striking realizations in behavioral ecology over the last two decades is that most animal populations contain individuals that parasitize the parental care provided to offspring by conspecifics. Parasitic exploitation of others occurs through both extra–pair mating and (in egg–laying species) intraspecific brood parasitism. Many studies have examined these reproductive tactics in various taxa, especially birds (e.g., Gladstone, 1979; Yom–Tov, 1980; Andersson, 1984; Rohwer & Freeman, 1989; Westneat et al., 1990; Field, 1992; Birkhead & Møller, 1992; Lyon, 1993; Brown & Brown, 1996, 1998, 2001; McRae, 1998). Much of this work has focused on determining the frequency, timing, and behavioral dynamics of these tactics in different populations and the socio–ecological contexts in which they occur. There is little information, however, on the

long–term fitness consequences associated with extra–pair mating or brood parasitism. For example, is brood parasitism a last–ditch tactic used by inferior individuals who cannot establish their own nest or provide acceptable levels of parental care, or is it an effective supplemental reproductive strategy used by superior individuals to enhance their inclusive fitness? Are there tradeoffs between increasing short– term reproductive success and suffering long–term survival costs that result from, for example, an increased energetic expenditure, greater exposure to sexually transmitted diseases, or increased vulnerability to predation? Answering these questions requires estimating components of fitness associated with extra–pair mating and intraspecific brood parasitism, such as annual survival probability.

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Extra–pair copulation Extra–pair copulation in Cliff Swallows occurs in two contexts: among birds away from nest sites while they gather mud for nest construction and among neighboring birds while at nests (Brown & Brown, 1996, 1998). Neighbors engage in extra– pair copulations when a male trespasses into a nearby nest during the male owner’s absence. Other males regularly patrol the mud–gathering sites and attempt extra–pair copulations with females who come there to collect nesting material. Most extra– pair copulation attempts seem to be initiated by males, and females may or may not resist. Extra– pair copulations are attempted both by males who are resident in a colony, maintaining a nest and mate of their own, and by nonresidents who circulate among colony sites (Brown & Brown, 1996). Because nonresidents are difficult to catch and color–mark, our data on survival came from resident males only. We made observations of extra– pair copulation attempts at two colonies where we could get close enough to the mud–gathering sites to identify color marks (paint stripes on the birds’ white forehead patches) of resident males that perpetrated the copulation attempts. Other observations were made on color–marked males while watching birds at their nests in these same colonies. Any color–marked male seen to engage in at least one extra–pair copulation attempt was designated an EPC male, whereas color–marked males in the same colonies not seen to engage in any extra–pair copulation attempts were classified as non–EPC males. For analyses involving extra–pair copulations (and brood parasitism, below), we used cohorts initially marked from 1983–1987 with recaptures extending through 1995. Annual survival was modeled with EPC males (n = 76 birds) and non–EPC males (n = 103 birds) as separate groups versus as a combined group. Recapture probability in all models was time–dependent. The best–fitting model (AIC of 634.8 compared to next lowest of 642.8) was one with EPC and non–EPC males as separate groups, and from this model we estimated annual survival probability (! SE) as 0.413 (! 0.040) for EPC males and 0.614 (! 0.055) for non–EPC males (Brown & Brown, 1998). Thus, annual survival of males engaging in extra–pair copulations was only about two–thirds that of males not seen to mate with extra–pair females. There appeared to be no other phenotypic differences (such as body mass) between the two classes of males that might have accounted for these results (Brown & Brown, 1998). The lower annual survival probability for males that engaged in extra–pair copulations might mean that extra–pair mating is costly for males, perhaps through increased risk of sexually transmitted diseases (Sheldon, 1993; Lockhart et al., 1996). More likely, however, the difference in survival probability reflects inherent male quality (Brown & Brown, 1998). If so, females who mated with these males via extra–pair copulations did so with relatively inferior partners. This result is in contrast to both

the widely held view that extra–pair copulations represent a way for females to achieve matings with males of high genetic quality (e.g., Westneat et al., 1990; Birkhead & Møller, 1992; Wagner, 1993; Jennions & Petrie, 2000) and field data on other species showing that longer–lived, more experienced, or "better" males are more likely to achieve extra–pair copulations (e.g., Kempenaers et al., 1992; Wagner et al., 1996; Weatherhead & Boag, 1995). In Cliff Swallows, extra–pair mating may be a "best–of–a–bad–job" strategy for inferior males, with deleterious consequences for females who participate either willingly or unwillingly (Brown & Brown 1998). This insight was made possible only by relating the behavior to annual survival probability using mark–recapture, and this study remains one of the few (if not only one) to measure long–term survival of males who do and do not exhibit this alternative reproductive tactic. Intraspecific brood parasitism Cliff Swallows commonly brood–parasitize nests with up to 20% or more in some colonies containing an egg laid by another female. All known cases of brood parasitism have been by females who were resident in the colony and maintained nests of their own, and parasitized nests are usually within a five–nest radius of the parasite’s own nest (Brown & Brown, 1989, 1991). Cliff Swallows parasitize nests in two ways: by laying eggs in nests during the host’s laying period and by physically moving eggs from the parasite’s nest to a host’s. Parasites time their laying or transfer of eggs to coincide with the host’s own laying stage, such that parasitic eggs hatch synchronously with the host’s. By observing color– marked birds at their nests and determining which individuals were consistently associated with a particular nest (thus owning it), we designated parasitic females as those seen laying or transferring eggs into a nest not their own. Host females were those in the same colonies whose nests were seen being parasitized by another bird or whose nests were found to be parasitized based on nest–check data. Daily or bi–daily checks of nest contents allowed us to infer instances of parasitic laying as cases where two or more eggs appeared per 24–hour period, and egg transfer as cases where an egg appeared in a nest during the host’s incubation period but hatched synchronously with the host’s eggs (Brown & Brown, 1998). Birds designated as neither hosts nor parasites were those color–marked individuals who were not observed to either parasitize others or be parasitized and whose nests showed no evidence of brood parasitism from nest checks. Additional details on how we designated the different classes of females are provided in Brown & Brown (1998). Annual survival was modeled with four groups of females: those known to be parasites (n = 17), host females parasitized by laying (n = 32), host females parasitized by transfer (n = 25), and females not known to be parasites or hosts (n = 65). A model with both survival and recapture probability depend-

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ent on female status was the best fit, with a lower AIC value (435.9) than a model with no effect of female status on survival (AIC = 438.1; Brown & Brown, 1998). From this, we estimated annual survival probability (! SE) as 0.761 (! 0.055) for parasitic females, 0.289 (! 0.082) for host females parasitized by laying, 0.595 (! 0.076) for host females parasitized by transfer, and 0.686 (! 0.045) for females that were neither parasites nor hosts (Brown & Brown, 1998). Parasitic females thus had the highest annual survivorship and host females who had parasitic eggs laid in their nests the lowest. These results suggest either that brood parasitism is a reproductive tactic used by high–quality females who are likely to live longer for that reason, or that parasitizing others reduces the cost of parental care to the extent that survival of the parasitic females is increased. The latter seems less likely because parasitic females maintain nests of their own and raise normal–sized clutches in their own nests (Brown & Brown, 1998). High–quality individuals will be those who have the necessary resources (time, energy) to parasitize their neighbors at little cost to themselves and at the same time have higher annual survivorship. Host individuals, on the other hand (especially those parasitized by laying), may be inferior individuals, and it is perhaps for this reason that they are parasitized. Parasitism only occurs when a nest is left unattended momentarily, and if host females are, for example, inherently inferior foragers, they may more often leave their nests unguarded because they take more time to find food. In addition to higher personal survival of the parasitic females, we found that offspring from the parasites’ own nests had higher first–year survival (as measured to their first breeding season) than birds raised in all other nests (Brown & Brown, 1998). With more of the young that they themselves rear surviving and with their own breeding lifespan being longer, the consequence is higher fitness for parasitic females. These females have an estimated lifetime reproductive success almost twice that of any other class of females (hosts, non–parasites, non–hosts; Brown & Brown 1998). Brood parasitism, at least as a supplemental reproductive strategy, would presumably spread in the Nebraska population if it wasn’t regulated by extensive nest–guarding by most individuals. The cost of being parasitized and thus raising an unrelated chick seems to have selected for intense nest–guarding in Cliff Swallows. With such high levels of nest–guarding, potential parasites often simply do not have the opportunity to parasitize nests because relatively few are ever left unattended by an owner. These results represent one of the few attempts to measure the long–term fitness consequences of intraspecific brood parasitism in any bird, and, as with those on extra–pair copulation in Cliff Swallows, they tend to go against conventional wisdom. Intraspecific brood parasitism is sometimes thought to represent a last–ditch strategy used by inferior females who were not successful in competing for

nest sites or territories (e.g., Lyon, 1993; McRae, 1998; Sandell & Diemer, 1999). To the contrary, brood parasitism in Cliff Swallows is an effective tactic used by superior individuals to enhance their fitness. Using mark–recapture to estimate annual survival of different classes of individuals in Cliff Swallows has provided insights into the evolution of alternative reproductive tactics that would not have been possible from behavioral observations alone. The evolution of clutch size A major paradox in behavioral ecology is that in many species of birds, females typically lay fewer eggs and thus have fewer offspring than they can actually rear. Experiments have shown that the most productive clutch size, that is, the one yielding the most offspring surviving to fledge, is often not the most common (Klomp, 1970; Stearns, 1992; VanderWerf, 1992), and the most common clutch size is often smaller than the most productive. The most popular explanation for this paradox was that of Lack (1947, 1954), who argued that selection should favor birds that lay the most productive clutch size, and that the most common clutch size is in fact the most productive. This is because the larger clutches, while perhaps yielding more offspring to fledging, do not produce more eventual recruits into the breeding population because the chicks fledging from those large clutches may be in poorer shape and less likely to survive their first year. Furthermore, parents who lay and tend the larger clutches may themselves be less likely to survive to breed again, owing to the stress and additional work involved in raising a larger clutch. Thus, their fitness will be reduced, and there will be selection against laying the larger clutch sizes. Lack’s (1947, 1954) views on clutch–size evolution have been popular despite relatively little empirical support. In part, this has been because most studies, while often measuring fledging success associated with different clutch sizes and frequently in an experimental context, have not followed birds over multiple years to estimate either first–year recruitment or parental survival, and the few that have attempted this have not used modern mark– recapture statistical methods. We studied clutch size in Cliff Swallows by doing daily or bi–daily nest checks at colonies throughout the study area, using a dental mirror and small flashlight inserted through each nest’s mud neck. We defined clutch size as the maximum number of eggs ever recorded in a nest. Nests were monitored throughout incubation until hatching. Once hatching date was determined, we returned to the nest when the nestlings were 10 days old. At that time they were banded; their subsequent survival in later years was monitored through our long–term, extensive mist–netting at different colony sites in the study area. We recorded clutch–size data for 8,835 nests distributed among colonies of all sizes. Further details are provided in Brown & Brown (1999).

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50 40 Percentage

Cliff Swallows most often lay clutches of 4 eggs, but 3 eggs are also common (fig. 2). Five–egg clutches are uncommon (and those of 6 eggs so rare that they are not considered further). This overall clutch–size distribution (fig. 2), taken from all nests in our population during 10 years between 1982 and 1997, was similar to that for each year separately (Brown & Brown, 1999). Cliff Swallows exhibit the typical clutch–size paradox, with the average number of young surviving to fledge increasing steadily across the clutch size range of from 1 to 5 eggs in both fumigated nests (where ectoparasites had been removed) and nests exposed to natural levels of ectoparasites (Brown & Brown 1999; also see fig. 3). Thus, if more young are fledged on average from clutch size 5, why are clutches of 5 eggs so uncommon (fig. 2)? One fitness component associated with clutch size is the number of offspring recruited as breeders in the year(s) following fledging. This can be determined by knowing how many chicks fledge from nests with a given brood size and estimating the average first–year survival probability associated with that brood size (Cliff Swallows breed as yearlings). Multiplying the number fledged by the probability of surviving the first–year gives an index of annual reproductive success (Brown & Brown, 1999). For birds under natural conditions (exposed to ectoparasites), an age–stratified (age 1 vs all others), fully time–dependent model with three groups corresponding to brood sizes of 1–2, 3–4, and 5 best described first–year survival; this model had an AIC value of 5313.2, compared to an AIC value of 5322.4 for an otherwise equivalent model but without a brood–size effect (Brown & Brown, 1999). The time–dependence in first–year survival probabilities indicates yearly variation in the payoffs potentially associated with laying different numbers of eggs; we found that birds from brood sizes 1–2 had the highest first–year survival in three years, birds from brood sizes 3–4 had the highest first– year survival in four years, and birds from brood size 5 had the highest first–year survival in four years (Brown & Brown, 1999). Annual reproductive success, as measured by the number of young surviving to the following season, varied with clutch size depending on each season’s climatic conditions. In relatively cool seasons (as measured by daily high temperatures in June, the month of most brood–rearing in the study area), growth of ectoparasitic swallow bug populations in nests is slowed, as generation time is temperature– dependent. In such years, birds rearing broods of 5 had markedly higher annual reproductive success than those with smaller brood sizes (fig. 3A). However, in a more "average" year, climatologically, the advantage of a brood of 5 disappeared, with broods of 4 and 5 doing equally well (fig. 3B), and in a warm year, birds with broods of 5 did worse than those with either broods of 3 or 4 (fig. 3C). Interestingly, in all years, broods of 5 fledged the most offspring (fig. 3), contributing to the apparent clutch–size paradox had only fledging success been measured. Yet when

30 20 10 0

3 4 Clutch size

Fig. 2. Percentage distribution of Cliff Swallow clutch sizes, pooled across all years, n = 8,835 nests (Brown & Brown, 1999). Fig. 2. Distribución porcentual de los tamaños de puesta de las golondrinas de frente canela, obtenidas en todos los años, n = 8.835 nidos (Brown & Brown, 1999).

longer–term survival was measured through mark– recapture and related to climatic conditions, the perennial advantage of clutch size 5 disappeared. The greater uncertainty associated with larger clutches is probably the key to explaining the observed clutch–size distribution in Cliff Swallows (fig. 2). Whenever birds cannot predict climatic conditions during brood–rearing at the time they lay eggs (which is likely the case with Cliff Swallows), the safest strategy is a risk–averse one of going with the clutch size that, on average, is likely to be best. Laying a clutch of 5 eggs potentially results in a relatively large payoff if the season turns out to be cooler than average, but if it is warmer than average, there is a serious fitness cost associated with that clutch size. In an average year, clutches of 4 do as well as those of 5, so it appears that the least risky strategy is to lay 4 eggs, the most common clutch size. A similar conclusion was reached by Boyce & Perrins (1987), who found that the reduced fitness for Great Tits (Parus major) rearing large clutches in the occasional bad years was enough to select for smaller average clutches than could be produced in good or average years. Others have also suggested that higher variance in fitness associated with larger clutches will result in selection for smaller clutches (Mountford, 1973; Yoshimura & Shields, 1992; DeWitt, 1997). These analyses illustrate the utility of mark–recapture in studying the age–old clutch–size paradox. Experimental manipulations of clutch sizes provide evidence on the capability of parents to rear different numbers of chicks, but unless these offspring (and

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A 1.6

38 216

1.2 Estimated ARS

Number of young fledged/nest

143

2 58 1

0.8 0.4

0.0

0 1

3 4 Brood size

B 18

0.5 0.4

3 3

Estimated ARS

Number of young fledged/nest

0.2 0.1

5 0

0.3

0 1

3 4 Brood size

2 3 4 Brood size

C 0.5

16 3

0.4 Estimated ARS

Number of young fledged/nest

2 108 1 0

0.3 0.2 0.1

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Fig. 3. Mean (! SE) number of nestling Cliff Swallows fledged and estimated annual reproductive success (ARS) in relation to brood size in: A. 1982, a cool season; B. 1990, an average season; C. 1988, a warm season, during June while broods were being reared. Annual reproductive success was estimated by multiplying the average number of chicks fledged by first–year survival probability for each brood size. Numbers above error bars indicate number of nests studied (Brown & Brown, 1999). Fig. 3. Promedio (! EE) del número de pollos que salen del nido de golondrinas de frente canela y éxito reproductivo anual (ARS) estimado según el tamaño de la nidada en: A. 1982, una estación fresca; B. 1990, una estación media; C. 1988, una estación cálida, durante junio, mientras se criaba la nidada. El éxito reproductivo anual se estimó multiplicando el promedio del número de pollos que salen del nido por la probabilidad de supervivencia durante el primer año de vida para cada tamaño de nidada. Los números situados encima de las barras de error indican el número de nidos estudiados (Brown & Brown, 1999).

Brown & Brown

the parents) are followed over multiple seasons and their survival estimated, the long–term tradeoffs of clutch size cannot be known. The number of studies that have done this is still quite small. Prospectus As with all mark–recapture studies, there are challenges inherent in the application of mark–recapture methods to behavioral questions, and two stand out as particularly relevant to behavioral ecology. One is distinguishing mortality from dispersal. This is always a problem with open populations, because individuals permanently departing from a study area will be classified as dead. Various methods exist to estimate the fraction of individuals missed (Barrowclough, 1978; Payne, 1990; Marshall et al., 2004), and this is a particularly serious problem if the research objective is to determine absolute survival probabilities, perhaps for management or conservation purposes. In some cases, extent of dispersal might be related to the behavioral variable of interest; for example, individuals who do not attract mates might travel more widely than ones who are mated. However, if permanent emigration does not differ among the groups being compared, one can still achieve estimates of relative survival probabilities among groups of individuals that differ with respect to the behavioral covariate of interest. One way to determine whether emigration differs among groups is to examine the extent of movement of individuals with different behavioral traits within a study area of moderate size, as, for example, we did in a survival study of Cliff Swallows whose ectoparasites had and had not been removed the previous year (Brown et al., 1995). Since there was no evidence that the two groups differed in extent of dispersal, mark– recapture provided appropriate relative measures of survival that could be related to extent of ectoparasitism. The other challenge for behavioral ecologists is coping with uncertainty in categorizing covariates associated with each individual. For example, we designated male Cliff Swallows as engaging in extra–pair copulations or not based on whether we saw them do it. Similarly, Cam et al. (2002) designated Black– legged Kittiwakes (Rissa tridactyla) as squatters or non–squatters based on whether individuals were seen squatting. In each case, however, since the animals could not be monitored continuously, some may have been misassigned as not having exhibited the behavior of interest when in fact they did show the behavior but it was undetected. Statistical methods exist for the modeling of unobservable and misclassified states (Kendall & Nichols, 2002; Kendall, 2004). Another solution is to consider the testing for differences among groups as conservative since it is "polluted" by misclassifications (Lank et al., 1990; Brown & Brown, 1998); if a difference is found, it is likely of sufficient strength to overcome the misclassifications. Despite these potential challenges, mark–recapture studies and their associated statistical analyses have great potential applicability to behavioral ecology. Rigorous estimates of survival in natural

populations enable us to specify the fitness consequences associated with a particular behavior in a direct and evolutionarily meaningful way that does not require (usually untested) assumptions about indirect correlates of fitness. This requires, of course, a long–term approach in which individuals are followed over multiple seasons and too often the surmounting of financial or other logistical obstacles to conducting long–term field studies (e.g., Tinkle, 1979; Nisbet, 1989; Malmer & Enckell, 1994). The insights that result, however, can often be worth the time and effort, as we have found for Cliff Swallows. The use of multistate models (Nichols & Kendall, 1995; Lebreton & Pradel, 2002) to infer probabilities of movement also offers great potential for behavioral ecology. Estimating the likelihood of an animal moving from one site to another is essentially a measure of dispersal, and can reveal patterns of space use that are otherwise difficult to discern. We used multistate movement probabilities, for example, to study colony choice in Sociable Weavers (Philetairus socius), finding that individuals showed preferences to settle in colonies of size similar to that they had used previously (Brown et al., 2003). This enabled us to go beyond anecdotal observations on where certain banded individuals were found and allowed the testing of explicit hypotheses about the decision rules these birds use to choose colonies. We hope that these sorts of approaches will become more common in behavioral ecology. Acknowledgments For field assistance, we thank S. Aldridge, C. Anderson, C. Brashears, K. Brazeal, A. Briceno, K. Brown, R. Budelsky, B. Calnan, S. Carlisle, B. Chasnoff, M. Chu, K. Cornett, Z. Deretsky, L. Doss, K. Edelmann, J. Thomson Fiorillo, E. Flescher, J. Grant, W. Hahn, L. Hatch, A. Hing, J. Hoskyn, S. Huhta, L. Jackson, D. Johnson, V. Johnson, J. Klaus, M. Kostal, E. Landay, J. Leonard, L. Libaridian, B. MacNeill, J. Malfait, K. Miller, C. Mirzayan, L. Molles, L. Monti, S. Narotum, C. Natunewicz, V. O’Brien, C. Ormston, C. Patenaude, B. Raulston, G. Redwine, C. Richman, K. Rodgers, S. Rosenberg, A. Rundquist, T. Scarlett, R. Sethi, M. Shaffer, M. Shanahan, L. Sherman, K. Van Blarcum, E. Westerman, and Z. Williams. The School of Biological Sciences, University of Nebraska–Lincoln, allowed use of the facilities of the Cedar Point Biological Station. For financial support, we thank the National Science Foundation (most recently through grants DEB–0075199, IBN–9974733), the National Geographic Society, the Erna and Victor Hasselblad Foundation, the American Philosophical Society, Princeton University, Yale University, the University of Tulsa, the Chapman Fund of the American Museum of Natural History, the National Academy of Sciences, Sigma Xi, and Alpha Chi. Emmanuelle Cam and an anonymous reviewer provided helpful comments on the manuscript. We are grateful to Juan Carlos Senar for the invitation to present this paper as a plenary address at the EURING 2003 technical meeting.

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References Andersson, M., 1984. Brood parasitism within species. In: Producers and scroungers: strategies of exploitation and parasitism: 195–228 (C. J. Barnard, Ed.). Croom Helm, London. Arneberg, P., Skorping, A., Grenfell, B. & Read, A. F., 1998. Host densities as determinants of abundance in parasite communities. Proceedings of the Royal Society of London B, 265: 1283–1289. Barrowclough, G. F., 1978. Sampling bias in dispersal studies based on finite area. Bird–Banding, 49: 333–341. Birkhead, T. R. & Møller, A. P., 1992. Sperm competition in birds. Academic Press, London. Boyce, M. S. & Perrins, C. M., 1987. Optimizing Great Tit clutch size in a fluctuating environment. Ecology, 68: 142–153. Brown, C. R. & Brown, M. B., 1986. Ectoparasitism as a cost of coloniality in Cliff Swallows (Hirundo pyrrhonota). Ecology, 67: 1206–1218. – 1989. Behavioural dynamics of intraspecific brood parasitism in colonial Cliff Swallows. Animal Behaviour, 37: 777–796. – 1991. Selection of high–quality host nests by parasitic Cliff Swallows. Animal Behaviour, 41: 457–465. – 1995. Cliff Swallow (Hirundo pyrrhonota). In: Birds of North America: no. 149 (A. Poole & F. Gill, Eds.). Academy of Natural Sciences, Philadelphia, and American Ornithologists’ Union, Washington, D.C. – 1996. Coloniality in the Cliff Swallow: the effect of group size on social behavior. Univ. of Chicago Press, Chicago. – 1998. Fitness components associated with alternative reproductive tactics in Cliff Swallows. Behavioral Ecology, 9: 158–171. – 1999. Fitness components associated with clutch size in Cliff Swallows. Auk, 116: 467–486. – 2001. Avian coloniality: progress and problems. Current Ornithology, 16: 1–82. – 2004. Empirical measurement of parasite transmission between groups in a colonial bird. Ecology, 85: 1619–1626. Brown, C. R., Brown, M. B. & Danchin, E., 2000. Breeding habitat selection in Cliff Swallows: the effect of conspecific reproductive success on colony choice. Journal of Animal Ecology, 69: 133–142. Brown, C. R., Brown, M. B. & Rannala, B., 1995. Ectoparasites reduce long–term survival of their avian host. Proceedings of the Royal Society of London B, 262: 313–319. Brown, C. R., Covas, R., Anderson, M. D. & Brown, M. B., 2003. Multistate estimates of survival and movement in relation to colony size in the Sociable Weaver. Behavioral Ecology, 14: 463–471. Brown, C. R., Komar, N., Quick, S. B., Sethi, R. A., Panella, N. A., Brown, M. B. & Pfeffer, M., 2001. Arbovirus infection increases with group size. Proceedings of the Royal Society of London B, 268: 1833–1840.

Cam, E., Cadiou, B., Hines, J. E. & Monnat, J. Y., 2002. Influence of behavioural tactics on recruitment and reproductive trajectory in the Kittiwake. Journal of Applied Statistics, 29: 163–185. Cliff, A. D., Hagget, P., Ord, J. K. & Versey, G. R., 1981. Spatial diffusion: an historical geography of epidemics in an island community. Cambridge Univ. Press, Cambridge. Côté, I. M. & Poulin, R., 1995. Parasitism and group size in social animals: a metaanalysis. Behavioral Ecology, 6: 159–165. Davies, C. R., Ayres, J. M., Dye, C. & Deane, L. M., 1991. Malaria infection rate of Amazonian primates increases with body weight and group size. Functional Ecology, 5: 655–662. DeWitt, T. J., 1997. Optimizing clutch size in birds. Trends in Ecology and Evolution, 12: 443. Field, J., 1992. Intraspecific nest parasitism as an alternative reproductive tactic in nest–building wasps and bees. Biological Reviews, 67: 79–126. Gladstone, D. E., 1979. Promiscuity in monogamous colonial birds. American Naturalist, 114: 545–557. Grafen, A., 1988. On the uses of data on lifetime reproductive success. In: Reproductive success: studies of individual variation in contrasting breeding systems: 454–471 (T. H. Clutton–Brock, Ed.). Univ. of Chicago Press, Chicago. Grenfell, B. T., Bjørnstad, O. N. & Kappey, J., 2001. Travelling waves and spatial hierarchies in measles epidemics. Nature, 414: 716–723. Hamilton, W. D., 1964. The genetical evolution of social behavior. Journal of Theoretical Biology, 7: 1–52. Hieber, C. S. & Uetz, G. W., 1990. Colony size and parasitoid load in two species of colonial Metepeira spiders from Mexico (Araneae: Araneidae). Oecologia, 82: 145–150. Hoogland, J. L., 1995. The black–tailed prairie dog: social life of a burrowing mammal. University of Chicago Press, Chicago. Jennions, M. D. & Petrie, M., 2000. Why do females mate multiply? A review of the genetic benefits. Biological Reviews, 75: 21–64. Kempenaers, B., Verheyen, G. R., Van den Broeck, M., Burke, T., Van Broeckhoven, C. & Dhondt, A. A., 1992. Extra–pair paternity results from female preference for high quality males in the Blue Tit. Nature, 357: 494–496. Kendall, W. L., 2004. Coping with unobservable and mis–classified states in capture–recapture studies. Animal Biodiversity and Conservation, 27.1: 97–107. Kendall, W. L. & Nichols, J. D., 2002. Estimating state–transition probabilities for unobservable states using capture–recapture/resighting data. Ecology, 83: 3276–3284. Klomp, H., 1970. The determination of clutch–size in birds: a review. Ardea, 58: 1–124. Krasnov, B., Khokhlova, I. & Shenbrot, G., 2002. The effect of host density on ectoparasite distribution: an example of a rodent parasitized by fleas. Ecology, 83: 164–175. Lack, D., 1947. The significance of clutch size. Ibis,

89: 302–352. – 1954. The natural regulation of animal numbers. Clarendon Press, Oxford. Lank, D. B., Rockwell, R. F. & Cooke, F., 1990. Frequency–dependent fitness consequences of intraspecific nest parasitism in Snow Geese. Evolution, 44: 1436–1453. Lebreton, J. D. & Pradel, R., 2002. Multistate recapture models: modeling incomplete individual histories. Journal of Applied Statistics, 29: 353–369. Lockhart, A. B., Thrall, P. H. & Antonovics, J., 1996. Sexually transmitted diseases in animals: ecological and evolutionary implications. Biological Reviews, 71: 415–471. Loehle, C., 1995. Social barriers to pathogen transmission in wild animal populations. Ecology, 76: 326–335. Lyon, B., 1993. Conspecific brood parasitism as a flexible female reproductive tactic in American Coots. Animal Behaviour, 46: 911–928. Malmer, N. & Enckell, P. H., 1994. Ecological research at the beginning of the next century. Oikos, 71: 171–176. Marshall, M. R., Diefenbach, D. R., Wood, L. A. & Cooper, R. J., 2004. Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: life–history implications and study designs that may help. Animal Biodiversity and Conservation, 27.1: 59–72. McRae, S. B. 1998. Relative reproductive success of female Moorhens using conditional strategies of brood parasitism and parental care. Behavioral Ecology, 9: 93–100. Monath, T. P., Lazuick, J. S., Cropp, C. B., Rush, W. A., Calisher, C. H., Kinney, R. M., Trent, D. W., Kemp, G. E., Bowen, G. S. & Francy, D. B., 1980. Recovery of Tonate virus (“Bijou Bridge” strain), a member of the Venezuelan equine encephalomyelitis virus complex, from Cliff Swallow nest bugs (Oeciacus vicarius) and nestling birds in North America. American Journal of Tropical Medicine and Hygiene, 29: 969–983. Moore, J., Simberloff, D. & Freehling, M., 1988. Relationships between Bobwhite quail social– group size and intestinal helminth parasitism. American Naturalist, 131: 22–32. Mountford, M. D., 1973. The significance of clutch size. In: The mathematical theory of the dynamics of biological populations: 315–323 (M. S. Bartlett & R. W. Hiorns, Ed.). Academic Press, New York. Nichols, J. D. & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. Journal of Applied Statistics, 22: 835–846. Nisbet, I. C. T., 1989. Long–term ecological studies of seabirds. Colonial Waterbirds, 12: 143–147. Payne, R. B., 1990. Natal dispersal, area effects, and effective population size. Journal of Field Ornithology, 61: 396–403. Pradel, R., Hines, J. E., Lebreton, J.–D. & Nichols, J. D., 1997. Capture–recapture survival models taking account of transients. Biometrics, 53: 60–72.

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Roff, D. A., 2002. Life history evolution. Sinauer, Sunderlund, Massachusetts. Rohwer, F. C. & Freeman, S., 1989. The distribution of conspecific nest parasitism in birds. Canadian Journal of Zoology, 67: 239–253. Rubenstein, D. I. & Hohmann, M. E., 1989. Parasites and social behavior of island feral horses. Oikos, 55: 312–320. Sandell, M. I. & Diemer, M., 1999. Intraspecific brood parasitism: a strategy for floating females in the European Starling. Animal Behaviour, 57: 197–202. Scott, T. W., Bowen, G. S. & Monath, T. P., 1984. A field study of the effects of Fort Morgan virus, an arbovirus transmitted by swallow bugs, on the reproductive success of Cliff Swallows and symbiotic House Sparrows in Morgan County, Colorado, 1976. American Journal of Tropical Medicine and Hygiene, 33: 981–991. Sheldon, B. C., 1993. Sexually transmitted disease in birds: occurrence and evolutionary significance. Philosophical Transactions of the Royal Society of London, B 339: 491–497. Stearns, S. C., 1992. The evolution of life histories. Oxford Univ. Press, Oxford. Swinton, J., Harwood, J., Grenfell, B. T. & Gilligan, C. A., 1998. Persistence thresholds for phocine distemper virus infection in Harbour Seal Phoca vitulina metapopulations. Journal of Animal Ecology, 67: 54–68. Tinkle, D. W., 1979. Long–term field studies. Bioscience, 29: 717. VanderWerf, E., 1992. Lack’s clutch size hypothesis: an examination of the evidence using meta– analysis. Ecology, 73: 1699–1705. Wagner, R. H., 1993. The pursuit of extra–pair copulations by female birds: a new hypothesis of colony formation. Journal of Theoretical Biology, 163: 333–346. Wagner, R. H., Schug, M. D. & Morton, E. S., 1996. Condition–dependent control of paternity by female Purple Martins: implications for coloniality. Behavioral Ecology and Sociobiology, 38: 379–389. Weatherhead, P. J. & Boag, P. T., 1995. Pair and extra–pair mating success relative to male quality in Red–winged Blackbirds. Behavioral Ecology and Sociobiology, 37: 81–91. Westneat, D. F., Sherman, P. W. & Morton, M. L., 1990. The ecology and evolution of extra–pair copulations in birds. Current Ornithology, 7: 331–369. White, A., Begon, M. & Bowers, R. G., 1996. Host– pathogen systems in a spatially patchy environment. Proceedings of the Royal Society of London B, 263: 325–332. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46: S120–S139. Yom–Tov, Y., 1980. Intraspecific nest parasitism in birds. Biological Reviews, 55: 93–108. Yoshimura, J. & Shields, W. M., 1992. Components of uncertainty in clutch–size optimization. Bulletin of Mathematical Biology, 54: 445–464.

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Effects of spring conditions on breeding propensity of Greater Snow Goose females E. T. Reed, G. Gauthier & J.–F. Giroux

Reed, E. T., Gauthier, G. & Giroux, J.–F., 2004. Effects of spring conditions on breeding propensity of Greater Snow Goose females. Animal Biodiversity and Conservation, 27.1: 35–46. Abstract Effects of spring conditions on breeding propensity of Greater Snow Goose females.— Breeding propensity, defined as the probability that a sexually mature adult will breed in a given year, is an important determinant of annual productivity. It is also one of the least known demographic parameters in vertebrates. We studied the relationship between breeding propensity and conditions on spring staging areas (a spring conservation hunt) and the breeding grounds (spring snow cover) in Greater Snow Geese (Chen caerulescens atlantica), a long distance migrant that breeds in the High Arctic. We combined information from mark–recapture, telemetry, and nest survey data to estimate breeding propensity over a 7– year period. True temporal variation in breeding propensity was considerable (mean: 0.574 [95% CI considering only process variation: 0.13 to 1.0]). Spring snow cover was negatively related to breeding propensity ( snow = –2.05 ± 0.96 SE) and tended to be reduced in years with a spring hunt ( = –0.78 ± 0.35). Nest densities on the breeding colony and fall ratios of young:adults were good indices of annual variation in breeding propensity, with nest densities being slightly more precise. These results suggest that conditions encountered during the pre–breeding period can have a significant impact on productivity of Arctic–nesting birds. Key words: Breeding propensity, Capture–recapture, Chen caerulescens atlantica, Environmental stochasticity, Greater Snow Goose, Reproduction. Resumen Efectos de las condiciones primaverales en la propensión reproductora de las hembras de ánsar nival.— La propensión reproductora, definida como la probabilidad de que un adulto sexualmente maduro se reproduzca en un determinado año, constituye un importante determinante de la productividad anual. También es uno de los parámetros demográficos menos conocidos de los vertebrados. Estudiamos la relación entre la propensión reproductora y las condiciones reproductoras en áreas de acumulación primaveral de nutrientes (cacería primaveral orientada a la conservación) y emplazamientos de reproducción (capa de nieve durante la primavera) en el ánsar nival (Chen caerulescens atlántica), un ave migratoria que recorre grandes distancias y que se reproduce en el Alto Ártico. Combinamos la información de marcaje–recaptura, telemetría y datos de supervivencia en el nido para estimar la propensión reproductora durante un período de siete años. La variación temporal verdadera en la propensión reproductora fue significativa [media: 0,574 (95% CI, considerando únicamente la variación en los procesos: de 0,13 a 1,0)]. La capa de nieve primaveral se relacionó negativamente con la propensión reproductora ( snow = – 2,05 ± 0,96 EE), tendiendo a reducirse en los años en que se había producido una cacería primaveral ( = –0,78 ± 0,35). Las densidades de los nidos de la colonia reproductora y la tasa de jóvenes:adultos en otoño constituyeron buenos índices de la variación anual en la propensión reproductora, siendo las densidades de los nidos ligeramente más precisas. Tales resultados sugieren que las condiciones presentes durante el período previo a la reproducción pueden tener un impacto significativo en la productividad de las aves que nidifican en el Ártico. Palabras clave: Propensión reproductora, Captura–recaptura, Chen caerulescens atlantica, Estocasticidad medioambiental, Ánsar nival, Reproducción. Eric T. Reed* & Gilles Gauthier, Dépt. de Biologie and Centre d’Études Nordiques, Univ. Laval, Sainte–Foy, QC G1K 7P4 Canada.– Jean–François Giroux, Dépt des Sciences Biologiques, Univ. du Québec à Montréal, Succursale Centre ville, C. P. 8888, Montréal, QC, H3C 3P8 Canada. * Present address: Migratory Birds Population Analyst, Canadian Wildlife Service, Environment Canada, 351 St. Joseph Blvd, Gatineau, QC K1A 0H3 Canada. E–mail: eric.reed@ec.gc.ca

ISSN: 1578–665X

Reed et al.

Introduction Breeding propensity, defined as the probability that a sexually mature female breeds in a given year (i.e. lays at least one egg, whether it is successfully or not), has a strong impact on the number of young produced. This parameter is thus of considerable interest in population dynamics, especially in long–lived species that may be prone to skip a breeding year when conditions are not appropriate (Tickell & Pinder, 1967; Chastel, 1995; Nur & Sydeman, 1999). Unfortunately, breeding propensity is difficult to estimate in most species because non–breeders are often less conspicuous or simply absent from breeding colonies (Spendelow & Nichols, 1989; Chastel, 1995). Thus, it is probably one of the least known demographic parameters in vertebrates. Absence of an individual from the breeding site due to breeding failure is a form of temporary emigration. However, other situations may also lead to temporary emigration, and can thus be easily confounded with breeding propensity: for instance, home range of some individuals may not be completely enclosed in the study area, so that that they are outside a sampling frame in some years; or the capture process itself may cause an individual to temporarily leave the study area (e.g. Pradel et al., 1995). Breeding propensity may be especially variable for species in unpredictable and highly heterogeneous environments such as the Arctic. In arcticnesting geese, it has been suggested that a large proportion of individuals may fail to breed in years of late snowmelt on the breeding grounds (Barry, 1962; Prop & de Vries, 1993; Reed et al., 1998). Delayed snowmelt prevents access to nesting sites and can impair the acquisition of some nutrients for egg–formation (Choinière & Gauthier, 1995; Ganter & Cooke, 1996). Recent estimates of breeding propensity in waterfowl indicate high values for this parameter (0.74–0.90 of mature adults breeding in a given year: Kendall & Nichols, 1995; Lindberg et al., 2001; Sedinger et al., 2001). In contrast to previous suggestions, these studies detected small annual variation in breeding propensity (see also Cooch et al., 2001). In addition to late snowmelt on nesting areas, events during spring migration may also influence breeding propensity in long–distance migrants. Nutrient reserves accumulated during spring staging are used to meet energy costs of spring migration and reproduction (Ankney & MacInnes, 1978; Gauthier et al., 1992; Choinière & Gauthier, 1995; Gauthier et al., 2003). Factors such as reduced food availability due to drought, or high disturbance due to spring hunting that negatively affect spring fattening may thus result in reduced breeding effort (Davies & Cooke, 1985; Mainguy et al., 2002; Féret et al., 2003). In open population mark–recapture models, detection probabilities confound the probability of being present in the sampled area and the prob-

ability of being detected given presence (Kendall & Nichols, 1995). When sampling is limited to a single site and provides one capture occasion per time period (e.g. Lebreton et al., 1992), separation of these two probabilities is not possible, as only their product is estimable. To circumvent this problem, ad hoc and maximum likelihood estimators of temporary emigration based on the robust design (Pollock, 1982) were developed by Kendall et al. (1997). In this design, multiple subsamples (secondary samples) within primary sampling periods are used to estimate probability of detection, conditional on animal presence; these detection probabilities are then used to adjust estimates from standard mark-recapture techniques, assuming a closed population during secondary samples (Pollock, 1982; Pollock et al., 1990). Recently, it was recognized that this approach could be used to estimate probability of breeding when presence is synonymous with breeding (Kendall & Nichols, 1995; Kendall et al., 1997; Lindberg et al., 2001). In geese, this condition is met when non-breeders undertake a molt migration away from breeding sites before banding (e.g Salomonsen, 1968; Abraham, 1980; Reed et al., 2003a). We estimated annual variation in breeding propensity in Greater Snow Geese (Chen caerulescens atlantica), a long–lived migrant that breeds in the High Arctic, a highly variable environment. Building on a long–term capture–mark–recapture study at a major breeding site, we combined information from recaptures, radio–tracking, and nest monitoring to estimate breeding propensity of adult female Greater Snow Geese over a 7–year period. Our objective was to investigate temporal variation of breeding propensity of adult females and evaluate the influence of 1) snow cover at the onset of nesting in the Arctic, and 2) a conservation hunt recently implemented on the spring staging area (Mainguy et al., 2002). We hypothesized that breeding propensity would be lower in years of extensive spring snow cover in the Arctic and in years with a spring conservation hunt on the staging area. We also sought to find out if other variables easier to sample were correlated with breeding propensity in order to use them as an index. Study area Data were collected at the Bylot Island breeding colony, Sirmilik National Park, Nunavut Territory, Canada. This area supports the largest concentration of breeding Greater Snow Geese, representing ca. 15% (55,000 breeding adults in 1993) of the world breeding population (Reed et al., 1998). Field work was concentrated at two main study sites, separated by ~30 km: the Base–camp valley, an important brood–rearing area where weather data collection and banding occurred, and the Camp–2 area where most geese nest and where nesting data were collected (Bêty et al., 2001). Most families that use the Base–camp area

Animal Biodiversity and Conservation 27.1 (2004)

for brood–rearing moved from camp–2 within 6 days after hatching (Mainguy, 2003). We defined the superpopulation (sensu Kendall et al., 1997) as females that had nested on Bylot Island’s south plain in at least one year of the study. Methods Capture and marking of geese Flightless geese were captured in corral traps during molt with the help of a helicopter and by personnel on foot over a 5 to 8–day period in early August from 1994 to 2001 (Menu et al., 2001). Captured birds were mostly successful parents with their young since non–breeders and failednesters left the island for distant molting sites or had regained flight ability by the time of banding (Reed et al., 2003a). This is referred to as the main banding operation. All birds captured for the first time were fitted with a metal U.S. Fish and Wildlife Service leg band and a sample of adult females was fitted with individually coded plastic neck bands. Recaptures were noted systematically. Because we had evidence that neck bands reduced female breeding propensity (Reed, 2003), we restricted our analysis to females marked only with leg bands. Females that had lost neck bands before recapture, however, were included, their first capture without the neck band being considered their initial capture. Similarly, leg–banded females that were subsequently fitted with a neck band were censored by considering their last capture as a leg–banded female as a loss on capture. From 1995 to 1999, captures of individual families were also made at the same site to fit radiotransmitters on neck bands (all years) or harnesses (1995 only) to adult females (Demers et al., 2003). This normally was done shortly before the main banding operation but overlapped in some years. In 2000–2001, incubating females were captured on their nest using bow traps (Mainguy, 2003) and also marked with radio–transmitters on neck bands. Presence of these females on Bylot Island in late summer was ascertained through aerial radio–tracking (see Reed et al., 2003a) to determine the number of radio–marked females located in the banding area. Detection range of radios was approximately 3–5 km from the ground and 10–15 km from the air (Bêty, 2001).

10 km). Patches of suitable brood–rearing habitat extended ~65 km to the south of the base-camp valley and were limited to one valley ~5 km to the north because of high mountains. Banding area was thus calculated as the distance between the most southerly and northerly capture sites each year (in km). In 1999, a year of almost complete reproductive failure (Mainguy et al., 2002), we covered the entire south plain of Bylot Island for banding. We used the distance between inter–annual recaptures to estimate of brood–rearing site fidelity (denoted as Fi). Mainguy (2003) found that 42.5 % of females were recaptured < 5 km, 17.5% 6–10 km, 10.0% 11–15 km, 15.0% 16–20 km, and 15.0% 21–30 km of their previous capture location (n = 40 females). We estimated the proportion of females on Bylot Island that were within our banding area (Fi) as the proportion of females recaptured in the 5 km distance class (described above) that included our median distance between extreme capture locations each year. For example, if the distance between our most northerly and most southerly capture during banding was 6 km (median = 3 km), we used the proportion of radio– marked females that were recaptured < 5 km from their previous capture location (i.e. the smallest class of Mainguy, 2003) as measure of site fidelity. Variance of site fidelity was estimated as:

where i refers to year and n is the sample size (40 females). Nest survival Greater Snow Goose nests were found by systematic searches during incubation in 1995, and egg laying or early incubation from 1996 to 2001 at the Camp–2 area (Bêty et al., 2001). Fate of nests was determined by revisiting them in the first half of incubation, during hatching and after goslings had left the nest. Nesting parameters are not biased by our visits (Bêty & Gauthier, 2001). We used the Mayfield method to calculate daily nest survival and the product method to evaluate nest survival probability for the whole nesting period (denoted as Sin Johnson, 1979). Nest survival for the years 1996– 2000 were taken from Bêty et al. (2002). Temporary emigration model

Extent of banding area and brood–rearing site fidelity Background and notation The area covered by the main banding operation varied among years in response to goose densities. We always covered the Base–camp valley before moving onto the adjacent plateau and other valleys (see map in Lesage & Gauthier, 1998). The study area was bounded to the west by Navy Board Inlet and to the east by glaciers, so geese were constrained to a north–south corridor (ca.

We estimated temporary emigration using the sampling design described by Kendall et al. (1997), i.e. individuals are captured during secondary samples (closed population) nested within primary periods (open population). Our model requires combination of different sources of information from Cormack–Jolly–Seber models to esti-

mate temporary emigration. Model notation follows Lebreton et al. (1992) and Kendall et al. (1997). First we define the following terms: i. Probability that an individual survives and does not permanently emigrate from primary period i to i+1, (i = 1,2,…,k–1); pi0. Probability that an individual is caught in primary period i, given that the animal is alive and present in the superpopulation at period i, (i = 1,2,…,k); pi*. Probability that an individual is captured in at least one of the l i secondary samples of primary period i, given that the individual is located in the sampled area during period i; pij. Probability that an individual is captured in secondary sample j of primary period i, given that it is alive and present in the sampled area during period i; i. Probability that a marked individual could not be captured during primary period i (i.e. is a temporary emigrant), given that it is in the superpopulation, but outside of the sampled area. Kendall et al. (1997) described an ad hoc estimator of temporary emigration from the sampling area (banding area in our case). Temporary emigration from our study area is mainly due to nonbreeding and nest failure (Reed et al., 2003a), but also incomplete fidelity to brood–rearing areas that we sampled. Our goal was to estimate breeding propensity (1 – i ) of females from the Bylot Island breeding population, so we modified Kendall et al.’s (1997) estimator to take into account temporary emigration of breeding females due to nest failure (1 – Sin) and change in brood–rearing site (1 – Fi):

An appropriate variance estimate based on the delta method (Seber, 1982: 7) would be:

We could ignore covariance terms because our samples were independent. Primary period capture probabilities–CJS modeling We used as a base model the Cormack–Jolly– Seber (CJS) model (Cormack, 1964; Jolly, 1965; Seber 1965) where survival ( i) and capture (pi0) probabilities are time–specific (model t, pt). We first tested the fit of this model using the goodness–of–fit (GOF) tests of program RELEASE (Burnham et al., 1987). Once we had a general model that provided a good fit to data, we proceeded to assess the effect of time on survival and capture probabilities. We used Akaike’s Information Criterion modified for small sample size (AICc) to select the best approximating model (low-

Reed et al.

est AICc value; Burnham & Anderson, 1998). Other models were ranked relative to deviations from the best model ( AICc). We used program MARK v2.1 (White & Burnham, 1999) for model selection and parameter estimation. We also used AICc weights (*AICc), which represent the weight of evidence in support of each model in the candidate set (Burnham & Anderson, 1998). Secondary sample capture probabilities–radio–tracking Capture probabilities for secondary sampling period (pi*) could not be estimated directly from the mark–recapture data because we avoided multiple recaptures of individuals in a given year. We used instead an independent sample of radiomarked females to estimate closed population capture probabilities. We assumed that radiomarked females, whose number was known, represented a random sample of the geese present in the study area. Even though we sampled the banding area systematically rather than randomly, we did not search for radio–marked females during the main banding operations and our selection of banding sites was thus independent from these birds being present or absent at the site. Few adults not marked with radios were captured more than once in the same year (3% of adult females between 1994 and 2001) so we can view our sampling as simple random sampling without replacement. Mean daily capture probabilities over secondary samples ( ) were estimated as the number of radio–marked females caught during the main banding operation over the sum of the number of radio–marked females present each day of capture over the entire banding period (i.e. radio– days). Once a female was captured, she was removed from the sample of birds available for capture (only one radio–marked female was caught twice in the same year) and new females could be added when radio–marking overlapped the main banding operation. The variance was computed as:

The probability that a female was captured in at least one of the secondary periods (pi*) was calculated as:

where k was the number of days of banding in a given year. The variance was estimated as:

In 1999, breeding was a general failure and less than 2,000 geese were on Bylot Island during banding operations (Gauthier, personal observation). The main banding operation and marking of females with radios took place at the same time, which prevented us from using radio–marked fe-

Animal Biodiversity and Conservation 27.1 (2004)

males to estimate pi* in that year. Based on visual observations, we estimated that about 85% of geese present on Bylot Island were captured, but because of uncertainty on the exact number we added a SE of 0.05 to this estimate (thus assuming that between 0.75 and 0.95 of the geese were captured). Covariates of breeding propensity We tested whether breeding propensity varied among years using the sum of squares derived from estimates of breeding propensity probabilities inversely weighted by their total variance and the covariances of open population capture rates (pi0, other variables in the calculation of breeding propensity being independent). The statistic for the null hypothesis (homogeneity across years) follows a chi–square distribution with n – 1 degrees of freedom (where n is the number of breeding propensity rates) (Sauer & Williams, 1989). This analysis was performed with program CONTRAST (Hines & Sauer, 1989). Total variance of breeding propensity estimates is contaminated by sampling error, which can lead to an overestimation of temporal variation in this parameter. Using a variance–components approach, we partitioned the total variability of estimated breeding propensity and calculated the percentage of the total variation that was accounted for by the sampling variation and covariation (Gould & Nichols, 1998). We then proceeded to test the effect of snow cover on the breeding grounds and the implementation of a spring conservation hunt on breeding propensity. Each year, a visual estimate of snow cover in the base–camp valley was made on 5 June (Lepage et al., 1996; Reed et al., 2003b). From 1997 to 2001, a simultaneous assessment of snow cover was made at Camp–2, which showed that data from our Base–camp valley was representative of the situation at Camp–2. Snow cover on 5 June averaged 71% but was variable among years (range: 40% to 85%). In Quebec, a spring conservation hunt (hereafter called spring hunt) on Greater Snow Geese was allowed in 1999–2001. We categorized years with a binary variable (with or without spring hunt). We used a weighted least squares approach to estimate regression coefficients (± SE) with the covariates (Lebreton, 1995; Link, 1999). Thus, variances of the annual estimates of breeding propensity, as well as covariances of pi0 were accounted for in the regression analysis. We fitted main effects only and tested for a negative effect of the covariates, i.e. < 0.0 ( snow = slope of the relationship between snow cover and breeding propensity; hunt = difference in breeding propensity between years without and with a spring hunt) with one–tailed z–tests. The method used here to estimate breeding propensity is difficult and expensive in the field (e.g. requires large number of radio-marked birds),

so we looked for a simple, reliable index of breeding propensity that could be used for monitoring purposes. Two indices were examined: productivity surveys in fall and nest densities at the main breeding colony. Productivity surveys on the fall staging areas of this population have been conducted by the Canadian Wildlife Service (CWS) along the St. Lawrence River Estuary in Quebec, Canada since 1973. Proportion of young in the fall flight varied from 0.02 to 0.37 between 1995 and 2001 (Reed et al., 1998, A. Reed CWS unpublished data). Nest density (nests per 50 ha) was estimated as the number of nests found divided by the area of the search zone (data from Bêty et al., 2002). We transformed these estimates as nests per ha (1996: 1.0, 1997: 2.64, 1998: 6.52, 1999: 1.1, 2000: 1.64). We used the weighted least squares approach to test for positive relations between these indices and breeding propensity (β > 0.0) with one–tailed z–tests. We also computed the total sums of squares (SSt) of the breeding propensity estimators as well as the residual sums of squares (SSr) from each model. We then calculated a coefficient of correlation (R2 = (SSt – SSr) / SSt)) for both indices to determine which fitted best the pattern of temporal variation in breeding propensity. Years 1996 to 2000 were used due to lack of precision of point estimates for 1995 and 2001 (see below). Results From 1994 to 2001, we marked 1646 adult females with leg bands and subsequently recaptured 227 of these birds. The goodness of fit test of our general model indicated a good fit ((232 = 26.82, P = 0.14). Our best model had constant apparent survival and time dependent capture probabilities ( pt) and this model was strongly supported by the data (*AICc = 0.98). The second best model was t pt ( AICc = 8.2). Apparent survival estimates from our best model were 0.87 ± 0.04 and capture probabilities conditional on presence in the superpopulation (pi0) ranged from 0.02 to 0.07 (table 1). Between 1995 and 2001, 42 radio–marked females were recaptured during the main banding operations out of 1,147 radio–days and these data were used to calculate an annual (1995: 2/ 64; 1996: 20/328; 1997: 4/304; 1998: 13/390; 2000: 2/24; 2001: 1/37). Estimated capture probabilities given presence in the sampled area (pi*) was thus highly variable and ranged from 0.09 to 0.46 (table 1). We estimated nest survival for between 179 and 326 nests each year. Nest survival was highly variable among years, being lowest in 1995 and 1999 and highest in 1997 and 2000 (table 1). Fidelity to the brood–rearing area was also variable and inversely related to the size of the sampled area. The extent of the banding area was 10.5 km in 1995, 12.3 in 1996, 17.2 in 1997, 6.1 in 1998, the entire south plain of 70 km in 1999,

Reed et al.

Table 1. Estimate of capture probability, given presence in the superpopulation (pi0 ), from the open population CJS model ( pt); capture probabilities given presence in the sampling area (pi*) obtained from radio–marked females; Mayfield estimate of nest survival (Sin) for the entire nesting period; fidelity to brood–rearing sites (Fi) estimated as the proportion of individuals expected to be present in the banding area; and probability of temporary emigration from the superpopulation of Greater Snow Geese on Bylot Island ( i) and breeding propensity of adults (1 – i). Values are mean ± SE. See methods for details of calculations: a No radio–marked females were present on Bylot Island during banding; we visually estimated that 85% (range 75–95%) of the geese present on Bylot Island were captured; b Data from Bêty et al. (2002). Tabla 1. Estimación de la probabilidad de captura, dada una presencia en la suprapoblación (pi0 ), obtenida a partir del modelo CJS de población abierta ( pt); probabilidades de captura dada una presencia en el área de muestreo (p i*), obtenidas a partir de las hembras marcadas con radiotransmisores; la estimación de Mayfield de la supervivencia de los nidos (Sin) para la totalidad del período de nidificación; fidelidad a los emplazamientos de cría (Fi), estimada como la proporción de individuos que se espera que estén presentes en el área de anillamiento; y probabilidad de emigración transitoria a partir de la suprapoblación del ánsar nival en la isla de Bylot ( i) y propensión reproductora por parte de los adultos (1 – i). Los valores son medias ± EE. Para detalles sobre los cálculos, consultar los métodos: a En la isla de Bylot se observaron hembras marcadas con radiotransmisores durante el anillamiento; se capturaron aproximadamente un 85% (variación 75–95%) de los ánsares de la isla de Bylot; b Datos de Bêty et al. (2002).

0.18 ± 0.03

0.60 ± 0.08

–1.28 ± 1.62

Year

pi 0

pi*

Sin

1995

0.05 ± 0.01

0.20 ± 0.13

1–

2.28 ± 1.62

1996

0.05 ± 0.01

0.40 ± 0.07

0.62 ± 0.03

0.60 ± 0.08

0.65 ± 0.11

0.35 ± 0.11

1997

0.04 ± 0.01

0.09 ± 0.04

0.85 ± 0.02b

0.60 ± 0.08

0.19 ± 0.45

0.81 ± 0.45

1998

0.07 ± 0.01

0.18 ± 0.05

0.83 ± 0.02b

0.43 ± 0.08

–0.09 ± 0.39

1.09 ± 0.39

0.02 ± 0.01

1.00 ± 0.00

0.83 ± 0.06

0.17 ± 0.06

1999

0.85 ± 0.05

0.15 ± 0.02

2000

0.07 ± 0.02

0.46 ± 0.23

0.83 ± 0.03

0.43 ± 0.08

0.56 ± 0.26

0.44 ± 0.26

2001

0.05 ± 0.01

0.10 ± 0.10

0.52 ± 0.03

0.60 ± 0.08

–0.64 ± 1.63

1.64 ± 1.63

7.7 km in 2000, and 12.6 km in 2001. Thus, the probability that a female present on Bylot Island during banding was also present in our banding area varied from 0.43 in 1998 and 2000 to 1.00 in 1999 (table 1). Estimated temporary emigration ( i ) from Bylot Island’s south plain varied considerably among years (table 1). The large negative values for 1995 and 2001 were associated with a very poor precision as shown by the large variance of these estimates and their confidence intervals, which included 0 (95% CI: [–4.45 to 1.89] and [–3.83 to 2.55] for 1995 and 2001 respectively). We thus excluded 1995 and 2001 from further analyses, although they did not have a large weight given their large variance. Breeding propensity (1 – i ) varied across years ((24 = 9.56, P = 0.049) ranging from 1 in 1998 to 0.17 in 1999 (table 1). We estimated total variance at 0.137 and true temporal variance at 0.051, indicating that sampling variation accounted for 62% of the total variation. Average breeding propensity for these 5 years was 0.574 [95% CI considering only process variation: 0.131 to 1.017]. Breeding

propensity was negatively related to snow cover on 5 June (fig. 1; snow: –2.05 ± 0.96, z = –4.14, P = 0.02) and was reduced in years with a spring hunt ( hunt = –0.78 ± 0.35, z = –2.19, P = 0.01). Both nest density and young/adult ratio in the fall flight provided good indices of breeding propensity for the 1996–2000 period (fig. 2). However, nest density (nest/ha) was a better predictor of breeding propensity ( nest density: 0.59 ± 0.11, R2 = 0.97) than was young/adult ratio in the fall flight ( young/adult: 5.02 ± 1.06, R2 = 0.89). Discussion Factors affecting breeding propensity Breeding propensity of adult female Greater Snow Geese varied considerably between 1996 and 2000, ranging from 0.17 to 1. True temporal variation in breeding propensity was large (95% CI: 0.13–1.0), with an average of 57% of surviving adult females breeding (successfully or not) in any given year. These results confirm that intermit-

Animal Biodiversity and Conservation 27.1 (2004)

1.2 Breeding propensity

1998

1.0 1997

0.8 0.6 0.4

2000 1996

0.2

1999

0.0 40

50 60 70 Percent snow cover

Fig. 1. Relationship between snow cover on 5 June on Bylot Island and estimated breeding propensity (± SE) of female Greater Snow Geese. The solid line and dark dots represents years without a spring hunt and the dashed line and grey dots represent years with a spring hunt. Lines are predicted values of breeding propensity from model: logit (breeding propensity) = 0.79 [± 0.67] – 2.05 [± 0.96] * snow cover –0.78 [± 0.35] * (0: no spring hunt; 1: spring hunt). Only estimates for years 1996–2000 are used due to imprecision of the 1995 and 2001 estimates. Fig. 1. Relación entre la capa de nieve observada en la isla de Bylot el 5 de junio y propensión reproductora estimada (± EE) del ánsar nival. La línea contínua y los puntos oscuros representan los años sin cacería primaveral, mientras que la línea discontinua y los puntos grises representan los años con cacería primaveral. Las líneas son valores previstos de la propensión reproductora a partir del modelo: logit (propensión reproductora) = 0,79 [± 0,67] – 2,05 [± 0,96] * capa de nieve – 0,78 [± 0,35] * (0: sin cacería primaveral; 1: con cacería primaveral). Sólo se emplean estimas para los años 1996–2000, debido a la imprecisión de las estimas correspondientes a los años 1995 y 2001.

tent breeding is common in this population. High rates of temporary non-breeding are well documented in seabirds (Tickell & Pinder, 1967; Chastel, 1995; Nur & Sydeman, 1999) but results are more variable in waterfowl. Studies in Black Brant (Branta bernicla nigricans) and Lesser Snow Geese (Chen caerulescens caerulescens) found high breeding probabilities with no detectable temporal variation (Cooch et al., 2001; Sedinger et al., 2001), whereas studies in Canvasback (Aythya valisineria) and Barnacle Geese (Branta leucopsis) found important temporal variation in breeding probability (Anderson et al., 2001; Prop & De Vries, 1993). Several studies in seabirds have suggested that temporal variation in breeding probability may be due to variations in prey availability (e.g. Chastel, 1995; Nur & Sydeman, 1999), whereas for prairie waterfowl habitat availability in relation to drought cycles may be a key determinant (Anderson et al., 2001). However, the mechanism governing the decision to initiate breeding in herbivores such as geese is less obvious. Our results indicate that spring snow cover is an important determinant of breeding propensity (see also Prop & De Vries, 1993). The other goose studies that showed no

variation in breeding propensity were conducted in low Arctic regions, where environmental stochasticity is presumably less than at higher latitudes. Our results suggest that, in the High Arctic where the summer is very short, climatic conditions in spring are a key determinant of breeding propensity, with most females (> 80%) breeding when snow cover is low and few (< 30%) breeding when snow cover is extensive. Our results thus support early suggestions that reproductive effort of geese can be quite variable in the high Arctic (Prop & De Vries, 1993), with widespread breeding failure in years of late snowmelt (Barry, 1962). Snow cover at the time of nesting is governed by both winter snow accumulation and the speed of snow melt in spring. Thus, snow accumulation and air temperature, which influences the rate of snowmelt, may both influence breeding propensity. At our study site, we lacked an appropriate estimate of snow fall over the entire study area and spring air temperature was highly correlated to snow cover on 5 June, so we could not test these two variables separately (E. T. Reed, unpublished data). Climatic conditions could influence breeding propensity of herbivorous birds by limiting availability

Reed et al.

of food or nest sites. Geese feed intensively during the interval between arrival on the breeding grounds and egg–laying (Gauthier & Tardif, 1991), and this nutrient intake contributes significantly to the energy invested in egg production (Bromley & Jarvis, 1993; Choinière & Gauthier, 1995; Ganter & Cooke, 1996; Gauthier et al., 2003). Snow cover can thus limit access to high quality foraging sites in lowlands, because primary production in the first snow–free areas (mountain and ridges) is low due to wind exposure and good soil drainage (Gauthier, 1993; Prop & De Vries, 1993). Early nesting is also important in seasonal environments because date at which young hatch strongly influences their growth rate (Larsson & Forslund, 1991; Lepage et al., 1998) and ultimately their survival and recruitment prospects (Spear & Nur, 1994; Lepage et al., 2000; Prévot– Julliard et al., 2001; Reed et al., 2003b). Although geese are flexible in the choice of their nest site when snowmelt is late (Lepage et al., 1996), a large proportion of adults may refrain from breeding if survival prospects of the young are too low at the date where nest sites become available. In arctic–nesting geese, endogenous nutrient reserves acquired on the spring staging grounds are an important fuel source for the northward migration and contribute to breeding success (Ankney & MacInnes, 1978; Ebbinge,1989; Gauthier et al., 1992; Ebbinge & Spaans, 1995). Davies & Cooke (1983) and Alisauskas (2002) provided evidence that events occurring on spring staging areas influenced reproduction of arctic–nesting geese. In the first two years of the spring hunt (1999–2000), Féret et al. (2003) found a marked reduction in nutrient accumulation of Greater Snow Geese during spring staging due to heavy hunting disturbance (Béchet et al., 2003). This reduction of body condition was also detected in laying geese (Mainguy et al. 2002) and most reproductive parameters were negatively affected in years with spring hunt at our study site: geese laid later and had smaller clutch size than in previous years, and radio–marked females showed a marked reduction in breeding effort (Mainguy et al., 2002; Bêty et al., 2003). In this study, we found an overall reduction in breeding propensity during years with a spring hunt (1999 to 2001). Breeding propensity was at a record low value in 1999, although it was not especially low in 2000 when considering snow cover and it could not be satisfactorily estimated in 2001. Even though we only had two precise estimates, evidence suggests that the spring hunt negatively impacted breeding propensity in Greater Snow Geese. Overall, our results suggest that spring climatic conditions in the Arctic and nutrient reserves acquired during spring staging may be important determinants of reproductive effort in Greater Snow Geese (see also Bêty et al., 2003). The effect of conditions encountered at arrival should increase as one moves from low to high Arctic, since environmental stochasticity should increase with lati-

tude. Moreover, the amount of endogenous nutrients remaining upon arrival in the Arctic should be reduced as the length and cost of the spring migration increases with latitudinal range (Choinière & Gauthier, 1995). Methodological considerations Our results showed that temporary emigration could be used to estimate breeding propensity when only breeding individuals are subject to capture. In the case where some breeders are not at risk of capture, temporary emigration must be corrected as was done in this study using information from radio–marked birds. In our study population, most non–breeders and failed nesters undertake a molt migration from our sampling area (Reed et al., 2003a), and those that remain on the island have regained flight capacity and are thus not sampled at banding. We could not rule out the possibility that some geese considered as non-breeders actually nested outside the study area, but a large proportion of adult females (including non–breeders) are present on Bylot Island from the pre–laying to the incubation period before leaving the island for molting, too late for a breeding attempt (Reed et al. 2003a). Also, given the high level of breeding philopatry to Bylot Island previously found (Reed et al., 2003b), it seems unlikely that an important segment of the breeding population could temporarily settle elsewhere to breed. Capture probabilities, given presence on the sampled area, varied considerably among years. Thus, estimates of capture probabilities from open population models (pi0 ) do not represent a valid index of breeding propensity. Because breeding propensity represents the probability that an adult will breed, irrespective of breeding success, we also had to correct our capture probability for individuals that leave Bylot Island following a nest failure. Furthermore, we did not sample all broodrearing areas on Bylot Island but only specific ones, and the size of the sampling area varied in response to goose density. We therefore had to correct our estimates for incomplete fidelity of families to specific brood–rearing areas (Mainguy, 2003). However, biases may be associated with these correction factors, especially the brood– rearing site fidelity one which was rather crude. The ad hoc method that we used was the best suited for estimation of breeding propensity and associated variance in our system. However, our first (1995) and last (2001) estimate of were ^ 0, because pi0 p (pi* SinFibr). In 1995, this result may be due to a negative bias in nest survival estimation resulting from nests being found late during incubation and followed over a short period of time. We do not know which component(s) of the equation led to poor estimation of breeding propensity of the last encounter period (2001). However, negative estimates are not unusual when is close to 0.

Animal Biodiversity and Conservation 27.1 (2004)

A Breeding propensity

1.2 1998

1.0 1997

0.8 0.6 0.4

2000 1996

0.2

1999

0.0 1.0

1.5 2.0 2.5 Nest density (nest / ha)

3.0

3.5

B Breeding propensity

1.2 1998

1.0 1997

0.8 0.6

2000

0.4 0.2

1996 1999

0.0 0.05

0.10 0.15 0.20 0.25 0.30 Proportion of young in fall flight

0.35

Fig. 2. Relationship between estimates of breeding propensity (± SE) of female Greater Snow Geese at Bylot Island between 1996 and 2000 and: A. Nest density (nest / ha) in the main nesting colony on Bylot Island (Bêty et al., 2002); B. Proportion of young in the fall flight in the St. Lawrence River estuary (Reed et al., 1998; unpublished data). Lines are predicted values of breeding propensity from model: A. Logit (breeding propensity) = 0.59 [± 0.33] * nest density, –1.95 [± 0.58], R2 = 0.97; B. Logit (breeding propensity) = 5.03 [± 1.24] * young/adult ratio in the fall flight –1.97 [± 0.32], R2 = 0.89. Fig. 2. Relación entre estimas de propensión reproductora (± EE) en hembras del ánsar nival en la isla de Bylot entre 1996 y 2000, y: A. Densidad de los nidos (nidos / ha) en la principal colonia de nidificación en la isla de Bylot (Bêty et al., 2002); B. Proporción de jóvenes en el vuelo otoñal en el estuario del río San Lorenzo (Reed et al., 1998; datos no publicados). Las líneas representan los valores predichos de propensión reproductora extraídos del modelo: A. Logit (propensión reproductora) = 0,59 [± 0,33] *densidad de nidos, –1,95 [± 0,58], R2 = 0,97; B. Logit (propensión reproductora) = 5,03 [± 1,24] * relación de jóvenes/adultos en el vuelo otoñal –1,97 [± 0,32], R2 = 0,89.

Predictors of breeding propensity and consequences for population dynamics Nest density at the main nesting colony and young/ adult ratio in the fall flight were positively related to breeding propensity. Based on their respective coefficient of correlation, nest density provided a better index of breeding propensity than young/ adult fall ratios. The former is less sensitive to variations in nest success (when nests are found in the early phases of nesting), and insensitive of

pre– and post fledging survival. Breeding effort of geese has often been inferred by fall or winter age ratios (e.g. Ebbinge, 1989; Ebbinge & Spaans, 1995). Our results indicate that fall age ratios can provide a reliable index of temporal variation in breeding propensity in this population. However, we believe that breeding ground indices should be preferred, as they are not influenced by factors occurring during the 3–month interval between the end of the egg–laying stage and arrival on the fall staging areas.

The proportion of young in the fall flight in 1999 (2.1%) was the lowest ever recorded since the inception of the productivity surveys in the St. Lawrence river estuary in 1973 (Reed et al., 1998). The combined occurrence of low breeding propensity and high predation pressure during the nesting period thus resulted in the near complete loss of this cohort. In contrast, most experienced females nested in 1998 and nest survival was high, resulting in a particularly strong cohort. Such variation across cohorts has been described for other species of waterfowl (Anderson et al., 2001). Other factors, such as gosling survival during their first year of life (Owen & Black, 1989; Francis et al., 1992a), also contribute significantly to variations in productivity of arctic–nesting geese. The frequency of occurrence of ‘bad’ and ‘good’ years of reproduction has important consequences on the growth rate of this population (Gauthier & Brault, 1998). We showed that breeding propensity was negatively correlated to spring snow cover, and thus this variable could be considered in future population models of Greater Snow Geese, or other arctic–nesting birds that are likely to be affected by late snowmelt. The use of satellite imagery could be a useful monitoring tool for predicting breeding effort of arctic goose populations from information on snow–cover on the breeding grounds in spring (Reeves et al., 1976). Acknowledgements Funding was provided by a Natural Sciences and Engineering Research Council of Canada (NSERC) grant to G. Gauthier, the Arctic Goose Joint Venture (Canadian Wildlife Service), The Fonds pour la Formation des Chercheurs et l’Aide à la Recherche (FCAR, Ministère de l’Éducation du Québec), and the Department of Indian and Northern Affairs Canada. E.T. Reed received financial support from the FCAR, la Fondation de l’Université Laval, le Centre d’Études Nordiques, the Dennis Raveling Scholarship Fund, and the Fonds Richard Bernard. The Polar Continental Shelf Project (Natural Resources Canada) generously provided logistic support. The Hunters and Trappers Association of Pond Inlet, Nunavut Territory, kindly provided assistance and support. Thanks to the many people who participated in the field work, particularly J. Bêty, J. Mainguy, S. Menu, A. Ootoovak, J. Ootoovak, G. Picard, and A. Reed. We also wish to thank J.–D. Lebreton for statistical advice. R. T. Alisauskas, S. Côté, M. Lindberg, and J. D. Nichols provided valuable comments on earlier drafts of the manuscript. This is PCSP contribution no. 007–04 References Abraham, K. F., 1980. Moult migration of Lesser Snow Geese. Wildfowl, 31: 89–93. Alisauskas, R. T., 2002. Arctic climate, spring nutri-

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tion, and recruitment in mid–continent Lesser Snow Geese. Journal of Wildlife Management, 66: 181–193. Anderson, M. G., Lindberg, M. S. & Emery, R. B. 2001. Probability of survival & breeding for juvenile female Canvasbacks. Journal of Wildlife Management, 65: 385–397. Ankney, C. D., & MacInnes, C. D., 1978. Nutrient reserves & reproductive performance of female Lesser Snow Geese. Auk, 95: 459–471. Barry, T. W., 1962. Effects of late seasons on Atlantic Brant reproduction. Journal of Wildlife Management, 26: 19–26. Béchet, A., Giroux, J.–F., Gauthier, G., Nichols, J. D. & Hines, J. E., 2003. Spring hunting changes the regional movements of migrating greater snow geese. Journal of Applied Ecology, 40: 553–564. Bêty, J., 2001. Interactions trophiques indirectes, prédation et stratégies de reproduction chez l’Oie des neiges nichant dans le Haut–Arctique. Ph. D. Thesis, Univ. Laval, Qc, Canada. Bêty, J. & Gauthier, G., 2001. Effects of nest visits on predator activity and predation rate in a greater snow goose colony. Journal of Field Ornithology, 72: 573–586. Bêty, J., Gauthier, G. & Giroux, J.–F., 2003. Body condition, migration and timing of reproduction in snow geese: a test of the condition–dependent model of optimal clutch size. American Naturalist, 162: 110–121. Bêty, J., Gauthier, G., Giroux, J.–F. & Korpimäki, E., 2001. Are goose nesting success and lemming cycles linked? Interplay between nest density and predators. Oikos, 93: 388–400. Bêty, J., Gauthier, G., Korpimäki, E. & Giroux, J.–F., 2002. Shared predators and indirect trophic interactions: lemming cycles and arctic-nesting geese. Journal of Animal Ecology, 71: 88–98. Bromley, R. G. & Jarvis, R. L., 1993. The energetics of migration and reproduction of dusky Canada geese. Condor, 95: 193–210. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference: a practical informationtheoretic approach. Springer–Verlag, New York, New York, USA. Burnham, K. P., White, G. C., Brownie, C. & Pollock, K. H., 1987. Design and analysis methods for fish survival experiments based on release– recapture. American Fisheries Society Monographs., 5. Chastel, O., 1995. Influence of reproductive success on breeding frequency in four southern Petrels. Ibis, 137: 360–363. Choinière, L., & Gauthier, G., 1995. Energetics of reproduction in female and male greater snow geese. Oecologia, 103: 379–389. Cooch E. G., Rockwell, R. F. & Brault, S., 2001. Retrospective analysis of demographic responses to environmental change: a Lesser Snow Goose example. Ecological Monographs, 71: 377–400. Cormack, R. M., 1964. Estimates of survival from

Animal Biodiversity and Conservation 27.1 (2004)

the sighting of marked animals. Biometrika, 51: 429–438. Davies, J. C. & Cooke, F., 1983. Annual nesting productivity in snow geese: prairie drought and arctic springs. Journal of Wildlife Management, 47: 291–296. Demers, F., Giroux, J.–F., Gauthier, G. & Bêty, J., 2003. Effects of collar–attached transmitters on behavior, pair bond, and breeding success of snow geese. Wildlife Biology, 9: 161–170. Ebbinge, B. S., 1989. A multifactorial explanation for variation in breeding performance of Brent Geese Branta bernicla. Ibis, 131: 196–204. Ebbinge, B. S. & Spaans, B., 1995. The importance of body reserves accumulated in spring staging areas in the temperate zone for breeding dark–bellied Brent Geese Brante b. bernicla in the high Arctic. Journal of Avian Biology, 26: 105–113. Féret, M., Gauthier, G., Béchet, A. & Hobson, K. A., 2003. Effect of a spring hunt on nutrient storage by Greater Snow Geese in Southern Quebec. Journal of Wildlife Management, 67: 796–807. Francis C. M., Richards, M. H., Cooke, F. & Rockwell, R. F., 1992. Changes in survival rates of lesser snow geese with age and breeding status. Auk, 109: 731–747. Ganter, B. & Cooke, F., 1996. Pre–incubation feeding activities and energy budgets of Snow Geese: can food on the breeding grounds influence fecundity? Oecologia, 106: 153–165. Gauthier, G., 1993. Feeding ecology of nesting greater snow geese. Journal of Wildlife Management, 57: 216–223. Gauthier, G., Bêty, J. & Hobson, K., 2003. Are greater snow geese capital breeders? new evidence from a stable isotope model. Ecology, 84: 3250–3264. Gauthier, G. & Brault, S., 1998. Population model of the Greater Snow Goose: projected impacts of reduction in survival on population growth rate. In: The Greater Snow Goose: report of the Arctic Goose Habitat Working Group: 65–80 (B. D. J. Batt, Ed.). Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington, D.C. and Canadian Wildlife Service, Ottawa, Ontario. Gauthier, G., Giroux, J.–F. & Bédard, J., 1992. Dynamics of fat and protein reserves during winter and spring migration in greater snow geese. Canadian Journal of Zoology, 70: 2077–2087. Gauthier, G. & Tardif, J., 1991. Female feeding and male vigilance during nesting in Greater Snow Geese. Condor, 93: 701–711. Gould, W. R. & Nichols, J. D., 1998. Estimation of temporal variability of survival in animal population. Ecology, 79: 2531–2538. Hines, J. E. & Sauer, J. R., 1989. Program CONTRAST – A general program for the analysis of several survival or recovery rate estimates. Washington: U.S. Fish and Wildlife Technical Report, 24. Johnson, D. H., 1979. Estimating nest success: the Mayfield method and an alternative. Auk, 96:

651–661. Jolly, G. M., 1965. Explicit estimates from capturerecapture data with both dead and immigrationstochastic model. Biometrika, 52: 225–247. Kendall, W. L. & Nichols, J. D., 1995. On the use of secondary capture–recapture samples to estimate temporary emigration and breeding proportions. Journal of Applied Statistics, 22: 751–762. Kendall, W. L., Nichols, J. D. & Hines, J. E., 1997. Estimating temporary emigration using capturerecapture data with Pollock’s robust design. Ecology, 78: 563–578. Larsson, K. & Forslund, P., 1991. Environmentally induced morphological variation in the Barnacle Goose, Branta leucopsis. Behavioral Ecology, 2: 116–122. Lebreton, J.–D., 1995. The future of population dynamic studies using marked individuals: a statistician’s perspective. Journal of Applied Statistics, 22: 1009–1030. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118. Lepage, D., Gauthier, G. & Menu, S., 2000. Reproductive consequences of egg–laying decisions in snow geese. Journal of Animal Ecology, 69: 414–427. Lepage, D., Gauthier, G. & Reed, A., 1996. Breeding–site infidelity in Greater Snow geese: a consequence of constraints on laying date? Canadian Journal of Zoology, 74: 1866–1875. – 1998. Seasonal variation in growth of greater snow goose goslings: the role of food supply. Oecologia, 114: 226–235. Lesage, L., & Gauthier, G., 1998. Effect of hatching date on body and organ development in greater snow geese. Condor, 100: 316–325. Lindberg, M. S., Kendall, W. L., Hines, J. E. & Anderson, M. G., 2001. Combining band recovery data and Pollock’s Robust Design to model temporary and permanent emigration. Biometrics, 57: 273–281. Lindberg, M. S., Sedinger, J. S., Derksen, D. V. & Rockwell, R. F., 1998. Natal and breeding philopatry in a Black Brant, Branta bernicla nigricans, metapopulation. Ecology, 79: 1893–1904. Link, W. A., 1999. Modeling pattern in collections of parameters. Journal of Wildlife Management, 63: 1017–1027. Mainguy, J., 2003. Déplacement des familles de la grande oie des neiges durant la période d’élevage, Ile Bylot, Nunavut. M. Sc. Dissertation, Université Laval. Mainguy, J., Bêty, J., Gauthier, G. & Giroux, J.–F., 2002. Are body condition and reproductive effort of laying Greater Snow Geese affected by the spring hunt? Condor, 104: 156–162. Menu, S., Gauthier, G. & Reed, A., 2001. Survival of juvenile Greater Snow Geese immediately after banding. Journal of Field Ornithology, 72:

282–290. Nur, N. & Sydeman, W. J., 1999. Survival, breeding probability, and reproductive success in relation to population dynamics of Brandt’s cormorants. Bird Study, 46S: 92–103. Owen, M. & Black, J. M., 1989. Factors affecting the survival of barnacle geese on migration from the breeding grounds. Journal of Animal Ecology, 58: 603–617. Pollock, K. H., 1982. A capture–recapture design robust to unequal probability of capture. Journal of Wildlife Management, 46: 757–760. Pollock, K. H., Nichols, J. D., Brownie, C. & Hines, J. E., 1990. Statistical inference for capture– recapture experiments. Wildlife Monographs, 107: 1–97. Pradel, R., Cooch, E. G. & Cooke, F., 1995. Transient animals in a resident population of snow geese: local emigration or heterogeneity? Journal of Applied Statistics, 22: 695–710. Prévot–Julliard, A.–C., Pradel, R., Julliard, R., Grosbois, V. & Lebreton, J.–D., 2001. Hatching date influences age at first reproduction in the Black–headed gull. Oecologia, 127: 62–68. Prop, J. & De Vries, J., 1993. Impact of snow and food conditions on the reproductive performance of barnacle geese Branta leucopsis. Ornis Scandinavica, 24: 110–121. Reed, A., Giroux, J.–F. & Gauthier, G., 1998. Population size, productivity, harvest and distribution. In: The Greater Snow Goose: report of the Arctic Goose Habitat Working Group: 5–31 (B. D. J. Batt, Ed.). Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington, DC and Canadian Wildlife Service, Ottawa, Ontario. Reed, A., Hughes, R. J. & Gauthier, G., 1995. Incubation behavior and body mass of female greater snow geese. Condor, 97: 993–1001. Reed, E. T., 2003. Coûts des soins parentaux et effets des conditions environnementales sur la reproduction de la grande oie des neiges. Ph. D. Thesis, Université Laval. Reed, E. T., Bêty, J., Mainguy, J., Giroux, J.–F. &

Reed et al.

Gauthier, G., 2003a. Molt migration in relation to breeding success in Greater Snow Geese. Arctic, 56: 76–81. Reed, E. T., Gauthier, G., Pradel, R. & Lebreton, J.–D., 2003b. Age and environmental conditions affect recruitment in greater snow geese. Ecology, 84: 219–230. Reeves, H. M., Cooch, F. G. & Munro, D. E., 1976. Monitoring Arctic habitat and goose production by satellite imagery. Journal of Wildlife Management, 40: 532–541. Rohwer F. C. & Anderson, M. G., 1988. Femalebiased philopatry, monogamy, and the timing of pair formation in migratory waterfowl. Current Ornithology, 5: 187–221. Salomonsen, F., 1968. The moult migration. Wildfowl, 19: 5–24. Sauer, J. R. & Williams, B. K., 1989. Generalized procedures for testing hypotheses about survival or recovery rates. Journal of Wildlife Management, 53: 137–142. Seber, G. A. F., 1965. A note on the multiple-recapture census. Biometrika, 52: 249–259. – 1982. The estimation of animal abundance and related parameters. Second edition. MacMillan, New York, N.Y. Sedinger, J. S., Lindberg, M. S. & Chelgren, N. D., 2001. Age–specific breeding probability in black brant: effects of population density. Journal of Animal Ecology, 70: 798–807. Spear, L. & Nur, N., 1994. Brood size, hatching order and hatching date: effects on four lifehistory stages from hatching to recruitment in western gulls. Journal of Animal Ecology, 63: 283–298. Spendelow, J. A. & Nichols, J. D., 1989. Annual survival rates of breeding adult Roseate Terns (Sterna dougalii). Auk, 106: 367–374. Tickell, W. L. N. & Pinder, R., 1967. Breeding frequency in the albatrosses Diomeda melanophris and D. chrysostoma. Nature, 213: 315–316. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46: S120–139.

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Assessing senescence patterns in populations of large mammals J.–M. Gaillard, A. Viallefont, A. Loison & M. Festa–Bianchet

Gaillard, J.–M., Viallefont, A., Loison, A. & Festa–Bianchet, M., 2004. Assessing senescence patterns in populations of large mammals. Animal Biodiversity and Conservation, 27.1: 47–58. Abstract Assessing senescence patterns in populations of large mammals.— Theoretical models such as those of Gompertz and Weibull are commonly used to study senescence in survival for humans and laboratory or captive animals. For wild populations of vertebrates, senescence in survival has more commonly been assessed by fitting simple linear or quadratic relationships between survival and age. By using appropriate constraints on survival parameters in Capture–Mark–Recapture (CMR) models, we propose a first analysis of the suitability of the Gompertz and the two-parameter Weibull models for describing aging–related mortality in free–ranging populations of ungulates. We first show how to handle the Gompertz and the two–parameter Weibull models in the context of CMR analyses. Then we perform a comparative analysis of senescence patterns in both sexes of two ungulate species highly contrasted according to the intensity of sexual selection. Our analyses provide support to the Gompertz model for describing senescence patterns in ungulates. Evolutionary implications of our results are discussed. Key words: Gompertz model, Two–parameter Weibull model, Ungulates, Survival, Life history, Sexual selection. Resumen Evaluación de pautas de senescencia en poblaciones de grandes mamíferos.— Por lo general, para estudiar el papel que desempeña la senescencia en la supervivencia, ya sea en humanos, en animales de laboratorio o en animales cautivos. Se emplean modelos teóricos, como los de Gompertz y Weibull. En el caso de las poblaciones silvestres de vertebrados, dicho papel tiende a evaluarse ajustando relaciones lineales o cuadráticas simples entre la supervivencia y la edad. En el presente estudio proponemos —a partir de la aplicación de constricciones apropiadas en los parámetros de supervivencia empleados en los modelos de captura–marcaje–recaptura (CMR)— un primer análisis de la idoneidad del modelo de Gompertz y del modelo de dos parámetros de Weibull para describir la mortalidad relacionada con el envejecimiento en poblaciones de ungulados criadas en régimen de pasto libre. En primer lugar indicamos cómo emplear el modelo de Gompertz y el modelo de dos parámetros de Weibull en el contexto de los análisis de CMR, para seguidamente llevar a cabo un análisis comparativo de las pautas de senescencia en dos especies de ungulados de sexos opuestos, altamente contrastadas según la intensidad de la selección sexual. Nuestro análisis apoya el modelo de Gompertz para la descripción de pautas de senescencia en ungulados. Se discuten las implicaciones evolutivas de los resultados obtenidos. Palabras clave: Modelo de Gompertz, Modelo de dos parámetros de Weibull, Ungulados, Supervivencia, Historia vital, Selección sexual. J.–M. Gaillard & A. Loison, Unité Mixte de Recherche N°5558 "Biométrie et Biologie Evolutive", Bâtiment 711, Univ. Claude Bernard Lyon 1, 43 Blvd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France.– A. Viallefont, Equipe de Recherche en Ingénierie des Connaissances, Bâtiment L, Univ. Lumière Lyon 2, 5 Av Pierre Mendès– France, 69676 Bron Cedex, France.– M. Festa–Bianchet, Groupe de recherche en écologie, nutrition et énergétique, Dépt. de Biologie, Univ. de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1. Corresponding author: J.–M. Gaillard. E–mail: gaillard@biomserv.univ-lyon1.fr ISSN: 1578–665X

Introduction Senescence, usually defined as a decrease in reproductive output and/or survival with increasing age (Partridge & Barton, 1993), occurs in a large range of organisms ranging from nematodes to humans (Finch, 1990). However, from an evolutionary standpoint, the persistence of senescence over generations despite counter– selective pressures is paradoxical and is yet to be explained. Two main non–exclusive theories have been proposed: the mutation accumulation that involves mutations with deleterious effects during late life (Medawar, 1952) and the antagonistic pleiotropy that involves pleiotropic genes with advantageous effects during early life but detrimental effects during late life (Williams, 1957). Of the two, the theory of antagonistic pleiotropy has received more support (Partridge & Gems, 2002 for a review) although Hughes et al. (2002) have recently provided clear support for the mutation accumulation theory. Both theories assume that there are alleles that are deleterious in old but not in young individuals. However, only the theory of antagonistic pleiotropy assumes that there is a trade–off between performance in early and late life. So far, empirical studies have failed to show whether such a trade–off is likely to occur in natural populations of vertebrates except in humans where the theory of antagonistic pleiotropy has received some support (e. g., Westendorp & Kirkwood, 1998). Senescence is indeed especially difficult to study in the field because it requires a large number of animals to be monitored from birth to death. Thus while senescence has been reported in most captive populations of birds and mammals studied so far (e. g., Ricklefs, 2000; Ricklefs & Scheuerlein, 2001), whether senescence is pervasive among free–ranging populations of the same species has long been a matter of discussion (e. g., Promislow, 1991; Wooler et al., 1992; Gaillard et al., 1994). Comfort (1979) had even suggested that most free–ranging animals die as young or prime– age adults because adverse conditions do not let them survive until old age. However, the increasing number of studies based on long–term monitoring of known–aged free–ranging animals and the increasing availability of methods to reliably estimate age–specific survival (especially capture–mark–recapture techniques, Schwarz & Seber, 1999) have recently allowed biologists to accumulate empirical evidence showing that senescence is also common in natural populations of large mammals such as ungulates (Loison et al., 1999; Gaillard et al., 2003b). Such a finding has important consequences for our understanding of population dynamics and life history evolution, as neglecting senescence could bias estimates of adult survival (e.g., Festa–Bianchet et al., 2003). Therefore, we need to know how to model senescence, an issue that has rarely been

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addressed in population studies of vertebrates. Senescence is most often modelled by using either the Gompertz (e. g., Finch & Pike, 1996) or the Weibull (e. g., Ricklefs, 2000) model, but (1) their use has been limited because of the lack of possibilities to implement these models in the analysis of survival data, and (2) consequently, comparative analyses of the adequacy of these models to survival data from free–ranging vertebrates are lacking. Such comparisons are required because Gompertz and Weibull models imply different patterns of senescence (i. e., the effects of senescence on mortality multiply (Gompertz) versus add to (Weibull) the initial mortality rate, Ricklefs, 1998). The following work is intended to fill this gap. We analysed age–specific variation in survival of both males and females in two contrasted species of large mammals. Indeed we studied roe deer (Capreolus capreolus), a long–lived forest dwelling ungulate that is slightly dimorphic in size and weakly polygynous, and bighorn sheep (Ovis canadensis), a long–lived mountain–dwelling ungulate that is highly dimorphic in size and strongly polygynous. We performed our comparative analysis of senescence including the two parameter Weibull and Gompertz models within the framework of CMR methods based on two long–term studies (> 20 years). This procedure allowed us to work on unbiased estimates of survival (i. e., taking account of capture rates less than 1, Nichols, 1992) and to benefit from updated procedures of model selection based on information criterion (AIC, Burham & Anderson, 1998). We first developed a new method permitting to estimate the parameters of the Gompertz model and those of the two–parameter Weibull model, directly from CMR data in wild animal populations, based on the use of specific constraints on the survival parameters. According to our current knowledge of senescence patterns in ungulates we expected that (1) senescence would fit the Gompertz model better than the two–parameter Weibull model in both sexes of both species because ungulate mortality rates have generally been reported to increase exponentially with age (Calder, 1982); (2) males would have both lower initial survival and higher senescence rates than females in both species because the sex ratio among adults usually decreases with increasing age in free– ranging ungulates (Clutton–Brock, 1991); (3) bighorns of both sexes would show higher initial survival than roe deer of both sexes because bighorns are larger than roe deer and survival increases allometrically with body size among mammals (Peters, 1983; Calder, 1984); and (4) bighorn males would show higher senescence rates than roe deer males because the intensity of sexual selection is more acute on strongly polygynous bighorn than on weakly polygynous roe deer, the intensity of sexual selection leading to survival cost (Promislow, 1992).

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Material and methods Study areas and populations studied Bighorn sheep were studied at Ram Mountain (52º N, 115º W), Alberta, from 1975 to 1997. We did not include data collected after 1997 because some ewes were removed that year and mortality was affected by an increase in cougar predation. The number of bighorn sheep in June ranged from 94 to 232. All adult ewes were marked in all years of the study, and resighting probability was over 99%. Over 98% of adult rams were marked, and annual resighting probability exceeded 95% (Jorgenson et al., 1997). From 1972 to 1980, yearly removals of 12–24% of adult ewes (Jorgenson et al., 1993) kept the population at 94– 105 sheep. After 1980, ewe removals were discontinued and the population increased. Some males aged 4 years and older were shot by hunters (range 0–6, average 2.4/year), both during and after the period of ewe removals (Jorgenson et al., 1993). Ages of all individuals were known exactly because they were first captured when aged 4 years or younger (almost all as lambs or yearlings), when age can be accurately determined from horn annuli (Geist, 1966). Potential predators of adult bighorns included cougars (Puma concolor), wolves (Canis lupus) and black bears (Ursus americanus). Roe deer were studied at Chizé (46º N, 0º E), France, from 1978 to 2003. The population is in a 26 km2 enclosure and about 70% of the adults are marked. Each year, about 50% of adult deer are captured with drive nets (Gaillard et al., 1993). Ages of almost all marked roe deer born during the study were known because they were caught as fawns. Each year, some unmarked deer were removed for release elsewhere in France. Changes in the number of deer removed from the study area led to population size estimates varying from 157 to 569 deer older than one year in March (therefore excluding fawns born the previous year) (Gaillard et al., 1993). There were no predators of adult roe deer. Modelling senescence with CMR models We analysed our data with recent developments of Capture–Mark–Recapture techniques. Because we were interested in natural mortality, all animals that died because of hunting or accidents, or were removed from the population, were excluded from our sample in the year of their death or removal (see Loison et al., 1999). As emigration was impossible at Chizé and extremely rare at Ram Mountain (Jorgenson et al., 1997; Loison et al., 1999), we therefore assumed that all disappearances were due to mortality. Recent survival analyses in these populations have shown that the assumptions of CMR models were fulfilled by our data sets, so that the time–dependent model, the so–called Cormack– Jolly–Seber model (Lebreton et al., 1992) can be used as a starting model (Festa–Bianchet et al.,

2003). Recapture probabilities for bighorn sheep were very high and did not vary among years (Jorgenson et al., 1997; Loison et al., 1999). For roe deer, recapture probabilities varied over time (e. g., Gaillard et al., 2003a). We did not include time in the models in either study because, based on earlier studies (e.g., Loison et al., 1999), we did not find any statistical evidence for time–specific variation in adult survival in the two sexes. We therefore started with the model ( , pt) for roe deer at Chizé and with the model ( , p) for bighorn sheep at Ram Mountain. Similarly to our previous work, the sexes were analysed separately (Gaillard et al., 1993 for roe deer, Jorgenson et al., 1997 for bighorn sheep). We then fitted the following models to assess the effects of senescence on survival: (1) a 4 age–class model distinguishing yearling (1 to 2 years), prime–aged (2 to 7 years), old adults (8 to 12 years) and senescent adults (13 years and older). Because only a few bighorn males survived past 12 years, we only considered 3 classes of adult males (yearling from 1 to 2 years, prime–aged from 2 to 7 years, and old males of 8 years and more); this model corresponds to the so–called Caughley model (Caughley, 1966; Gaillard et al., 2000) and fits our current knowledge of survival patterns in ungulate populations (Gaillard et al., 2000) and previous analyses on bighorn sheep and roe deer (Loison et al., 1999); we noted this model ( c). (2) The Gompertz model (see below) from 2 years onwards. We indeed excluded yearlings that often show lower survival than adults because of their higher susceptibility to environmental variation (see Gaillard et al., 2000, for a review); in this model we considered that senescence should begin at the age of first reproduction as assumed in evolutionary theories of senescence (Hamilton, 1966); this model corresponds to an evolutionary–based model; we noted this Gompertz model ( g). (3) The Gompertz model from 8 years onwards. We here excluded yearlings and prime–age adults from the senescence model to account for a possible delayed senescence in longlived species; we chose 8 years of age as a threshold because in both species a marked tooth wear in animals older than 7 allows people to identify them as old individuals; this model corresponds to an empirically–based model; we noted this Gompertz model ( g8). (4) The two–parameter Weibull model (see below) from 2 years onwards; we noted this Weibull model ( w). (5) The two–parameter Weibull model from 8 years onwards. We noted this Weibull model ( w8). And (6) the complete age–dependent model involving a specific survival at each age; we noted this model ( a). We used the Akaike Information Criterion (AIC) to select the best parsimonious model (Burnham & Anderson, 1998) at each stage of the analysis. Because all models had a high information/parameter ratio (> 30), we chose to use AIC instead of AICc (Burnham & Anderson, 1998). All parameter estimates are given ± 1 SE.

Gaillard et al.

Method: introducing the two–parameter Weibull and Gompertz models These two models suppose that the hazard function, i.e., the function of instantaneous risk of death, changes with time following a specific shape. The general shape of the hazard rate for the two– parameter Weibull model is given by the function:

where is a positive real number representing the shape parameter, and is a positive real number called the scale parameter. When > 1, the hazard rate increases with time, whereas it decreases if < 1. For = 1, the hazard rate is constant, thus the Weibull model is equivalent to the exponential model (see fig. 1 for the shape of the hazard rate function for various values of and ). For 1 < < 2, the rate of increase of the hazard rate in this model is larger in the early periods than later; for = 2, the rate of increase of the hazard rate is constant over time; and for > 2, the rate of increase of the hazard rate is smaller in the early periods than later. It is important to notice that when the two– parameter Weibull model is used with an increasing hazard rate, the initial hazard rate is null, corresponding to a survival rate equal to one. This is a very strong constraint for the models, since the mortality is naturally never null at any time or age. This will constrain the estimation of , such as to obtain estimates of survival at early ages that are reasonable. Thus this may induce low estimates of ( < 2), whereas the biological hypotheses are more in favour of an increasing increase in the hazard rates with age ( > 2). To relax this constraint on the initial survival rates, we would have to use the 3–parameter Weibull model, which is more complicated to implement, and less easy to compare to the (two–parameter) Gompertz model. The general shape of the hazard rate for the Gompertz model is given by the function: h(t) = .exp( .t), where and are strictly positive real numbers. In this model, the increase in mortality is exponential, i.e. the ageing process is more and more important with time. The parameter represents the instantaneous risk at time 0, and regulates the intensity and delay of the increase in mortality: the higher is, the more intense the aging process (fig. 2). Method: discrete–time two–parameter Weibull and Gompertz models In CMR experiments, the most usual situation is one when the time is discretised in equal intervals separated by successive capture occasions. Thus, applying Weibull or Gompertz models to CMR data requires the survival to be expressed per time period (usually annual) in function of the model parameters. In the following, S(ai) represents the probability that an animal survives until age ai at least, and (ai) represents the probability that an

animal alive at age ai survives until age ai+1. The function S is called the survival function, whereas is the conditional survival per time unit. We have:

and

Let us denote hi the hazard rate at age ai. We assume that this risk is constant over the interval [ai; ai+1]. Then (ai) = exp(–hi (ai+1–ai)). For the two– parameter Weibull model, we have:

thus

Thus Ln(–Ln( (ai))) = Ln(ai+1–ai) + Ln( ) – Ln( ) + ( –1) Ln(ai). If we assume that ai+1–ai = 1 (time unit between two successive occasions of capture), we can write: Loglog( (ai)) = Ln( ) –

Ln( ) + ( – 1) Ln(ai) (1)

with Loglog(x) = –Ln(–Ln(x)) For the Gompertz model, we have

thus

Thus Ln(–Ln( (ai))) = Ln(ai+1 – ai) + Ln( ) + ai Assuming that ai+1–ai = 1, we can write: Loglog( (ai)) = Ln( ) + .ai

(2)

Method: implementation of the two–parameter Weibull and Gompertz models into CMR program design Computer programs that can produce survival estimates from capture–recapture data usually allow to use generalized linear constraints on the parameters. The equation for such constraints is: f(V) = D.B, where f is a link function (a strictly monotonous continuous real function), V is the vector of the k parameters to be constrained, D is a k×p–matrix of variables (called the "design matrix" in a software like MARK, White & Burnham, 1999), and B is the p–vector of the "mathematical parameters", i.e. the parameters that are really estimated. The use of such constraints in capture–recapture was first introduced by Clobert & Lebreton (1985) and Pradel et al. (1990). Implementing the two–parameter Weibull or Gompertz model into CMR programs is fairly

Animal Biodiversity and Conservation 27.1 (2004)

Hazard rate

30 25 20 15

= 0.5;

= 1.5;

= 0.5;

= 0.8

= 2.5;

= 2.5 = 1.5

= 0.5;

= 1;

= 1.5

0 0

10 Age

Fig. 1. Hazard function of the two–parameter Weibull model for various values of

and .

Fig. 1. Función de riesgo del modelo de dos parámetros de Weibull para diversos valores de

y .

160 140 Hazard rate

120 100

= 0.5; = 0.5; = 0.3;

= 0.5 = 0.4 = 0.5

60 40 20 0 0

10 Age

Fig. 2. Hazard function of the Gompertz model for various values of

and

Fig. 2. Función de riesgo del modelo de Gompertz para diversos valores de

straightforward when using this framework. The vector V is the vector of the survival parameters according to age (i.e. the i–th element of V is the probability of surviving from age i to age i+1, (ai)). For the two–parameter Weibull model we have: Loglog( (ai)) = Ln( ) –

Ln( ) + ( – 1) Ln(ai) (1)

that we can also write: f( (ai)) = B0 + B1 Ln(ai)

(3)

where the link function is f = Loglog, B0 = Ln( ) – Ln( ), and B1 = – 1.

. y

Thus, to fit the two–parameter Weibull model we must use the link function Loglog, and a matrix D constituted of a column of 1’s (intercept), and a column of the Ln(ai)’s. In other words the second column of D is the vector of the natural logarithms of ages. Fitting a model with this constraint will produce the estimated parameters and , from which we can reconstitute the estimates of the parameters of the Weibull model, as:

Gaillard et al.

Table 1. Model selection for the senescence of adult bighorn sheep males at Ram Mountain, Canada. Selected models are (2) and (4). Capture probability is constant in all models (p). Models fitted include: (1) a two age class dependent survival ( , yearling and older); (2) a yearling survival and a Gompertz model of senescence from 2 years onwards ( g); (3) a yearling survival, a prime–age adult survival and a Gompertz model of senescence from 8 years onwards ( g8); (4) a yearling survival and a Weibull model of senescence from 2 years onwards ( w); (5) a yearling survival, a prime–age survival and a Weibull model of senescence from 8 years onwards ( w); (6) a three age class dependent survival ( c, yearling, 2 to 7, and older) that correspond to the Caughley model; and (7) a complete age–dependent survival ( a); AIC corresponds to the difference of AIC between a given model and the selected model (i.e., 0 for the selected model). Tabla 1. Selección de modelos para el estudio de la senescencia en el cordero cimarrón macho, de edad adulta, de Ram Mountain, Canadá. Modelos seleccionados son (2) y (4). La probabilidad de captura es constante en todos los modelos (p). Los modelos ajustados incluyen: (1) una supervivencia dependiente de dos clases de edad ( , de 1 año de edad y más); (2) una supervivencia de 1 año de edad y un modelo de senescencia de Gompertz que abarca desde los 2 años de edad en adelante ( g); (3) una supervivencia de 1 año de edad, una supervivencia en el período de plenitud de la edad adulta y un modelo de senescencia de Gompertz que abarca desde los 8 años de edad en adelante ( g8); (4) una supervivencia de 1 año de edad y un modelo de senescencia de Weibull que abarca desde los 2 años de edad en adelante ( w); (5) una supervivencia de 1 año de edad, una supervivencia en el período de plenitud la edad adulta y un modelo de senescencia de Weibull que abarca desde los 8 años de edad en adelante ( w); (6) una supervivencia dependiente de tres clases de edad ( c: de 1 año de edad, de 2 a 7 años, y más), correspondiente al modelo de Caughley; y (7) una supervivencia dependiente por completo de la edad ( a); AIC corresponde a la diferencia de AIC entre un modelo determinado y el modelo seleccionado (es decir, 0 para el modelo seleccionado).

Model

Parameter

Deviance

AIC

(1) ( , p)

393.404

1.990

(2) ( g, p)

389.480

0.066

(3) (

390.603

3.189

, p)

(4) ( w, p)

389.414

(5) (

390.870

3.456

(6) ( c, p)

391.710

2.296

(7) ( a, p)

385.616

14.202

, p) w8

For the Gompertz model we have: Loglog( (ai)) = Ln( ) + .ai

(2)

that we can also write: f( (ai)) = B0 + B1.ai

(4)

where the link function is f = Loglog, B0 = Ln( ), and B1 = . Thus, to fit the Gompertz model we must use the link function Loglog, and a matrix D constituted of a column of 1’s (intercept), and a column of the ages ai’s. Fitting a model with this constraint will produce and , from which we the estimated parameters can reconstitute the estimates of the parameters of the Gompertz model, as:

The variances of the reconstituted parameters ( and for the Weibull model, and for the Gompertz model) can finally be computed from those of the estimated parameters B0 and B1, using the "delta–method" (e. g., Seber 1982, pp. 8–9). For the Gompertz model the obtained formulas are quite simple: ,

Method: application to real data For computational reasons the link function called "Loglog" in software MARK (White & Burnham, 1999) is in fact the opposite of the function Loglog: f(x) = – Ln(–Ln(x)), which is an increasing function,

Animal Biodiversity and Conservation 27.1 (2004)

Survival

A 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

5 6 7 Age

Survival

C 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Age D

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Age

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Age

Fig. 3. Age–dependent survival modelled by using a complete age–dependent model (filled symbols with ± 1 SE), a Gompertz function (full line) and a Weibull function (dotted line) in: A. Bighorn sheep males at Ram Mountain, Canada; B. Bighorn sheep females at Ram Mountain, Canada; C. Roe deer males at Chizé, France; D. Roe deer females at Chizé, France. Fig. 3. Supervivencia dependiente de la edad, modelada empleando un modelo dependiente por completo de la edad (símbolos complementados con ± 1 EE), una función de Gompertz (línea continua) y una función de Weibull (línea discontinua) en: A. Cordero cimarrón macho de Ram Mountain, Canadá; B. Cordero cimarrón hembra de Ram Mountain, Canadá; C. Corzo macho de Chizé, Francia; D. Corzo hembra de Chizé, Francia.

contrary to the "true" Loglog that is decreasing. Thus when using this software it is important to know that the parameter estimates obtained are in fact (– ) and (– ). The following steps are identical to those described above. Results Male bighorn sheep (table 1, fig. 3A) The Caughley model provided a similar fit as the constant survival model (i.e., models [ c , p] and [ , p] with very close AIC). As expected survival first slightly increased from yearlings (0.837 ± 0.025) to prime–aged adults (0.850 ± 0.014) and then decreased from prime– aged adults to old ones (0.752 ± 0.082). This model

accounted for age–dependence in survival (i.e., model [ c, p] with a much lower AIC than model [ a, p]). Both the Weibull and the Gompertz models performed better than the Caughley model (i.e., models [ w, p] and [ g, p] with a lower AIC than model [ c, p]), indicating that senescence of male bighorn is better fitted by using a continuous model than a threshold model. Modelling senescence from 2 years onwards provided a better fit than modelling senescence from 8 years. For the same number of parameters, both evolutionary– based models provided the same fit. According to our first prediction, we selected the Gompertz model from 2 years onwards to describe senescence of male bighorns. From this model, the initial mortality ( ) was estimated to be 0.121 (± 0.024) and the rate of senescence ( ) to be 0.105 (± 0.052).

Gaillard et al.

Table 2. Model selection for the senescence of adult bighorn sheep females at Ram Mountain, Canada. The selected model is (2). Capture probability is constant in all models (p): P. Parameter; D. Deviance. (See table 1 for more information on models fitted.)

Table 3. Model selection for the senescence of adult roe deer males at Chizé, France. The selected models are (2), (3) and (5). Capture probability is time–dependent in all models (pt): P. Parameter; D. Deviance. (See table 1 for more information on models fitted.)

Tabla 2. Selección de modelos para el estudio de la senescencia en el cordero cimarrón hembra, de edad adulta, de Ram Mountain, Canadá. El modelo seleccionado es el (2). La probabilidad de captura es constante en todos los modelos (p): P. Parámetro; D. Desviación. (Para más información sobre los modelos ajustados, ver tabla 1.)

Tabla 3. Selección de modelos para el estudio de la senescencia en el corzo macho, de edad adulta, de Chizé, Francia. Los modelos seleccionados son el (2), (3) y ( 5 ) . L a p r o b a b i l i d a d d e c a p t u r a es dependiente del tiempo en todos los modelos (p t ): P. Parámetro; D. Desviación. (Para más información sobre los modelos ajustados, ver tabla 1.)

Model

(1) ( , p)

314.543

33.886

(1) ( , pt)

(2) ( g, p)

278.657

AIC

950.840

17.160

(2) ( g, pt)

931.993

0.313

930.027

0.347

935.391

3.711

929.680

(3) (

, p)

279.736

3.079

(3) (

(4) (

, p) w

284.015

5.358

(4) (

, pt) w

(5) (

280.932

4.275

(5) (

(6) ( c, p)

281.297

4.640

(6) ( c, pt)

934.766

5.086

(7) ( a, p)

267.495

18.838

(7) ( a, pt)

919.100

9.420

, p)

, p t) , p t)

Female bighorn sheep (table 2, fig. 3B)

Male roe deer (table 3, fig. 3C)

The Caughley model provided a much better fit than the constant survival model (i.e., model [ c, p] with a much lower AIC than model [ , p]). Survival first increased from yearlings (0.813 ± 0.025) to prime–aged adults (0.940 ± 0.008) and then decreased first from prime–aged adults to old ones (0.832 ± 0.025), and then from old adults to senescent ones (0.776 ± 0.062). This model accounted for age–dependence in survival (i.e., model [ c, p] with a much lower AIC than model [ a, p]). While the Weibull model did not outperform the Caughley model (i.e., model [ c, p] with a lower AIC than model [ w, p]), the Gompertz model did (i.e., model [ g, p] with a much lower AIC than model [ c, p] or [ w, p]). The Gompertz model with the onset of senescence at 2 years of age outperformed the Gompertz model with the onset of senescence at 8 years of age (i. e., model [ g, p] with a lower AIC than model [ g8, p]) and was finally selected to model senescence of female bighorns. From this model, the initial mortality ( ) was estimated to be 0.039 (± 0.007) and the rate of senescence ( ) to be 0.161 (± 0.025).

The Caughley model provided a much better fit than either the constant or the complete age–dependent model (i.e., model [ c, pt] with a much lower AIC than both models [ , pt] and [ a, pt]). Survival first increased from yearlings (0.826 ± 0.037) to prime– aged adults (0.887 ± 0.017) and then decreased first from prime–aged adults to old adults (0.804 ± 0.046), and then from old to senescent adults (0.383 ± 0.167). Both the Weibull and the Gompertz models performed better than the Caughley model (i.e., models [ w, p] and [ g, p] with a lower AIC than model [ c, p]), indicating that senescence of male roe deer is better fitted by using a continuous model than a threshold model. For the same number of parameters, the Gompertz model provided a much better fit than the Weibull model when senescence was assumed to occur from 2 years of age whereas both models performed very well and similarly when senescence occurred from 8 years of age. From a statistical viewpoint, we cannot choose among both evolutionary– and empirically–based Gompertz models and the evolutionary–based Weibull model (differences of AIC among these

Animal Biodiversity and Conservation 27.1 (2004)

Table 4. Model selection for the senescence of adult roe deer females at Chizé, France. The selected models are (2) and (3). Capture probability is time–dependent in all models (p t ): P. Parameter; D. Deviance. (See table 1 for more information on models fitted.) Tabla 4. Selección de modelos para el estudio de la senescencia en el corzo hembra, de edad adulta, de Chizé, Francia. Los modelos seleccionados son (2) y (3). La probabilidad de captura es dependiente del tiempo en todos los modelos (p t ): P. Parámetro; D. Desviación. (Para más información sobre los modelos ajustados, ver tabla 1.)

Model

AIC

(1) ( , pt)

1483.662

39.638

(2) ( g, pt)

1442.024

(3) (

1440.580

0.556

(4) (

, p t)

1449.892

7.868

(5) (

, p t) w8

1444.284

4.260

(6) ( c, pt)

1447.714

7.690

(7) ( a, pt)

1421.735

9.711

, pt)

three models within 1). For comparative purposes, we selected the evolutionary–based Gompertz to model senescence of male roe deer. From this model, the initial mortality ( ) was estimated to be 0.064 (± 0.017) and the rate of senescence ( ) to be 0.171 (± 0.039). Female roe deer (table 4, fig. 3D) As for males, the Caughley model provided a much better fit than the constant survival model (i.e., model [ c, p] with a much lower AIC than model [ , p]). Survival first increased from yearlings (0.818 ± 0.032) to prime–aged adults (0.957 ± 0.010) and then decreased first from prime–aged adults to old adults (0.886 ± 0.023), and then from old adults to senescent ones (0.725 ± 0.054). This model accounted for age– dependence in survival (i. e., model [ c, p] with a much lower AIC than model [ a, p]). While the Weibull model did not outperform the Caughley model (i.e., model [ c, p] with similar AIC as model [ w, p]), the Gompertz model did (i.e., model [ g, p] with a much lower AIC than model [ c, p]). The Gompertz model with the onset of senescence at 2 years of age performed closely to the Gompertz model with the onset of senescence at 8 years of age (i.e., model [ g, p] with similar AIC as model

[ g8, p]). We selected the evolutionary–based Gompertz to model senescence of female roe deer. From this model, the initial mortality ( ) was estimated to be 0.021 (± 0.006) and the rate of senescence ( ) to 0.199 (± 0.031). Discussion The results reported here on senescence of survival in both sexes of two contrasted ungulate populations monitored over the long–term using a new methodology revealed some consistent patterns that allow us to discuss whether or not the predictions we set either from theoretical or empirical knowledge so far accumulated are supported. Comparison with the existing methodology Ad hoc methods mostly based on life tables have often been used to assess senescence in mammals (e.g., Nesse, 1988; Promislow, 1991). By allowing to account for capture rates less than one (Nichols, 1992), CMR methods offer a promising way to reliably estimate the rate of senescence and thereby to test current theories about senescence patterns. To date, CMR analyses have aimed to assess whether or not survival decreases with increasing age but no study to date has yet tried to fit various theoretical models of senescence directly from capture–recapture data, and thus to make comparisons between these models. However, previous CMR studies have already approximated the Gompertz model either by fitting a logit–linear relationship between estimates of survival and age of ungulates (Loison et al., 1999), a linear relationship between the logarithm of the "instantaneous force of mortality" (estimated by the Kaplan–Meier method) and age of captive animals (McDonald et al., 1996), or by regressing the Log of mortality estimates (1 – (ai)) on age in ungulates (Gaillard et al., 2003b). These approaches were in fact very close to the models presented here. The (–Loglog) link used here is almost equal to the Logit link used by Loison et al. (1999) when survival rates are greater than 0.75 as is usually observed for large mammals such as ungulates. In the model used by McDonald et al. (1996), is defined as: l – Ln( (ai)), which means a Loglog relationship between and age, exactly like in the Gompertz model implemented here. Lastly, the approximation used by Gaillard et al. (2003b) is also very close to ours, especially when survival is high, as can be seen by comparing the graphical representations of functions "complementary–Log" (f( ) = Ln(1– )) and of the –Loglog functions. However, when dealing with CMR data, the method described in this paper, fitting the model with a built–in constraint on the parameters, should be preferred because the estimates obtained for various ages in an unconstrained CMR model are not independent of each other, thus not suitable for the estimation of regression parameters (Lebreton et al., 1992). Moreover, our procedure also allowed us

to implement the two–parameter Weibull model for the first time in a CMR context. By allowing us to compare between senescence models, the approach presented here appears as the most efficient and reliable to date to assess senescence from CMR data. Modelling senescence by Gompertz or Weibull models In females of both species, the Gompertz model provided the best fit whereas in males, both the two–parameter Weibull and Gompertz models provided the same fit. We therefore found support to our first empirically–based prediction that mammalian senescence can be reliably modelled by using Gompertz model. The multiplicative effects of senescence on mortality might be caused by a degeneration of vital functions when ageing, predicted by both theories of senescence ("mutation accumulation" and "antagonistic pleiotropy"). That the Gompertz model is suitable to describe senescence in ungulates supports previous works performed on more usual material for studying senescence such as drosophilae, nematodes, medfly and humans (Finch & Pike, 1996; Olshanski & Carnes, 1997). However, these works did not provide information on the suitability of other senescence models. The apparent difference between ungulates (that seem to follow better the Gompertz model, this study, Gaillard et al., 2003) and birds (that seem to follow better the Weibull model, Ricklefs, 2000; Ricklefs & Scheuerlein, 2001) in the shape of the senescence curve could account for the much longer longevity of birds despite similar rates of initial mortality (typically less than 0.10). However, such a hypothesis would warrant further investigation. Both models predict an increase of mortality rates with increasing age and only differ according to the shape of the increase (additive vs multiplicative). To address the question, Ricklefs (1998) used regression on survival estimates to fit a three–parameter version of the Gompertz model, and the three parameter Weibull model. Ricklefs & Scheuerlain (2002) reported a reasonably good fit of both Gompertz and Weibull models. In our present analyses, we failed to conclude whether the Gompertz or the two–parameter Weibull model better describe senescence for males of both species for which data spanned over a lower range of ages because of shorter lifespan compared to females. Distinguishing the Gompertz model from the three–parameter Weibull model would probably be more appropriate, but it would not be an easy task because of (1) the usual lack of data for very old individuals, and (2) the difficulty in implementing the three–parameter Weibull model in capture–recapture software. Our modelling did not allow us to assess whether antagonistic pleiotropy or mutation accumulation better accounted for observed senescence patterns in the two ungulate populations we studied. Such an issue is however of major importance and has not yet been satisfactorily solved, especially in

Gaillard et al.

natural populations. We feel that having reliable methods for modelling senescence is a prerequisite for any progress in this area. Between–sex differences in initial mortality and rate of senescence Senescence patterns reported for bighorn sheep partly supported our second prediction based on current theories of sexual selection. As expected, initial mortality was higher in males than in females in both species. Costly mating tactics of males and between–sex differences in the timing of energy expenditures might account for these results. Highly polygynous bighorn males compete intensively during the mating period at the end of the autumn and almost stop to feed (Ruckstuhl, 1998). At the same time, females spend most time feeding in order to recover body condition after the costly lactation period (Festa–Bianchet et al., 1998). Therefore, males enter winter in much poorer condition than females and are more susceptible to mortality factors (winter severity, predation or diseases, Jorgenson et al., 1997). Results reported for roe deer also supported higher vulnerability of males for initial mortality. Despite slight sexual dimorphism in size and low level of polygyny (Andersen et al., 1998), roe deer males are much more vulnerable to harsh conditions than females. Territory defence over half the year by male roe deer could be involved. Contrary to the second prediction, however, senescence rates were not higher in males than in females in either species, females showing an even slightly higher rate. Low numbers of old males could be involved in this surprising result. Allometric effects in senescence patterns We did not find any support for the third prediction based on allometry according to which larger bighorn should enjoy lower initial mortality than smaller roe deer. Contrary to the expectations, initial mortality rates were higher in bighorn than in roe deer in both sexes. Survival of adult ungulates is high (typically around 0.90, Gaillard et al., 2000) in all species irrespective of their size, likely because of a risk adverse strategy of adults (Gaillard & Yoccoz, 2003). Such a life history tactic is expected to dampen among–species variation in survival. Alternatively, sampling error of survival estimates might mask any existing allometric effect. Between–species differences in male senescence rate As expected from our fourth prediction based on current theories of sexual selection, male bighorn sheep tended to show a higher rate of senescence than male roe deer. Such a result supports the hypothesis that males of highly polygynous and dimorphic species exhibit a high risk–high benefit tactic from birth onwards: they usually are born

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heavier, grow faster, have higher juvenile mortality, allocate more energy to growth and reproduction than to maintenance, and spend more time in aggressive contests with conspecifics than females (Clutton–Brock, 1991; Short & Balaban, 1994). Although male roe deer differ from females by spending time to defend territories against conspecifics, they are born at the same size, have similar growth rates, and similar juvenile survival as females (Gaillard et al., 1998), likely accounting for the lower rate of senescence observed in male roe deer. Acknowledgments We are very grateful to Jim Hines and an anonymous referee for greatly helping us to improve this work. References Andersen, R., Gaillard, J. M., Liberg, O. & San Jose, C., 1998. Variation in life history parameters. In: The European Roe deer: The Biology of Success: 285–307 (R. Andersen, J. D. C. Linnell & P. Duncan, Eds.). Scandinavian University Press, Oslo. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference: a practical information– theoretic approach. Springer Verlag, Berlin. Calder, W. A. III., 1982. The relationship of the Gompertz constant and maximum potential lifespan to body mass. Experimental Gerontology, 17: 383–385. – 1984. Size, function and life history. Harvard University Press, Harvard, U.S.A. Caughley, G., 1966. Mortality patterns in mammals. Ecology, 47: 906–918. Clobert, J. & Lebreton, J.–D., 1985. Dépendance de facteurs de milieu dans les estimations de taux de survie par capture–recapture. Biometrics, 41: 1031–1037. Clutton–Brock, T. H., 1991. The evolution of parental care. Princeton University Press, Princeton, U.S.A. Comfort, A., 1979. The biology of senescence. Third Edition. Churchill Livingstone, Edinburgh, UK. Festa–Bianchet, M., Gaillard, J. M. & Jorgenson, J. T., 1998. Mass– and density–dependent reproductive success and reproductive costs in a capital breeder. The American Naturalist, 152: 367–379. Festa–Bianchet, M., Gaillard, J. M. & Cote, S. D. 2003. Variable age structure and apparent density–dependence in survival of adult ungulates. Journal of Animal Ecology, 72: 640–649. Finch, C. E., 1990. Longevity, senescence, and the genome. Chicago University Press, Chicago. Finch, C. E. & Pike, M. C., 1996. Maximum life span predictions from the Gompertz mortality model. Journal of Gerontology Series A– Biological Sciences and Medical Sciences, 51: 183–194. Gaillard, J. M., Allaine, D., Pontier, D., Yoccoz, N.G. & Promislow, D. E., 1994. Senescence in natural

populations of mammals: a reanalysis. Evolution, 48: 509–516. Gaillard, J. M., Andersen, R., Delorme, D. & Linnell, J., 1998. Family effects on growth and survival of juvenile roe deer. Ecology, 79: 2878–2889. Gaillard, J. M., Delorme, D., Boutin, J. M., Van Laere, G., Boisaubert, B. & Pradel, R., 1993. Roe deer survival patterns: a comparative analysis of contrasting populations. Journal of Animal Ecology, 62: 778–791. Gaillard, J. M., Duncan, P., Delorme, D., Van Laere, G., Pettorelli, N., Maillard, D. & Renaud, G., 2003a. Effects of Hurricane Lothar on the population dynamics of European roe deer. J. Wildl. Manage., 67: 767–773. Gaillard, J. M., Festa–Bianchet, M., Yoccoz, N. G., Loison, A. & Toigo, C., 2000. Temporal variation in fitness components and population dynamics of large herbivores. Annual Review of Ecology and Systematics, 31: 367–393. Gaillard, J.–M., Loison, A., Festa–Bianchet, M., Yoccoz, N. G. & Solberg, E., 2003b. Ecological Correlates of Life Span in Populations of Large Herbivorous Mammals. Population and Development Review, 29: 39–56. Gaillard, J. M. & Yoccoz, N. G., 2003. Temporal variation in survival of mammals and its impact on fitness: a case of environmental canalization? Ecology, 84: 3294–3306. Geist, V., 1966. Validity of horn segment counts in aging bighorn sheep. Journal of Wildlife Management, 30: 634–646. Hamilton, W. D., 1966. The moulding of senescence by natural selection. Journal of Theoretical Biology, 12: 12–45. Hughes, K. A., Alipaz, J. A., Drnevich, J. M. & Reynolds, R. M., 2002. A test of evolutionary theories of aging. Proceedings of the National Academy of Sciences of the United States of America, 99: 14286–14291. Jorgenson, J. T., Festa–Bianchet, M. & Wishart, W. D., 1993. Harvesting bighorn ewes: consequences for population size and trophy ram production. Journal of Wildlife Management, 57: 429–435. Jorgenson, J. T., Festa–Bianchet, M., Gaillard, J. M. & Wishart, W. D., 1997. Effects of age, sex and density on natural survival of yearling and adult bighorn sheep. Ecology, 78: 1019–1032. Lebreton, J. D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118. Loison, A., Festa–Bianchet, M., Gaillard, J.–M., Jorgenson, J. T. & Jullien, J. M., 1999. Agespecific survival in five populations of ungulates: evidence of senescence. Ecology, 80: 2539–2554. McDonald, D. B., Fitzpatrick, J. W. & Woolfenden, G. E., 1996. Actuarial senescence and demographic heterogeneity in the Florida scrub jay. Ecology, 77: 2373–2381. Medawar, P. B., 1952. An unsolved problem of biology. Lewis, London, U.K.

Nesse, R. M., 1988. Life table tests of evolutionary theories of senescence. Experimental Gerontology, 23: 445–453. Nichols, J. D., 1992. Capture–recapture models. Bioscience, 42: 94–102. Olshansky, S. J. & Carnes, B. A., 1997. Ever since Gompertz. Demography, 34: 1–15. Partridge, L. & Barton, N. H., 1993. Optimality, mutation and the evolution of ageing. Nature, 362: 305–311. Partridge, L. & Gems, D., 2002. Mechanisms of ageing: Public or private? Nature Reviews Genetics, 3: 165–175. Peters, R. H., 1983. The ecological implications of body size. Cambridge Studies in Ecology, Cambridge, U.K. Pradel, R., Clobert, J. & Lebreton, J.–D., 1990. Recent developments for the analysis of capture–recapture multiple data sets. An example concerning two blue tit populations. The Ring, 13: 193–204. Promislow, D. E. L., 1991. Senescence in natural populations of mammals: a comparative study. Evolution, 45: 1869–1887. – 1992. Cost of sexual selection in natural populations of mammals. Proceedings of the Royal Society of London Series B, 247: 203–210. Ricklefs, R. E., 2000. Intrinsic aging–related mortality in birds. Journal of Avian Biology, 31: 103–111. Ricklefs, R. E. & Scheuerlein, A., 2001. Comparison of aging–related mortality among birds and mammals. Experimental Gerontology, 36: 845–857.

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– 2002. Biological implications of the Weibull and Gompertz models of aging. Journal of Gerontology Series A – Biological Sciences and Medical Sciences, 57: 69–76. Ruckstuhl, K. E., 1998. Foraging behaviour and sexual segregation in bighorn sheep. Animal Behaviour, 56: 99–106. Schwarz, C. J. & Seber, G. A. F., 1999. A review of estimating animal abundance. III. Statistical Science, 14: 1–134. Seber, G. A. F., 1982. The estimation of animal abundance and related parameters. Second Edition. Charles W. Griffin, London, U.K. Short, R. V. & Balaban, E., 1994. The differences between sexes. Cambridge University Press, Cambridge, U.K. Vaupel, J. W., Manton, K. G. & Stallard, E., 1979. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16: 439–454. Westendorp, R. G. J. & Kirkwood, J. B. L., 1998. Human longevity at the cost of reproductive success. Nature, 396: 743–746. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 supplement: 120–138. Williams, G. C., 1957. Pleiotropy, natural selection and the evolution of senescence. Evolution, 11: 398–411. Wooler, R. D., Bradley, J. S. & Croxall, J. P., 1992. Long–term population studies of seabirds. Trends in Ecology and Evolution, 7: 111–114.

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Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: study designs that may help M. R. Marshall, D. R. Diefenbach, L. A. Wood & R. J. Cooper

Marshall, M. R., Diefenbach, D. R., Wood, L. A. & Cooper, R. J., 2004. Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: study designs that may help. Animal Biodiversity and Conservation, 27.1: 59–72. Abstract Annual survival estimation of migratory songbirds confounded by incomplete breeding site–fidelity: study designs that may help.— Many species of bird exhibit varying degrees of site–fidelity to the previous year’s territory or breeding area, a phenomenon we refer to as incomplete breeding site–fidelity. If the territory they occupy is located beyond the bounds of the study area or search area (i.e., they have emigrated from the study area), the bird will go undetected and is therefore indistinguishable from dead individuals in capture–mark–recapture studies. Differential emigration rates confound inferences regarding differences in survival between sexes and among species if apparent survival rates are used as estimates of true survival. Moreover, the bias introduced by using apparent survival rates for true survival rates can have profound effects on the predictions of population persistence through time, source/sink dynamics, and other aspects of life–history theory. We investigated four study design and analysis approaches that result in apparent survival estimates that are closer to true survival estimates. Our motivation for this research stemmed from a multi–year capture–recapture study of Prothonotary Warblers (Protonotaria citrea) on multiple study plots within a larger landscape of suitable breeding habitat where substantial inter–annual movements of marked individuals among neighboring study plots was documented. We wished to quantify the effects of this type of movement on annual survival estimation. The first two study designs we investigated involved marking birds in a core area and resighting them in the core as well as an area surrounding the core. For the first of these two designs, we demonstrated that as the resighting area surrounding the core gets progressively larger, and more "emigrants" are resighted, apparent survival estimates begin to approximate true survival rates (bias < 0.01). However, given observed inter–annual movements of birds, it is likely to be logistically impractical to resight birds on sufficiently large surrounding areas to minimize bias. Therefore, as an alternative protocol, we analyzed the data with subsets of three progressively larger areas surrounding the core. The data subsets provided four estimates of apparent survival that asymptotically approached true survival. This study design and analytical approach is likely to be logistically feasible in field settings and yields estimates of true survival unbiased (bias < 0.03) by incomplete breeding site–fidelity over a range of inter–annual territory movement patterns. The third approach we investigated used a robust design data collection and analysis approach. This approach resulted in estimates of survival that were unbiased (bias < 0.02), but were very imprecise and likely would not yield reliable estimates in field situations. The fourth approach utilized a fixed study area size, but modeled detection probability as a function of bird proximity to the study plot boundary (e.g., those birds closest to the edge are more likely to emigrate). This approach also resulted in estimates of survival that were unbiased (bias < 0.02), but because the individual covariates were normalized, the average capture probability was 0.50, and thus did not provide an accurate estimate of the true capture probability. Our results show that the core–area with surrounding resight–only can provide estimates of survival that are not biased by the effects of incomplete breeding site–fidelity. Key words: Apparent survival, Site–fidelity, Dispersal, Emigration, Cormack–Jolly–Seber model, Migratory birds. ISSN: 1578–665X

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Resumen Estimación de la supervivencia anual de aves canoras migratorias bajo el efecto de una fidelidad incompleta al área de reproducción: diseños de estudios que pueden resultar de utilidad.— Numerosas especies de aves presentan distintos grados de fidelidad al territorio o área de reproducción del año anterior, fenómeno que denominamos fidelidad incompleta al lugar de reproducción. Si el territorio que ocupan las aves está situado más allá del área de estudio o investigación (es decir, si las aves han emigrado del área de estudio), el ave no podrá ser detectada y, por consiguiente, en los estudios de captura–marcaje–recaptura, no podrá distinguirse de los individuos muertos. Si se emplean las tasas de supervivencia aparente como estimaciones de la supervivencia real, las tasas de emigración diferencial sesgan las distintas inferencias sobre variaciones en supervivencia entre sexos y entre especies. Además, el sesgo introducido por el empleo de tasas de supervivencia aparente en lugar de tasas de supervivencia real puede repercutir significativamente en las predicciones de la persistencia poblacional a través del tiempo, la dinámica de fuente/sumidero, y otros aspectos de la teoría sobre historias vitaels. Investigamos cuatro enfoques de diseños de estudios y análisis que proporcionan estimaciones de supervivencia aparente más próximas a las estimaciones de supervivencia real. Esta investigación es fruto de un estudio multianual de captura–recaptura de reinitas cabecidoradas (Protonotaria citrea) en múltiples parcelas de estudio incluidas en un paisaje más amplio de hábitats de reproducción adecuados, en los que se documentaron los movimientos interanuales más importantes entre distintas parcelas de estudio adyacentes por parte de individuos marcados. Nuestro objetivo era cuantificar los efectos de este tipo de movimiento en la estimación de la supervivencia anual. Los dos primeros diseños de estudio que investigamos consistían en el marcaje de aves en un área central, para posteriormente volverlas a avistar, tanto en dicha área como en un área adyacente a la misma. Por lo que respecta al primero de estos dos diseños, demostramos que cuando el área de reavistaje que rodea al área central se va ampliando y el número de "emigrantes" reavistados aumenta, las estimaciones de supervivencia aparente empiezan a aproximarse a las tasas de supervivencia real (sesgo < 0,01). Sin embargo, teniendo en cuenta los movimientos interanuales de las aves observados, lo más probable es que, desde un punto de vista logístico, no resulte práctico reavistar aves en áreas adyacentes que sean lo suficientemente amplias como para minimizar el sesgo. Por consiguiente, como protocolo alternativo, analizamos los datos con subconjuntos de tres áreas adyacentes al área principal, que se iban ampliando de forma progresiva. Los subconjuntos de datos proporcionaron cuatro estimaciones de supervivencia aparente que abordaban asintóticamente la supervivencia real. Lo más probable es que, desde un punto de vista logístico, este diseño de estudio y enfoque analítico resulte viable en estudios de campo, además de producir estimaciones de supervivencia real no sesgadas (sesgo < 0,03) por fidelidad incompleta al área de reproducción en un rango de patrones de movimiento territorial interanual. El tercer enfoque que investigamos empleaba una serie de datos de un diseño robusto de toma de datos y un enfoque analítico. Este enfoque proporcionó estimaciones de supervivencia que, si bien no eran sesgadas (sesgo < 0,02), resultaban muy imprecisas, por lo que probablemente no proporcionarían estimaciones fiables en situaciones de campo. El cuarto enfoque utilizaba un tamaño de área de estudio fijo, pero modelaba la probabilidad de detección como una función de la proximidad de las aves al límite de la parcela de estudio (es decir, las aves situadas más cerca del borde presentan más probabilidades de emigrar). Este enfoque también produjo estimaciones de supervivencia no sesgadas (sesgo < 0,02), pero debido a que las covarianzas individuales se normalizaron, la probabilidad de captura media era de 0,50, por lo que no proporcionaba una estimación precisa de la probabilidad de captura real. Nuestros resultados demuestran que el hecho de combinar el área principal con áreas adyacentes dedicadas exclusivamente al reavistaje puede proporcionar estimaciones de supervivencia que no resulten sesgadas por los efectos de una fidelidad incompleta al área de reproducción. Palabras clave: Supervivencia aparente, Fidelidad al área de reproducción, Dispersión, Emigración, Modelo de Cormack–Jolly–Seber, Aves migratorias. Matthew R. Marshall, School of Forest Resources, Pennsylvania State Univ., University Park, Pennsylvania, U.S.A.– Duane R. Diefenbach, U.S. Geological Survey, Pennsylvania Cooperative Fish and Wildlife Research Unit, Pennsylvania State Univ., University Park, Pennsylvania, U.S.A.– Larry A. Wood, Wildlife Investigations, LLC Georgetown, South Carolina, U.S.A.– Robert J. Cooper, School of Forest Resources, Univ. of Georgia, Athens, Georgia, U.S.A. Corresponding author: D. R. Diefenbach. E–mail: drd11@psu.edu

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Introduction Biologists are able to estimate annual survival rates for many species of migratory songbirds via traditional capture–mark–recapture methodology largely because many surviving individuals are faithful to the breeding (or wintering) area of the previous year. Birds are captured and marked during one breeding season and recaptured (typically, color– banded individuals are resighted) during subsequent breeding seasons. However, not all surviving individuals return to precisely the same breeding territory the following year, a phenomenon we refer to as incomplete breeding site–fidelity. If the territory they occupy in a subsequent year is located beyond the bounds of the study area, the bird will go undetected. Researchers usually cannot determine how many of these non–returning birds are dead and how many are alive somewhere outside the study area. This inability to distinguish between death and dispersal poses a problem for obtaining accurate survival rate estimates with potentially profound effects on the predictions of population persistence through time, source/sink dynamics, and other aspects of life–history theory and conservation ecology. The effect of incomplete breeding site–fidelity on annual survival estimation using a Cormack–Jolly– Seber (CJS) model (Cormack, 1964; Jolly, 1965; Seber, 1965) has long been recognized; the survival parameter, , represents those birds that survive and remain on the study area (Lebreton et al., 1992). Hence, terms such as "apparent survival" and "local survival" are used to describe the survival parameter. The use of apparent survival estimates are clearly superior to simple return rates because a detection probability is incorporated (Lebreton et al., 1992; Martin et al., 1995). Furthermore, if the objective of a study is to compare demographic rates among different treatments, management options, habitats, or other factors, apparent survival is a meaningful parameter because it reflects the multiple ways in which the population may be affected by the factor in question (e.g., birds may have lower survival or abandon a treated area). Hypotheses regarding the effects of a particular treatment can still be tested as long as it is recognized that the response variable includes both mortality and permanent emigration from the study area (Marshall et al., 2000). However, if the study questions involve aspects of life–history theory, such as sex–specific survival or the costs of reproduction, then it is required that survival be estimated. If apparent survival estimates are used in these cases (i.e., as estimates of true survival) it must be assumed or demonstrated that permanent emigration does not occur. If permanent emigration does occur, then, by definition, apparent survival will underestimate true survival by some unknown amount and may vary among groups (among age/ sex classes and/or species). Recent studies of migratory songbirds have examined the degree to which apparent survival rate

estimates potentially underestimate true survival rates because of emigration from the study area (Cilimburg et al., 2002; M. R. Marshall, unpublished data), whereas others have begun to elucidate the ecological mechanisms explaining why individuals exhibit varying degrees of site–fidelity (Haas, 1998; Hoover, 2003). For example, Cilimburg et al. (2002) found that permanent emigration of Yellow Warblers (Dendroica petechia) from their study sites was common (30% of resighted birds were found outside their original study site in one year of the study), and survival probabilities increased by 0.07–0.23 with the inclusion of these capture events. Our motivation for this study stemmed from a multi–year capture–recapture study of Prothonotary Warblers (Protonotaria citrea) on multiple study plots within a larger, homogenous, landscape of suitable breeding habitat (Wood, 1999). We (L. A. Wood, unpublished data) detected substantial inter–annual movements of marked individuals among neighboring study plots and found that these movements could have resulted in underestimates of true survival of 0.17 for males and 0.19 for females had these movements not been detected (M. R. Marshall, unpublished data). Underestimates of survival were greater for females than males because of more frequent and longer movements of females, a pattern expected given a female bias in between–year breeding site–fidelity (Clark et al., 1997), especially among females with lower reproductive success (Hoover, 2003). These studies (Cilimburg et al., 2002; M. R. Marshall, unpublished data) serve as cautionary notes on the interpretation of apparent survival rate estimates (i.e., the bias introduced by emigration if apparent survival estimates were used as true survival rate estimates), but also highlight the fact that true survival rate estimation (often the parameter of interest when addressing predictions of population persistence through time, source/sink dynamics, and other aspects of life–history theory) will remain problematic until sampling designs and analytic approaches are developed that account for incomplete site–fidelity. The objectives of this study were to investigate, via simulations parameterized with empirical data, several study area designs and analytical approaches intended to reduce or eliminate the effect of incomplete site–fidelity on estimates of survival using a CJS model. Our goal was to identify study designs and methods of estimating annual survival for migratory songbirds that would provide estimates of true survival in the presence of incomplete breeding site–fidelity. Study designs Core with resight–only area I The first study design we investigated consisted of a study site with a core area where birds were both marked and resighted in following years and a

surrounding resight–only area where marked individuals were resighted but no new individuals were marked (fig. 1). The motivation behind this study design was to centralize the area where birds were marked relative to study area edges to increase the chances that birds that switch territory location between years would be detected in the resight– only area. We investigated increasingly larger resight–only areas to evaluate what size area would be required to minimize or eliminate bias in apparent survival rates because of emigration when true survival rates are desired. We investigated bias in survival estimates using two different core area sizes (16 and 36 territories) and two different movement patterns ("males" and "females") and progressively increased the larger resight–only areas until the complete habitat patch simulated (2,500 territories; see Simulations below) was considered part of the resight–only study area. We used capture histories for all birds marked and resighted on both the core and resight–only areas in a traditional CJS model in software MARK 3.1 (White & Burnham, 1999) to estimate apparent survival ( ). All assumptions of traditional CJS models apply (e.g. Lebreton et. al., 1992) and we also assume that (1) capture probability in the resight–only area equals that of the core area, (2) survival of birds that leave the core area is the same for birds that remain on the core area, and (3) the habitat is homogenous such that progressively larger resight–only areas will increase the proportion of individuals resighted. CJS survival estimates that include capture history information from the resight–only areas will have less bias because birds that move off the core study area can be resighted in the resight–only area. Core with resight–only area II—subsets analysis The second study design we investigated used the same core area and resight–only area design for collecting data, but the estimation of survival was performed differently. The motivation behind this study design arose from the recognition that in most situations it is logistically impractical for researchers to resight marked birds in the very large resight–only areas described heretofore. Here again we used the traditional CJS model, but instead estimated on subsets of the data. The smallest subset was the capture histories of birds marked and resighted on the core area only. The next subset included the capture histories of birds on the core area as well as resightings that occurred in a resight–only area surrounding the core area. The resight–only area was one territory width surrounding the core area. For example, a 16 territory core area (simulated by a 4 × 4 territory area) with a one territory larger resighting area would result in a 6 × 6 territory area (36 territories total; 16 territories in the core area with a ring of 20 territories surrounding the core that make up the resight–only area). The next two subsets increased the size of resight–only area by one additional territory width,

Marshall et al.

respectively. We investigated this study design starting with core areas of 16, 36, and 64 territories each with progressively larger resight–only areas up to three territories in width. This resulted in study area sizes of 100, 144, and 196 territories, respectively. Thus, we were able to obtain four estimates of for each area based on capture histories that resulted from progressively larger subsets (i.e., larger resight–only areas) that were inclusive of the preceding subsets. The four estimates of obtained from these subsets of the data had diminishing bias (see Results) because each subset had an increasingly larger area to resight birds. Thus, we used these four estimates of in an asymptotic nonlinear model to obtain an unbiased estimate of survival (S). We used the following model to estimate S:

where A is the size of the study area (number of territories or hectares) and k is a nuisance parameter. We used SAS (PROC NLIN; SAS Institute, Cary, North Carolina, U.S.A.) to estimate the parameters of the nonlinear model (S, k) and a bootstrap procedure to estimate the 90% CI. In the bootstrap procedure we used the estimates of SE( ) to generate 1,000 random–normal deviates for each of the four estimates of . We used these randomly–generated values in PROC NLIN to obtain 1,000 estimates of S. We used the 5th and 95th percentiles of the bootstrapped S estimates as the 90% CI limits. Robust design The third study design we investigated was based on the robust design approach (Pollock, 1989; Kendall & Nichols, 1995; Kendall et al., 1995, 1997). The motivation behind this study area design and analysis approach was the recognition that the between year movements of territory locations did not necessarily result in permanent emigration (L. A. Wood, unpublished data) and, thus, constituted a form of temporary emigration. Here, we used a single study area of 100 territories in which we captured and resighted birds during primary (between year) and secondary (within year) sampling occasions. We simulated three secondary sampling occasions within each primary sampling period, in which the probability of capturing a bird $1 time during a secondary sampling period was equivalent to the annual capture probabilities of other CJS simulations. Data were analyzed in software MARK 3.1 using the robust design with Huggins’ estimator (Huggins, 1989, 1990; Alho, 1990) to estimate S. All assumptions of traditional robust design models applied (e.g., Kendall et. al., 1997). Individual covariates In the fourth study design we used a fixed–size study area but modeled capture probabilities as a

Animal Biodiversity and Conservation 27.1 (2004)

Territory Resight–only area (r) Core area (h)

Habitat patch (H) Fig. 1. Illustration of the habitat patch (H) composed of H × H individual bird territories, with a core area (h × h territories) for marking and resighting and a surrounding resight–only area (r × r territories) used in the simulations (see Methods). Fig. 1. Ilustración de la parcela de hábitat (H), compuesta por territorios de aves individuales H × H, con un área central (territorios h × h) para marcaje y reavistaje, y un área adyacente dedicada exclusivamente al reavistaje (territorios r × r), empleada en las simulaciones (véase Methods).

function of distance from the boundary of the study area. During the simulation we recorded the distance from the nearest study area boundary where each bird was first captured. We used the same input parameters for males and females as used in the other study designs, but specified a fixed study area size of 196 territories for females and 100 territories for males. We analyzed the data as a CJS model with the capture probabilities modeled as a function of distance from the study area boundary as an individual covariate with a logit link function. We did not include an intercept term and normalized the individual covariates. Simulations We wrote a simulation program in FORTRAN (Digital Visual Fortran 6.0, Digital Equipment Corporation, Maynard, Massachusetts, U.S.A.) to generate capture histories under the study designs described heretofore. User–specified parameter values generated capture histories under a specified study design. The following user–specified input parameters were required: H. Habitat patch width (H), where H × H = number of possible territories in habitat patch; each cell representing one territory (fig. 1); h. Core study area width (h), where h × h = area where birds can be marked and resighted, h [ H (fig. 1); r. Distance surrounding the core study area where birds only can be resighted, (i.e., resight–only area) 0 [ r [ H – h

(fig. 1); L. Number of capture intervals (i.e., years; primary sampling intervals in robust design), L = 1,2,3,…; l. Number of independent capture events within a primary sampling occasion (i.e., secondary sampling intervals in robust design), l = 1,2,…,5; Si – Survival rate between primary sampling intervals (i.e., annual survival), 0 [ Si [ 1; pi. Capture probability each year (under the robust design the capture probability for each secondary sampling event was defined as lªpi), 0 < pi < 1; D. Maximum distance a bird could move its territory between years, D < H; . Mean distance moved according to a negative exponential distribution, ^ D; m. Proportion of individuals who do not move their territory according to a negative exponential distribution between primary sampling intervals, 0 [ m [ 1; N0. Initial number of birds occupying territories, 0 < N0 < H × H The simulation performed the following sequence of events: 1. Randomly assigned N0 individuals to territories within the habitat patch; 2. For each individual within the study area (h × h) if a random number U(0,1) [ pi then the bird was captured. If l > 1 (i.e., robust design) then the bird was captured if a random number U(0,1) [ lªpi for each of l occasions; 3. Birds that were captured (or resighted in later capture occasions) were assigned a "1" to their capture history and "0" if not captured (or resighted); 4. A bird died if a random number U(0,1) > Si; 5. A bird did not move if a random number U(0,1) [ m; 6. Of those birds that could potentially move (1 – m), they moved terri-

tory locations according to a negative exponential distribution (0 = ) in a random direction, U(0,1) (although the movements were modeled from continuous distributions for distance and direction, the coordinate of the new territory location was the truncated integer x–y values; if the new territory location was already occupied, or the movement would take the bird outside the habitat patch, then the process was repeated); 7. (1 – Si) × N0 new birds were randomly placed in the habitat patch in unoccupied territories; 8. For each individual within the study area (h × h), and the r territories surrounding the study area, if a random number U(0,1) < pi then the bird was captured or resighted. If l > 1 (i.e., robust design) then the bird was captured or resighted if a random number U(0,1) < lªpi ; 9. The process was repeated beginning with Step #3 for each of the remaining L × l capture occasions. For each simulation scenario, we conducted 100 simulations (200 simulations for females in study designs 2 and 3) in which survival and capture probabilities were constant over time and no heterogeneity existed among birds. The output from the FORTRAN program was formatted for input into software MARK to estimate under the CJS model and S under the Huggins estimator robust design model. The CJS model specified a constant and p, and the robust design model specified a constant survival rate (S), immigration and emigration rate ( ' and ''), p, recapture probability (c), and p = c. We based our simulations on empirical data from a 5–year capture–mark–resight study of 423 Prothonotary Warblers (Protonotaria citrea) at the White River National Wildlife Refuge, Arkansas, U.S.A., 1994–1999 that provided estimates of the distribution of movement distances, survival and capture probabilities, density of individuals in the landscape, and territory size (Wood, 1999; L. A. Wood & R. J. Cooper, unpublished data; M. R. Marshall, unpublished data). The White River National Wildlife Refuge is a 60,000–ha tract of bottomland hardwood forest, one of the largest contiguous bottomland hardwood forests remaining in the United States (Harris & Gosselink, 1990), much of it suitable breeding habitat for Prothonotary Warblers. From 1994–1999 we recorded the territory locations of all marked birds and calculated the distance between territory centers for all birds that were resighted in subsequent years. Due to the spatial arrangement of six 50–ha study plots, we were able to detect between–year movements up to 3 km. Not all movements had an equal probability of detections (Koenig et al., 1996), thus, we utilized the method of Baker et al. (1995) to eliminate the bias in movement distributions inherent in studies conducted within a finite area. The corrected movement distributions indicated that inter–annual movements of territories differed between males and females with females moving more frequently between years and longer distances, but both could be described with a negative exponential distribution (fig. 2). For males, the shape of the negative exponential distribution that

Marshall et al.

reflected the empirical, corrected movement distributions was accurately described with a maximum distance moved (D) of 12 territories, an average movement ( ) of 3.87 territories, and 35% of males not moving between years (m, in addition to the movements of zero territories under the negative exponential model). For females, we described movements with a maximum distance moved (D) of 22 territories, an average movement ( ) of 3.87 territories, and 8% of females not moving between years (m, in addition to the movements of zero territories under the negative exponential model). In our simulations, we used an annual survival rate of 0.69 for males and 0.57 for females and capture probability of 0.95 for males and 0.68 for females. All simulations began with H = 50, N0 = 1,875 (75% of all possible territories occupied), and an average territory size of 125 m in diameter (L. A. Wood & M. R. Marshall, unpublished data). Our model was spatially–explicit in terms of the location of birds on the landscape (i.e., the habitat patch, H; fig. 1), but we expressed distances in terms of territories in which each unit of distance represented 1 territory. Thus, our simulation results should be comparable across species if scaled to an average breeding territory size. In our simulations we assumed a homogeneous habitat (i.e., contiguous patches of habitat suitable for the species of interest) large enough to contain the study area designs of interest. Each set of simulations for each of the four study area designs represented a 5–year capture–mark–recapture scenario and was conducted separately for males and females. Consequently, we were able to evaluate each design with one group (males) that was less likely to move territory locations between years and moved shorter distances and another group (females) that moved territory locations more often and moved greater distances when movement occurred (M. R. Marshall, unpublished data). We evaluated the accuracy and precision of each study design by comparing estimated parameters ( and ) to the true value used to simulate the data, estimating the SD of estimated parameters, and calculating the average square root of mean square error ( ). Results of simulations Core with resight–only area I Simulation results from the first study area design indicate that the negative bias in apparent survival rate estimates relative to true values decreases for both males and females as the resight–only area increases in size (fig. 3). With progressively larger resight–only area size, an increasing number of marked birds that move territory locations between years are resighted in the resight–only area and bias approaches zero. Females, which move more frequently and far-

Animal Biodiversity and Conservation 27.1 (2004)

0.55 Male Female

Proportion of individuals

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Number of territories moved

Fig. 2. Distribution of movements used in simulations (distance between inter–annual placement of breeding territories) for male and female Prothonotary Warblers expressed as the number of territories moved between years. Fig. 2. Distribución de los movimientos utilizados en las simulaciones (distancia entre la ubicación interanual de los territorios de reproducción) para reinitas cabecidoradas macho y hembra, expresados como el número de territorios recorridos entre un año y otro.

ther distances than males, and were, therefore, more likely to move beyond the bounds of the resight–only areas, exhibited a greater bias in both the 16 and 36 territory core area design at smaller resight–only areas. Bias for both sexes began to converge at the 20 × 20 resight–only size and bias was negligible at the 30 × 30 resight– only area size.

Robust design

Core with resight–only area II–subsets analysis

Individual covariates

Study areas containing 196 territories (64 territory core area) exhibited little bias in estimates of survival (Ö ) for both males and females (table 1) and estimates had reasonable precision. Study areas of 144 territories (36 territory core area) in size exhibited a slight negative bias, and study areas of 100 territories (16 territory core area) provided estimates of survival with poor precision. If capture probabilities were high (e.g., males, table 1), the smallest study area size provided reasonable survival estimates, but when capture probabilities were low (e.g., females, table 1) the survival estimates were extremely variable. Using a bootstrap approach to incorporate variability from both the CJS model and the asymptotic model of nested survival estimates provided confidence intervals close to the nominal value; 93.5–95.0% of 90% CIs encompassed the true value.

The survival estimates when capture probability was modeled as a function of distance from the study area boundary were precise and unbiased (S males = 0.69, Ö males = 0.69, SE males = 0.027; Sfemales = 0.55,Öfemales = 0.53, SEfemales = 0.028). Because the individual covariates were normalized and the logit of capture probability was modeled without an intercept parameter, the average estimated capture probability was 0.50, and thus did not provide an accurate estimate of the true capture probability if birds were available to be captured on the study area. Although these models provided unbiased survival estimates, they were always inferior (AIC weight < 0.01) to standard CJS models. Consequently, when analyzing data in which true survival is not known, it is unlikely AIC would indicate selection of the model with unbiased survival estimates, thus limiting the utility of this approach.

Results from the robust design analysis indicated that the average survival and capture probability estimate from simulations, for both males and females, exhibited little bias (table 2). However, the precision of survival estimates was poor (males, range 0.51–1.00; females, range 0.35–1.00).

Marshall et al.

0.25

4 x 4 core area

• = female

0.20

• = male

Bias (survival–apparent survival)

0.15 0.10 0.05 0.00 0

0.25 0.20

6 x 6 core area

• = female • = male

0.15 0.10 0.05 0.00

0 5 10 15 20 25 30 35 40 45 50 Resight–only study area size diameter (number of territories)

Fig. 3. Results of simulations of the "Core with resight–only area I" design for a 16 territory core area (top) and a 36 territory core area (bottom). Bias approaches zero as the resight–only area size increases. Fig. 3. Resultados de las simulaciones en las que se utilizó el diseño de combinación del área central con el área I dedicada exclusivamente al reavistaje, para un área central compuesta por 16 territorios (arriba) y un área central compuesta por 36 territorios (abajo). El sesgo se aproxima a cero a medida que el tamaño del área dedicada exclusivamente al reavistaje aumenta.

Discussion We investigated four study design and analysis approaches that resulted in estimates of survival that were not biased by the effects of incomplete breeding site–fidelity (some proportion of marked individuals moving beyond the bounds of the study area and going undetected). The first two involved a smaller core area where all marking of birds takes place and progressively larger resight–only areas surrounding the core where researchers search for marked birds. For the first of these two designs, we demonstrated that as the resighting areas get progressively larger, and therefore incorporates more "emigrants", apparent survival estimates begin to approximate true survival rates. Given observed inter–annual movements of birds, however, it is likely to be logistically impractical to resight birds on a sufficiently large resight–only area to minimize bias. For example, a resight– only area of at least 400 territories (20 × 20 terri-

tory study area; fig. 3) would be needed to eliminate the relative difference in bias between males and females because of greater movements of females, whereas a resight–only area of approximately 900 territories (30 × 30 territory study area) would be needed to eliminate the bias for both sexes completely. Whether study areas of 400–900 territories are logistically feasible depends on the average territory size of a species. We believe any study area larger that 196 territories (14 × 14 territory study area) would be logistically impractical in a field setting for Prothonotary Warblers (L. A. Wood & M. R. Marshall, personal observation). A 196 territory study area for Prothonotary Warblers would be > 300 ha where marked birds would have to be resighted (table 3), whereas a 900 territory area would be approximately 1,400 ha. Therefore, we conclude that increasing the resighting area to an area large enough to eliminate bias from inter–annual territory relocations is logistically impractical for most species.

Animal Biodiversity and Conservation 27.1 (2004)

Table 1. Mean estimated parameters from simulations using the "Core with resight–only area II– subsets analysis" design to estimate S: N. Number of simulations; SAS. Study area size (units are number of territories, size of core area —no. territories— where unmarked birds can be captured); n. Mean number of birds marked over 5 year period. Tabla 1. Parámetros promedio estimados, obtenidos a través de simulaciones en las que para estimar S se empleó el diseño de análisis de subconjuntos–combinación del área central con el área II dedicada exclusivamente al reavistaje: N. Número de simulaciones; SAS. Tamaño del área de estudio, (las unidades son el número de territorios, tamaño del área central —número de territorios— donde se pueden capturar aves no marcadas); n. Promedio de número de aves marcadas en un periodo de 5 años. Sex Male

SAS 100(16)

N 100

n 38

S 0.69

0.64

D( ) 0.0769

0.0917

Male

144(36)

Male

196(64)

Female

100(16)

200

p 0.95

100

0.69

0.67

0.0520

0.0557

0.95

0.94

0.0641

100

126

0.69

0.0391

0.95

0.0381

0.57

0.55

0.2214

0.2223

0.68

0.75

0.3449

0.93

D( ) 0.0796

Female

144(36)

200

0.57

0.56

0.1287

0.1291

0.68

0.71

0.2442

Female

196(64)

200

130

0.57

0.0848

0.68

0.70

0.1170

Table 2. Mean estimated parameters from simulations for a robust design model for a study area of 100 territories in size, 5 primary sampling periods (e.g., years), and 3 secondary sampling periods (e.g., within–year sampling). For males S = 0.69 and p = 0.95 and for females S = 0.57 and p = 0.68: N. Number of simulations; Probability. Probability captured m 1 time within a primary sampling period. Tabla 2. Parámetros promedio estimados, obtenidos a través de simulaciones para un modelo de diseño sólido para un área de estudio compuesta por 100 territorios, cinco periodos de muestreo primario (años, por ejemplo) y tres periodos de muestreo secundario (muestreo interanual, por ejemplo). Para los machos, S = 0,69 y p = 0,95, y para las hembras S = 0,57 y p = 0,68: N. Número de simulaciones; Probability. Probabilidad de ser capturado m 1 durante el primer periodo de muestreo.

Survival Sex

Immigration

Emigration

Probability

D( )

Male

100

0.71

0.176

0.49

0.469

0.18

0.177

0.95

0.011

Female

200

0.55

0.178

0.43

0.438

0.21

0.188

0.68

0.022

The second approach utilized the same study area design, but estimated for three progressively larger resight–only areas and modeled the ’ s as asymptotically approaching true survival. This study design and analytical method provided estimates of S with little bias and reasonable precision. This approach is likely to be logistically feasible in field settings because of the smaller study area sizes required (100–200 territories in size). Furthermore, the number of marked birds used in the simulations is comparable to what could realistically be marked in most field situations.

The choice of study area size using this core area and resight–only area study design and subset analysis approach depends on several factors: capture probabilities, territory sizes, and the distribution of the distances between inter–annual territory locations. For example, compared to females, male Prothonotary Warblers had high capture probabilities (p = 0.95), fewer birds moved their territories between years, and they moved shorter distances when they did move. This resulted in survival (table 1) even estimates with reasonable when study areas were 156 ha (100 territories) in

Marshall et al.

Table 3. Approximate study area size in hectares for two species of migratory songbird given average territory size. Territory size used in conversion equals: A. Prothonotary Warblers, 1.45 ha (125 m diameter) (L. A. Wood, unpublished data); B. Red–eyed Vireos, 0.50 ha (71 m diameter) (Vireo olivaceus; Marshall & Cooper, 2004). Tabla 3. Tamaño aproximado del área de estudio, expresado en hectáreas, para dos especies de aves cantoras migratorias dado el tamaño territorial medio. El tamaño territorial empleado en la conversión equivale: A. Reinitas cabecidoradasa, 1,45 ha (125 m diámetro) (datos no publicados de L. A. Wood); B. Vireos ojirrojos, 0,50 ha (71 m diámetro) (Vireo olivaceus; Marshall & Cooper, 2004).

Study Area Size (ha) Territories

16 (4 x 4)

25 (5 x 5)

36 (6 x 6)

49 (7 x 7)

64 (8 x 8)

100

81 (9 x 9)

126

100 (10 x 10)

156

121 (11 x 11)

189

144 (12 x 12)

225

169 (13 x 13)

264

196 (14 x 14)

306

size. Comparable precision of survival estimates for females required study areas 306 ha (196 territories) in size. In general, greater capture probabilities and shorter inter–annual movements of breeding territories result in greater precision of survival estimates. Study area sizes also will be greater for species with larger territories possibly affecting the precision of survival estimates because of the logistical difficulties of capturing and resighting birds over these larger areas. We present study area sizes for two species of migratory songbird that translate the territory units of the simulations into actual field dimensions (table 3). The Red–eyed Vireo was arbitrarily chosen to represent those songbirds with territory sizes smaller than a Prothonotary Warbler. This example assumes that both species have the same distribution of movements even though male Red–eyed Vireos tend to be more site–faithful and move shorter distances than Prothonotary Warblers (Marshall et al., 2002). Decisions as to the feasibility of resighting birds over these areas will be dependent on multiple factors, but we believe data collection on study areas of the presented size is possible. Moreover, an example analysis was provided (table 4) with a 64 territory core area (8 × 8 territory study area) and three progressively larger resight–only areas resulting in the largest resight area of 196 territories (14 × 14 territory study area). Assuming researchers can feasibly mark a similar number of birds on the core area as were marked in the simulations as well as resight birds in a study area with these dimension, the results of this example (table 4) demonstrate that the survival estimates generated with this approach were unbiased and had reasonable precision. Lastly, table 5 presents analyses investigating the effects of different distributions of the distances between inter–annual territory locations (fig. 5) on the utility of a 64 territory core area and three progressively

Table 4. Results of an example analysis using the "Core with resight–only area II–subsets analysis" design with a 64 territory (8 x 8) core area and three progressively larger (one territory width each) resight–only areas (196 territories for largest resight–only area): n. Number of birds marked over 5 year period. Tabla 4. Resultados de ejemplo de análisis en el que se utilizó el diseño de análisis de subconjuntos– combinación del área central con el área II dedicada exclusivamente al reavistaje, en un área central (8 x 8) compuesta por 64 territorios y tres áreas dedicadas exclusivamente al reavistaje (cada una de ellas con una anchura equivalente a un territorio), progresivamente más grandes (196 territorios para el área más grande dedicada exclusivamente al reavistaje): n. Número de aves marcadas a lo largo de un periodo de 5 años.

Sex

Males

140

0.69

0.73

Females

128

0.57

0.50

E( )

90%CI ( )

0.074

0.62–0.87

0.95

0.097

0.37–0.68

0.68

E( )

90% CI( )

0.90

0.035

0.84–0.96

0.83

0.331

0.54–1.27

Animal Biodiversity and Conservation 27.1 (2004)

Table 5. Results of analyses using the "Core with resight–only area II–subsets analysis" design with a 64 territory (8 x 8) core area and three progressively larger (one territory width each) resight– only areas (196 territories for largest resight–only area) over a range of between–year territory movement distributions as shown in figure 5: . The value of the mean from a negative exponential distribution describing bird movement patterns; 90% Cl cov. Proportion of 90% CI coverage that encompass the true value. Tabla 5. Resultados de los análisis en los que se utilizó el diseño de análisis de subconjuntos– combinación del área central con el área II dedicada exclusivamente al reavistaje, en un área central (8 x 8) compuesta por 64 territorios y tres áreas dedicadas exclusivamente al reavistaje (cada una de ellas con una anchura equivalente a un territorio) (196 territorios para el área más grande dedicada exclusivamente al reavistaje), en un rango de distribuciones de movimientos territoriales interanuales, según se indica en la figura 5: . El valor de la media a partir de una distribución exponencial que describe los patrones de movimineto de las aves; 90% Cl cov. Proporción del 90% Cl que incluye el valor real.

Core–area only

Core–area with resight–only areas

D( )

90% CI cov

1.0

0.70

0.64

0.07

0.72

0.05

0.91

0.08

2.0

0.70

0.58

0.12

0.71

0.05

0.92

0.09

3.0

0.70

0.53

0.17

0.71

0.06

0.97

0.10

6.0

0.70

0.43

0.27

0.66

0.09

0.94

0.17

0.35 Proportion of individuals

Observed movements 0.30

All movements

0.25 0.20 0.15 0.10 0.05 0.00 0

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Number of territories moved

Fig. 4. Distribution of observed movements of marked birds (i.e., resighted individuals) closely matched the distribution of movements for all birds in the habitat patch. Values are based on females and a 64 territory core area with a 196 territory resight–only area. Fig. 4. La distribución de los movimientos observados realizados por aves marcadas (es decir, individuos reavistados) concordó estrechamente con la distribución de los movimientos de todas las aves en la parcela de hábitat. Los valores se basan en hembras y en un área central compuesta por 64 territorios, con un área dedicada exclusivamente al reavistaje compuesta por 196 territorios.

Marshall et al.

0.70

= = = =

0.65

Proportion of individuals

0.60 0.55

1 2 3 6

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 1

4 5

8 9 10 11 12 13 14 15 16 17 Number of territories moved

18 19 20

Fig. 5. Range of between–year territory movement distributions used to evaluate the "Core with resight–only area II–subsets analysis" design with a 64 territory (8 x 8) core area and three progressively larger (one territory width each) resight–only areas (196 territories for largest resight–only area). Results of analyses presented in table 5. Fig. 5. Rango de distribuciones de movimientos territoriales interanuales utilizado para evaluar el diseño del análisis de subconjuntos–combinación del área central con el área II dedicada exclusivamente al reavistaje, con un área central (8 x 8) compuesta por 64 territorios y tres áreas dedicadas exclusivamente al reavistaje (cada una de ellas con una anchura equivalente a un territorio) progresivamente mayores, (196 territorios para el área más grande dedicada exclusivamente al reavistaje). Resultados de los análisis presentados en la tabla 5.

larger resight–only areas approach. These results indicate that the approach would yield unbiased estimates of survival with reasonable precision that would allow comparisons in survival estimates between sexes of a species (or among multiple species) with different fidelity/movement distributions. This is in contrast to the erroneous conclusions that would be drawn from evaluating apparent survival estimates from the core area only (i.e., a typical fixed study area approach; table 5). Furthermore, the error (as defined by ) associated with from the core–resight area approach is equal to or less than that associated with from the core–area only because of the reduction in bias. It appears that this study area design and analysis approach can yield reliable results over a range of capture probabilities, territory sizes, and movement patterns. Data on these basic biological parameters is required to effectively design a study area for a given species or suite of species that may vary in patterns of site–fidelity and between–year movements.

The third approach we investigated was a robust design (Kendall et al., 1995). We investigated the robust design because even though the majority of movements that resulted in emigration observed for Prothonotary Warblers were permanent, not all were (L. A. Wood, unpublished data). That is, some individuals moved off the study area in one year, only to return to the study area in a subsequent year. This constitutes a form of temporary emigration that could be estimated with the robust design. However, so little temporary emigration was observed that parameter estimation was unreliable. This may not be true for other species where inter–annual movements may more typically resemble temporary emigration. Another problem with the robust design simulations is that the model investigated did not consider the heterogeneity that existed in the probability of immigrating or emigrating; birds closer to the border of the study area had a greater probability of leaving (entering) the study area. However, we believe further research on the use of the robust design is warranted.

Animal Biodiversity and Conservation 27.1 (2004)

The fourth analysis modeled capture probabilities as a function of distance from the study area boundary. Although this approach (unlike the robust design approach) does incorporate some of the heterogeneity in capture probabilities due to the proximity of birds to the study area edges, it cannot account for birds, for example, that are first captured in the center of the study area in one year and in a subsequent year move to a territory near the study area boundary. Although birds with these types of movements have a high probability of staying on the study area in the first year, they have a much lower probability in subsequent years. Furthermore, the individual covariate value is unknown for years in which a bird is not recaptured. Finally, the greatest problem with this approach is that use of the AIC criterion failed to select the model with the most accurate survival estimate because of the biased estimates of capture probability. An additional avenue of future research could involve constructing a model that explicitly incorporates the probability that a bird present on the study area in one year will remain within the study area in subsequent years. For example, the simulations indicated that the distribution of observed movements of marked birds (i.e., individuals resighted on the study area) closely matched the distribution of movements for all birds in the habitat patch (fig. 4; values were based on females and a 64 territory core area with a 196 territory resight–only area). If one could model the probability of movement of all individuals based only on the observed data (e.g., Rodriguez, 2002), and had spatially explicit data on territory locations, one could estimate the probability that a bird at location x, y in year i will be within the study area in year i + 1. If so, one could potentially construct a model that estimates , p, and the probability of remaining on the study plot, where approximates S. The study designs and analysis approaches presented here need to be field–tested and may be limited by the logistical difficulties of data collection. However, we believe a core area with surrounding resight–only areas (subsets analysis) has the potential to provide estimates of survival rates that can be used to evaluate aspects of life–history theory, population viability and other aspects of conservation ecology that apparent survival estimates confounded with incomplete breeding site– fidelity cannot. Acknowledgments We would like to thank Mike Conroy, Bill Kendall, Clint Moore, Jim Nichols, Larkin Powell, and Randy Wilson for earlier discussions, ideas, and insights on this topic. Thanks also to the School of Forest Resources at Penn State University for supporting MRM on a Post–Doctoral Teaching and Research Scholarship during this project. We would also like to thank Charles Brown, David Thompson, Jeffrey Hoover and one anonymous reviewer for careful and critical reviews of this manuscript.

References Alho, J. M., 1990. Logistic regression in capture– recapture models. Biometrics, 46: 623–635. Baker, M., Nur, N. & Geupel, G. P., 1995. Correcting biased estimates of dispersal and survival due to limited study area: theory and an application using wrentits. Condor, 97: 663–674. Cilimburg, A. B., Lindberg, M. S., Tewksbury, J. J. & Hejl, S. J., 2002. Effects of dispersal on survival probability of adult yellow warblers (Dendroica petechia). Auk, 119: 778–789. Clark, A. L., Saether, B. E. & Roskaft, E., 1997. Sex biases in avian dispersal: a reappraisal. Oikos, 79: 429–438. Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438. Haas, C. A., 1998. Effects of prior nesting success on site fidelity and breeding dispersal: An experimental approach. Auk, 115: 929–936. Hoover, J. P., 2003. Decision rules for site fidelity in a migratory bird, the Prothonotary Warbler. Ecology, 84(2): 416–430. Huggins, R. M., 1989. On the statistical analysis of capture–recapture experiments. Biometrika, 76: 133–140. – 1991. Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics, 47: 725–732. Jolly, G. M., 1965. Explicit estimates from capture– recapture data with both death and immigration stochastic model. Biometrika, 52: 225–247. Kendall, W. L., Nichols, J. D. & Hines, J. E., 1997. Estimating temporary emigration using capture– recapture data with Pollock’s robust design. Ecology, 78: 563–578. Kendall, W. L. & Nichols, J. D., 1995. On the use of secondary capture–recapture samples to estimate temporary emigration and breeding proportions. Journal of Applied Statistics, 22: 751–762. Kendall, W. L., Pollock, K. H. & Brownie, C., 1995. A likelihood–based approach to capture–recapture estimation of demographic parameters under the robust design. Biometrics, 51: 293–308. Koenig, W. D., Vuren, D. V. & Hooge, P. N., 1996. Detectability, philopatry, and the distribution of dispersal distances in vertebrates. Trends in Ecology and Evolution, 11: 514–517. Lebreton, J. D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case–studies. Ecological Monographs, 62: 67–118. Marshall, M. R. & Cooper, R. J., (In press). Territory size of a migratory songbird in response to caterpillar density and foliage structure. Ecology, 85(1). Marshall, M. R., Cooper, R. J., Strazanac, J. S. & Butler, L., 2002. Effects of experimentally reduced prey abundance on the breeding ecology of the Red–eyed Vireo. Ecological Applications, 12: 261–280. Marshall, M. R., Wilson, R. R. & Cooper, R. J.,

2000. Estimating survival of Neotropical–Nearctic migratory birds: are they dead or just dispersed? U.S. Forest Service General Technical Report RMRS–P–16. Martin, T. E., Clobert, J. & Anderson, D. R., 1995. Return rates in studies of life history evolution: Are biases large? Journal of Applied Statistics, 22: 863–875. Rodrigues, M. A., 2002. Restricted movement in stream fishes: The paradigm is incomplete, not

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lost. Ecology, 83:1–13. Seber, G. A. F., 1965. A note on the multiple recapture census. Biometrika, 52: 249–259. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study 46 Supplement: 120–138. Wood, L. A., 1999. Short–term effects of timber management on Prothonotary Warbler (Protonotaria citrea) breeding biology. M. S. Thesis, Univ. of Georgia.

Animal Biodiversity and Conservation 27.1 (2004)

Testing the additive versus the compensatory hypothesis of mortality from ring recovery data using a random effects model M. Schaub & J.–D. Lebreton

Schaub, M. & Lebreton, J.–D., 2004. Testing the additive versus the compensatory hypothesis of mortality from ring recovery data using a random effects model. Animal Biodiversity and Conservation, 27.1: 73–85. Abstract Testing the additive versus the compensatory hypothesis of mortality from ring recovery data using a random effects model.— The interaction of an additional source of mortality with the underlying "natural" one strongly affects population dynamics. We propose an alternative way to test between two forms of interaction, total additivity and compensation. In contrast to existing approaches, only ring–recovery data where the cause of death of each recovered individual is known are needed. Cause–specific mortality proportions are estimated based on a multistate capture–recapture model. The hypotheses are tested by inspecting the correlation between the cause– specific mortality proportions. A variance decomposition is performed to obtain a proper estimate of the true process correlation. The estimation of the cause–specific mortality proportions is the most critical part of the approach. It works well if at least one of the two mortality rates varies across time and the two recovery rates are constant across time. We illustrate this methodology by a case study of White Storks Ciconia ciconia where we tested whether mortality induced by power line collision is additive to other forms of mortality. Key words: Additive mortality, Compensatory mortality, Ring recoveries, White stork, Variance components, Power line collision. Resumen Estudio comparativo entre la hipótesis de la mortalidad aditiva y la hipótesis de la mortalidad compensatoria mediante el empleo de un modelo de efectos aleatorios basado en datos de recuperación de anillas.— La interacción de una fuente adicional de mortalidad con la fuente subyacente "natural" incide de forma considerable en la dinámica poblacional. Proponemos un método alternativo para comprobar los dos tipos de interacción: la aditividad total y la compensación. A diferencia de lo que sucede con los modelos empleados actualmente, en este caso sólo se precisan datos de recuperación de anillas de cada uno de los individuos recuperados cuando se conoce la causa que ha provocado su muerte. Los porcentajes de mortalidad inducida por una causa específica se estiman a partir de un modelo de captura–recaptura multiestado. Las hipótesis se comprueban examinando la correlación existente entre los porcentajes de mortalidad inducida por una causa específica. Posteriormente, se efectúa una descomposición de varianza a fin de obtener una estimación apropiada de la verdadera correlación del proceso. La estimación de los porcentajes de mortalidad provocada por una causa específica representa el punto más crítico de este planteamiento. Funciona adecuadamente si por lo menos una de las dos tasas de mortalidad varía con el tiempo y las dos tasas de recuperación se mantienen constantes en el tiempo. Para ilustrar esta metodología, presentamos un estudio de la cigüeña blanca Ciconia ciconia, en el que verificamos si la mortalidad inducida por colisiones con los tendidos eléctricos se suma a otras formas de mortalidad. Palabras clave: Mortalidad aditiva, Mortalidad compensatoria, Recuperación de anillos, Cigüeña blanca, Componentes de varianza, Colisión con tendidos eléctricos. Michael Schaub, Schweizerische Vogelwarte, 6204 Sempach, Switzerland; Zoological Inst., Conservation Biology, Univ. Bern, Baltzerstrasse 6a, 3012 Bern, Switzerland.– Jean–Dominique Lebreton, CEFE/CNRS, 1919 Route de Mende, 34293 Montpellier Cedex 5, France. Corresponding author: M. Schaub. E–mail: michael.schaub@vogelwarte.ch

ISSN: 1578–665X

Introduction Many animal populations are subjected to man– induced sources of mortality. These include, e.g., harvesting and hunting, collisions with vehicles or objects such as power lines, or contamination with pesticides. All of this man–induced mortality can be viewed as a form of population exploitation. In the context of harvesting the exploitation is direct and intentional, in other contexts it may be indirect and unintentional. In any case, exploitation is an additional source of mortality to which the population is submitted. Determining the impact of exploitation in the broad sense on the dynamics of the population is a key question. Examples of this question are the determination of a harvesting rate which does not result in a population crash (e.g. Nichols et al., 2001), assessment of the long–term population persistence when an additional mortality cause emerges (e.g. Tavecchia et al., 2001), or the evaluation of pest control strategies (e.g. Brooks & Lebreton, 2001). Central for the evaluation of all these examples and in general is the knowledge of how the additional mortality interacts with the natural mortality. Two extreme hypotheses about the interaction between natural and an additional mortality rate can be formulated: the totally additive and the completely compensatory hypothesis (Anderson & Burnham, 1976; Burnham & Anderson, 1984). The totally additive hypothesis of mortality assumes that deaths due to a specific mortality cause represent an additional component of mortality in the population. Hence individuals that die due to this mortality cause would, if this mortality cause wouldn’t have existed, not have died during the time interval considered. If this hypothesis is true, the overall natural survival rate drops by the amount of the additional mortality rate (fig 1). Under the completely compensatory hypothesis of mortality, deaths due to the additional mortality cause would be compensated for by lowering the natural mortality rate. Hence, individuals that die due to the additional mortality cause would, if this mortality cause wouldn’t have existed, have died because of another reason within the time interval considered. If this hypothesis is true, an increase of the additional mortality rate does not reduce the overall survival rate (fig. 1). Complete compensation is only possible when the additional mortality rate is lower or equal to the overall mortality rate in the absence of the additional mortality cause (Anderson & Burnham, 1976; fig. 1). Between these two extreme hypotheses any degree of partial compensation is possible (fig. 1). Under partial compensation the overall survival rate decreases when animals are subjected to an additional mortality rate, but the decrease is lower than the value of the additional mortality rate. Complete or partial compensation of mortality can occur as a result of density–dependent mortality or of heterogeneity in survival among indi-

Schaub & Lebreton

viduals (Burnham & Anderson, 1976). However, Lebreton (in press) showed by means of simple calculations that the resulting compensation must generally be weak even under strong density–dependence or heterogeneity. Deciding between additivity or compensation based on empirical data has been proven to be difficult (Williams et al., 2002; Lebreton, in press). Anderson & Burnham (1976) and Burnham & Anderson (1984) were the pioneers in formulating these hypotheses and in establishing methods to test them. Since then, there has been little effort to refine the existing or to develop further methods. The basic principle approach proposed by Anderson & Burnham (1976) is to estimate the overall survival rate and the mortality rate induced by the additional mortality cause (called kill rate), and then to estimate the slope of survival against kill rate while taking account of the sampling variation. The complete compensation hypothesis is supported if this slope does not differ from 0. The critical step in this approach is the estimation of the kill rate. Because only recoveries of animals that died from the particular mortality cause are considered, an independent estimate of the recovery rate (and crippling loss rate) is required to work out the kill rate. Reward band experiments can help to obtain these independent estimates (Henny & Burnham, 1976; Nichols et al., 1991). Another approach is to test whether overall survival rate is a function of the mortality intensity due to the cause in question (e.g. harvest rate) using an ultrastructural model (e.g. Smith & Reynolds, 1992; Sedinger & Rexstad, 1994; Gauthier et al., 2001). The complete compensation hypothesis is supported if overall survival is not a function of the varying mortality intensity. Both approaches need information independently from the capture–recovery data. Because the independent variable is estimated with some sampling variance (and even bias), the slope of the regression line which serves to test the hypotheses will be biased to some degree (Lebreton, in press). Lebreton (in press) pointed out that the potential bias or uncertainty in this information tends to bias the additivity test towards the alternative hypothesis, i.e., compensation, which is quite an undesirable property of a statistical test. The case of seasonal compensation is addressed by Boyce et al. (1999). Here we attempt to develop an alternative approach for testing the total additivity hypothesis which does not need additional independent information. Rather, this approach uses knowledge about the cause of death of each recovered, marked animal. Schaub & Pradel (2004) showed recently that it is possible to estimate separately the overall survival and the proportions of different mortality causes from capture–recovery data when the cause of death of each recovered individual is known. We use a different parameterisation of their model to estimate directly two mortality rates ("natural" and "kill" rate). We develop a random

Animal Biodiversity and Conservation 27.1 (2004)

S0 Overall survival rate

effects model, in order to estimate in a similar way as Burnham & Anderson (1976) the correlation between the two mortality rates which serves as a test for the two opposing hypotheses. The additive hypothesis is supported if the correlation does not differ from zero. We illustrate our approach with a case study of White Storks (Ciconia ciconia), where we tested whether the mortality induced by power line collisions is completely additive or compensated for by other forms of mortality. Finally we discuss advantages, drawbacks and perspectives of this approach.

Cc Pc Ta Kill rate

Methods The proposed approach The data needed for our approach are capture– mark–recovery data where the cause of death of each recovered individual is known. We then allocate all individuals that died because of the mortality cause under consideration to cause A, all other dead individuals to cause B. A multistate capture– history is then constructed for each individual, in which resightings, recoveries due to mortality cause A and recoveries due to mortality cause B are coded differently. Over a defined time interval (usually one year) an individual has three possible fates: it may survive with probability S, it may die because of cause A with probability MA, or it may die because of cause B with probability MB. Conditional on the three fates the individual may be observed with resighting probability p (probability to resight a marked individual that is alive), with recovery probability rA (probability that an animal that has died because of cause A is recovered and reported) and with recovery probability rB (probability that an animal that has died because of cause B is recovered and reported), respectively. A three–states capture–recapture model serves to estimate the unknown parameters. Written with a transition matrix (departure states are written in rows, arriving states in columns, states from top to down and from left to right are "alive", "dead due to cause A" and "dead due to cause B") and a vector of recapture probabilities, the model is,

(1),

where, subscript t of the matrix and the vector denote time–dependence. In fact this model would contain a fourth state "dead for at least one year", but as it is absorbing and non–observable it is not necessary to consider it explicitly (Lebreton et al., 1999). An alternative notation of this model is {MA(t), MB(t), p(t), rA(t), rB(t)}. Originally, Schaub & Pradel (2004) used another parameterisation of this model. Instead of directly estimating MA and MB, they estimated the propor-

Fig. 1. Simple illustration of the complete compensatory, partial compensatory and totally additive hypotheses of mortality. S0 is the survival that would be observed in the absence of the additional mortality cause (kill rate = 0). Complete compensation can occur maximally up to the threshold given by c = 1 – S0: Cc. Complete compensation; Pc. Partial compensation; Ta. Total additivity. Fig. 1. Ilustración simple de las hipótesis de mortalidad compensatoria total, compensatoria parcial y aditiva total. S0 es la supervivencia que se observaría ante la ausencia de la causa de mortalidad adicional (tasa de mortalidad = 0). La compensación completa sólo puede darse como máximo hasta el umbral indicado por c = 1 – S0: Cc. Compensación total; Pc. Compensación parcial; Ta. Aditividad total.

tion ( ) of animals that have died due to cause A among all animals that have died in the specified time interval, and the overall survival rate (S). These parametrisaitons are equivalent, since linked by: MA = (1 – S) and MB = (1 – S)(1 – ). Schaub & Pradel (2004) pointed out that the indentifiablity of their model depends on the model structure. Using formal calculus software (Catchpole & Morgan, 1997; Catchpole et al., 2002; Gimenez et al., 2003), we tested the intrinsic identifiablity of several models with different complexity regarding time–dependence of the parameters. The models were intrinsically identifiable (i.e., not parameter redundant) when at least one of the two mortality rates is time–dependent and the two recovery rates are constant across time, or when only one mortality rate (e.g. MA) and the recovery rate associated with the other cause of death (rB) are time–dependent (table 1). As the model with time–constant mortality and recovery rates {M A(.), M B(.), p(t), rA(.), rB(.)}, is not identifiable, parameter estimation using identifiable models can nevertheless be negatively affected.

Schaub & Lebreton

This is because the non–identifiable model {MA(.), MB(.), p(t), rA(.), rB(.)} is a nested submodel of identifiable models. An inadequate performance can be made apparent by unrealistic estimates of some parameters and very large or zero standard errors. Catchpole et al. (2001) provide a thorough examination of the same problem found in a different model. For testing the additivity hypothesis we estimated the correlation between the two mortality rates. If the mortality rate due to cause A were totally additive to the mortality rate due to cause B, the two mortality rates would vary independently from each other over time, and hence their correlation would be zero. However, as mortality events of the two types compete over a non negligible time period, the numbers at risk of mortality over time are affected by both sources of mortality. As a consequence, even under the assumption of additivity, the proportions dying from the two causes of mortality will be slightly negatively correlated (see appendix). However this correlation will be small in absolute value (Burnham & Anderson, 1984; Lebreton, in press) and the null hypothesis of a correlation equal to 0 remains a good approximation. In contrast, if the mortality rate due to cause A would be compensated by decreasing mortality rate due to cause B, their correlation would be negative (–1, if compensation is complete). The correlation of the two estimated mortality rates cannot be used directly for this purpose, because it is affected by sampling correlation to an unknown degree. Instead we have to estimate and decompose the different variance components, i.e. the true process variance and the sampling variance. The covariation over time (indexed by i) between MA and MB is examined using a random effect model according to: MA(i) =

+ UA(i) + V(i)

(2)

MB(i) =

+ UB(i) + V(i)

(3)

where UA(i), UB(i) and V(i) are independent and normally distributed random variables with respective variances, independent of i, A2, B2, and 2. The true process correlation between MA and MB can then be calculated as

(4)

The null (additive) hypothesis H0 corr (MA, MB) = 0 translates then into H0 var (V) = 0. It can thus be tested simply by a Wald test once estimates of the variance component 2 and of its standard error have been obtained. In practice, the components of variance have to be estimated based on estimates and obtained from the multistate capture–recapture model. We used the following general procedure:

Let be a vector of parameters in a probabilistic model for which maximum likelihood estimates are available together with an estimate of their covariance matrix . The maximum likelihood estimates are normally distributed and asymptotically it follows that . When the number of parameters has been reduced by some model selection, this approximation will be quite valid (Besbeas et al., 2002), even considering as known without uncertainty. Then let us assume that is modeled as mixed models with fixed effects described by a design matrix X and components of variance being part of a covariance matrix W, as: (5). It follows that: (6). The likelihood of this overall mixed model can then be easily maximized to find MLEs of and of the variance components in W. Maximum likelihood is among the standard methods for fitting mixed models and appears as a good competitor to more sophisticated methods such as REML (Searle et al., 1992, ch. 6). Otis & White (2004) showed that variance components are estimated accurately from band recovery data. Obviously, a Bayesian model could also be used. This simple two–step maximum likelihood approach was used for estimating the components of variance in the model with the two sources of mortality and to test for var(V) = 0, i.e., for additivity. Application to data: the White Stork and power line collisions To illustrate this approach, we consider capture– recovery data of White Storks from Switzerland. A significant source of mortality in White Storks is collision with overhead powerlines (Riegel & Winkel, 1971; Schaub & Pradel, 2004). Evaluation of how strongly the population dynamics of Swiss White Storks are affected by power line accidents is of conservation relevance. Reconstruction of power lines is an efficient conservation option if mortality due to power lines would be additive, but less so, if power line mortality would be compensated for by other forms of mortality. From 1984 to 1999 2912 nestlings have been ringed, of which 61 were later resighted at the breeding sites, 195 were recovered as due to power line collision and 221 as due to other sources of mortality (table 2). According to a priori knowledge we constructed our candidate models in the following way. The resighting effort was low and highly variable between the study years, therefore we always kept the resighting probability (p) time–dependent. White Storks start to breed at age 3 to 4 years before this age they may return to the breeding colonies without breeding or they may stay elsewhere. To reduce heterogeneity, we only considered resightings of

Animal Biodiversity and Conservation 27.1 (2004)

Table 1. Test results of intrinsic identifiability of constant and time–dependent mortality causes models obtained by computer algebra methods (Gimenez et al., 2003). The parameters in the model are MA (mortality rate due to cause A), MB (mortality rate due not cause B), p (resighting rate), rA (recovery rate due to cause A), and rB (recovery rate due to cause B). t denotes time–dependence, and k is the number of capture occasions. Tabla 1. Resultados de la identificabilidad intrínseca de los modelos de causas de mortalidad constantes y dependientes del tiempo obtenidos mediante el empleo de métodos algebraicos asistidos por ordenador (Giménez et al., 2003). Los parámetros utilizados en el modelo son MA (tasa de mortalidad inducida por la causa A), MB (tasa de mortalidad no inducida por la causa B), p (tasa de reavistaje), rA (tasa de recuperación debida a la causa A), y rB (tasa de recuperación debida a la causa B). t indica la dependencia del tiempo y k es el número de casos de captura.

Model

Separately identifiable parameters

Number of estimated quantities

MA(t), MB(t), p(t), rA(t), rB(t)

p2,...,pk–1

5k–10

MA(t), MB(t), p(t), rA(t), rB(.)

p2,...,pk–1

4k–5

MA(t), MB(t), p(t), rA(.), rB(.)

All

3k–1

MA(t), MB(.), p(t), rA(.), rB(.)

All

2k+1

MA(.), MB(.), p(t), rA(.), rB(.)

p2,...,pk

k+2

MA(.), MB(.), p(t), rA(.), rB(t)

p2,...,pk

MA(.), MB(.), p(t), rA(t), rB(t)

p2,...,pk

3k–2

MA(t), MB(.), p(t), rA(t), rB(t)

p2,...,pk–1

4k–5

MA(t), MB(.), p(t), rA(.), rB(t)

All

3k–1

storks older than 4 years and fixed the resighting probabilities of the younger storks to zero. The two mortality rates (M E. Electrocution mortality; MN. Natural mortality) were always considered to be age– (two age classes, the first refer to the first year of life, the second to all later years) and time–dependent. Time–dependence was required to test the hypotheses. An age structure was enforced because we know that overall mortality strongly differs between young and adult storks (Lebreton, 1978; Barbraud et al., 1999; Doligez et al., 2004). The recovery rate (rE) associated with electrocuted White Storks is unlikely to be age–dependent, but may be constant or time– dependent. In contrast, the recovery rate (r N) associated with all other mortality causes may be age–dependent, as it compromises different sources of mortality to which young and adult storks may be differently sensitive. In addition this recovery rate may vary over time or may be constant. In summary, we used eight candidate models, that differ only in the complexity of the two recovery rates. We tested the intrinsic identifiability of all candidate models using formal calculus (Gimenez at al., 2003). Compared to Schaub & Pradel (2004), who made a similar analysis of the data, we only considered storks ringed as nestlings and did not include natal dispersal in the model. This made the model sim-

pler. Since Schaub & Pradel (2004) did not find significant temporal variation in natal dispersal, its omission is unlikely to have altered the estimated temporal pattern of the two mortality rates. A goodness–of–fit test for multistate models including nonobservable states doesn’t currently exist (Pradel et al., 2003). In order to have some indication of the goodness–of–fit we used the following ad hoc approach. We only considered the recovery data but did not distinguish between different causes of death. According to Brownie et al. (1985) we compared the observed number of dead storks for each cohort and year to the expected value under model {S(t), r(t)}. This model fitted the data well ( 232 = 37.11, P = 0.25). Compared to the model we would like to test, it makes very strong assumptions, e.g., it does not allow for different recovery rates due to the mortality causes or for age–dependence of the recovery rates. We argue that because the simple model fitted the data, the more complicated model which accounts for more heterogeneity would also fit. This goodness–of–fit test does not consider the resighted storks. However the bulk of the data are the recoveries and the few resightings are therefore unlikely to significantly induce lack of fit. We used program M– SURGE (Choquet et al., 2003) to fit the different models and to estimate the parameters and their associated variance–covariance matrix.

Schaub & Lebreton

Table 2. Capture–recovery data for White Storks from Switzerland summarized in m–array format. White Storks can be encountered dead due to power line collision (E), encountered dead due to a natural cause (N), or can be resighted alive (R). All White Storks are initially released as juveniles (Rj), but when resighted, they are "released" again as adult (Ra): Y. Year; NR. Number of releases. Tabla 2. Datos de captura–recuperación correspondientes a la cigüeña blanca de Suiza, resumidos en formato de matriz m. Las cigüeñas blancas se pueden encontrar muertas por haber colisionado con tendidos eléctricos (E), por causas naturales (N), o se pueden reavistar vivas (R). En un principio, todas las cigüeñas blancas se liberan siendo jóvenes (Rj), pero cuando son reavistadas, se "liberan" de nuevo como adultas (Ra): Y. Año; NR. Número de liberaciones.

Encounter period 1985 Y 1984 1985 1986 1987 1988 1989 1990 1991

E N R

E NR

E N R

EN R

0 1 1

0 0 0

0 1 0

Ra = 0

0 0 0

Rj = 100

7 5 0

0 0 0

0 1 0

0 0 1

0 1 0

Ra = 0

0 0 0

6 6 0

0 1 0

0 0 0

1 0 0

1 1 0

Ra = 0

0 0 0

Rj = 123

8 8 0

1 0 0

0 1 0

0 1 3

Ra = 0

0 0 0

Rj = 162

9 12 0

1 0 0

0 2 0

Ra = 1

0 0 0

Rj = 140

19 1 0

0 0 0

Ra = 1

0 0 1

0 0 0

Rj = 229

16 7 0

Ra = 2

0 0 0

Rj = 151 Rj = 265 Rj = 211 Rj = 131 Rj = 117 Rj = 300 Rj = 334 Ra = 6

1998

Rj = 337 Ra = 3

1999

Rj = 80

Ra = 14 1997

1991

0 2

Ra = 16 1996

1990

E N R

Ra = 5 1995

1989

0 0 0

Ra = 16 1994

1988

9 6 0

Ra = 9 1993

E N R

1987

Rj = 101

Ra = 3 1992

E NR

1986

Rj = 131 Ra = 9

Animal Biodiversity and Conservation 27.1 (2004)

Encounter period 1992

1993

1994

1995

1996

1997

1998

1999

2000

E N R

0 0

0 1 1

E N R

1 1 0

0 0 0

E N R

E NR

0 0 0

0 1 0

0 0

0 0 0

0 0

0 1 1

0 1 0

0 0 0

0 1 0

0 0 0

0 0

0 0 0

0 1

0 0 0

0 0 1

0 0 0

0 0

0 0 0

0 2

1 0 0

0 0 0

1 0 1

0 0 1

0 0 0

0 0

0 0 0

1 4

0 0 3

0 0 1

0 0 2

0 0 0

1 0 0

0 0 1

0 0

0 0 0

1 0

0 1 5

0 0 1

0 2 0

1 0 0

0 1 0

0 0 0

0 0 1

0 0 0

0 0

0 0 0

2 0

0 2 0

0 2 2

0 1 5

0 2 2

1 1 0

0 0 0

0 0

0 0 1

0 0 0

2 0

2 1 0

1 1 0

1 0 2

0 1 1

0 0 0

0 0 1

0 0 0

0 2

0 0 0

24 11 0

1 6 0

2 2 0

1 2 5

1 0 2

0 1 1

0 0 1

0 0 0

0 0 5

0 0 0

0 0 1

0 0 0

9 18 0

0 0 0

0 1 4

0 0 0

0 0 1

0 0 4

0 0 0

0 0 1

1 0 0

1 7 0

0 2 0

0 1 0

0 0 0

0 1 1

0 0 0

0 0 2

0 0 0

4 4 0

2 1 0

0 0 0

0 2 0

1 0 3

0 0 0

6 20 0

1 1 0

2 0 0

0 1 0

0 0 1

0 1 0

23 17 0

1 1 0

1 2 0

0 0 0

0 0 2

0 0 0

14 21 0

1 2 0

0 0 1

0 0 0 2 11 0 0 0 1

Schaub & Lebreton

Table 3. Selection among different recovery models of Swiss White Storks. rE represents the recovery rate of storks killed by power lines and rN denote the recovery rate of storks that died because of other causes. The expression in parentheses denote whether the parameter is constant (.), time– dependent (t), age–dependent (a2), or age– and time–dependent (a2*t). The other parameters in the models, the mortality rate due to power line collision (ME), the mortality rate due to other causes (MN) and recapture rate (p) were always kept age and time–dependent and time–dependent, respectively (ME(a2*t), MN(a2*t), p(t)). Tabla 3. Selección entre los diferentes modelos de recuperación de cigüeña blanca de Suiza. rE representa la tasa de recuperación de cigüeñas muertas tras haber colisionado con tendidos eléctricos, mientras que rN revela la tasa de recuperación de cigüeñas que murieron por otras causas. La expresión entre paréntesis indica si el parámetro es constante (.), dependiente del tiempo (t), dependiente de la edad (a2), o dependiente de la edad y del tiempo (a2*t). El resto de parámetros empleados en los modelos, tasa de mortalidad debida a la colisión con tendidos eléctricos (ME), tasa de mortalidad debida a otras causas (MN) y tasa de recaptura (p), siempre se mantuvieron dependientes de la edad y del tiempo y dependientes del tiempo, respectivamente (ME(a2*t), MN(a2*t), p(t)). Model

Deviance

Parameters

AIC

rE(.), rN(a2)

4481.38

0.00

0.60

rE(.), rN(.)

4486.50

1.13

0.34

rE(t), rN(a2)

4462.65

5.29

0.04

rE(.), rN(t)

4467.88

8.52

0.01

rE(t), rN(.)

4470.26

8.89

0.01

rE(.), rN(a2*t)

4460.25

100

20.88

0.00

rE(t), rN(t)

4464.13

105

34.77

0.00

rE(t), rN(a2*t)

4459.40

118

56.03

0.00

AIC–weight

Table 4. Test results of intrinsic identifiability of the mortality causes models used in the case study (table 2) as obtained by computer algebra methods (Gimenez et al., 2003). t denotes time–dependence, and k is the number of capture occasions. See table 2 for a description of the model notations. Tabla 4. Resultados de ensayo de la identificabilidad intrínseca de los modelos de causas de mortalidad empleados en nuestro estudio (tabla 2), obtenidos mediante el empleo de métodos algebraicos asistidos por ordenador (Giménez et al., 2003). t indica la dependencia del tiempo, mientras que k es el número de casos de captura. Ver tabla 2 para una descripción de las anotaciones sobre los modelos.

Model ME(a2*t), MN(a2*t), p(t), rE(t), rN(t)

Separately identifiable parameters

Number of estimated quantities

All, but the last in all parameters

7k–8

ME(a2*t), MN(a2*t), p(t), rE(t), rN(a2)

All

6k–4

ME(a2*t), MN(a2*t), p(t), rE(t), rN(.)

All

6k–5

p2,...,pk–2

7k–8

ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2*t) ME(a2*t), MN(a2*t), p(t), rE(.), rN(t)

All

6k–5

ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2)

All

5k–2

ME(a2*t), MN(a2*t), p(t), rE(.), rN(.)

All

5k–3

Animal Biodiversity and Conservation 27.1 (2004)

1 0.9 0.8

Mortality

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 B

0.6

Collision Natural

0.5

Mortality

0.4 0.3 0.2 0.1 0 84

Fig. 2. Mortality rates due to collisions with overhead power lines (filled dots) and due to natural causes (open dots) in juvenile (A) and adult (B) Swiss White Storks estimated with the most parsimonious model {ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2)}. The vertical lines show the range of the 95% confidence interval. Fig. 2. Tasas de mortalidad por colisiones con tendidos eléctricos aéreos (círculos negros) y por causas naturales (círculos blancos) en cigüeñas blancas de Suiza jóvenes (A) y adultas (B), estimadas mediante el empleo del modelo más moderado {ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2)}. Las líneas verticales indican el rango del intervalo de confianza del 95%.

Results Model selection revealed no evidence that the recovery rates varied over time (table 3). There was some uncertainty about whether the recovery rate due to other causes than collision with power lines was age–dependent. The best model with age– dependent recovery rate had 1.76 times more support than the model with constant recovery rates (table 3). Still, for the presentation of the results and the calculations that follow we considered only the most parsimonious model.

Six of the eight candidate models are intrinsically identifiable (table 4), including the most parsimonious one. A presumption that the estimates from that model are suitable is therefore fulfilled. Both mortality rates were higher in juveniles than in adults (fig. 2). Power line kill rate in both age classes was usually lower than the mortality rate due to other causes. The confidence intervals of the estimates were rather wide, resulting either from the possible over–parameterisation of the model (no model selection was performed for the mortality rates) and/or from the near non–identifiability of the

Schaub & Lebreton

Natural mortality rate

Table 5. Estimated variance components and their standard errors. E2 and N2 are the temporal variances of the independent components of powerline and natural mortality rates, respectively. 2 is the variance of their common component. A non–null value for 2 results in a negative correlation over time between mortality rates (see text for further explanations). All variance components are not statistically different from zero (P > 0.05): P. Parameter; E. Estimate; SE. Standard Error.

r = –0.961

0.8

0.6

0.4

0.2

0 0

0.2 0.4 0.6 0.8 Power line kill rate

Natural mortality rate

0.2

r = –0.071

0.15

Tabla 5. Componentes de varianza estimados y sus errores estándar. E2 y N2 son las varianzas temporales de los componentes independientes de las tasas de mortalidad por colisión con tendidos eléctricos y las tasas de mortalidad por causas naturales, respectivamente. 2 es la varianza de su componente común. Un valor de no nulidad para 2 se traduce en una correlación negativa a lo largo del tiempo entre las tasas de mortalidad (para más detalles al respecto, ver el texto). Ningún componentes de varianza difiere estadísticamente de cero (P > 0.05): P. Parámetro; E. Estimado; SE. Error estándar.

P 0.1

0.028915

0.03297

0.040161

0.02911

0.021737

0.01368

0.000000

Not available

0.000000

Not available

0.000000

Not available

Juveniles E N

0.05

Adults 0

0.05 0.1 0.15 Power line kill rate

0.2

Fig. 3. Correlation of the mortality rates due to power line collision and the mortality rate due to natural causes in Swiss White Storks in juvenile (A) and adult (B) (estimated using model {ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2)}). The correlations are subject to sampling and true process correlation, and thus not suited to reject or support the additive hypothesis of mortality causes. Fig. 3. Correlación de las tasas de mortalidad por colisiones con tendidos eléctricos y por causas naturales en la cigüeña blanca de Suiza en juveniles (A) y adultos (B) (estimadas mediante el empleo del modelo {ME(a2*t), MN(a2*t), p(t), rE(.), rN(a2)}). Las correlaciones están sujetas a muestreo y a la verdadera correlación del proceso, por lo que no resultan apropiadas para desestimar o defender la hipótesis aditiva de las causas de mortalidad.

N 2

model. There was a strong negative correlation between the two mortality rates ME and MN in juveniles, but not in adults (fig. 3). This suggest that there is some form of compensation in the juveniles. However, the observed correlation results from correlation between real (i.e. parameter) values and sampling correlation. The variance components of the two mortality rates were not different from zero in juveniles (table 5; P > 0.05), thus variation over time was small. Yet the correlation between the two mortality rates was negative [corr(ME, MN) = –0.3882]. This suggest that power line mortality is slightly compensated for by other forms of mortality in juveniles. In the adults the variation of the two mortality rates over time was very low, rendering the estimation of the variance components and the correlation between the two mortality causes impossible. Consequently the hypothesis could not be tested in the adults.

Animal Biodiversity and Conservation 27.1 (2004)

Discussion The population growth rate of the long–lived White Stork is much more sensitive to changes in adults survival than to changes in juvenile survival (Schaub et al., 2004). The weak compensation of the power line induced mortality in juveniles may therefore not have a very strong significance for the population dynamics. Based on the data at hand we could not test whether power line mortality of adults is compensated for by other forms of mortality, which is a pity, because this evaluation would be more important for the population dynamics. However, if compensation in the adults would occur, it would presumably be weak because the overall natural mortality rate is low (Lebreton, in press). Hence our conservative and preliminary conclusion is that power line collision is likely to have a negative impact on White Stork survival rates. We encourage other studies using data from other areas or time–periods to get more conclusive results. The main advantage of our approach to test the additive versus the compensatory hypothesis of mortality is that there is no need to have data other than ring recoveries. This is in contrast to traditional approaches. Most of them need additional, independent information (kill rate intensity, reporting rate, crippling loss rate) resulting in biased test results (see Otis & White, 2003 for an exception). This seems particularly relevant for studies on other mortality causes than hunting, where usually information about "intensity" is completely lacking. The test results based on the new approach are more rigorous. Finally, this methodology allows the use of data which are widely available. For example in the EURING data base the mortality cause of all recovered birds has been stored routinely since years (Speek et al., 2001). Hence it is possible to test whether particular mortality risks have changed over time and whether they have significantly affected population dynamics. Testing the additive versus the compensatory hypothesis of mortality is hampered by the need for temporal variation in at least one of the two mortality rates. The reasons are twofold. First, it is impossible to study the interaction of two mortality rates or the survival and the kill rate if there is no temporal variation. This is true for all approaches attempting to test these hypotheses. Second, and specific to our approach, the estimation of the two mortality rates only works properly when there is temporal variation in at least one of the mortality rates. As pointed out, the reason for this difficulty is the non–identifiability of the simplest submodel (see Catchpole el al., 2001). Poor estimates result when the underlying true parameter values follow the non–identifiable model, e.g. when the two mortality rates are not or only slightly variable over time or when the two recovery rates are strongly variable over time. Hence, the estimation will work accurately with some data, but not with others. We recommend the approach be used with care — parameter estimates and their variances must be

checked in order to decide whether the results are sound. Another problem for testing the additive hypothesis is the fairly strong negative correlation between the two mortality rates that arose from competing risks (see appendix). Hence, the proper null hypothesis H0 would not be corr(MA, MB) = 0, but rather corr(MA, MB) = –x, where x has an unknown value. The difficulty of formulating the proper H0 exists also in other approaches and appears as a general difficulty in testing the additive and compensatory hypotheses of mortality. What could be done to render the described approach more generally applicable? First, for a wider use of this method it would be very valuable to conduct a simulation study. Such a study could evaluate how accurately the parameters can be estimated depending on different degrees of temporal variation in the two mortality or recovery rates, and how strongly the outcome of the hypothesis test is compromised by inaccurate estimates of mortality rates. Second, it is worthwhile to explore how additional information could be used to stabilise the estimation of the parameters. The use of Bayesian priors for the recovery rates is certainly a promising possibility to explore. Acknowledgements We are greatly indebted to the organisation Storch Schweiz and the Swiss Ornithological Institute, Sempach, for providing the data, and to Gary C. White and an anonymous reviewer for providing valuable comments on the paper. References Anderson, D. R. & Burnham, K. P., 1976. Population ecology of the Mallard. VI The effect of exploitation on survival. United States Fish and Wildlife Service Resource Publication, 128: 1–66. Barbraud, C., Barbraud, J.–C. & Barbraud, M., 1999. Population dynamics of the White Stork Ciconia ciconia in western France. Ibis, 141: 469–479. Besbeas, P., Lebreton, J.–D. & Morgan, B. J. T., 2002. The efficient integration of abundance and demographic data. Journal of the Royal Statistical Society C (Applied Statistics), 52: 95–102. Boyce, M. C., Sinclair, A. R. E. & White, G. C., 1999. Seasonal compensation of predation and harvesting. Oikos, 87: 419–426. Brooks, E. N. & Lebreton, J.–D., 2001. Optimizing removal to control a metapopulation: application to the yellow legged herring gull (Larus cachinnans). Ecological Modelling, 136: 269–284. Brownie, C., Anderson, D. R., Burnham, K. P. & Robson, D. S., 1985. Statistical inference from band–recovery models —a handbook. 2nd ed. U.S. Fish and Wildlife Service, Resource Publication 156, Washington DC, U.S.A.

Burnham, K. P. & Anderson, D. R., 1984. Tests of compensatory vs. additive hypotheses of mortality in Mallards. Ecology, 65: 105–112. Catchpole, E. A., Kgosi, P. M. & Morgan, B. J. T., 2001. On near–singularity of models for animal recovery data. Biometrics, 57: 720–726. Catchpole, E. A. & Morgan, B. J. T., 1997. Detecting parameter redundancy. Biometrika, 84: 187–196. Catchpole, E. A. & Morgan, B. J. T. & Viallefont, A., 2002. Solving problems in parameter redundancy using computer algebra. Journal of Applied Statistics, 29: 625–636. Choquet, R., Reboulet, A. M., Pradel, R., Gimenez, O. & Lebreton, J.–D., 2003. User’s manual for M– SURGE 1.0. Mimeographed document, CEFE/ CNRS, Montpellier. ftp://ftp.cefe.cnrs-mop.fr/biom/Soft-CR/ Doligez, B., Thomson, D. L. & Van Noordwijk, A., 2004. Population dynamics of the White Stork in the Netherlands: assessing life–history and behavioural traits using data collected at large spatial scales. Animal Biodiversity and Conservation, 27.1: 387–402. Gauthier, G., Pradel, R., Menu, S. & Lebreton, J.– D., 2001. Seasonal survival of greater snow geese and effect of hunting under dependence in sighting probability. Ecology, 82: 3105–3119. Gimenez, O., Choquet, R. & Lebreton, J.–D., 2003. Parameter redundancy in multistate capture–recapture models. Biometrical Journal, 45: 704–722. Henny, C. J. & Burnham, P. K., 1976. A Mallard reward band study to estimate band reporting rates. Journal of Wildlife Management, 40: 1–14. Lebreton, J.–D. (in press). Dynamical and statistical models for exploited populations. – 1978. Un modèle probabiliste de la dynamique des populations de cigogne blanche (Ciconia ciconia L.) en Europe occidentale. In: Biométrie et Ecologie: 277–343 (J. M. Legay & R. Tomassone, Eds.). Société Française de Biométrie, Paris, France. Lebreton, J.–D., Almeras, T. & Pradel, R., 1999. Competing events, mixtures of information and multistratum recapture models. Bird Study, 46 (suppl.): 39–46. Mood, A. M., Graybill, F. & Boes, D. C., 1974. Introduction to the theory of statistics. 3rd edition. McGraw–Hill, New–York. Nichols, J. D., Blohm, R. J., Reynolds, R. E., Trost,

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R. E., Hines, J. E. & Bladen, J. P., 1991. Band reporting rates for Mallards with reward bands of different dollar values. Journal of Wildlife Management, 55: 119–126. Nichols, J. D., Lancia, R. A. & Lebreton, J.–D., 2001. Hunting statistics: what data for what use? An account of an international workshop. Game and Wildlife Science, 18: 185–205. Otis, D. & White, G. C., 2004. Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations. Animal Biodiversity and Conservation, 27.1: 157–173. Pradel, R., Wintrebert, C. M. A. & Gimenez, O., 2003. A proposal for a goodness–of–fit test to the Arnason–Schwarz multisite capture–recapture model. Biometrics, 59: 43–53. Riegel, M. & Winkel, W., 1971. Über Todesursachen beim Weissstorch (C. ciconia) an Hand von Ringfundangaben. Vogelwarte, 26: 128–135. Schaub, M. & Pradel, R., 2004. Assessing the relative importance of different sources of mortality from recoveries of marked animals. Ecology, 85: 930–938. Schaub, M., Pradel, R. & Lebreton, J.–D., 2004. Is the reintroduced White Stork (Ciconia ciconia) population in Switzerland self–sustainable? Biological Conservation, 119: 105–114. Searle, S. R., Casella, G. & McCulloch, E. C., 1992. Variance components. Wiley. New York, U.S.A. Sedinger, J. S. & Rexstad, E. A., 1994. Do restrictive harvest regulations result in higher survival rates in Mallards? A comment. Journal of Wildlife Management, 58: 571–577. Smith, G. W. & Reynolds, R. E., 1992. Hunting and Mallard survival, 1979–88. Journal of Wildlife Management, 56: 306–316. Speek, G., Clark, J. A., Rohde, Z., Wassenaar, R. D. & Van Noordwijk, A. J., 2001. The EURING exchange–code 2000. Heteren. ISBN 90–74638– 13–9. Tavecchia, G., Pradel, R., Lebreton, J.–D., Johnson, A. R. & Mondain–Monval, J.–Y., 2001. The effect of lead exposure on survival of adult mallards in the Camargue, southern France. Journal of Applied Ecology, 38: 1197–1207. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and Management of Animal Populations. Academic Press. San Diego, U.S.A.

Animal Biodiversity and Conservation 27.1 (2004)

Appendix. The correlation between two competing sources of mortality. Apéndice. Correlación entre dos causas de mortalidad competitivas.

The starting point is the approximate equation for survival: S = 1 – MN – ME l S0 – S0ME form which one deduces: MN l (1 – S0) (1 – ME) The three terms in this equation are random variables changing from year to year. The property X and Y independent implies E(XY) = E(X)E(Y) (e.g., Mood et al., 1974, p.181) and leads then to: E(MN) l (1 – E(S0))(1 – E(ME)) E(MN ME) l (1 – E(S0)) E(ME) – (1 – E(S0))E(ME2) Hence: E(MN ME) – E(MN) E(ME) l (1 – E(S0)) E(ME) – (1 – E(S0))E(ME2) – (1 – E(S0))E(ME) + (1 – E(S0)E(ME)2 cov(MN,ME) l –(1 – E(S0)) var(ME)

i.e.,

This first result implies a negative correlation between MN and ME even with additivity of instantaneous sources of mortality. The next step is to calculate the correlation. First, using the formula for the variance of a product of independent random variables (Mood et al. 1974, p. 181): var(MN) l var((1 – S0) (1 – ME)) = (1 – E(S0))2 var(ME) + (1 – E(ME))2 var(S0) + var(ME) var(S0) Then, using the various results above, with

which simplifies to:

or still:

For the first year White Stork with E(S0) l 0.65, E(ME) l 0.25, var(S0) l 0.04, and var(ME) l 0.03, we expect the correlation between MN and ME to be corr(MN,ME) = –0.36 also if the two mortality causes are completely additiv.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

Individual heterogeneity and identifiability in capture–recapture models W. A. Link

Link, W. A., 2004. Individual heterogeneity and identifiability in capture–recapture models. Animal Biodiversity and Conservation, 27.1: 87–91. Abstract Individual heterogeneity and identifiability in capture–recapture models.— Individual heterogeneity in detection probabilities is a far more serious problem for capture–recapture modeling than has previously been recognized. In this note, I illustrate that population size is not an identifiable parameter under the general closed population mark–recapture model Mh. The problem of identifiability is obvious if the population includes individuals with pi = 0, but persists even when it is assumed that individual detection probabilities are bounded away from zero. Identifiability may be attained within parametric families of distributions for pi, but not among parametric families of distributions. Consequently, in the presence of individual heterogeneity in detection probability, capture–recapture analysis is strongly model dependent. Key words: Capture–recapture, Detection probability, Heterogeneity, Identifiability, Population estimation. Resumen Heterogeneidad individual e identificabilidad en modelos de captura–recaptura.— La heterogeneidad individual en las probabilidades de detección representa un problema para la modelación del procedimiento de captura–recaptura mucho más serio de lo que previamente se había reconocido. En este artículo se demuestra que el tamaño de la población no constituye un parámetro identificable en el modelo general Mh que emplea técnicas de marcaje–recaptura de poblaciones cerradas. El problema de la identificabilidad resulta evidente si la población incluye individuos con pi = 0, pero sigue persistiendo aun cuando se presuponga que las probabilidades de detección individual se han alejado de cero. La identificabilidad puede conseguirse en familias paramétricas de distribuciones para pi, pero no entre familias paramétricas de distribuciones. Por consiguiente, si se da una heterogeneidad individual en la probabilidad de detección, el análisis de captura–recaptura depende considerablemente del modelo considerado. Palabras clave: Captura–recaptura, Probabilidades de detección, Heterogeneidad, Identificabilidad, Estimación de la población. William A. Link, USGS Patuxent Wildlife Research Center, 11510 American Holly Drive, Laurel, Maryland 20708 U.S.A.

ISSN: 1578–665X

Introduction Let Xi, i = 1,2,...,N be independent binomial random variables, with common index T and success rates pi sampled independently from distribution g(p). Further, let

where 1(·) is the indicator function. Having observed f c = (f1,f2,...,fT), the problem is to estimate N, or equivalently, to predict f0. This is the closed population capture–recapture model Mh: N is the unknown population size, Xi is the number of times animal i is captured in T sampling occasions, fj is the number of animals captured exactly j times. Numerous methods for estimating N exist, ranging from the jackknife method of Burnham & Overton (1978), to finite mixture models (Norris & Pollock, 1996; Pledger, 2000), and including parametric models such as the logit normal (Coull & Agresti, 1999) and beta models (Dorazio & Royle, 2003). Given the restrictions on g(p) implicit to these methods, estimation of N is usually successful. However, as will be demonstrated here and elsewhere (Link, 2003), N is not identifiable without untestable model assumptions restricting the set of distributions g(p). One example of this difficulty is well known. If the population consists of N1 individuals with pi = 0, and N2 individuals with pi > 0, an analyst of model Mh can at best estimate N2, rather than N = N1 + N2. This circumstance is generally dismissed with the assertion that "we’re only estimating the observable portion of the population." But what of animals with low but nonzero detection probabilities? These are clearly the ones which present the challenge to capture–recapture analysis. Huggins (2001), seeking to identify restrictions on the collection of distributions that would ensure identifiability, focused his attention on removing difficulties associated with low detection probabilities. The condition he considered was that g(p) places no mass on values of p < 1 – (1 – )1/T for a fixed value c (0,1). This means that every individual has probability of at least > 0 of being captured on one of the T sampling occasions. Huggins concluded that if the converse of his Theorem 3 were true (he describes this as "difficult to establish" and "an open question") then the condition would be sufficient to ensure identifiability. The restriction is not sufficient, as is demonstrated by example1, below. It is possible to construct 2 distinct distributions, g1(p) g g2(p), each with support bounded away from zero, the two distributions producing identical sampling distributions for the observed data f c, but leading to contradictory inference about f0. Stronger restrictions, or at least different restrictions are required, to ensure identifiability of N. For example, Burnham’s (1972) thesis includes a demonstration that restricting attention to beta distrib-

Link

uted heterogeneity leads to identifiability of N. Thus we can feel confident dealing with model Mh if we are confident that g(p) is a beta distribution. But what if, unbeknownst to us, g(p) is a logit normal distribution? It can be demonstrated by example that the sampling distribution of fc induced by a beta distribution can be very closely approximated by the sampling distribution of f c induced by a logit normal distribution, but with substantially different inferences about N. (See example 2, below.) The inferences are distinct, but there is no way, on the basis of data f c to decide which is correct (except with vast sample sizes). Since it is unlikely that one will have epistemological grounds for assuming the beta distribution over the logit normal (or other distributions, such as the log gamma; see Link, 2003), it seems faint comfort to learn that N is identifiable within any one of these classes. My third example, below, shows that if nature is perverse in its selection of g(p), the sampling distribution of f c can be strongly and misleadingly suggestive of a particular form for g(p), even for a variety of values for T. Additional notation Let n denote the number of distinct animals ever sighted, i.e., n = f1 + f2 +...+ fT . I refer to the data f c as the observed frequency distribution, and to f = (f0,f1,...,fT) as the complete frequency distribution. The vectors f and f c are multinomial random variables with indices N and n, respectively, and jth cell probabilities designated by (j) and C(j), respectively. These are related by C(j) = (j)/1 – (0). Under model Mh, we have (1) for g(p) in (1), one Substituting an estimate , j = 0,1,2,...,n. obtains estimates It is easily verified that (2) Thus, it is natural to predict the number of individuals not seen by

and to predict the unknown population size by =n+ . Example 1 Suppose that T = 6, that g1(p) corresponds to a uniform distribution on (0.008512, 0.76), and that g2(p) corresponds to a 3–point mixture placing masses {0.350739, 0.414090, 0.235172} on values {0.161937, 0.449089, 0.692734}. The minimum value of p attainable under either model is 0.008512 = 1 – (1 – )1/T, for = 0.050; every animal has at

Animal Biodiversity and Conservation 27.1 (2004)

Table 1. Cell probabilities for Example 1.

(x) and

(x) for complete and observed frequency distributions of

Tabla 1. Probabilidades de cada celda para y observadas del Ejemplo 1.

Unif (a,b)

(x)

(x) de las frecuencias de distribución completas

.179048

.189616

.188057

.178371

.147685

.089382

.027840

(x)

–

.230971

.229072

.217273

.179895

.108876

.033912

(x)

.133294

.200184

.198538

.188312

.155916

.094364

.029392

–

.230971

.229072

.217273

.179895

.108876

.033912

3 pt. mixture

(x) y

(x)

least a 5% chance of being caught on one or more sampling occasions. The cell probabilities for f and f c are given in table 1. Note that the sampling distribution of the data f c is identical for the two distributions g(p), but that the predicted value of f0 is nearly half again as large under the uniform distribution as under the two–point mixture: with n = 100, the prediction of f0 under the uniform specification is 100 (0.179) /(1 – 0.179) . 22, while the prediction of f0 under the 3–point specification is 100(0.133)/ (1 – 0.133) . 15. I describe the method used for constructing Example 1 in presenting Example 3, below. Example 1 may be of special interest to analysts, since the two distributions correspond to models that could be fit based on observations from T = 6 sampling periods. It is worth mentioning that the problem of identifiability does not depend on both models being fittable, a point to which I return in presenting Example 3. Example 2 Let T = 5, and g1(p) represent the distribution resulting from the assumption that logit(p) has a normal distribution with mean of –1.75 and standard deviation of 2.00. I calculated the sampling distribution for f and f c by numeric integration over a grid of 100,000 points. Next, I minimized the Kullback–Leibler (KL) distance from the distribution of f c induced by g1(p) to the distribution of f c induced by g(p) in the beta family of distributions (for details on the KL–distance; see Agresti, 1990: p. 241). The resulting beta distribution has parameters a = 0.2512 and b = 1.1300. This beta distribution and the logit normal distribution described in the previous paragraph are plotted in figure 1. The observed and complete frequency distributions are given in table 2. Note that while the observed frequency distributions are not identical, they are close enough to be virtually indistinguish-

able except with extremely large samples. The discrepancy in predictions of f0 is substantial: based on n = 100, the predictions are 83 (for the logit– normal model) and 156 (for the beta model). Example 3 Expanding the term (1 – p)T–j in equation (1) by means of the binomial theorem, it is seen that the values (x) are linear combinations of the first T moments of distribution g(p), for x = 1,2,...,T. The same is true for

Let mg(j) denote the jth moment of g(p). If we could construct a distribution h(p) with moments mh(j) = cmg(j), for some c g 1, and for j = 1,2,...,T, the observations in the previous paragraph, and the relation C(j) = (j)/(1 – (0)) lead to the conclusion that distributions g(p) and h(p) will induce the same values C(j), but different values for (0). Using subscripts g and h to distinguish the values of (0), we obtain the relation (3) Thus these distinct distributions of heterogeneity will lead to the same sampling distributions for observed data, but different predictions for f0, and consequently, for N. If g(p) is the uniform distribution, mg(j) = 1 / (j + 1). Consider the distribution function h(p) = 1/ 15 ( 114 – 4950 p + 79200 p 2 – 600600 p 3 + 2522520 p 4 – 6306300 p 5 + 9609600 p 6 – 8751600 p7 + 4375800 p8 – 923780 p9). This distribution is plotted along with the uniform distribution in figure 2. Straightforward calculation shows that the moments of h(p) are mh(j) = c / (j + 1), with c = 14 / 15, for j = 1,2,...,9. Thus for a study involving any number of sampling occasions up to T = 9, the data produced with heterogeneity distribution h(p) will be

Link

0.0

0.05

0.10

0.20

0.30 0.400.50 0.701.00

Fig. 1. Density functions used in Example 2. Dashed line represents beta distribution with parameters a = 0.2512 and b = 1.1300; solid line is distribution of p corresponding to logit(p) having a normal distribution with mean of –1.75 and standard deviation of 2.00. Note that x–axis has been distorted to accentuate the differences between the two densities. Fig. 1. Funciones de densidad utilizadas en el Ejemplo 2. La línea discontinua representa la distribución beta con los parámetros a = 0,2512 y b = 1,300; la línea continua es la distribución de p correspondiente al logit(p) que presenta una distribución normal con una media de –1,75 y una desviación estándar de 2,00. Nótese que el eje x se ha distorsionado para acentuar las diferencias entre las dos densidades.

Table 2. Cell probabilities for Example 2.

(x) and

(x) for complete and observed frequency distributions of

Tabla 2. Probabilidades de cada celda para y observadas del Ejemplo 2.

Logit–normal

(x)

(x) de las frecuencias de distribución completas

0.454

0.208

0.126

0.090

0.070

0.052

(x)

–

0.381

0.231

0.165

0.128

0.095

(x)

0.609

0.149

0.090

0.065

0.050

0.037

–

0.381

0.231

0.166

0.127

0.095

Beta

(x) y

(x)

indistinguishable from data produced under a uniform distribution of heterogeneity. However, the predictions of f0 will differ substantially. Since under the uniform distribution g(0) = 1/(T + 1), we find from (3) and (2) that the prediction of f 0 based on the uniform distribution will be smaller than the prediction based on h(p) by a factor of 14 / (T + 15). Some might dismiss this example on the grounds that they would never even consider fitting a distribution that looks like h(p); to this, I reply "That’s exactly my point!" If nature perversely selects h(p) as the distribution of heterogeneity, and we are (excusably) misled into assuming a uniform distri-

bution, our predictions of f0 will be too small by a factor of 14 / (T + 15), and we’ll never know the difference. Conclusions and discussion Population size N is not an identifiable parameter under model Mh, except under the imposition of untestable model assumptions. Thus estimation of population size, in the presence of individual heterogeneity in detection is inevitably model based, much the same as the analysis of oft–reviled count survey data.

Animal Biodiversity and Conservation 27.1 (2004)

8 7 6 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 2. Density functions used in Example 3. The first nine moments of the nonuniform density are precisely 14/15th’s the size of the corresponding moments of the uniform density. Fig. 2. Funciones de densidad utilizadas en el Ejemplo 3. Los primeros nueve momentos de la densidad no uniforme equivalen precisamente a 14/15 del tamaño de los momentos homólogos de la densidad uniforme.

It is worth considering what the further implications of this finding are, particularly for open population models. Some early indications (Link, 2003) are that while estimates of population sizes may be biased in a manner similar to that described here for closed population estimation, survival estimates may be less sensitive to heterogeneity in detection rates. On the other hand, since capture–mark–recapture experiments essentially create "populations" of marked animals that are closed except to mortality, it is possible that time variation in detection rates might induce bias in survival estimates. The problems presented here should come as no surprise. Indeed, without specific parametric models for the heterogeneity in p, we find ourselves in the unpleasant circumstances described in the classic paper of Kiefer & Wolfowitz (1956) which demonstrated, among other things, that maximum likelihood estimates of parameters of interest may be asymptotically biased, and badly so, if the number of nuisance parameters is allowed to increase without bound. This is precisely the situation under model Mh, with individual detection probabilities in the role of nuisance parameters. References Agresti, A., 1990. Categorical Data Analysis. New York, Wiley. Burnham, K. P., 1972. Estimation of population size

in multiple capture–recapture studies when capture probabilities vary among animals. Ph. D. Thesis, Oregon State Univ., Corvallis. Burnham, K. P. & Overton, W. S., 1978. Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika, 65: 625–633. Coull, B. A. & Agresti, A., 1999. The use of mixed logit models to reflect heterogeneity in capture– recapture studies. Biometrics, 55: 294–301. Dorazio, R. M. & Royle, J. A., 2003. Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics, 59: 351–364. Huggins, R., 2001. A note on the difficulties associated with the analysis of capture–recapture experiments with heterogeneous capture probabilities. Statistics and Probability Letters, 54: 147– 152. Kiefer, J. & Wolfowitz, J., 1956. Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Annals of Mathematical Statistics, 27: 887–906. Link, W. A., 2003. Nonidentifiability of population size from capture–recapture data with heterogeneous detection probabilities. Biometrics, 59: 1125–1132. Norris, J. L. & Pollock, K. H., 1996. Nonparametric MLE under two closed–capture modelswith heterogeneity. Biometrics, 52: 639–649. Pledger, S., 2000. Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics, 56: 434–442.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

Multi–state models: metapopulation and life history analyses A. N. Arnason & E. Cam

Arnason, A. N. & Cam, E., 2004. Multi–state models: metapopulation and life history analyses. Animal Biodiversity and Conservation, 27.1: 93–95. Multi–state models are designed to describe populations that move among a fixed set of categorical states. The obvious application is to population interchange among geographic locations such as breeding sites or feeding areas (e.g., Hestbeck et al., 1991; Blums et al., 2003; Cam et al., 2004) but they are increasingly used to address important questions of evolutionary biology and life history strategies (Nichols & Kendall, 1995). In these applications, the states include life history stages such as breeding states. The multi–state models, by permitting estimation of stage–specific survival and transition rates, can help assess trade–offs between life history mechanisms (e.g. Yoccoz et al., 2000). These trade–offs are also important in meta– population analyses where, for example, the pre–and post–breeding rates of transfer among sub– populations can be analysed in terms of target colony distance, density, and other covariates (e.g., Lebreton et al. 2003; Breton et al., in review). Further examples of the use of multi–state models in analysing dispersal and life–history trade–offs can be found in the session on Migration and Dispersal. In this session, we concentrate on applications that did not involve dispersal. These applications fall in two main categories: those that address life history questions using stage categories, and a more technical use of multi–state models to address problems arising from the violation of mark–recapture assumptions leading to the potential for seriously biased predictions or misleading insights from the models. Our plenary paper, by William Kendall (Kendall, 2004), gives an overview of the use of Multi–state Mark– Recapture (MSMR) models to address two such violations. The first is the occurrence of unobservable states that can arise, for example, from temporary emigration or by incomplete sampling coverage of a target population. Such states can also occur for life history reasons, such as dormancy or the inability to capture non–breeders and in these cases, the rates of transition to and from the unobservable state provide life history insights. The second failure Kendall considers is the misclassification of states (for example in models involving states for age, sex, breeding condition, etc. where these cannot be determined without error). He reviews solutions for these that encompass three approaches: constraints on parameters to ensure identifiability (the least desireable solution); incorporating additional information; and the use of sub– sampling that leads to the multi–state application of the Robust design. In passing, Kendall makes reference to what are probably the 3 most significant developments in the area of multi–state models since the last Euring meeting: (1) the incorporation of tag–recovery data in addition to recapture data in MSMR models; (2) a comprehensive methodology for goodness–of–fit testing and assessing parameter identifiability in MSMR models; and (3) the development of new software to make these methods accessible. Much of (2) and (3) is based on the landmark thesis of Olivier Gimenez (Gimenez, 2003). Two further presentations in this session followed up the plenary theme of unobservable and misclassified states. The presentation by Roger Pradel (Pradel, in press), represented in these proceedings as a brief abstract only, dealt with the problem of errors in sexing animals; an example of what Kendall refers to as bidirectional misclassification. The presentation by Marc Kéry (Kéry & Gregg, 2004) is, we think, the first

A. Neil Arnason, Dept. of Computer Science, Univ. of Manitoba, Winnipeg, Manitoba R3T 2N2 Canada. E–mail: arnason@cs.umanitoba.ca Emmanuelle Cam, Lab. Evolution et Diversité Biologique, UMR–CNRS 5174, Bâtiment 4R3, Univ. P. Sabatier–Toulouse III, 118 route de Narbonne, 31062 Toulouse Cedex 04, France. E–mail: emmacam@cict.fr ISSN: 1578–665X

Arnason & Cam

occurrence in the Euring proceedings of an application of mark–recapture to plants. This presentation, represented in these proceedings by an extended abstract with complete references, is an example of dormancy as an unobservable state. Even though the non–dormant states are observed with probability 1, MSMR permits reliable estimation of the proportion of dormant plants in the presence of state ambiguity (dormant or dead?) and can permit assessing the influence of environmental covariates on this proportion. The presentation by Christophe Barbraud (Barbraud & Weimerskirk, 2004), included in these proceedings as an extended abstract, takes the life–history trade–off application of MSMR models a step further by taking into account environmental and individual covariates on survival. The trade–off considered is between survival and transitions among several states describing breeding experience of long–lived petrels and how it is affected by harsh climate conditions. The study is a showcase for the powers of the new software (U–Care and M–Surge). The paper by Senar and Conroy (Senar & Conroy, 2004) is a novel application of MSMR models to animal epidemiology. States included age, sex and infected state and the model permits estimation of survival, infection, and recovery rates for birds during an outbreak of Serin avian pox. The use of a MSMR model permits estimation of the prevalence rate unconfounded by differences in capture rates of infected and non–infected birds. Here too there is a potential for ambiguous states in that the uninfected state might include both immune post–infection animals and susceptible pre–infection animals and these groups would likely have different survival rates. The authors are able to deal with this because of the length of the study and the availability of data outside the main outbreak. Finally, this session includes a paper by Jamieson and Brooks (Jamieson & Brooks, 2004) that appears to lie outside the MSMR theme of this session but which was included because of its relevance to metapopulation analyses. Our call for papers for this session also invited papers illustrating multi– population meta–analysis and use of Bayesian methods. By these criteria, their paper is no outlier. It addresses the longstanding question of density dependence in game bird survival; a question of great interest to theoretical biologists and of vital importance to wildlife managers. The controversy arises because the density dependence revealed in estimates may not reflect the underlying density dependence mechanism in the population parameters. The Bayesian analysis presented here circumvents this problem by fitting a 2 stage model: a time series model for the true population sizes Nt allowing for density dependence and then for the distribution of the estimates given the Nt. The use of data– intensive sampling methods to fit this model neatly sidesteps the insurmountable problem for a purely frequentist (likelihood) approach of having to integrate out the Nt from the likelihood. (As a historical note, this problem confronted George Jolly when he developed the original Jolly Seber model and he handled it by simply fixing the unobservable marked pool sizes Mt a at their maximum likelihood values… a procedure that biases the s.e. of the estimates). This paper is also instructive as the Bayesian methodology provides a straightforward means of accounting for model uncertainty in parameter estimates and model predictions. In summary, the session was a gratifying and useful mix of overview and case studies. The case studies are valuable for their sophisticated use of multi–state and Bayesian models and the subtlety and care with which inferences are drawn from the model fitting results. References Barbraud, C. & Weimerskirch, H., 2004. Modelling the effects of environmental and individual variability when measuring the costs of first reproduction. Animal Biodiversity and Conservation, 27.1: 109–111 (Extended abstract). Blums, P., Nichols, J. D., Lindberg, M. S., Hines, J. E. & Mednis, A., 2003. Estimating natal dispersal movement rates of female European ducks with multistate modeling. Journal of Animal Ecology, 72: 1027–1042. Breton, A. R., Diamond, A. W. & Kress, S. W., 2005. Encounter, survival, and movement probabilities from an Atlantic puffin (Fratercula arctica) metapopulation. Ecological Monographs, in review. Cam, E., Oro, D., Pradel, R. & Jimenez, J., 2004. Assessment of hypotheses about dispersal in a long–lived seabird using multistate capture–recapture models. Journal of Animal Ecology, 73: 723–736 Gimenez, O., 2003. Estimation et tests d’adéquation pour les modèles de capture–recapture multiétats. Thèse présentée pour obtenir le titre de Docteur de l’Université des sciences et techniques de Languedoc, Montpellier, France. Hestbeck, J. B., Nichols, J. D. & Malecki, R. A., 1991. Estimates of movement and site fidelity using mark– resight data of wintering Canada geese. Ecology, 72: 523–533. Jamieson, L. E. & Brooks, S. P., 2004. Density dependence in North American ducks. Animal Biodiversity and Conservation, 27.1: 113–128. Kéry, M. & Gregg, K. B., 2004. Demographic estimation methods for plants with dormancy. Animal Biodiversity and Conservation, 27.1: 129–131 (Extended abstract). Lebreton, J.–D., Hines, J. E., Pradel, R., Nichols, J. D. & Spendelow, J. A., 2003. Estimation by capture–

Animal Biodiversity and Conservation 27.1 (2004)

recapture of recruitment of dispersal over several sites. Oikos, 101: 253–264. Nichols, J. D. & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. Journal of Applied Statistics, 22: 835–846. Pradel, R., Gimenez, O. & Lebreton, J.–D. (in press). Principles and interest of GOF tests for multistate capture–recapture models. Animal Biodiversity and Conservation. Senar, J. C. & Conroy, M. J., 2004. Multi–state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): the importance of estimating recapture rates. Animal Biodiversity and Conservation, 27.1: 133–146. Yoccoz, N. G., Erikstad, K. E., Bustnes, J. O., Hanssen, S. A. & Tveraa, T., 2002. Costs of reproduction in common eiders (Somateria mollissima): an assessment of relationships between reproductive effort and future survival and reproduction based on observational and experimental studies. Journal of Applied Statistics, 29: 57–64.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

Coping with unobservable and mis–classified states in capture–recapture studies W. L. Kendall

Kendall, W. L., 2004. Coping with unobservable and mis–classified states in capture–recapture studies. Animal Biodiversity and Conservation, 27.1: 97–107. Abstract Coping with unobservable and mis–classified states in capture–recapture studies.— Multistate mark– recapture methods provide an excellent conceptual framework for considering estimation in studies of marked animals. Traditional methods include the assumptions that (1) each state an animal occupies is observable, and (2) state is assigned correctly at each point in time. Failure of either of these assumptions can lead to biased estimates of demographic parameters. I review design and analysis options for minimizing or eliminating these biases. Unobservable states can be adjusted for by including them in the state space of the statistical model, with zero capture probability, and incorporating the robust design, or observing animals in the unobservable state through telemetry, tag recoveries, or incidental observations. Mis–classification can be adjusted for by auxiliary data or incorporating the robust design, in order to estimate the probability of detecting the state an animal occupies. For both unobservable and mis–classified states, the key feature of the robust design is the assumption that the state of the animal is static for at least two sampling occasions. Key words: Temporary emigration, Robust design, Auxiliary data, Metapopulation, Breeding probability. Resumen Cómo abordar los estados inobservables y clasificados incorrectamente en los estudios de captura– recaptura.— Los métodos de marcaje–recaptura de estados múltiples brindan un excelente marco conceptual para considerar la estimación en los estudios de animales marcados. Los métodos tradicionales incluyen las dos hipótesis siguientes: (1) cada uno de los estados que ocupa un animal es observable; (2) el estado se asigna correctamente en cada momento. Fallos con cualquiera de estas dos hipótesis pueden traducirse en estimaciones sesgadas de parámetros demográficos. El presente estudio analiza las opciones de diseño y análisis para minimizar o eliminar estos sesgos. Los estados inobservables pueden ajustarse incluyéndolos en el espacio de estados del modelo estadístico, con cero probabilidades de captura, e incorporando el diseño robusto u observando los animales en estado inobservable mediante telemetría, recuperaciones de marcas u observaciones fortuitas. La clasificación errónea puede ajustarse mediante datos auxiliares o incorporando el diseño robusto, con objeto de estimar la probabilidad de detectar el estado que ocupa un animal. Tanto para los estados inobservables como para los clasificados erróneamente, la característica clave del diseño robusto se basa en la hipótesis de que el estado del animal es estático como mínimo en dos muestreos. Palabras clave: Emigración temporal, Diseño robusto, Datos auxiliares, Metapoblación, Probabilidad de reproducción. William L. Kendall, USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, U.S.A.

ISSN: 1578–665X

Kendall

Introduction The development of multi state mark–recapture (MSMR) methods dates back to the late 1950’s (Darroch, 1961) for short–term studies and the early 1970’s (Arnason, 1972, 1973) for longer term studies. MSMR methods saw little use until the late 1980’s (good ideas often take a while to catch on, and this usually requires usable software). At that time Hestbeck et al. (1991) used a maximum likelihood approach and program SURVIV (White, 1983) to estimate annual survival and movement probabilities for wintering Canada Geese (Branta canadensis). Whereas Arnason’s work modeled movement probability as a first–order Markov process, Hestbeck et al. (1991) utilized a memory model, where an animal’s movement depended not only on its current location, but on its location in the previous time period. In Nichols et al. (1992) and Nichols et al. (1994), respectively, state transitions are not geographic movements, but transitions between phenotypic states (weight classes or breeding states, respectively). Schwarz et al. (1993) provided a fuller treatment of the theory for the Arnason model for recaptures and tag recoveries (Schwarz et al., 1988, had first addressed estimating movement from tag recoveries), and Brownie et al. (1993) provided the theory for the memory model, as well as relatively user–friendly software, MSSURVIV (Hines, 1994). The basic MSMR model (without memory), commonly called the Arnason–Schwarz model, can be viewed as a multi–state extension of the Cormack– Jolly–Seber (CJS) model, where the state an animal occupies changes stochastically from time period to time period (Williams et al., 2002, section 17.3). These states can be geographic (e.g., breeding, wintering, or stopover areas) or phenotypic (e.g., size classes, breeding states, disease states), and are discrete (e.g., weight which is continuous is partitioned into classes). Some geographic and phenotypic states either do not change over time (e.g., sex), or change completely deterministically (e.g., age), and therefore MSMR methods have not been necessary for accounting for them (Pollock, 1981; Lebreton et al., 1992). However, to exploit emerging methods for testing goodness of fit (Pradel et al., 2003; Pradel et al., in press) and new software (Choquet et al., 2004), Lebreton et al. (1999) found it convenient to consider all mark–recapture models as special cases of MSMR models. Previous Euring proceedings have included reviews of MSMR modeling. Nichols et al. (1993) and Nichols & Kaiser (1999) reviewed their use in estimating movement. Nichols & Kendall (1995) and Viallefont et al. (1995) demonstrated their usefulness in testing hypotheses in evolutionary ecology. Lebreton & Pradel (2002) and Williams et al. (2002, section 7.3) provided more recent thorough reviews. The assumptions of the Arnason–Schwarz model include the following: (1) each animal in state r at time i has the same probability of surviving to time i + 1 (Sir), of transitioning, given it survives, to any

state s just before time i + 1 ( irs), and of being observed at time i + 1 (psi + 1), given it is present; (2) marks do not affect the survival or behavior of the animal, are not lost, and are recorded correctly; (3) each animal is independent with respect to survival, transitions, and detection probability; (4) each animal is available for detection at every capture occasion, and (5) the state of each animal is assigned without error at each capture occasion. In this paper I will focus on violations of the last two assumptions, reviewing what can be done to adjust for unobservable or mis–classified states. Discussion in subsequent sections will benefit from a quick review of the structure of MSMR models. Survival and transition among two states from time i to i + 1 is characterized in figure 1. Two sample encounter histories for a three–period study of this population are presented below, along with the probability structure for these histories conditional on first release at time 1: AAA A0A For the second history the animal could have been in either state at time 2. Whereas with the CJS model an interior zero in a capture history implies the animal was there but not detected, with MSMR models the probability structure must acknowledge uncertainty about where the animal is at that time. This model becomes more complex as the number of time periods or states grows, and its full expression is more easily presented in terms of transition matrices and capture probability vectors (Schwarz et al., 1993; Brownie et al., 1993). Software developed to implement this model includes MSSURVIV (Hines, 1994; also includes the memory model), MARK (White & Burnham, 1999), and M–SURGE (Choquet et al., 2004). Barker et al. (in review) and Kendall et al. (in review) developed models to combine MSMR data with tag recoveries. The latter paper included estimation of movement from capture to recovery states, and found that all parameters are estimable if the number of recovery areas does not exceed the number of capture states, and if recovery occurs shortly after recapture (as is assumed in North American waterfowl studies). Barker’s model is available in program MARK and the Kendall et al. model is available from J. Hines in program MSSRVrcv (www.mbr-pwrc.usgs.gov/software). Unobservable states In many cases there are members of a population or meta–population that are not available for capture each time sampling occurs in a mark–recapture study. In some cases this occurs simply because there are areas of an animal’s territory or home range that are not covered by sampling effort. In other cases this unobservable state

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S iA

SiU

SiA

UU i

AU i

U SiU

UA i

Fig. 1. Diagram of movements of animals between study areas A and U (e.g., breeding colonies, subpopulations), with associated probabilities of survival (Sri) for area r and movement from area r to area s ( irs) for time period i. Fig. 1. Diagrama de movimientos de animales entre las áreas de estudio A y U (es decir, colonias de reproducción, subpoblaciones), con probabilidades de supervivencia asociadas (Sri) para el área r y de movimiento desde el área r hasta el área s ( irs) para el periodo de tiempo i.

(Lebreton et al., 1999; Pradel & Lebreton, 1999; Kendall & Nichols, 2002) has an ecologically interesting interpretation. For example, in some studies conducted on breeding colonies only breeders are available to capture or sight. By definition an unobservable state implies the detection probability for that state is zero, at least by capture or sighting during the sampling interval. I will briefly discuss the bias induced for demographic parameters associated with the observable state when this unobservable state is ignored. I will then discuss what can be done to minimize this bias, through study design or modeling. Single observable state, single unobservable state Most of the work to date in dealing with unobservable states is for the case where there is one observable and one unobservable state (i.e., where there is a single site, single state study). This scenario is well represented in figure 1 by assuming that state A represents the study area and state U is the unobservable state. Because the unobservable state for marked animals is caused by movement out of the study area, the term "temporary emigration" has often been used to describe this scenario (e.g., Burnham, 1993; Kendall et al., 1997; Schwarz & Stobo, 1997; Barker, 1997; Fujiwara & Caswell, 2002a; Schaub et al., 2004). Alternatively, movement out of the study area after capture has sometimes been characterized as permanent by definition, such as with natal dispersal. Burnham (1993) pointed out that when movement in and out of the observable state is completely random (i.e., each individual in the population is equally likely to be available for detection in a given sampling period), the CJS estimator for capture

probability actually estimates the product of capture probability and the probability the animal is in the study area. In this case estimation of survival probability is unaffected. However, the Jolly–Seber estimator for population size actually estimates the size of the super–population, or population that would potentially use the study area in any given time period (Kendall et al., 1997), if unmarked animals including new recruits have the same probability of being available for capture as marked animals (Barker, 1997). When emigration is by definition permanent, then the CJS estimator for survival probability estimates the product of true survival and fidelity to the study area. Kendall et al. (1997) evaluated the bias in CJS estimators for the case of Markovian movement to and from the unobservable state for two scenarios ( iAU < iUU and iAU > iUU ). CJS estimators were (1) negatively biased in each case for piA, and (2) negatively and positively biased, respectively, for SiA. The first step to properly account for an unobservable state is to include it in the model. The use of MSMR models in this regard, as in figure 1, is a logical approach, both conceptually and in terms of computing tools (Lebreton et al., 1999; Pradel & Lebreton, 1999; Kendall & Nichols, 2002; Schaub et al., 2004; Choquet et al., 2004). As one might expect, however, the unobservable state causes parameter redundancy problems in estimation. With no additional information it is possible to estimate parameters when (1) some are set equal across time and (2) either there is partial determinism in state transitions (e.g., breeders become obligate non–breeders for one or two years, Fujiwara & Caswell, 2002a; Kendall & Nichols, 2002) or parameters are set equal across groups (e.g., sex, Schaub et al., 2004).

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Pradel & Lebreton (1999) also used partial determinism when they put the model of Clobert et al. (1994) into a MSMR context. In this case hatching– year birds are released and become unobservable until they return to breed. Transition from pre– breeder to breeder was modeled as age–dependent, and there was an assumed age at which all birds that had not yet bred would breed with probability 1.0. Once a bird bred, it was assumed to breed each year thereafter. To be forced to assume a priori that parameters are equal over time or group is unsatisfactory. In fact, testing that hypothesis might be of interest. Of course the most direct way to relax this assumption is to expand the mark–recapture study to eliminate the unobservable state altogether. However, this is often not practical. I will discuss two other basic ways to account for the unobservable state with less restrictive assumptions. The first involves some kind of sampling in the unobservable state. The best solution here is to use telemetry on a subset of the animals released, and track the animal where it is observable and where it is unobservable by other means such as capture or direct observation. If detection probability is 1.0 for those animals with a telemetry device, movement probabilities in and out of the study area can be monitored directly. Powell et al. (2000) used this approach in a study of Wood Thrush (Hylocichla mustelina), maintaining a search area for birds with radios that encompassed the capture study area. In addition, if mortalities can be partitioned from censoring (e.g., if a radio stops moving is it because the animal died or because the radio fell off?), information on survival for the unobservable state can be directly obtained. However, even if mortalities cannot be detected or detection probability for birds with radios is < 1.0, movement probabilities can be estimated by conditioning on first and last detection of the bird and modelling its detection history in between. Another potential source of information for birds and fish, especially when movement out of the study area is permanent by definition, is ring recoveries. Burnham (1993) demonstrated that, assuming there is no unobservable state with respect to recoveries (i.e., no matter where the bird dies it can be found and reported), recoveries provide information on survival, whereas recaptures provide information on apparent survival, the product of survival and fidelity probabilities. Therefore, if and , where A is the observable state, and and are estimates for survival probability computed from ring recovery or recapture data, respectively, then a reasonable estimator for fidelity becomes

A key assumption is that survival probability is independent of state (SiU = SiA = Si). This potentially restrictive assumption will come up again. Barker

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(1997) demonstrated that the same approach can be applied when there are incidental observations of marked animals wherever they occur. In this case these observations can be viewed as a recovery where the bird is released again. He also showed that these observations can be used to estimate Markovian movement in and out of the study area, but this requires setting parameters equal across time. Another source of information for estimating parameters in the face of an unobservable state is subsampling. Each period of interest would consist of at least two formal capture occasions (fig. 2), where each animal present in the study area at each occasion is exposed to capture effort. This robust design was first suggested by Pollock (1982). At that time unobservable states had not been considered. Pollock proposed that sub–samples within each primary period i should be sufficiently close in time that population closure could be assumed within primary period. In that way the full array of closed population capture–recapture models (see Otis et al., 1978) could be employed to estimate population size robustly, while the CJS model could be used to estimate apparent survival probability. Kendall et al. (1997) demonstrated that under a model of completely random movement in and out of the observable state A (i.e., UA = iAA = i.A), the CJS estimator for detection i probability pi* (the probability an animal is captured in at least one subsample within primary period i) actually estimates the product i.A pi* (Burnham, 1993). However, closed model analysis yields an unbiased estimate of

where pij = probability of detection in sample j of primary period i, given it is present. From this development algebra yields an ad hoc estimator for transition probability:

This idea is illustrated in figure 2. When transitions are Markovian an ad hoc approach is not possible, but Kendall et al. (1997) developed likelihood approaches to both models and J. Hines programmed them in RDSURVIV (Kendall & Hines, 1999). Program MARK includes a conditional and unconditional (population size is estimated directly) version of this model. Under this model, all parameters are estimable except iAA and iUA for the last time interval of the study. Lindberg et al. (2001) combined Pollock’s robust design with band recoveries to estimate probabilities of temporary and permanent emigration simultaneously. They applied this model to Canvasback ducks (Aythya valisineria), where temporary emigration implied a breeder skipping a breeding season, and permanent emigration implied dispersal to another breeding population. This model can be implemented in program RDSURVIV or MARK.

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·A i

p 2*

SiA Year

Samples

...

p2 * Fig. 2. Diagram of Pollock’s robust design, where each of k primary occasions consists of li closely spaced capture occasions. As indicated, information on detection probability (pi*) is derived from capture occasions within a primary period, and survival (SiA) and the product of capture and transition probability ( i·A) is derived across primary periods. Fig. 2. Diagrama del diseño robusto de Pollock, donde cada una de las principales ocasiones k consiste en ocasiones de captura li próximas. Tal y como se indica, la información relativa a la probabilidad de detección (pi*) se deriva de ocasiones de captura dentro de un periodo principal, mientras que la probabilidad de supervivencia (SiA) y la del producto de la captura y transición ( i·A) se deriva a partir de varios periodos principales.

There are two other variations on the robust design, which can be compared against the closed robust design of Pollock (1982) by referring to figures 3, 4, and 5. For the closed robust design (fig. 3) complete demographic and geographic closure with respect to the study area are assumed across sampling occasions within each primary period (although Kendall, 1999 identified some exceptions). Figure 4 represents the open robust design (Schwarz & Stobo, 1997; Kendall & Bjorkland, 2001). This design was motivated by breeding seals or sea turtles, where breeders arrive at (before sample j + 1 within primary period i with probability ij) and depart from (after sample j with probability ij) breeding beaches in a staggered fashion. An analogous application for birds would be a staging or stopover area. The statistical model used by Schwarz & Stobo (1997) and Kendall & Bjorkland (2001) within each primary period is the Schwarz & Arnason (1996) parameterization of the Jolly–Seber model. The principal modification is in the interpretation of the recruitment and survival parameters. In this case they represent arrival and departure probabilities. Therefore under this model geographic closure within each primary period is partially relaxed (only one entry and exit is permitted), but demographic closure is maintained. Schwarz & Stobo (1997) wrote computer code for their version of the robust design, and Kendall & Bjorkland (2001) modified RDSURVIV to create ORDSURVIV (www.mbr-pwrc.usgs.gov/software). Program Mark can now also run this model. Figure 5 represents a "gateway" robust design, developed by Bailey et al. (2004). Here animals are

captured as they enter a breeding area, indicating they have decided to breed. They are captured again as they leave the study area at the conclusion of their breeding season. The authors applied it to pond–breeding amphibians, where a drift net surrounds the pond and individuals are caught on either side of the fence, and released on the opposite side. One can envision at least one other application: spawning fish who enter a river system and pass through one or more dams. With this design, mortality is permitted to occur between entry and exit, but only one entry and exit is permitted per primary period. As with the other approaches to the robust design, survival probability for those in the unobservable state must be set equal to its counterpart for the observable state. However, the gateway robust design provides some flexibility, because seasonal survival probabilities are estimated (i.e., SAi1 for the time spent in the study area and SAi2 for the rest of the year). For the tiger salamander (Ambystoma tigrinum tigrinum) data set analyzed by Bailey et al. (2004), this was an advantage. When this animal is not breeding it is in terrestrial habitat surrounding the pond. Therefore SAi2, which is estimable, refers to survival of breeders from the time they leave the pond to the time of entry the next year, during which they are in terrestrial habitat. Therefore it makes sense to constrain both SUi1 and SUi2 equal to SAi2, although to do so for the former requires an adjustment for differences in time scale. No fully efficient computer software has yet been written for this model. To be fully efficient, when an

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p21

p22

p23

p24

Fig. 3. Sampling process for the closed robust design. Mortality (1 – Sir) and transitions ( irs) occur between primary periods, whereas geographic and demographic closure are assumed within the set of capture occasions in each primary period. Fig. 3. Proceso de muestreo correspondiente al diseño robusto cerrado. La mortalidad (1 – Sir) y las transiciones ( irs) se dan entre periodos principales, mientras que la acotación geográfica y demográfica se presuponen en el conjunto de ocasiones de captura en cada periodo principal.

animal is first captured as it exits the study area, the fact that it entered the study area but was not captured should be modeled. However, Bailey et al. (2004) conditioned on first capture and used program MSSURVIV. Program MARK or M–SURGE could also have been used. The three versions of the robust design presented in figures 3, 4, and 5 are fairly different in the way they model probability structure within each primary period. They vary from an assumption of complete geographic and demographic closure, to partial relaxation of geographic closure with respect to the study area, to partial relaxation of demographic closure. However, they share the assumption that state remains static within each primary period. As indicated in each figure, the decision about state transition is made between primary periods. Therefore in each of these designs an animal in the observable state in primary period i is exposed to capture effort at least twice within that primary period. It is this feature of the robust design that permits estimation of transition probabilities to and from the unobservable state. This is accomplished via estimation of the effective or pooled capture probability for that primary period (pi*). This value is defined for the closed robust design above. For the open robust design

where i1 = i0, and i,j+1= ij (1 – pij) ij + ij. For the gateway robust design, pi* = pi1 + (1 – pi1) SAi1 pi2. In conclusion, with some assumptions or additional sources of information, such as the robust

design, demographic parameters can be estimated in the face of an unobservable state. The most biologically restrictive assumption required under most of these approaches is that survival probability is equal for the observable and unobservable states. This might be unrealistic, and impedes the testing of some interesting hypotheses. For example, in studies of a single breeding population where non–breeders are unobservable, under the robust design a trade–off between current and future breeding could be evaluated, but a trade–off in terms of future survival probability could not. As noted above, the solution to this problem is simple: sample animals in the unobservable state, either through reliable telemetry or through formal capture or resighting periods. Multiple observable states, one or more unobservable states In some cases, even where capture effort is applied to multiple states, there still could be one or more unobservable states. A question arises about how directly applicable are the single observable state results derived above to the case of multiple observable states. The level of complexity grows with multiple states, making it difficult to predict estimability or to develop a comprehensive guideline such as Kendall & Nichols (2002) and Schaub et al. (2004). As with the case of a single observable state, the solution to parameter redundancy problems will be telemetry, the robust design, universal recoveries/resightings, and partial determinism in transitions.

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p21

p22

p23

p24

Fig. 4. Sampling process for the open robust design. Mortality (1 – Sir) and transitions ( irs) occur between primary periods. Demographic closure is assumed within the set of capture occasions in each primary period, but one entry to ( ij) and one exit from ( ij) the study area is permitted per primary period. Fig. 4. Proceso de muestreo correspondiente al diseño robusto abierto. La mortalidad (1 – Sir) y las transiciones ( irs) se dan entre periodos principales. La acotación demográfica se presupone en el conjunto de ocasiones de captura en cada periodo principal, pero para cada periodo principal se permite una entrada en ( ij) y una salida de ( ij) del área de estudio.

p21

S12

OU 2

p22

S21

S22

Fig. 5. Sampling process for the gateway robust design. Transitions ( irs) occur between primary periods. Geographic closure is assumed within the two capture occasions in each primary period, but mortality is permitted within (1 – Sri1) or between (1 – Sri2) primary periods. Fig. 5. Proceso de muestreo para el diseño robusto gateway. Las transiciones ( irs) se dan entre periodos principales. La acotación geográfica se presupone en las dos ocasiones de captura en cada uno de los periodos principales, pero se permite la mortalidad en (1 – Sri1) o entre (1 – Sri2) periodos principales.

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There are several examples of work underway involving > 2 states. Kery & Gregg (in review) consider the case of an orchid with two above–ground life stages and a below–ground (unobservable) dormant stage, where detection probability is 1.0 for the above–ground states. Forcada et al. (in review) consider a special case of partial determinism for a breeding population of albatross. Breeders are partitioned into successful and unsuccessful states, and successful breeders are assumed to skip at least one year of breeding with certainty. Kendall et al. (in review) and Barker et al. (in review) consider multiple observable capture states combined with ring recoveries, with permanent or temporary movement to an unobservable capture state. Lebreton et al. (2003) presented a special case of partial determinism that extends Clobert et al. (1994) and Pradel & Lebreton (1999) to multiple observable states. Here a hatching year bird is released in one of three colonies, makes a decision about at which of the three it will eventually breed, and remains in the unobservable pre–breeder state for that colony with some probability. There is no robust design, but there is an age at which breeding probability matches that of the observable adults, and accession to breeding is modelled as age–dependent, whereas survival and detection probability are modeled as time–dependent. Mis–classified and unknown states When the state of an animal that is captured is mis–classified or unknown the potential for bias in transition probabilities as well as all other parameters arises. Differences in survival between states could be underestimated. As with other biases, those due to mis–classification could certainly bias projections of population change from matrix population models (Caswell, 2001). Lebreton & Pradel (2002) outlined the problem of mis–classified states, and pointed out that without additional information parameter redundancy problems would arise. Fujiwara & Caswell (2002b) modeled mis–classification and adjusted for it by incorporating fixed mis–classification probabilities derived outside the capture–mark–recapture modeling process. Kendall et al. (2003) and Kendall et al. (2004) considered a two–state case where mis–classification can only occur in one direction. The problem was motivated by a study of adult female Florida manatees (Trichechus manatus latirostris) and their calves. A female is determined to have bred by the presence of an attendant young calf. In some cases a calf that is present is not seen by the photographer that documents the cow by scar pattern. By being conservative about assigning a calf to a female, breeders can be mis–classified as non–breeders but not vice versa. The probability structure for this model can be illustrated with the example capture histories below, where C indicates with calf (breeder) and N indicates without calf (non–breeder):

CC CN NC where piC , piC(1 – ) = probability a breeder is detected in sampling period i and her calf is or is not detected, respectively, piN = probability a non– breeder is detected in period i, and i = probability a cow seen without a calf in period i is indeed a non–breeder. To adjust for mis–classification, Kendall et al. (2003) partitioned the season into two sampling occasions, producing a robust design. Because there were two opportunities to sight each female and determine if she had a calf, the detection probability for a cow (pijC or pijN) and any calf conditional on detecting its mother ( ijC) could be estimated. The detection parameters for each primary period (piC , piC(1 – ), and piN) are then functions of these parameters at the sub sample scale. Whereas Kendall et al. (2003) relied on a subset of known non–breeders to estimate parameters, Kendall et al. (2004) did not require this. I present example sighting histories and their probabilities for primary period 2 below: NN

where and In this case i = probability that an adult female in the population and available for detection in year i is a breeder. There is an interesting additional benefit of the structure described above, unrelated to mis–classification. Assuming breeders and non–breeders to be equally likely to be in the study area, i is interpreted as the proportion of females that breed in a given year. This implies that the conditional and unconditional probabilities of breeding are in the same model. More generally, survival or transition probabilities for a given state could be modelled as a function of the proportion of the population currently in that state. This would permit investigators to consider hypotheses about frequency– dependent dynamics while properly accounting for sampling variance and covariance. The approach to mis–classification listed above could certainly be useful in other applications. Cam et al. (2002) identified such a mis–classification problem with pre–breeding Kittiwakes. Some pre– breeders can be classified as squatters, where they practice nesting behavior at unattended nests. However, for a given squatter the actual behavior is not observed each time it is sighted within a season.

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Mis–classification certainly arises in disease studies, and in this case misclassification can occur in either direction. Some sick animals do not show clinical signs, and some animals that have recovered have residual clinical symptoms. The MSMR probability structure that accounts for mis–classification is straightforward to write, but the misclassification probabilities must be derived from another source of data. Nichols et al. (2004) considered the case where a static state is not mis–classified, but is unknown for a given period of time. The motivating problem was that of sexually monomorphic birds, where sex is indeterminate until they display sex–specific behavior. They showed that naively back–dating sex assignment to the original capture occasion produces positively biased estimates of survival probability for both sexes (because the longer they live the greater the chance their sex will be assigned). Instead they modeled the animals that were never assigned to sex as a mixture of males and females, and also modelled the probability of being assigned to sex at each detection occasion. It is a similar approach to the manatee case of a dynamic state above, with an important distinction. Whereas the robust design was necessary above in order to estimate the probability of assigning state correctly, Nichols et al. (2004) used cross–period information for the same purpose because sex is a static state. Pradel et al. (in press) explored estimability issues where sex is assigned each time but never known with certainty. Discussion I have reviewed various examples of unobservable and mis–classified states, and shown how the MSMR modelling framework, combined with partial determinism or additional sources of data can be used to estimate demographic parameters in the face of these phenomena. These models have performed well in many examples. Although MSMR models require more parameters, the need for this additional structure could outweigh that disadvantage. For a data set of limited size, putting the problem in a MSMR context permits one to use model selection to see where constraints should be imposed. For example, parsimony might dictate that it is better to give up time dependency in survival probability than to ignore the transitions to the unobservable state. Nevertheless there is a cost to having unobservable or mis–classified states. For the former, one is required to assume a priori that the survival probability for the unobservable state is equal to that for an observable state. The validity of this assumption, and therefore the ability to test it, is of great interest biologically. Mis–classification, even if adjusted for properly, causes reduction in precision. Therefore, as much as possible, design should be used to reduce problems. Unobservable states should be made observable as much as possible. For example, the use of telemetry to supplement capture or resighting stud-

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ies where possible. Researchers should look for opportunities to include multiple sampling occasions per period of interest, so that robust design methods can be utilized. Behavioral cues should be collected each time an animal is detected, to be able to estimate the probability of mis–classification. A buffer zone around a study area can be useful in minimizing temporary emigration, or to avoid confusing nuisance movements with meaningful ones. For example, in a study of breeding colonies where study plots are inserted in larger groups (e.g., albatross), breeders on the edge of the plot might breed the next year outside the plot. By conventional design these birds become unobservable, as are those that do not breed and are therefore not on any nest. Estimates of transitions to the unobservable state would include both types of movement and therefore bias tests of hypotheses about breeding probability. By creating a buffer zone around the plot where marked birds are searched for, this nuisance movement can be reduced or eliminated. Much work has been done on unobservable and mis–classified states. Nevertheless there is plenty of opportunity for future work. The issue of multiple observable and unobservable states has barely been considered. Given the potential complexity of multi state models, the computer algebra methods of Catchpole et al. (2002), Gimenez et al. (2003), and Gimenez et al. (2004) will be especially useful in evaluating parameter redundancy. I suggest that these methods be used to determine which parameters are estimable in theory, then simulate data or use expected value methods for reasonable parameter values and sample sizes to determine if these parameters are estimable in practicality. Goodness of fit issues also deserve attention in the case of unobservable states. A generic Pearson test is conducted by programs MSSURVIV, RDSURVIV, and ORDSURVIV, but no test is available in MARK. Pradel et al. (2003) has provided a more detailed test for fit of multi state models, which has been implemented in program U–CARE. However, this test does not permit the empty cells inherent with an unobservable state, although this could be partially remedied (R. Pradel, pers. comm.). Other future work that is needed is the capacity to combine an arbitrary number of sources of information (Barker & White, 2004). Mis–classification correction could be incorporated into models that involve more than two states, whether or not those additional states involve mis–classification directly. In addition, more work needs to be done on bi– directional mis–classification. Finally, none of what I have reviewed here has involved hierarchical models. This option should be explored. Acknowledgments I thank Emmanuelle Cam and Neil Arnason for the invitation to present this paper, and Richard Barker and Olivier Gimenez for their thoughtful reviews of the

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manuscript. The juxtaposition of topics presented here was partly inspired by stimulating discussions with Jim Nichols, Larissa Bailey, and Evan Cooch. References Arnason, A. N., 1972. Parameter estimates from mark–recapture experiments on two populations subject to migration and death. Researches on Population Ecology, 13: 97–113. – 1973. The estimation of population size, migration rates, and survival in a stratified population. Researches in Population Ecology, 15: 1–8. Bailey, L. L., Kendall, W. L., Church, D. R. & Wilbur, H. M., 2004. Estimating survival and breeding probability for pond–breeding amphibians: a modified robust design. Ecology, 85: 2456–2466. Barker, R. J., 1997. Joint modelling of live–recapture, tag–resight, and tag–recovery data. Biometrics, 53: 666–677. Barker, R. J. & White, G. C., 2004. Towards the mother of all models: customized construction of the mark–recapture likelihood function. Animal Biodiversity and Conservation, 27.1: 177–185. Barker, R. J., White, G. C. & McDougal, M., in review. Movement of paradise shelducks between molt sites. Journal of Wildlife Management. Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H. & Hestbeck, J. B., 1993. Capture–recapture studies for multiple strata including non–Markovian transitions. Biometrics, 49: 1173–1187. Burnham, K. P., 1993. A theory for combined analysis of ring recovery and recapture data. In: Marked individuals in the study of bird population: 199– 214 (J.–D. Lebreton & P. M. North, Eds.) Birkhäuser–Verlag, Basel, Switzerland. Cam, E., Cadiou, B., Hines, J. E. & Monnat, J. Y., 2002. Influence of behavioural tactics on recruitment and reproductive trajectory in the kittiwake. In: Statistical Analysis of Data from Marked Bird Populations: 163–186 (B. J. T. Morgan & D. L. Thomson, Eds.). Journal of Applied Statistics 29, nos 1–4. Caswell, H., 2001. Matrix population models, 2 nd edition. Sinauer, Sunderland, MA, U.S.A. Catchpole, E. A., Morgan, B. J. T. & Viallefont, A., 2002. Solving problems in parameter redundancy using computer algebra. In: Statistical Analysis of Data from Marked Bird Populations: 625–636 (B. J. T. Morgan & D. L. Thomson, Eds.). Journal of Applied Statistics, 29, nos 1–4. Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O. & Lebreton, J.–D., 2004. M–SURGE: an integrated software for multi state recapture models. Animal Biodiversity and Conservation, 27.1: 207–215. Clobert, J., Lebreton, J.–D., Allaine, D. & Gaillard, J. M., 1994. The estimation of age–specific breeding probabilities from recaptures or resightings in vertebrate populations: II. Longitudinal models. Biometrics, 50: 375–387. Darroch, J. N., 1961. The two–sample capture–

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recapture census when tagging and sampling are stratified. Biometrika, 48: 241–260. Fujiwara, M., & Caswell, H., 2002a. Temporary emigration in mark–recapture analysis. Ecology, 83: 3266–3275. – 2002b. Estimating population projection matrices from multi–stage mark–recapture data. Ecology, 83: 3257–3265. Gimenez, O., Choquet, R. & Lebreton, J.–D., 2003. Parameter redundancy in multistate capture–recapture models. Biometrical Journal, 45: 704–722. Gimenez, O., Viallefont, A., Catchpole, E. A., Choquet, R. & Morgan, B. J. T., 2004. Methods for investigating parameter redundancy. Animal Biodiversity and Conservation, 27.1: 561–572. Hestbeck, J. B., Nichols, J. D. & Malecki, R. A., 1991. Estimates of movement and site fidelity using mark–resight data of wintering Canada Geese. Ecology, 72: 523–533. Hines, J. E., 1994. MSSURVIV User’s Manual. USGS Patuxent Wildlife Research Center, Laurel, MD 20708–4017. Kendall, W. L., 1999. Robustness of closed capture–recapture methods to violations of the closure assumption. Ecology, 80: 2517–2525. Kendall, W. L. & Bjorkland, R., 2001. Using open robust design models to estimate temporary emigration from capture–recapture data. Biometrics, 57: 1113–1122. Kendall, W. L. & Hines, J. E., 1999. Program RDSURVIV: an estimation tool for capture–recapture data collected under Pollock’s robust design. Bird Study, 46 (supplement): S32–S38. Kendall, W. L., Hines, J. E. & Nichols, J. D., 2003. Adjusting multi–state capture–recapture models for misclassification bias: manatee breeding proportions. Ecology, 84:1058–1066. Kendall, W. L., Langtimm, C. A., Beck, C. A. & Runge, M. C., 2004. Capture–recapture analysis for estimating manatee reproductive rates. Marine Mammal Science, 20: 424–437. Kendall, W. L. & Nichols, J. D., 2002. Estimating state–transition probabilities for unobservable states using capture–recapture/resighting data. Ecology, 83: 3276–3284. Kendall, W. L., Nichols, J. D. & Hines, J. E., 1997. Estimating temporary emigration using capture– recapture data with Pollock’s robust design. Ecology, 78: 563–578. Kery, M. & Gregg, K. B., 2004. Demographic estimation methods for plants in the presence of dormancy. Oikos, in review. Lebreton, J.–D., Almeras, T. & Pradel, R., 1999. Competing events, mixtures of information and multi stratum recapture models. Bird Study, 46 (supplement): S39–S46. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118. Lebreton, J.–D., Hines, J. E., Pradel, R., Nicchols, J. D. & Spendelow, J. A., 2003. The simultane-

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ous estimation by capture–recapture of accession to reproduction and dispersal–fidelity in a multi–site system. Oikos, 101: 253–264. Lebreton, J–D. & Pradel, R., 2002. Multi state recapture models: modelling incomplete individual histories. In: Statistical Analysis of Data from Marked Bird Populations: 353–370 (B. J. T. Morgan & D. L. Thomson, Eds.). Journal of Applied Statistics 29, nos 1–4. Lindberg, M. S., Kendall, W. L., Hines, J. E. & Anderson, M. G., 2001. Combining band recovery data and Pollock’s robust design to model temporary and permanent emigration. Biometrics, 57: 273–282. Nichols, J. D., Brownie, C., Hines, J. E., Pollock, K. H. & Hestbeck, J. B., 1993. The estimation of exchanges among populations or subpopulations. In: Marked individuals in the study of bird population: 265–280 (J.–D. Lebreton & P. M. North, Eds.). Birkhäuser–Verlag, Basel, Switzerland. Nichols, J. D., Hines, J. E., Pollock, K. H., Hinz, R. L. & Link, W. A., 1994. Estimating breeding proportions and testing hypotheses about costs of reproduction with capture–recapture data. Ecology, 75: 2052–2065. Nichols, J. D. & Kaiser, A., 1999. Quantitative studies of bird movement: a methodological review. Bird Study, 46 (supplement): S289–S298. Nichols, J. D., & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. Journal of Applied Statistics, 22: 835–846. Nichols, J. D., Kendall, W. L., Hines, J. E. & Spendelow, J. A., 2004. Estimation of sex–specific survival from capture–recapture data when sex is not always known. Ecology, 85. Nichols, J. D., Sauer, J. R., Pollock, K. H. & Hestbeck, J. B., 1992. Estimating transition probabilities for stage–based population projection matrices using capture–recapture data. Ecology, 73: 306–312. Otis, D. L., Burnham, K. P., White, G. C. & Anderson, D. R., 1978. Statistical inference for capture data on closed animal populations. Wildlife Monographs, No. 62. Pollock, K. H., 1981. Capture–recapture models for age–dependent survival and capture rates. Biometrics, 37: 521–529. – 1982. A capture–recapture design robust to unequal probability of capture. Journal of Wildlife Management, 46: 757–760.

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Powell, L. A., Conroy, M. J., Hines, J. E., Nichols, J. D. & Krementz, D. G., 2000. Simultaneous use of mark–recapture and radio telemetry to estimate survival, movement and capture rates. Journal of Wildlife Management, 64: 302–313. Pradel, R., Gimenez, O. & Lebreton, J.–D. (in press). Principles and interest of GOF tests for multi–state models. Animal Biodiversity and Conservation. Pradel, R. & Lebreton, J.–D., 1999. Comparison of different approaches to the study of local recruitment of breeders. Bird Study, 46 (supplement): S74–S81. Pradel, R., Maurin–Bernier, O. & Gimenez, O. (in press). Determination of sex in Larus audouinii. A model incorporating a possibility of error. Pradel, R., Wintrebert, C. M. A. & Gimenez, O., 2003. A proposal for a goodness–of–fit test to the Arnason–Schwarz multi–site capture–recapture model. Biometrics, 59: 36–42. Schaub, M., Gimenez, O., Schmidt, B. R. & Pradel, R., 2004. Estimating survival and temporary emigration in the multistate capture–recapture framework. Ecology, 85. Schwarz, C. J. & Arnason, A. N., 1996. A general methodology for the analysis of capture–recapture experiments in open populations. Biometrics, 52: 860–873. Schwarz, C. J., Burnham, K. P. & Arnason, A. N., 1988. Post–release stratification in band–recovery models. Biometrics, 44: 765–785. Schwarz, C. J., Schweigert, J. F. & Arnason, A. N., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Schwarz, C. J. & Stobo, W. T., 1997. Estimating temporary migration using the robust design. Biometrics, 53: 178–194. Viallefont, A., Cooch, E. G. & Cooke, F., 1995. Estimation of trade–offs with capture– recapture models: a case study on the lesser snow goose. Journal of Applied Statistics, 22: 847–862. White, G. C., 1983. Numerical estimation of survival rates from band–recovery and biotelemetry data. Journal of Wildlife Management, 47: 716–728. White, G. C. & Burnham, K. P., 1999. Program MARK for survival estimation. Bird Study, 46 (supplement): 120–139. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, San Diego, CA, U.S.A.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Modelling the effects of environmental and individual variability when measuring the costs of first reproduction C. Barbraud & H. Weimerskirch

Barbraud, C. & Weimerskirch, H., 2004. Modelling the effects of environmental and individual variability when measuring the costs of first reproduction. Animal Biodiversity and Conservation, 27.1: 109–111. Extended abstract Modelling the effects of environmental and individual variability when measuring the costs of first reproduction.— How do animals balance their investment in young against their own chances to survive and reproduce in the future? This life–history trade–off, referred to as the cost of reproduction (Williams, 1966), holds a central place in life–history theory (Roff, 1992; Stearns, 1992; McNamara & Houston, 1996). Because individuals can only acquire a limited amount of energy, reproduction and survival as well as current and future reproduction are considered as functions competing for the same resources. In this framework, individuals may optimise life–history decisions. If the reproductive effort in one year leads to a loss in future reproductive output through decreased adult survival or reduced fecundity, then the optimal effort in the current season is less than the effort that would maximize the number of offspring produced in that season (Charnov & Krebs, 1974). There are at least two kinds of factors likely to confound the measurement of the costs of reproduction in the wild. First, there could be differences in the amount of energy individuals acquire and allocate to various functions. This phenotypic heterogeneity can mask or exacerbate individual allocation patterns when trends are averaged across a population (Vaupel & Yashin, 1985; McDonald et al., 1996; Cam & Monnat, 2000). Second, there could be variations in resource availability affecting energy acquisition and allocation. Theoretical models examining the optimal phenotypic balance between reproduction and survival under variable breeding conditions have investigated the influence of environmental stochasticity on the cost of reproduction in birds (Erikstad et al., 1998; Orzack & Tuljapurkar, 2001). However, there is little empirical evidence supporting these theoretical models. Here, we present analysis of the influence of experience, but also of the differential effects of environmental and individual variation on survival and future breeding probability. We address the question of the costs of reproduction using data from a 17–year study of individually marked blue petrels (Halobaena caerulea), a small (190 g) long–lived seabird breeding on sub–Antarctic islands. Data were analysed using multistate capture–recapture models (Brownie et al., 1993; Schwarz et al., 1993; Nichols et al., 1994). The most general model we started with was the conditional Arnason– Schwarz model (Schwarz et al., 1993). We used the following notation for states: 1. Nonbreeder that never previously bred; 2. First–time breeder; 3. Experienced breeder; and 4. Nonbreeder that previously bred. This general model was constrained since some parameters were not defined, given our definition of individual states. Using matrix notation, the parameters defined above can be summarized in matrices of survival, transition and capture probabilities:

Christophe Barbraud & H. Weimerskirch, Centre d’Etudes Biologiques de Chizé, CNRS UPR 1934, 79360 Villiers en Bois, France. Corresponding author: Christophe Barbraud, CEBC–CNRS, 79360 Villiers en Bois, France. E–mail: barbraud@cebc.cnrs.fr ISSN: 1578–665X

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, and

We examined the effect of two covariates that were suspected to affect survival and breeding probabilities: sea surface height representing oceanographic conditions at a regional scale, and body mass of birds during breeding. Covariates were tested through ultrastructural models in which survival probability is a function of sea surface height and/or body mass, following a linear–logistic function: , where is the intercept parameter, and is a slope parameters. Our selection of models for estimation was based on model goodness–of–fit (GOF) tests and a modified Akaike’s Information Criterion that takes into account sample sizes (AICc; see Akaike, 1973; Lebreton et al., 1992; Burnham & Anderson, 2002). We used program U–CARE (Choquet et al., 2003a) for GOF testing, and M–SURGE (Choquet et al., 2003b) for model selection and parameter estimation. The GOF test of our general model indicated a lack of fit and we used a variance inflation factor ( = 1.336) in the remaining analysis. Recapture probabilities varied with state. Recapture probability for breeders was extremely close to one. Experienced nonbreeders had higher recapture probabilities (0.528 ± 0.033) than inexperienced breeders (0.364 ± 0.019). First–time breeders had the lowest mean survival probabilities (0.775 ± 0.035), and experienced breeders had the highest mean survival probabilities (0.882 ± 0.035). Inexperienced and experienced nonbreeders had intermediate mean survival probabilities, indicating a cost of first reproduction for first time breeders. First–time breeders had a lower probability of breeding in the following year than experienced breeders, and nonbreeders had a lower probability of breeding in the following year than breeders. Among nonbreeders, inexperienced nonbreeders had a lower probability of breeding in the following year than experienced nonbreeders. A model where state survival probabilities were a function of sea surface height had the lowest QAICc. Survival of inexperienced individuals (both breeders and nonbreeders) was negatively affected by poor oceanographic conditions, whereas experienced birds seem to be only weakly affected by similar conditions. The costs of reproduction for first–time breeders were particularly marked during harsh climatic conditions. Body condition of experienced breeders was higher than the body condition of first–time and nonbreeders. Body condition of individuals seen only once was lower than body condition of those seen at least twice. At the individual level, there was no clear evidence for an increase in body condition across years. These results can be interpreted in the light of the selection hypothesis (Curio, 1983; Forslund & Pärt, 1995). The inferiority of inexperienced breeders may be linked to a higher proportion of lower–quality individuals in younger age classes. First reproduction may act as a filter selecting individuals of higher quality/body mass. The improvement of performance within individuals may contribute marginally to the observed patterns at the population level. Environmental stochasticity, and more particularly the variation in sea surface height reflecting resource availability is probably a major factor of selection. References Akaïke, H., 1973. Information theory and an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory: 267–281 (B. N. Petran & F. Csáki, Eds.). Akadémiai Kiadi, Budapest, Hungary. Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H. & Hestbeck, J. B., 1993. Capture–recapture studies for multiple strata including non–Markovian transition probabilities. Biometrics, 49: 1173–1187. Burnham, K. P. & Anderson, D. R., 2002. Model selection and multimodel inference. Springer–Verlag, New York. Cam, E. & Monnat, J.–Y., 2000. Apparent inferiority of first–time breeders in the kittiwake: the role of heterogeneity among age classes. Journal of Animal Ecology, 69: 380–394. Charnov, E. L. & Krebs, J. R., 1974. On clutch–size and fitness. Ibis, 116: 217–219. Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O. & Lebreton, J.–D., 2003a. U–Care user’s guide, Version 2.0. Mimeographed document, CEFE/CNRS, Montpellier. – 2003b. User’s manual for M–SURGE 1.0. Mimeographed document, CEFE/CNRS, Montpellier. Curio, E., 1983. Why do young birds reproduce less well? Ibis, 125: 400–404. Erikstad, K. E., Fauchald, P., Tveraa, T. & Steen, H., 1998. On the cost of reproduction in long–lived birds:

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the influence of environmental variability. Ecology, 79: 1781–1788. Forslund, P. & Pärt, T., 1995. Age and reproduction in birds – hypotheses and tests. Trends in Ecology and Evolution, 10: 374–378. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modelling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118. McDonald, D. B., Fitzpatrick, J. W. & Woolfenden, G. E., 1996. Actuarial senescence and demographic heterogeneity in the Florida scrub jay. Ecology, 77: 2373–2381. McNamara, J. M. & Houston, A. I., 1996. State–dependent life histories. Nature, 380: 215–221. Nichols, J. D., Hines, J. E., Pollock, K. H., Hinz, R. L. & Link, W. A., 1994. Estimating breeding proportions and testing hypotheses about costs of reproduction with capture–recapture data. Ecology, 75: 2052– 2065. Orzack, S. H. & Tuljapurkar, S., 2001. Reproductive effort in variable environments, or environmental variation is for the birds. Ecology, 82: 2659–2665. Roff, D. A., 1992. The evolution of life histories: data and analysis. Chapman and Hall, New York. Schwarz, C. J., Schweigert, J. F. & Arnason, A. N., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Stearns, S. C., 1992. The evolution of life histories. Oxford Univ. Press, Oxford. Vaupel, J. W. & Yashin, A. I., 1985. Heterogeneity’s ruses: some surprising effects of selection on population dynamics. American Statistician, 39: 176–185. Williams, G. C., 1966. Natural selection, the costs of reproduction, and a refinement of Lack’s principle. American Naturalist, 100: 687–690.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Density dependence in North American ducks L. E. Jamieson & S. P. Brooks

Jamieson, L. E. & Brooks, S. P., 2004. Density dependence in North American ducks. Animal Biodiversity and Conservation, 27.1: 113–128. Abstract Density dependence in North American ducks.— The existence or otherwise of density dependence within a population can have important implications for the management of that population. Here, we use estimates of abundance obtained from annual aerial counts on the major breeding grounds of a variety of North American duck species and use a state space model to separate the observation and ecological system processes. This state space approach allows us to impose a density dependence structure upon the true underlying population rather than on the estimates and we demonstrate the improved robustness of this procedure for detecting density dependence in the population. We adopt a Bayesian approach to model fitting, using Markov chain Monte Carlo (MCMC) methods and use a reversible jump MCMC scheme to calculate posterior model probabilities which assign probabilities to the presence of density dependence within the population, for example. We show how these probabilities can be used either to discriminate between models or to provide model–averaged predictions which fully account for both parameter and model uncertainty. Key words: Bayesian approach, Markov chain Monte Carlo, Model choice, Autoregressive, Logistic, State space modelling. Resumen Dependencia de la densidad en los ánades norteamericanos.— La existencia o ausencia de efectos dependientes de la densidad en una población puede acarrear importantes repercusiones para la gestión de la misma. En este artículo empleamos estimaciones de abundancia obtenidas a partir de recuentos aéreos anuales de las principales áreas de reproducción de diversas especies de ánades norteamericanos, utilizando un modelo de estados espaciales para separar los procesos de observación y los procesos del sistema ecológico. Este enfoque basado en estados espaciales nos permite imponer una estructura que depende de la densidad de la población subyacente real, más que de las estimaciones, además de demostrar la robustez mejorada de este procedimiento para detectar la dependencia de la densidad en la población. Para el ajuste de modelos adoptamos un planteamiento bayesiano, utilizando los métodos de Monte Carlo basados en cadenas de Markov (MCMC), así como un programa MCMC de salto reversible para calcular, por ejemplo, las probabilidades posteriores de los modelos que asignan probabilidades a la presencia de una dependencia de la densidad en la población. También demostramos cómo pueden emplearse estas probabilidades para discriminar entre modelos o para proporcionar predicciones promediadas entre modelos que tengan totalmente en cuenta tanto la incertidumbre referente a parámetros como a modelos. Palabras clave: Enfoque bayesiano, Métodos de Monte Carlo basados en cadenas de Markov, Autorregresivo, Logístico, Modelación de espacio de estados. Lara E. Jamieson & S. P. Brooks, The Statistical Laboratory – CMS, Univ. of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K.

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Introduction and model National and international legislation are increasingly putting pressure on local authorities to identify and protect key wildlife species and their habitats. This brings to the forefront the design and implementation of effective management strategies and makes them of paramount importance. In order to design an appropriate management strategy, key factors affecting the population must be understood. In particular, the identification of factors affecting survival and/or population size becomes an integral part of the management design process. The question as to whether or not population density affects population size is one of the first that must be addressed. See Nichols et al. (1995), Bulmer (1975) and Vickery & Nudds (1984), for example. Essentially density dependence within a population acts as a stabilising mechanism which tends to move the population size towards its mean level. When the population size is small, there is a natural pressure on the population, generally in the form of an abundance of resources, to increase its numbers and vice versa. Thus, density dependence increases the population’s ability to cope with "shocks" to the system (i.e., rapid increases or, more often, decreases in population size). In terms of management policy, the presence of density dependence within the population indicates an ability to resist destabilising perturbations to the system often induced by human activity (Nichols et al., 1984; Massot et al., 1992). Such activities might vary from direct effects of hunting to less obvious effects from changes in land use within the population’s natural habitat, for example. Most of the work in this area has focused upon simple hypothesis tests for the presence of density dependence. Though the data available are often rich and varied, unfortunately much of it is ignored in order to facilitate model fitting and the selection process due to analytic tractability. Data in the form of population estimates and standard errors are common in the population ecology literature and data collection authorities are often very protective of the raw data i.e., the original counts upon which the estimates are based. Though Kalman filter–based methods are available (Sullivan, 1992; Newman, 1998), most analyses simply ignore the standard errors and treat the population estimates as if they were the true population sizes. Such simplifying assumptions often

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compound the problem suggesting density dependence (Vickery & Nudds, 1984; Shenk et al., 1998). The false detection of density dependence can have a substantial effect upon the corresponding management strategy with possibly dire consequences for the species concerned. Management of population sizes feed into key policy–making initiatives for biodiversity and environmental preservation policies. The issuance of planning permits for work in key habitat areas and recreational hunting licences must consider the impact on population status. The "ideal" population level is fixed by the "North American Waterfowl Management Plan" which is, in itself, a valuable source of information. The data and population goals are summarised in fig. 1. The datasets under consideration comprise yearly population indices for ten duck breeding populations in North America from 1955 to 2002 (Williams, 1999). The ten time series are for Mallard (Anas platyrhynchos), Gadwall (Anas strepera), American Wigeon (Anas americana), Green– winged Teal (Anas crecca), Blue–winged Teal (Anas discors), Northern Shoveler (Anas clypeata), Northern Pintail (Anas acuta), Redhead (Aythya americana), Canvasback (Aythya valisineria) and Scaup (Greater–Aythya marila and Lesser–Aythya affinis). Of these ten species the latter three are diving ducks, the remainder are dabblers. Data comes in the form of annual population size estimates (based upon the raw counts and then "adjusted for biases" by comparing the aerial counts with samples on the ground), together with associated standard errors. The raw count data themselves are, unfortunately, not available. In order to make full use of the data available and to avoid this confounding effect, we seek to separate the observation error (associated with the data collection and population size estimation processes) from the system error (associated with the variability within the population itself) so that the density dependence model is fitted to the true underlying (but hidden) population size rather than the estimates provided. This can be most easily achieved through the development of a state space model separating the system and observation processes as follows. Let us now introduce the population model we consider. Denote the population abundance at time t as Nt = 1,2,...,T. We use the density dependence model of Dennis & Taper (1994) in which (1)

(in millions) together with approximate 95% confidence Fig. 1. The population index values intervals derived from the standard errors and the population goal (horizontal line). (en millones), junto con los intervalos de confianza Fig. 1. Valores de los índices poblacionales aproximados al 95%, derivados de los errores estándar y del objetivo poblacional (línea horizontal).

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A. American Wigeon

B. Blue Winged Teal

8 7 6 5 4 3 2 1 0

4 3 2 1 0 C. Canvasback

D. Gadwall 5

0.8 0.6

0.4

0.2

0 E. Green Winged Teal

F. Mallard 13 12 11 10 9 8 7 6 5 4 3 2 1 0

4 3 2 1 0 G. Northern Pintail

H. Northern Shoveler

12 11 10 9 8 7 6 5 4 3 2 1 0

5 4 3 2 1 0 I. Redhead

J. Scaup 10 9 8 7 6 5 4 3 2 1 0

1.4 1.2 1 0.8 0.6 0.4 0.2 0 1960

1970

1980

1990

2000

1960

1970

1980

1990

2000

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where Zt i N(0,1).There is considerable debate over the choice of a model for density dependence; Dennis & Taper (1994) discuss a number of models and motivate their model by considering the per–unit– abundance growth rate log Nt+1 – log Nt as a linear function with a Normal error term. For computational ease we use a log transformation, pt = logeNt, to give the additive model

The likelihood for our problem therefore comes in two parts: one corresponding to the system and the other to the observation equations. The model in (3) can be re–written in matrix form as = d – Yk bk where the elements t of have independent normal distributions with zero mean and variance 2,

(2) We shall refer to this model as the logistic model Dennis & Taper (1994). Clearly, setting b1 = 0 in (2) reduces the model to a simple linear trend with normal errors. The model may also be extended to include longer–range dependence by taking (3) for some suitable value k. Turchin (1990) and Turchin et al. (1991) argue for the prevalence of second– order density dependence in ecological populations, for example. Thus, the problem of determining the nature and extent of density dependence within the population therefore reduces to a model selection problem in which the model (indexed by k) is unknown. Berryman & Turchin (1991) discuss the inherent difficulty of determining the order of any observed density dependence and suggests the use of autocorrelation–based tests to determine the true order. Amongst other things, we will, in this paper, show how the Bayesian approach provides a very natural framework both for selecting the model order and averaging predictive inference across a range of models when the true model order is unclear. The Nt above denote the true population sizes, which are unknown. However, the data provide us with information as to what these values might be. In particular, the data provide us with estimates Nt of the true underlying population size together with associated standard errors st. The observation process, which relates the observed data to the true underlying population values, is then described by a simple Gaussian process whereby

or equivalently (4) for t = 1,...,T. For notational ease we refer to the full set of abundance values {pt, t = 1,2,...,T} as P and similarly the observed series and associated standard errors S. Let P(t) denote the series P without the point pt. Our time series model for P requires knowledge of previous values; consider for example p1 on the left–hand side of Equation (2), then the right–hand side references p0. Here, we use data from 1955–1960 to place priors on these values which are then imputed as part of the modelling process.

and

(5)

Hence, the so–called system likelihood, relating the latent process P = {p1,...,pT} to the parameters under model k, is given by

(6)

where Pk– = {p1–k,...,p0} denote additional parameters to be estimated and

Similarly, from (4), the so–called observation likelihood, which relates the sequence P to the observed data, is given by

(7)

Of course, the interpretation of Lsys as a likelihood is slightly misleading in that it contains no terms corresponding to the observed data. However, it is intended to represent the likelihood of the P vector had it been observed. An alternative interpretation is as a prior for the unobserved P, conditioning on the remaining parameters which is consistent with a Bayesian approach to the analysis of the data. In fact, a Bayesian approach is the only viable option here since, in order to undertake the analysis, the Nt values must be removed from the joint likelihood by integration so that the model is expressed entirely without reference to the Nt values. This is impossible analytically and even classical EM–based or Kalman Filtering techniques are prohibitively complex to apply in this non–linear setting. However, the Bayesian approach integrates out the Nt numerically as part of the MCMC simulation process that we shall describe in the next section.

Animal Biodiversity and Conservation 27.1 (2004)

We discuss the Bayesian approach to statistical inference in the next section before undertaking an analysis of our data. We then extend our range of models to include an alternative density–dependent dynamic and end with some discussion of our results and of their implication for the management of these important wildlife species. The Bayesian approach Bayesian analyses involve the combination of the likelihood with the priors to obtain the posterior distribution as the basis for inference. If we assume that the observed data are described by some model m with associated parameter vector c , then the Bayesian analysis is based upon the posterior distribution ( x ) where

117

Though posterior model probabilities provide a useful means of model comparison, formal model– checking methods are also required to ensure that the models fitted provide an adequate description of the data. There are two formal model checking procedures commonly used in the literature: the Bayesian p–value, and cross–validation. Bayesian p–values (Gelman & Meng, 1996) can be used to check the discrepancy between the sample values and the observed. The classical discrepancy statistic is to take

where x is the data or observed values and ej the expected; j indexes the vectors. In our case i is the sample P produced by the ith iteration of the MCMC sampler. Using the observed likelihood of (7), we calculate

( x ) } f ( x ) p( ), f ( x ) denotes the joint probability distribution function of the data given and p( ) denotes a prior distribution representing the analyst’s beliefs about the model parameters obtained independently from the data. Markov chain Monte Carlo (MCMC; Brooks, 1998) methods can be used to explore and summarise this posterior distribution. When the model itself is the subject of inference, the Bayesian posterior distribution can be extended to incorporate model as well as parameter uncertainty. By specifying a prior model probability, p(m) for models m c M, the corresponding posterior distribution becomes

( m,mx ) } fm( x m) pm( m) p(m), where fm( x m) denotes the joint probability distribution of the data under model m given parameter vector m c m, and pm( m) denotes the corresponding prior for m under model m. This more complex posterior distribution can be explored and summarised using reversible jump (RJ) MCMC methods (Green, 1995). Posterior inference is often summarised in the form of posterior moments under models of interest and marginal posterior model probabilities. These model probabilities may be used either to discriminate between competing models using Bayes factors (Kass & Raftery, 1995) or to provide model–averaged prediction for parameters that retain a coherent interpretation across models (Clyde, 1999; Madigan et al., 1996). We return to this later in the paper. The inference problem essentially reduces to the integration of the posterior density function over what is typically a large and complex parameter space. MCMC methods overcome this problem by simulating realisations from the posterior distribution so that empirical estimates can be calculated for any statistic of interest. These realisations are obtained by simulating a Markov chain with as its stationary distribution. See Gilks et al. (1996), Brooks (1998) and Brooks et al. (2000), for example.

We sample Ñ from the observed model using obtain

The Bayesian p–value is then the proportion of times that D(xi, i) > D(x, i) which, if the model describes the data well, should be close to 1/2. See Bayarri & Berger (1998) and Brooks et al. (2000) for further details. Cross–validation (Gelman et al., 1995; Carlin & Louis, 1996) involves treating observed values as if they were missing and investigating how well the model predicts this value. Suppose we treat observation Nt* as a missing value and update the remaining parameters as before. For iteration i sample Nit* using a Metropolis–Hastings step, excluding the term. We then sample Ñit* from the model i.e., N (N it*, s 2t*), and calculate the ratio Ñ it*/ . Repeating the same procedure for each t = 1,...,T we can calculate

which, if the model describes the data well, should be close to 1. The Bayesian analysis We begin our analysis by determining prior distributions for each of the parameters in the different models (and for the models themselves). We have full data from 1961 to 2002; additionally we also have population size estimates (and corresponding errors) for each of the duck species from 1955 to 1960. Thus we have informative priors for the latent subseries values Pk– = {p1–k,...,p 0}. Recall that the system likelihood essentially acts as a prior for the remaining log population levels, P. To

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Table 1. Prior parameter values used for the data analysis for the different duck species. Tabla 1. Valores de parámetros previos empleados para los análisis de datos correspondientes a diferentes especies de ánades.

Duck

2 0

Duck

2 0

Mallard

8.200

0.632

0.3

Northern Shoveler

2.000

0.308

2.0

Gadwall

1.500

0.274

2.0

Northern Pintail

5.600

0.707

0.5

American Wigeon

3.000

0.412

1.0

Redhead

0.640

0.141

5.0

Blue Winged Teal

4.700

0.592

3.0

Canvasback

0.540

0.130

4.0

Green Winged Teal

1.800

0.224

1.0

Scaup

6.400

0.742

0.5

obtain informative priors for b we also use the additional data to obtain a suitable range from a preliminary analysis, as follows. We first analyse the data 1955–1960 with vague priors on the unobserved , taking a Normal distribution based upon the North American Waterfowl Management Plan (Williams et al., 1999) population goal, so that

for all t = 1–k,...,0 with 0 and 20 given in table 1. This initial analysis provides an estimate of the population means and variances for b reflecting expert knowledge obtained independently from the data. Thus, for the main analysis we can take bi i N(0, 2b), = 0,...,k, where the 2b are given in table 1 for the different duck species. We use the observed data as our prior for Nt, t = 1–k,...,0 and, for the remaining parameters we take –2 = i ( , ), with = = 10–3, and adopt a uniform model prior k i U[0, kmax] where, here, we take kmax = 5, in order to cover a reasonable range of plausible density dependence models. Initial runs of the code were made to investigate convergence and mixing. By mixing we mean the extent and spread with which the parameter space is explored by the chain. Using widely dispersed starting values we observed that the chains converge quickly, within 5,000 iterations. Graphical checks were made on each of the parameters updated in the chain and other standard diagnostic techniques were used (Brooks & Roberts, 1998). Prior sensitivity was investigated, in particular the prior for 2. Although it has been common practise to use an inverse–gamma prior with both parameters set to 10–3 there is a growing body of evidence that suggests that this could be hazardous. To check this assumption we used a variety of parameters for the inverse–gamma as well as a U(0, 10,000). Our results did not alter substantially and we are therefore content with our choice of prior. Similar sensitivity

studies were undertaken for the remainder of the model parameters, with similar robustness observed. Finally, we tested our simulation algorithm using simulated data from a range of models and parameter values. The algorithm consistently identified the correct model and provided highly accurate parameter estimates (i.e., the posterior means were close the the values used to simulate the data). We therefore conclude that the algorithm performs well for a range of different data sets consistent with those observed. The analyses are based upon MCMC simulations comprising 2,020,000 iterations, with the first 20,000 discarded as burn–in. Table 2 provides the posterior model probabilities for the six possible models for each of the data sets, together with cross–validation and p–values. For many of the duck species the k = 0 model attracts the highest posterior model probability. However, there is some evidence to suggest the existence of density dependence for the Redhead, Canvasback and possibly the American Wigeon. Vickery & Nudds (1984) suggest that diving duck species tend to exhibit a greater degree of density dependence than dabblers and so it is interesting to note that the Redhead and Canvasback ducks are both divers whereas the rest (apart from Scaup) are all dabbling ducks. Thus, there does indeed appear to be some evidence for a distinction in behaviour between diving and dabbling ducks, though the Scaup and American Wigeon might perhaps be anomalous species. We shall return to this point later. Both the cross–validation and p–values suggest that most models provide an adequate fit for the data, though the p–values do appear to increase with the value of k for most species. The p– values also seem to be somewhat higher for the Green Winged Teal, Canvasback and especially the Scaup. However, all of the p–values lies within the (0.05, 0.95) range and therefore provide no evidence to suggest that any of the models perform poorly.

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Table 2. Posterior model probabilities (MP), cross–validation (CV) and p–values for the logistic models with k = 0,...,5. Tabla 2. Probabilidades posteriores de los modelos (MP), validación cruzada (CV) y valores p para los modelos logísticos con k = 0,...,5.

p–value

Mallard

p–value

Northern Shoveler

0.848

0.999

0.583

0.807

0.998

0.604

0.136

0.999

0.556

0.177

0.999

0.564

0.015

1.003

0.660

0.015

1.004

0.548

0.001

0.998

0.692

0.001

0.999

0.567

0.000

0.999

0.645

0.000

0.998

0.616

0.000

0.999

0.679

0.000

0.998

0.610

Gadwall

Northern Pintail

0.835

0.995

0.649

0.850

0.998

0.621

0.108

0.996

0.629

0.104

0.997

0.583

0.037

0.997

0.855

0.043

0.998

0.545

0.008

0.994

0.897

0.003

0.996

0.543

0.010

0.993

0.820

0.000

0.998

0.558

0.002

0.993

0.850

0.000

0.996

0.566

American Wigeon

Redhead

0.463

0.993

0.673

0.020

0.996

0.518

0.397

0.994

0.585

0.016

0.997

0.468

0.125

0.997

0.687

0.658

0.998

0.518

0.013

0.992

0.768

0.179

0.993

0.689

0.002

0.993

0.804

0.100

0.995

0.613

0.000

0.993

0.814

0.027

0.995

0.615

Blue Winged Teal

Canvasback

0.729

0.997

0.599

0.117

0.98

0.780

0.254

0.998

0.557

0.324

0.983

0.652

0.016

1.002

0.626

0.353

0.980

0.700

0.001

0.997

0.655

0.126

0.975

0.860

0.000

0.997

0.788

0.056

0.979

0.862

0.000

0.997

0.769

0.025

0.982

0.897

0.702

0.988

0.807

0.980

0.989

0.885

0.252

0.989

0.767

0.018

0.989

0.848

0.034

0.991

0.796

0.002

0.996

0.821

0.007

0.986

0.834

0.000

0.989

0.860

0.004

0.986

0.846

0.000

0.989

0.917

0.001

0.989

0.798

0.000

0.990

0.902

Green Winged

Scaup

Table 3 provides the posterior means and 95% HPDI’s for four typical duck species. In each case, the posterior mean for b0 is fairly close to zero for the k = 0 model reflecting the fairly constant nature

of the population level of these four species, though we note that the value for the Northern Pintail is negative reflecting the population decline in figure 1. For the k = 1 models, the value of b1 is

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Table 3. Parameter estimates and 95% HPDI’s for four duck species under the logistic models with k = 0,...,5. Tabla 3. Estimaciones de parámetros e intervalos de predicción de alta densidad (HDPI) al 95%, para cuatro especies de ánades con arreglo a los modelos logísticos con k = 0,...,5.

Blue Winged Teal 0

0.001(–0.045,0.047) 0.022(0.011,0.035)

0.226(0.023,0.437) 0.022(0.011,0.034) –0.051(–0.098,–0.006)

0.247(0.045,0.455) 0.020(0.008,0.033) –0.012(–0.101,0.077) –0.044(–0.132,0.044)

0.230(0.003,0.459) 0.019(0.008,0.032) –0.004(–0.102,0.097) –0.059(–0.217,0.089)

0.011(–0.100,0.124)

0.247(0.010,0.458) 0.014(0.002,0.029)

0.067(–0.085,0.070) –0.192(–0.495,0.070)

0.138(–0.106,0.425) –0.065(–0.215,0.074)

0.344(0.109,0.579) 0.013(0.003,0.025)

0.022(–0.083,0.137) –0.084(–0.282,0.080) –0.001(–0.161,0.202)

0.101(–0.074,0.254) –0.118(–0.226,–0.006)

Northern Pintail 0 –0.026(–0.085,0.034) 0.038(0.020,0.059) 1

0.126(–0.052,0.309) 0.038(0.021,0.058) –0.040(–0.084,0.005)

0.085(–0.100,0.269) 0.038(0.021,0.057) –0.101(–0.180,–0.021) 0.071(–0.007,0.147)

0.073(–0.114,0.263) 0.038(0.021,0.058) –0.108(–0.195,–0.023) 0.057(–0.042,0.153)

0.024(–0.062,0.107)

0.075(–0.117,0.265) 0.039(0.021,0.059) –0.106(–0.194,–0.020) 0.060(–0.044,0.162)

0.031(–0.067,??)

–0.012(–0.98,0.072)

0.086(–0.096,0.268) 0.034(0.018,0.053) –0.115(–0.200,–0.030) 0.069(–0.028,0.165)

0.055(–0.040,0.150)

0.037(–0.056,0.131) –0.073(–0.139,–0.006)

Redhead 0 0.013(–0.018,0.045) 0.010(0.004,0.018) 1

0.179(0.020,0.347) 0.010(0.004,0.018) –0.265(–0.528,–0.013)

0.194(0.070,0.324) 0.006(0.002,0.011)

0.358(–0.101, 0.734) –0.652(–1.031,–0.196)

0.171(0.070,0.280) 0.002(0.000,0.005)

0.834(–0.161, 1.803) –0.832(–2.678, 1.080) –0.278(–1.410, 0.746)

0.227(0.089,0.374) 0.002(0.000,0.006)

0.662(–0.382, 1.742) –1.034(–3.034, 1.037) 0.642(–1.592, 2.431) –0.640(–1.553,0.454)

0.227(0.041,0.446) 0.003(0.000,0.006)

0.594(–0.557, 1.698) –0.820(–2.858, 0.965) 0.435(–1.418, 2.406) –0.600(–2.255,1.194)

0.021(–0.008,1.085)

Canvasback 0 0.002(–0.027,0.030) 0.008(0.001,0.017) 1

0.334(0.039,0.669) 0.011(0.002,0.022) –0.625(–1.253,–0.074)

0.345(0.063,0.664) 0.009(0.002,0.019) –0.239(–1.254, 0.571) –0.407(–1.121,0.486)

0.277(0.052,0.567) 0.004(0.000,0.013)

0.305(–1.027, 1.932) –0.532(–2.726,1.137) –0.295(–1.249,0.820)

0.312(0.026,0.653) 0.004(0.000,0.012)

0.136(–1.287, 1.569) –0.374(–2.205,1.348)

0.031(–1.645,0.089) –0.382(–1.754,0.876)

5 0.340(–0.007,0.669) 0.003(0.000,0.010)

0.013(–1.222,1.524) –0.294(–2.188,1.366)

0.429(–1.499,2.260) –0.538(–2.173,1.310) 0.254(–1.623,1.039)} \\

always negative, as we would expect. Perhaps most interesting are the parameter estimates for the a posteriori most probable model for the Redhead duck. The second–order model is suggested here with a positive coefficient for the population size in the previous year and a negative (and larger) coefficient for the year before that. This suggests that the population size two years earlier has the greatest effect on the current population size. We shall return to discuss the underlying dynamics of the Redhead population later.

Figure 2 provides a model–averaged plot of the estimated population sizes across time and compares them with the observed data for four duck species. The Canvasback and Redhead plots are clearly much smoother than the other two since the posterior is dominated by density dependent models which have a natural smoothing effect. Note for example the occasional quite substantial differences between the population estimates and the posterior means for the Nt especially for the Canvasback data set. This is a result of the smooth-

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A. Blue Winged Teal

B. Canvasback

9 8 7 6 5 4 3 2 1 0

1 0.8 0.6 0.4 0.2 0 C. Northern Pintail

D. Redhead

12 11 10 9 8 7 6 5 4 3 2 1 0

1.4 1.2 1 0.8 0.6 0.4 0.2 0 1960 1970 1980 1990 2000 2010

1060 1970 1980 1990 2000 2010

Data Popn Goal

Prior MCMC

Fig. 2. Plot of observed population estimates (in millions) together with model–averaged posterior means and 95% HPDI’s for the Nt under the logistic models with k = 0,...,5 (i.e., six models per species). Posterior predictive values for the next fifteen years together with corresponding 95% HPDI’s are also included. Fig. 2. Representación gráfica de las estimaciones poblacionales observadas (en millones), junto con las medias posteriores de los promedios de predicciones según los modelos y los HPDI al 95% para Nt con arreglo a los modelos logísticos con k = 0,...,5 (es decir, seis modelos por especie). También se incluyen los valores posteriores de predicción para los próximos quince años, junto con los correspondientes HPDI al 95%.

ing effect of the density dependent system process which moderates the observed estimates removing any unlikely sharp changes in the population level. Figure 2 also provides the posterior means for the elements of Pk– together with 95% HPDI’s as well as a model–averaged posterior predictive plot for the true population size for the 15 years following the completion of the study. Though the 95% HPDI’s for the predicted population levels are fairly wide, the plots for both the Blue Winged Teal and the Canvasback suggest that the current management of these species is very much in keeping with the aim of the population goals set. However, the prediction for the Northern Pintails suggests

that the decline observed over recent years is likely to continue with very little chance of achieving the agreed population goal under the current management regime. On the other hand, the Redhead predictions suggest that the recent decline may soon end and that the population goal may well be achieved within the medium term. Thus, whilst the management of the Blue Winged Teal and Canvasbacks appears to be performing well, that of the Northern Pintail and Redhead might well be improved. In fact, similar predictive plots for the remaining duck species, suggests that the population levels for the Scaup and American Wigeon are also set to fall well below their popu-

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lation goals, whilst those for the Gadwall, Green Winged Teal and Northern Shoveler will continue to exceed them. Of course, these analyses are based upon the assumption that growth depends linearly upon population size. In the next section we challenge this assumption by considering an alternative model that assumes that the growth rate depends only logarithmically upon population size. An alternative density–dependent model There is considerable debate over the best model for describing density dependence in models of this sort. The model of Equation (1) is a recent addition to the literature and is based upon the assumption that the per–unit–abundance growth rate can be defined in discrete time as log Nt+1 – log Nt plus noise, see Dennis & Taper (1994). An alternative first–order population model was suggested by Reddingius (1971) and sets Nt = Nt–1 exp {b0 + b1 log Nt–1 +

Zt }

(8)

Using the log transformation introduced above, the model becomes pt = pt–1 + b0 + b1 pt–1 + = b0 + (1 + b1) pt–1 +

Zt = Zt

Trivial manipulations can be used to recast this model as a simple first–order autoregressive model which can be easily analysed using most standard statistical packages. We therefore refer to this model as the autoregressive model. Royama (1981) extends this model to consider higher order density dependence providing models equivalent to the general kth–order autoregressive process. Whilst this model has convenient statistical properties (e.g., with k = 1, taking x1 + b1x < 1 leads to a stationary model) and is easy to fit to observed data, it relies on the assumption that growth depends only logarithmically on population density and is therefore somewhat weaker than the logistic model. To consider the suitability of the autoregressive model for describing our data, we can repeat the analysis described in previous sections to fit this model to our data and to determine the order of the density dependence. We also extend our analysis to discriminate between the logistic and autoregressive models by calculating posterior model probabilities for suitable values of k under both models. As with the logistic model, we analysed the 1955–1960 data separately first to obtain priors for b for the subsequent analysis of the 1961–2002 data sets of primary interest. The remaining priors are unchanged. We ran the RJMCMC simulations for a total of 2,020,000 iterations, discarding the first 20,000 iterations as before.

Table 4 provides the posterior model probabilities for the logistic and autoregressive models of different orders, together with CV and p–values for the autoregressive models (the corresponding values for the logistic models are provided in table 2). (Note that when k = 0, the logistic and autoregressive models are identical and so they have identical CV and p–values.) The broad interpretation of the results of table 4 are similar to those of table 2 in that the majority of species show little evidence of being density dependent apart from the Redhead, Canvasback, American Wigeon and now also the Blue Winged Teal. The posterior model probabilities for the k = 0 model have generally decreased with the introduction of the autoregressive model since, for most species, the autoregressive models attract higher posterior model probabilities than the corresponding logistic models. This suggests that the majority of species are best modelled by the autoregressive model, though the Redhead and Canvasback species appear to be best described by the logistic model. In terms of density dependence, the Canvasback and Redhead species both have high posterior probability on density–dependent models, as do the American Wigeon and Blue Winged Teal. We note that for the American Wigeon and Blue Winged Teal, the posterior support for the first–order autoregressive model is considerably stronger than the support for the first–order logistic model which now dominates the zeroth order model. According to the CV and p–values, all models appear to fit reasonably well though, as before, the p–values seem to increase with k and are particularly high for the higher–order Canvasback and Scaup models. Table 5 provides the posterior means and 95% HPDI’s for the autoregressive models for our four key species. We first note the strong agreement between this and table 3 in terms of the system error, 2. We also see that the first–order models are all stationary with b1 negative. Though somewhat more complex to check it can also be shown that the a posteriori most probable model for the Blue Winged Teal (the k = 1 model) is also stationary (roots are 0.941 and 0.047; see Diggle, 1990) so that the population should neither explode nor die away but settle to a value around 4.4 million. Figure 3 provides the model–averaged predictive plots for the four key duck species. In fact there is very little difference between these plots and those provided in figure 2. This is because, in both figures, essentially the same models dominate the overall plot. The one exception is for the Blue Winged Teal, where the first–order autoregressive model now dominates the k = 0 model. Whereas the predictive plot for this species appears to be slowly rising in figure 2A (since b0 in the k = 0 model is positive), it appears to quickly settle to the value predicted above in figure 3A. Very similar predictive plots are obtained for the remaining duck species and so the management conclusions drawn from the original analysis remain largely unchanged with the introduction of the autoregressive models.

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Table 4. Posterior model probabilities (MP) for the logistic (LM) and autoregressive (AM) models for k = 0,...,5 and both with and without the state space element to the models i.e., the observation in the system process model. The cross– process model is removed and the Nt replaced by validation and p–values for the autoregressive state space models are also included. Tabla 4. Probabilidades posteriores de los modelos (MP) para los modelos logísticos (LM) y autorregresivos (AM) para k = 0,...,5 ambos con y sin el elemento de espacio de estados correspondiente a los modelos; es decir, el modelo de procesos de observación ha sido eliminado, mientras que Nt ha sido substituido por en el modelo de procesos del sistema. También se incluyen la validación cruzada y los valores p para los modelos de estado espaciales autorregresivos.

With state space MP k Mallard

Gadwall

American Wigeon

Blue Winged Teal

Green Winged Teal

Northern Shoveler

Autoregressive

Without state space

p–value

0.439

MP LM

AM 0.582

0.070

0.421

0.998

0.554

0.088

0.220

0.008

0.058

0.999

0.644

0.015

0.049

0.001

0.002

0.998

0.688

0.003

0.012

0.000

0.999

0.641

0.004

0.021

0.000

0.999

0.659

0.001

0.006

0.629

0.807

0.081

0.245

0.996

0.619

0.075

0.070

0.028

0.002

0.995

0.748

0.027

0.014

0.006

0.000

0.995

0.805

0.003

0.001

0.008

0.000

0.992

0.643

0.002

0.001

0.000

0.990

0.852

0.001

0.224

0.000 0.165

0.192

0.481

0.994

0.584

0.370

0.371

0.061

0.032

0.993

0.705

0.045

0.041

0.006

0.004

0.993

0.734

0.005

0.004

0.001

0.000

0.993

0.745

0.001

0.000

0.993

0.770

0.000

0.273

0.438

0.095

0.623

0.998

0.543

0.156

0.334

0.006

0.003

0.998

0.588

0.019

0.042

0.000

0.998

0.609

0.003

0.006

0.000

0.999

0.650

0.000

0.001

0.000

0.998

0.741

0.000

0.470

0.000 0.276

0.169

0.328

0.990

0.753

0.341

0.259

0.023

0.001

0.989

0.786

0.076

0.031

0.005

0.002

0.987

0.844

0.011

0.003

0.000

0.989

0.850

0.002

0.000

0.992

0.761

0.001

0.000

0.541

0.655

0.119

0.318

0.999

0.559

0.135

0.145

0.010

0.012

0.999

0.545

0.034

0.024

0.001

0.000

0.999

0.548

0.004

0.002

0.000

0.999

0.612

0.001

0.000

0.998

0.621

0.000

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Table 4. (Cont.)

With state space MP k Northern Pintail

Redhead

Canvasback

Scaup

Autoregressive

Without state space MP

p–value

0.530

AM 0.497

0.065

0.225

0.998

0.585

0.078

0.134

0.027

0.150

0.998

0.5437

0.076

0.146

0.002

0.001

0.998

0.541

0.017

0.036

0.000

0.999

0.612

0.003

0.007

0.000

0.997

0.558

0.002

0.019

0.015

0.013

0.997

0.629

0.031

0.171

0.000

0.096

0.026

0.006 0.360

0.464

0.366

0.186

0.995

0.63

0.055

0.016

0.995

0.641

0.007

0.001

0.000

0.995

0.602

0.006

0.001

0.000

0.996

0.589

0.002

0.000

0.095

0.017

0.262

0.186

0.983

0.631

0.562

0.295

0.285

0.004

0.977

0.809

0.079

0.030

0.102

0.000

0.977

0.835

0.011

0.003

0.045

0.000

0.979

0.861

0.002

0.000

0.020

0.000

0.982

0.906

0.001

0.896

0.000 0.420

0.017

0.083

0.989

0.850

0.085

0.173

0.002

0.003

0.989

0.826

0.051

0.151

0.000

0.989

0.855

0.019

0.078

0.000

0.991

0.881

0.003

0.017

0.000

0.989

0.894

0.001

0.004

Discussion In this paper we provide a detailed analysis of the North American duck census data collected over the past forty years. We analyse the data for ten separate species and investigate the possibility that each species exhibits density dependent fluctuations in population level. We discuss the two distinct density dependence models proposed in the literature and conclude that both provide adequate fits to our data. However, some species appear to be better described by one model rather than the other. The imputation of the true underlying population levels from the observed estimates has a significant impact on the results of the analysis. The rightmost columns of table 4 provide the corresponding posterior model probabilities when we treat the as if they were the true population levels i.e., removing the state space element of the

model and ignoring the estimated standard errors. These can be compared with the corresponding results under the state space models provided in the leftmost columns. Table 4 suggests that without the state space element to the model, the American Wigeon are equally well described by the logistic and autoregressive models and the Redhead population have significant posterior support for the model without density dependence with very little support for the second–order model. Also, without the state space element, the Canvasback population have high posterior support for the first–order logistic model with moderate support for the first–order autoregressive model as well. Though not provided, the corresponding CV and p–values have also been calculated for the non state–space models. Whilst most of the CV values remain high (at around 0.85), all of the models have extremely low p–values (all less than

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Table 5. Parameter estimates and 95% HPDI’s for four duck species under the autoregressive models with k = 0,...,5. Tabla 5. Estimaciones de parámetros y HDPI al 95% para cuatro especies de ánades con arreglo a los modelos autorregresivos con k = 0,...,5.

Blue Winged Teal 0

0.001(–0.045,0.047)

0.022(0.011,0.035)

0.383 (0.023,0.437)

0.022(0.011,0.034) –0.263(–0.098,–0.006)

0.247(0.045,0.455)

0.020(0.008,0.033) –0.012(–0.101, 0.077) –0.044(–0.132,0.044)

0.383(0.037,0.738)

0.020(0.009,0.033) –0.128(–0.534, 0.284) –0.175(–0.773,0.404)

0.040(–0.404,0.465)

0.401(0.027,0.781)

0.019(0.005,0.033) –0.073(–0.589, 0.546) –0.268(–1.293,0.494)

0.151(–0.570,0.974) –0.087(–0.567,0.373)

0.546(0.198,0.904)

0.014(0.004,0.026) 0.007(–0.414,0.427) –0.277(–0.907,0.281) –0.035(–0.558,0.562)

0.408(–0.191,0.940) –4.824(–0.894,–0.044)

Northern Pintail 0 –0.026(–0.085,0.034)

0.038(0.020,0.059)

0.157(–0.064,0.380)

0.038(0.021,0.059) –0.144(–0.311, 0.024)

0.110(–0.114,0.335)

0.038(0.021,0.057) –0.403(–0.727,–0.068) 0.292(–0.035,0.610)

3 0.097(–0.132,0.326)

0.038(0.021,0.058) –0.424(–0.775,–0.075) 0.222(–0.192,0.627)

0.100(–0.251,0.449)

0.103(–0.129,0.337)

0.038(0.021,0.059) –0.417(–0.760,–0.061) 0.232(–0.197,0.665)

0.135(–0.278,0.548) –0.055(–0.402,0.291)

5 0.127(–0.095,0.350)

0.035(0.019,0.053) –0.441(–0.779,–0.096) 0.269(–0.140,0.668)

0.210(–0.182,0.604)

0.169(–0.236,0.563) –0.320(–0.632,–0.002)

Redhead 0

0.013(–0.018, 0.045) 0.010(0.004,0.018)

1 –0.072(–0.160, 0.013) 0.010(0.004,0.018) –0.173(–0.336,–0.014) 2 –0.072(–0.114,–0.032) 0.002(0.000,0.004) 0.646 (0.311, 0.936) –0.795(–1.063,–0.477) 3 –0.076(–0.138,–0.016) 0.002(0.000,0.005) 0.579(–0.128, 1.259) –0.640(–1.817, 0.698) –0.951(–0.851,0.545) 4 –0.099(–0.184,–0.014) 0.002(0.000,0.006) 0.520(–0.480,1.660) –0.769(–2.995, 0.878) 0.344(–1.335,1.760) –0.292(–0.863,0.446) 5 –0.091(–0.212,0.003)

0.002(0.000,0.006) 0.501(–0.387,1.316) –0.696(–2.195,1.102)

0.381(–1.032,2.037) –0.492(–1.508,0.753)

0.126(–0.478,0.662)

Canvasback 0

0.002(–0.027, 0.030) 0.008(0.001,0.017)

1 –0.226(–0.449,–0.026) 0.011(0.002,0.023) –0.355(–0.699,–0.048) 2 –0.178(–0.371,–0.033) 0.005(0.000,0.016) 0.197(–0.611, 0.821) –0.474(–0.989,0.181) 3

0.192(–0.406,–0.029) 0.005(0.000,0.014) 0.111(–0.651, 0.943) –0.263(–1.414,0.682) –0.148(–0.685,0.476)

4 –0.214(–0.434,–0.016) 0.004(0.000,0.012) 0.014(–0.739, 0.723) –0.151(–1.167,0.832)

0.115(–0.817,1.202) –0.309(–1.037,0.454)

5 –0.234(–0.439,–0.022) 0.002(0.000,0.010) 0.049(–0.658, 1.414) –0.240(–1.949,0.734)

0.280(–0.883,1.207) –0.162(–1.089,1.453)

0.15 and most less than 0.05) suggesting that these models provide a particularly poor fit. A more practical comparison is obtained by comparing the predictive plots between our models and those without the state space element. Figure 4 provides the corresponding plot for the Redhead ducks and can be compared with figure 3D. Treating the population estimates as if they were the true values suggests a far more stable dynamic than the corresponding state space model and far greater uncertainty in future population levels. Of course,

0.285(–0.939,0.472)

the state space models are very difficult to fit within the classical paradigm which is, perhaps why they have often been ignored in the analysis of data of this sort. However, the Bayesian approach we adopt here easily deals with the imputation of the true underlying population level, providing a far better framework for population management which properly accounts for uncertainties in the data and provides a better model of the true underlying dynamic, essentially removing the observation process from the system model.

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A. Blue Winged Teal 9 8 7 6 5 4 3 2 1 0

B. Canvasback 1 0.8 0.6 0.4 0.2 0

C. Northern Pintail 12 11 10 9 8 7 6 5 4 3 2 1 0

D. Redhead 1.4 1.2 1 0.8 0.6 0.4 0.2

0 1960 1970 1980 1990 2000 2010 1060 1970 1980 1990 2000 2010 Data Prior Popn Goal MCMC

Fig. 3. Plot of observed population estimates (in millions) together with model–averaged posterior means and 95% HPDI’s for the Nt under the logistic and autoregressive models with k = 0,...,5 (i.e., 11 models per species). Posterior predictive values for the next fifteen years together with corresponding 95% HPDI’s are also included. Fig. 3. Representación gráfica de las estimaciones poblacionales observadas (en millones), junto con las medias posteriores de los promedios de predicciones según los modelos y los HPDI al 95% para Nt con arreglo a los modelos logísticos y autorregresivos con k = 0,...,5 (es decir, 11 modelos por especie). También se incluyen los valores posteriores de predicción para los próximos quince años, junto con los correspondientes HPDI al 95%.

An additional advantage of the Bayesian approach is that it can be directly extended to consider alternative observation and system processes. For example, over–dispersed Poisson or log–normal distributions might provide a better model of the observation process. Extensions of this sort are easy to implement by simply adjusting the relevant posterior parameter updates within the MCMC simulation. Within the classical paradigm, such extensions prevent the use of Kalman filtering equations that might otherwise be used for some of the models described within this paper. Our analyses suggest that the Northern Pintail, Redhead and Canvasback ducks exhibit some form of density dependence. Dynamics of this sort can be explained in any number of ways. For example the process of home–range establish-

ment during the breeding season; restricted resources such as food or prime habitat; or environmental factors such as weather or predator levels affecting mortality in a density dependent manner. The Redhead and Canvasback are both diving ducks and it is popularly believed that divers exhibit a greater tendency for density dependent behaviour than dabbling ducks (Bailey, 1981; Johnson & Grier, 1988; Viljugrein et al., 2004). However, these effects are often masked by the use of overly–simplistic models and, in particular, by treating the observed estimates of population levels as if they were the true underlying level. By separating the observation and system process, we have been able to detect a fairly strong density dependent signal in three duck species and little evidence for density dependence in the rest.

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1.4

127

Data Popn Goal Prior MCMC

1.2 1.0 0.8 0.6 0.4 0.2 0 1960

1970

1980

1990

2000

2010

Fig. 4. Plot of observed population estimates (in millions) for the Redhead duck, together with model–averaged posterior means and 95% HPDI’s for the Nt under the logistic and autoregressive models, with k = 0,...,5 (i.e., 11 models), but without the state space element to the models i.e., in the system process model. Posterior observation process model removed and Nt replaced by predictive values for the next fifteen years together with corresponding 95% HPDI’s are also included. (en millones) para el Fig. 4. Representación gráfica de las estimaciones poblacionales observadas porrón americano, junto con las medias posteriores de los promedios de predicciones según los modelos y los HPDI al 95% para Nt con arreglo a los modelos logísticos y autorregresivos, con k = 0,...,5 (es decir, 11 modelos), pero sin el elemento del estado espacial correspondiente a los modelos; es decir, el modelo en el modelo de procesos de observación ha sido eliminado, mientras que Nt ha sido substituido por de procesos del sistema. También se incluyen los valores posteriores de predicción para los próximos quince años, junto con los correspondientes HDPI al 95%.

By providing predictive plots of population levels, we are also able to assess the success or otherwise of the U.S. Fish and Wildlife Service’s management policy in maintaining their target population levels. For many species, predictions suggest that the management policy works extremely well. However, for others, it is clear that the current policy is leading to either a steady growth or decline in the population level. Perhaps the most alarming is the predicted rapid decline of the Northern Pintail and Scaup populations, with the lower limits of the Northern Pintail predicted intervals moving towards dangerously low levels within a very short period. Ideally, the management process would be integrated into the analysis presented here. Additional data providing information on reproductive and mortality rates together with, for example, hunting license records would enable us to extend this analysis to provide a more detailed picture of the population level dynamics and in particular, their dependence upon human activity. For example, different land–use or licensing policies could be investigated to determine their effect on the population level over

time. We hope that the analyses we present here highlight the importance of the integration of the models and methodology we describe here within the management policy process. References Bailey, R. O., 1981. A Theoretical Approach to Problems in Waterfowl Management. Transactions North American Wildlife and Natural Resources Conference, 46: 58–71. Bayarri, M. & Berger, J., 1998. Quantifying Surprise in the Data and Model Verification (with discussion). In: Bayesian Statistcs VI: 53–82 (J. Bernardo, J. Berger, A. Dawid & A. Smith, Eds.). Oxford Univ. Press, Oxford. Berryman, A. & Turchin, P., 2001. Identifying the Density–Dependent Structure Underlying Ecological Time Series. Oikos, 92: 265–270. Brooks, S. P., 1998. Markov Chain Monte Carlo Method and its Application. The Statistician, 47: 69–100. Brooks, S. P., Catchpole, E. A. & Morgan, B. J. T.,

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2000. Bayesian Animal Survival Estimation. Statistical Science, 15: 357–376. Brooks, S. P. & Roberts, G. O., 1998. Diagnosing Convergence of Markov Chain Monte Carlo Algorithms. Statistics and Computing, 8: 319–335. Bulmer, M. G., 1975. The Statistical Analysis of Density Dependence. Biometrics, 31: 901–911. Carlin, B. P. & Louis, T. A., 1996. Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall, London. Clyde, M. A., 1999. Bayesian Model Averaging and Model Search Strategies. In: Bayesian Statistics 6: 157–186 (J. M. Bernardo, A. F. M. Smith, A. P. Dawid and J. O. Berger, Eds.). Oxford Univ. Press, Oxford. Dennis, B. & Taper, M. L., 1994. Density Dependence in Time Series Observations of Natural Populations: Estimation and Testing. Ecological Monographs, 64: 205–224. Diggle, P. J., 1990. Time Series: A Biostatistical Introduction. Oxford Univ. Press, Oxford. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B., 1995. Bayesian Data Analysis. Chapman and Hall, London. Gelman, A. & Meng, X., 1996. Model Checking and Model Improvement. In: Markov Chain Monte Carlo in Practice: 189–201 (W. R. Gilks, S. Richardson & D. J. Spiegelhalter, Eds.). Chapman and Hall, London. Gilks, W. R., Richardson, S. & Spiegelhalter, D. J., 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall, London. Green, P. J., 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82: 711–732. Johnson, D. H. & Grier, J. W., 1988. Determinants of Breeding Distribution of Ducks. Wildlife Monographs, 100: 5–37. Kass, R. E. & Raftery, A. E., 1995. Bayes Factors. Journal of the American Statistical Association, 90: 773–795. Madigan, D. M., Raftery, A. E., Volinsky, C. & Hoeting, J., 1996. Bayesian Model Averaging. In: Integrating Multiple Learned Models (IMLM–96): 77–83 (P. Chan, S. Stolofo & D. Wolpert, Eds.). Massot, M., Clobert, J., Pilorge, T., Lecomte J. & Barbault, R., 1992. Density Dependence in the Common Lizard: Demographic Consequences of

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a Density Manipulation. Ecology, 73: 1742–1756. Newman, K. B., 1998. State–Space Modelling of Animal Movement and Mortality with Application to Salmon. Biometrics, 54: 1290–1314. Nichols, J. D., Conroy, M. J., Anderson, D. R. & Burnham, K. P., 1984. Compensatory Mortality in Waterfowl Populations: A Review of the Evidence and Implications for Research and Management. Transactions of the North American Wildlife Nature Reserve Conference, 49: 535–554. Nichols, J. D., Johnson, F. A. & Williams, B. K., 1995. Managing North–American Waterfowl in the Face of Uncertainty. Annual Review of Ecological Systems, 26: 177–199. Reddingius, J., 1971. Gambling for Existence: A Discussion of some Theretical Problems in Animal Population Ecology. Acta Biotheoretica, 20: 1–208. Royama, T., 1981. Fundamental Concepts and Methodology for the Analysis of Population Dynamics, with particular reference to univoltine species. Ecological Monographs, 51: 473–493. Shenk, T. M., White, G. C. & Burnham, K. P., 1998. Sampling–Variance Effects on Detecting Density Dependence from Temporal Trends in Natural Populations. Ecological Monographs, 68: 445– 463. Sullivan, P. J., 1992. A Kalman Filter Approach to Catch–at–Length Analysis. Biometrics, 48: 237– 257. Turchin, P., 1990. Rarity of density Dependence or Population Regulation with Lags? Nature, 344: 660–663. Turchin, P., Lorio Jr, P. L., Taylor, A. D. & Billings, R. F., 1991. Why do Populations of Southern Pine Beetles (Coleoptera: Scolytidae) Fluctuate? Environmental Entomology, 20: 401–409. Vickery, W. & Nudds, T., 1984. Detection of Density Dependent Effects in Annual Duck Censuses. Ecology, 65: 96–104. Viljugrein, H., Stenseth, N. C. & Steinbakk, G. H. (in press). Density Dependence in North American Ducks: A State–Space Modelling Approach. Ecology Letters. Williams, B. K., Koneff, M. D. & Smith, D. A., 1999. Evaluation of Waterfowl Management under the North American Waterfowl Management Plan. Journal of Wildlife Management, 63: 417–440.

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Demographic estimation methods for plants with dormancy M. Kéry & K. B. Gregg

Kéry, M. & Gregg, K. B., 2004. Demographic estimation methods for plants with dormancy. Animal Biodiversity and Conservation, 27.1: 129–131. Extended abstract Demographic estimation methods for plants with dormancy.— Demographic studies in plants appear simple because unlike animals, plants do not run away. Plant individuals can be marked with, e.g., plastic tags, but often the coordinates of an individual may be sufficient to identify it. Vascular plants in temperate latitudes have a pronounced seasonal life–cycle, so most plant demographers survey their study plots once a year often during or shortly after flowering. Life–states are pervasive in plants, hence the results of a demographic study for an individual can be summarized in a familiar encounter history, such as 0VFVVF000. A zero means that an individual was not seen in a year and a letter denotes its state for years when it was seen aboveground. V and F here stand for vegetative and flowering states, respectively. Probabilities of survival and state transitions can then be obtained by mere counting. Problems arise when there is an unobservable dormant state, i.e., when plants may stay belowground for one or more growing seasons. Encounter histories such as 0VF00F000 may then occur where the meaning of zeroes becomes ambiguous. A zero can either mean a dead or a dormant plant. Various ad hoc methods in wide use among plant ecologists have made strong assumptions about when a zero should be equated to a dormant individual. These methods have never been compared among each other. In our talk and in Kéry et al. (submitted), we show that these ad hoc estimators provide spurious estimates of survival and should not be used. In contrast, if detection probabilities for aboveground plants are known or can be estimated, capturerecapture (CR) models can be used to estimate probabilities of survival and state–transitions and the fraction of the population that is dormant. We have used this approach in two studies of terrestrial orchids, Cleistes bifaria (Kéry et al., submitted) and Cypripedium reginae (Kéry & Gregg, submitted) in West Virginia, U.S.A. For Cleistes, our data comprised one population with a total of 620 marked ramets over 10 years, and for Cypripedium, two populations with 98 and 258 marked ramets over 11 years. We chose the ramet (= single stem or shoot) as the demographic unit of our study since there was no way distinguishing among genets (genet = genetical individual, i.e., the "individual" that animal ecologists are mostly concerned with). This will introduce some non–independence into the data, which can nevertheless be dealt with easily by correcting variances for overdispersion. Using ramets instead of genets has the further advantage that individuals can be assigned to a state such as flowering or vegetative in an unambiguous manner. This is not possible when genets are the demographic units. In all three populations, auxiliary data was available to show that detection probability of aboveground plants was m 0.995.

Marc Kéry*, Patuxent Wildlife Research Center, U.S. Geological Survey, 11510 American Holly Drive, Laurel, MD 20708, U.S.A. Katharine B. Gregg, West Virginia Wesleyan College, 59 College Avenue, Buckhannon, WV 26201, U.S.A. *Present address: Swiss Ornithological Institute, Schweizerische Vogelwarte, 6204 Sempach, Switzerland. E–mail: marc.kery@vogelwarte.ch ISSN: 1578–665X

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Kéry & Gregg

We fitted multistate models in program MARK by specifying three states (D, V, F), even though the dormant state D does not occur in the encounter histories. Detection probability is fixed at 1 for the vegetative (V) and the flowering state (F) and at zero for the dormant state (D). Rates of survival and of state transitions as well as slopes of covariate relationships can be estimated and LRT or the AIC machinery be used to select among models. To estimate the fraction of the population in the unobservable dormant state, the encounter histories are collapsed to 0 (plant not observed aboveground) and 1 (plant observed aboveground). The Cormack–Jolly–Seber model without constraints on detection probability is used to estimate detection probability, the complement of which is the estimated fraction of the population in the dormant state. Parameter identifiability is an important issue in multi state models. We used the Catchpole–Morgan– Freeman approach to determine which parameters are estimable in principle in our multi state models. Most of 15 tested models were indeed estimable with the notable exception of the most general model, which has fully interactive state- and time-dependent survival and state transition rates. This model would become identifiable if at least some plants would be excavated in years when they do not show up aboveground. Our analyses for three analyzed populations of Cleistes and Cypripedium yielded annual ramet survival rates ranging from 0.86–0.96. Estimates of the average fraction dormant ranged from 0.02–0.30, but with up to half a population in the dormant state in some years. Ultrastructural modeling enables interesting hypotheses to be tested about the relationships of demographic rates with climatic covariates for instance. Such covariate modeling makes the CR approach particularly interesting for evolutionary– ecological questions about, e.g., the adaptive significance of the dormant state. Previous and foreseeable future applications of CR in plant ecology Since the paper by Alexander et al. (1997), it has become increasingly clear that CR models may be useful for demographic analysis of plant populations. In the future, we are likely to see increasing use of these methods that were originally developed for animal populations. Here is a summary about all previous applications that I have come across. I am grateful if readers point out to me any titles that I may have missed. If a reliable way to mark seeds can be devised, CR might indeed provide the analysis tool for tackling one of the ultimate frontiers in plant population ecology: the dynamics of the seed bank. Indeed, the first ever application of CR to plants that I have come across (Naylor, 1972) used a fluorescent dye to mark seeds and a Lincoln–Peterson–type estimator to estimate the seed bank size in an agricultural weed. The application of CR to plants with dormancy has been treated by Shefferson et al. (2001, 2003), Kéry et al. (submitted) and Kéry & Gregg (submitted). Population size, and survival rates of plants whose aboveground states are easily overlooked have been estimated for an elusive prairie plant (Alexander et al., 1997; Slade et al., 2003) and for a tropical savannah tree (Lahoreau et al., 2003). For plot–based plant demographic studies, we have shown previously that (not surprisingly) different life–states may have different detection probabilities, and that this may seriously bias inference from population modelling (Kéry & Gregg, 2003). It is somewhat astonishing that there still appear to be no applications of CR to the analysis of plant populations and communities. For instance, species richness, patch occupancy, population extinction rates, and species turnover in communities are all still based on adding up the raw data, even though the animal literature has plenty of papers showing more adequate ways of estimating these quantities (e.g., Boulinier et al. 1998; Nichols et al. 1998). I have submitted a note (Kéry, submitted) describing the use of the Cormack–Jolly–Seber model to estimate extinction probabilities for plant populations in a manner exactly analogous to patch occupancy models (MacKenzie et al., 2002, 2003). It is perhaps in plant community ecology where we will see most future applications of CR. References (This list contains all plant studies that have used CR, that I have come across by the end of October 2003. There are also a few non–plant papers mentioned in the text.) Alexander, H. M., Slade, N. A. & Kettle, W. D., 1997. Application of mark–recapture models to the estimation of the population size of plants. Ecology, 78: 1230–1237. Boulinier, T., Nichols, J. D., Sauer J. R., Hines J. E. & Pollock K. H., 1998. Estimating species richness: The importance of heterogeneity in species detectability. Ecology, 79: 1018–1028. Kéry, M. (in prep.) Extinction rate estimates for plant populations in revisitation studies: Importance of detectability. Submitted. Kéry, M. & Gregg, K. B., 2003. Effects of life–state on detectability in a demographic study of the terrestrial

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orchid Cleistes bifaria. Journal of Ecology, 91: 265–273. – (in prep.) Demographic analysis of dormancy in the terrestrial orchid Cypripedium reginae. Submitted. Kéry, M., Gregg, K. B. & Schaub, M. (in prep.) Demographic estimation methods for plants in the presence of dormancy. Submitted. Lahoreau, G., Gignoux, J. & Julliard, R., 2004. Exploring mark–resighting–recovery models to study savannah tree demographics. Animal Biodiversity and Conservation, 27.1: 549–560. MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Royle, J. A. & Langtimm, C. A., 2002 Estimating site occupancy rates when detection probabilities are less than one. Ecology, 83: 2248–2255. MacKenzie, D. I., Nichols, J. D., Hines, J. E., Knutson, M. G. & Franklin, A. B., 2003. Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology, 84: 2200–2207. Naylor, R. E. L., 1972. Aspects of the population dynamics of the weed Alopecurus myosuroides Huds. in winter cereal crops. Journal of Applied Ecology, 9: 127–139. Nichols, J. D., Boulinier, T., Hines, J. E., Pollock, K. H. & Sauer, J. R., 1998. Estimating rates of local species extinction, colonization, and turnover in animal communities. Ecological Applications, 8: 1213–1225. Shefferson, R. P., Proper, J., Beissinger, S. R. & Simms, E. L., 2003. Life history trade–offs in a rare orchid: The costs of flowering, dormancy, and sprouting. Ecology, 84: 1199–1206. Shefferson, R. P., Sandercock, B. K. Prope,r J. & Beissinger, S. R. 2001. Estimating dormancy and survival of a rare herbaceous perennial using mark–recapture models. Ecology, 82: 145–156. Slade, N. A., Alexander, H. M. & Kettl, W. D., 2003. Estimation of population size and probabilities of survival and detection in Mead’s milkweed. Ecology, 84: 791–797.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Multi–state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): the importance of estimating recapture rates J. C. Senar & M. J. Conroy

Senar, J. C. & Conroy, M. J., 2004. Multi–state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): the importance of estimating recapture rates. Animal Biodiversity and Conservation, 27.1: 133–146. Abstract Multi–state analysis of the impacts of avian pox on a population of Serins (Serinus serinus): the importance of estimating recapture rates.— Disease is one of the evolutionary forces shaping populations. Recent studies have shown that epidemics like avian pox, malaria, or mycoplasmosis have affected passerine population dynamics, being responsible for the decline of some populations or disproportionately killing males and larger individuals and thus selecting for specific morphotypes. However, few studies have estimated the effects of an epidemic by following individual birds using the capture–recapture approach. Because avian pox can be diagnosed by direct examination of the birds, we are here able to analyze, using multistate models, the development and consequences of an avian pox epidemic affecting in 1996, a population of Serins (Serinus serinus) in northeastern Spain. The epidemics lasted from June to the end of November of 1996, with a maximum apparent prevalence rate > 30% in October. However, recapture rate of sick birds was very high (0.81, range 0.37–0.93) compared to that of healthy birds (0.21, range 0.02– 0.32), which highly inflated apparent prevalence rate. This was additionally supported by the low predicted transition from the state of being uninfected to the state of being infected (0.03, SE 0.03). Once infected, Serin avian pox was very virulent with (15–day) survival rate of infected birds being of only 0.46 (SE 0.17) compared to that of healthy ones (0.87, SE 0.03). Probability of recovery from disease, provided that the bird survived the first two weeks, however, was very high (0.65, SE 0.25). The use of these estimates together with a simple model, allowed us to predict an asymptotic increase to prevalence of about 4% by the end of the outbreak period, followed by a sharp decline, with the only remaining infestations being infected birds that had not yet recovered. This is in contrast to the apparent prevalence of pox and stresses the need to estimate recapture rates when estimating population dynamics parameters. Key words: Avian pox, Epidemics, Serin, Serinus serinus, Survival, Capture–recapture. Resumen Análisis mediante modelos de multiestados del impacto de la viruela aviar sobre una población de Verdecillos (Serinus serinus): la importancia de estimar las tasas de recaptura.— Las enfermedades infecciosas son una de las fuerzas evolutivas que modulan a las poblaciones animales. Estudios recientes han puesto de manifiesto como epidemias como la viruela aviar, la malaria o la mycoplasmosis afectan a la dinámica de las poblaciones de passeriformes, siendo responsables de dramáticas reducciones en el tamaño de algunas poblaciones, o de la muerte desproporcionada de machos o de los individuos de mayor tamaño, seleccionando de ese modo en favor de determinados morfotipos. Sin embargo, pocos estudios han estimado los efectos de una epidemia mediante el seguimiento de los distintos individuos utilizando las técnicas de captura–recaptura. Debido al hecho de que la viruela aviar puede ser diagnosticada mediante el examen directo de los individuos, hemos podido analizar, utilizando modelos de multiestado, el desarrollo y consecuencias de una epidemia de viruela aviar que afectó en 1996, a una población de Verdecillos en el nordeste de España. La epidemia afectó a los Verdecillos desde junio hasta finales de noviembre, con una prevalencia aparente máxima de > 30% en octubre. Sin embargo, la tasa de recaptura de los individuos enfermos fue muy alta (0,81, rango 0,37–0,93), comparada con la de los individuos sanos (0,21, rango ISSN: 1578–665X

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0,02–0,32), lo cual exageraba en gran medida la tasa de prevalencia aparente. Este resultado estaba adicionalmente apoyado por la baja tasa estimada de transición del estado de no infectado al estado de infectado (0,03, SE 0,03). Una vez un Verdecillo quedaba infectado, la viruela aviar resultó muy virulenta, siendo la tasa de supervivencia (a 15 días) de los individuos enfermos de tan solo 0,46 (SE 0,17), comparada con la de los individuos no infectados (0,87, SE 0,03). La probabilidad de recuperación de la enfermedad, siempre y cuando el individuo hubiera sobrevivido las dos primeras semanas, fue sin embargo, muy alta (0,65, SE 0,25). Estos valores fueron utilizados para construir un modelo que permitió predecir el valor real de prevalencia de la enfermedad. Según el modelo, el porcentaje de individuos infectados después del brote debió incrementarse de forma asintótica hasta el 4%, manteniéndose en ese valor, hasta que se produjo una abrupta reducción en el número de individuos infectados al final de la epidemia, siendo estos los individuos que todavía no se habían recuperado de la enfermedad. Estos valores contrastan con los valores aparentes de prevalencia de la viruela y enfatiza la necesidad de estimar la tasa de recaptura cuando se realizan estimaciones de los distintos parámetros de dinámica de poblaciones. Palabras clave: Viruela aviar, Epidemia, Verdecillo, Serinus serinus, Supervivencia, Captura–recaptura. Juan Carlos Senar(1), Museu de Ciències Naturals, Psg. Picasso s/n., Parc de la Ciutadella, 08003 Barcelona, Spain.– Michael J. Conroy(2), USGS, Georgia Cooperative Fish and Wildlife Research Unit, Univ. of Georgia, Athens, GA, U.S.A. (1)

E–mail: jcsenar@mail.bcn.es

(2)

E–mail: conroy@fisher.forestry.uga.edu

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Introduction It is increasingly recognised that infectious disease may shape animal populations (May, 1983; Scott, 1988; Clayton & Moore, 1997; Newton, 1998; Hudson et al., 2001). In North America during 1994 mycoplasmosis was recorded to have spread throughout the east coast in just two years (Fischer et al., 1997; Hochachka & Dhondt, 2000) and to have killed some 225 million birds (Nolan et al., 1998). Avian pox and malaria were responsible for the decline of several Hawaiian bird populations (Ralph & Fancy, 1995; Van Ripper III et al., 2002; Benning et al., 2002) and of some continental bobwhite quail and wild turkey populations in the southeastern United States (Hansen, 2004), and recent models of metapopulation dynamics consider disease as an important factor to have into account for the conservation of endangered populations (Woodroffe, 1999; Gog et al., 2002; Hess, 2003). Recent studies have shown that epidemics like avian pox and mycoplasmosis have affected passerine population dynamics, disproportionately killing males and larger individuals and thus selecting for specific morphotypes (Thompson et al., 1997; Nolan et al., 1998; Brawner et al., 2000). Avian pox and mycoplasmosis are also responsible for shifts in mean plumage colour of whole populations, which may have important consequences on the strength of sexual selection in these populations (Thompson et al., 1997; Zahn & Rothstein, 1999). Given the importance of disease in animal populations, several national programs have been developed to follow up infection dynamics (e.g., House finch Conjunctivitis survey) (Dhondt et al., 1998; Hartup et al., 2001). However, most studies have to rely on the establishment of prevalence (% infected birds) of the disease and few studies have estimated the effects of an epidemic by following individual birds by capture–recapture techniques (e.g., Faustino et al., in press). This is of importance because sick birds may have higher or lower probabilities of capture/recapture (e.g., McClure, 1989; Faustino et al., in press), thus biasing estimates of prevalence of the infection (Williams et al., 2002). Avian pox is viral infection of birds caused by Poxvirus avium. The disease is worldwide in distribution (Hansen, 1987; Van Ripper III et al., 2002), and occurs in two forms; (1) most commonly, a skin form with warty lesions, mostly on the unfeathered body; and (2) a diptheritic form, which involves the mouth and respiratory tract. Transmission may occur either directly, by contact among infected birds, or with mechanical transfer via biting insects, especially mosquitoes (Hansen, 1987; Van Ripper III et al., 2002). Although usually not directly lethal, the disease may increase the vulnerability of birds to other risks, such as predation or secondary infections (Hansen, 1987; Van Ripper III et al., 2002; Gortazar et al., 2002). Infection is thought to confer immunity to the disease that lasts 12–18 months (Boch &

Schneidawind, 1994), although this figure may vary among species, sometimes immunity being permanent (Arnall & Keymer, 1975; Del Pino, 1977). Recovery time is variable, but generally within 25 days of exposure (Del Pino, 1977). In our study we used capture–mark–recapture (CMR) and multistate models, to model the development and consequences of an avian pox epidemic affecting a population of Serins (Serinus serinus) in northeastern Spain. Our approach allows the estimation of probabilities of infection and of disease recovering, and of survival rate of infected and uninfected birds, parameters that in natural populations are otherwise very difficult to estimate. Methods Field methods The study was carried out at the Desert de Sarria, a ringing station (3 Ha) within the suburban area of Barcelona (NE Spain). The area is formed by orchards, small pine woods (Pinus halepensis) and gardens, which conforms a typical Serin habitat (Senar, 1986). Serins have been trapped there since 1985 on a weekly basis using platform and funnel traps, clap–nets and mist nets, all of them except mist–nets, associated with baited feeders; the use of several trapping devices allowed to reduce biases in trapping probabilities of different sex and age classes, allowing to obtain a representative sample of birds in the population (Yunick, 1971; Domènech & Senar, 1997, 1998; Conroy et al., 1999; Domènech et al., 2001). In June 1996 we had an outbreak of avian pox. From this time forward we implemented special procedures for the handling of care infected birds. We employed dedicated containers and measuring devices for the infected birds, and wore clinical gloves, disinfecting hands with HalamidTM after handling birds. We also disinfected traps after each capture and several times during the following week. Avian pox typically causes discrete, warty and proliferative lesions on the skin of legs, feet, eyelids and the base of the beak, and so can easily be diagnosed by visual inspection of the birds (Hansen, 2004). We confirmed we were dealing with avian pox by histopathologic examination of a bird which showed the typical eosinophilic intracytoplasmic inclusions diagnostic of avian pox (Laboratorio de Diagnóstico Veterinario) (Gortazar et al., 2002; Hansen, 2004). Hence, here "infected" denotes exhibiting signs of the disease, i.e., symptomatic, and "uninfected" denotes the absence of symptoms. Other studies have found good correspondence between exhibition of lesions and actual prevalence of avian pox; for instance Van Riper III et al. (2002) confirmed presence of avian pox in 20 of 22 histopathological examination of tissue from birds exhibiting pox–like lesions. Nevertheless, we recognise that some infected birds may have been asymptomatic and thus our analysis potentially incorrectly classi-

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fies some birds as uninfected. In fact, some pox virus have developed strategies to minimise external appearance (Seet et al., 2003). Later, we discuss implications of misclassification for interpreting our results. Statistical analyses Incidence of pox was confined to Jun–Nov of 1996 (table 1); therefore, we confined our analyses to birds captured and or recaptured during May–December. To avoid issues of unidentified juvenile sex classes and transience (Conroy et al., 1999) analyses were confined to within–year recaptures, with the years constituting a grouping variable along with age and sex class. Within each year we identified 5 age–sex classes based on plumage characteristics: adult male (AM), adult female (AF), subadult male (SM), subadult female (SF), and juveniles (J), for which sex could not be identified. These were later grouped into 3 categories, in which adult and subadult classes were combined and identified by sex (M, F) and juveniles considered separately (J). For 1996, we constructed multistate recapture models using program MARK (White & Burnham, 1999). Captures and recaptures were grouped by 15–day periods from 1 May–31 December. At each sampling occasion individuals were classified as exhibiting active pox lesions (P) or not (N), and modelled state–specific survival, capture, and transition probabilities. As discussed below, the state N may actually be a mixture of birds who are susceptible to the disease, and those who have been infected and are now immune. Specifically, we used the multistate data structure, by 15–day period and stratified by age–sex categories, to estimate Sts,as the probability of 15–day survival for birds in state s (1 = uninfected or 2 = infected), age–sex as = AM, AF, SM, SF, or J over [t, t+1], t = 1,...,15; Pts,as, the probability of capture at occasion of t = 2,...,16 for birds state s and age–sex as; and ts,r,as the probability of movement to state r at t+1 for birds in state s at t, age–sex as, at sampling period t = 1,...,15. We employed 15–day periods as the shortest interval over which data could be grouped while providing sufficient data for estimation. Because we had no data on recovery times for pox–infected Serins, we desired this interval to be as short as possible so as to allow estimation of rates of recovery. Data from other similar species suggests recovery period to be of about 25 days (Del Pino, 1977), which validates our analysis interval in detecting rates of recovery. Unlike the CJS analysis (described below), this analysis focussed on modelling survival, capture, and transition processes within a single year. However, our a priori expectation was that age, sex, and state (infected or not) and other individual attributes (considered below) would account for greater variability in survival and capture rates, than would variation among 15–day periods. In addition, data were sparse, particularly captures and recaptures of infected birds. We thus constructed a number of models in which

additive age–sex and state effects were modelled, using the design matrix feature in MARK. We also attempted to fit a "global" model in which group effects (age–sex and year) interacted with time (recapture occasions), for comparison to constrained– parameter models. We used c as a measure of model fit/ over dispersion, estimated by =

/df,

where 2 is the deviance (–2 loge [likelihood]) statistic and df is computed as the number of independent multinomial cells minus the number of parameters estimated. However, sparse data render deviance– based statistics unreliable as measures of fit, and we therefore conducted 250 bootstrap simulations under a highly–parameterized ("global") model and compared the mean of the bootstrap estimate of c under this model to that under the corresponding estimated model to obtain as . Because bootstrap goodness of fit tests are not currently available in MARK, we developed a bootstrap program using a modification of the SAS code (simulate.sas) provided by G. White as part of MARK, integrated via a SAS macro with a batch version of MARK; this code is available from the second author at http://coopunit.forestry.uga.edu/ conroy/software/bootstrap.txt. We then used this adjustment in program MARK to compute QAICc (quasilikelihood–adjusted AIC, corrected for small sample size; (Burnham & Anderson, 2002) and QAICc, where QAICc(i) =

QAICc(i) –

QAICc(min)

and QAICc(min) was the model under consideration having the lowest value for QAICc. This statistic was in turn used to compute model weights (wi) for each competing model as ,

where R is the number of models in the set of candidate models. We then used model averaging (Burnham & Anderson, 2002) to obtain estimates of parameters by

and of unconditional standard errors by

are the estimates of and its where and conditional (sampling) variance under model i. These were used to create asymptotic normal confidence intervals by multiplication with the 0.05 and 0.95 standard normal deviates.

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Table 1. Monthly frequency of pox incidence for Serins captured and recaptured during 1996: Infection status determined by external examination (presence or absence of warty lesions characteristic of avian pox). "Not infected" birds may include some birds that have been previously infected and are likely immune (see text). a

Tabla 1. Incidencia mensual de viruela aviar en los Verdecillos capturados y recapturados durante 1996: a Estatus de infección determinado mediante inspección externa (presencia o ausencia de lesiones ulcerosas características de la viruela aviar). La categoría de "no infectado" puede incluir a algunas aves previamente infectadas pero que han desarrollado inmunidad (ver texto).

Not infecteda Month

Infected

Male

Female

Juvenile

Male

Female

Juvenile

January

February

March

111

April

May

June

116

July

August

September

128

October

November

December

283

172

953

Total

To provide background estimates of survival rate and of annual variation in within–year survival, and to aid in the interpretation of state transitions for asymptomatic ("1") birds (see below), we conducted a Cormack–Jolly–Seber (CJS) analyses of captures and recaptures at 15–day intervals from 1 May to 31 December in all years except 1996. As with the multistate analysis, we wished to avoid issues of unidentified juvenile sex classes and transience. Thus, the CJS analyses were confined to within–year recaptures, with the years constituting a grouping variable along with age and sex class. Within each year we identified 5 age–sex classes as above, later grouped into males, females, and juveniles. The parameters of the CJS model were survival probabilities tas,y and capture probabilities ptas,y where t = 1,...,16 sampling occasions correspond to the 15–day intervals, as = 1,...,5 are the age–sex categories, and y = 1,...,10 are the years (1990–2000 excluding 1996). Because we anticipated greater year– to–year variation in survival probability than variation among 15–day intervals within year, we constructed a number of constrained–parameter models using MARK. We were particularly interested in modelling year–to–year variation in age–sex

specific survival for comparison to survival during the year of pox epidemic. We therefore constructed a number of models in which additive age–sex and year effects were modelled, using the design matrix feature in MARK. We also attempted to fit a "global" model in which group effects (age–sex and year) interacted with time (within–year recapture occasions), for comparison to constrained– parameter models. Because of sparse data, goodness of fit based either on RELEASE or on deviance statistics were unreliable. We therefore conducted 500 bootstrap simulations under a "global" model and compared the mean of the bootstrap deviance to compute statistics under this model to the deviance under the corresponding estimated model to obtain as described above, with computations performed within MARK. We were also interested in the possible relationship of individual covariates (measured upon first capture) to state–specific survival and to the probability of transition between states. In particular, we identified mass, and body size as measured by wing length, tail length, and length of P3, as potentially influencing one or both of these rates. We used the design matrix feature of MARK to incorporated predictive relationships of the form

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, and , where Sis is predicted survival over [t, t+1] for individual i in state s, is is predicted transition over [t, t+1] to the alternate state (s') for individual i in state s at time t, 0s, 0s are state–specific effects on survival and transition, j = 1,...,k are individual covariates, j, j are coefficients including dummy variables expressing levels for age–sex and (potentially) capture occasion, and slopes for the covariate effects. For covariates we used standardized, individual mass, wing length, tail length, and P3 length and interactions of these with grouping variables (age–sex, state) as appropriate. Standardization was within age–sex class, with resulting predictors represent deviation from the–within–class mean. Because individual covariates were not taken for all individuals, we formed 3 subsets of the data, in which (1) mass, (2) mass and wing length, and (3) all covariates were recorded. For each subset, we selected the top–ranked multistate model (covariates absent) as a baseline model, and modified this model to incorporate covariate effects. We performed model evaluation and selection as above, with based on the bootstrap results from the multistate, no covariate model. Results During May–December 1996 we captured 428 individual Serins for a total of 1,470 capture–recapture events. We captured birds in the "infected" state on 62 occasions, representing 42 individuals (total prevalence of 9.8%), and the majority (52 captures of 37 birds) were of juvenile birds (table 1). Avian pox appeared from June to November. When stratifying by 15–day periods, apparent prevalence raised by the second half of October to 33% of birds trapped being infected (fig. 1). Birds differed in the part of the body infected: 13% of birds had legs infected, 34% the eyelids and 53% the base of the beak, with 10 individuals having both infected eyelids and the base of the beak (n = 42). Data on the 1,470 capture–recapture events was used in the multistate modelling. Due to sparse data we were unable to estimate parameters under a global model incorporating time and group effects for all parameters. We instead used model Spoxppox pox+as incorporating state and age–sex effects for survival and transition, and additive state and time effects for capture. We compared for this model (11.28) to the mean from 250 bootstrap simulations (4.65) to obtain an estimate of = 2.43 for use in model evaluation and comparison (table 2). We selected model Spoxppox+t pox as the best candidate model, allowing for state–specific effects on survival, capture, and transition, and additive time effects on capture prob-

abilities. Several other models had non–negligible QAICc weights; thus we used model averaging to obtain estimates and unconditional confidence intervals of state–specific survival and transition and of state and time–specific capture probabilities (table 3). These results show, first, a notably higher survival rate for birds captured as "uninfected" ( = 0.868, SE = 0.025) than "infected" ( = 0.458, SE = 0.17) and second, a higher rate of transition from "infected to uninfected" than the reverse ( = 0.654, SE = 0.254 vs. = 0.032, SE = 0.030, table 3). The CJS analysis confirms that birds in the "infected" state had an unusually low probability of survival, taking into account yearly variation in these rates. We selected model as+y+t pas+y+t, incorporating age–sex, year–to–year, and within–year time variation on survival and capture probabilities; all other models had negligible credibility ( QAICc > 36). Because we were interested in comparison of our point estimate of state–specific survival to yearly variation in group–specific survival, we computed annual estimates of survival as the average over the 15 within–year capture occasions, with confidence intervals based on the ordinary variance among these empirical estimates. The confidence intervals thus include components of both sampling and temporal variation in within–year survival. We plotted these estimates over the years of the CJS analysis, together with the state–specific estimates of survival for 1996 from the multistate analysis. Birds captured in the "infected" state had a clearly lower probability of survival, lower than even the most extreme (early) years of the CJS analysis (fig. 2). We note that both the CJS and multistate models likely underestimate "true" survival, in that permanent emigration is confounded in these estimates of "apparent survival"; nevertheless we take these analyses as strong evidence of a state–specific influence on survival. However, these results are affected both by 1) potential misclassification of states (N or P) and 2) the fact that some asymptomatic (state N) birds may have been immune, due to previous infection. We discuss both of these issues in more detail below. For the covariate analysis, we used recaptures of 417 birds for which mass was recorded; 333 birds for which both mass and wing length were recorded, and 323 birds for which all covariates (mass, wing length, tail, and P3) were recorded. These data subsets were used to fit series of covariate models, summarized in table 4. Although several of the covariate models are close competitors to the "baseline" (no covariate) model, the baseline model was the top–ranked model in 2 of the 3 data subsets. Model averaging resulted in unconditional estimates of covariate effects with large standard errors and confidence intervals widely dispersed near the origin, indicative of weak effects (table 5). Furthermore, coefficient signs for comparable models sometime differed among data subsets positive for the first and third but negative for (e.g., the second subset). We conclude that the evidence for covariate effects on survival and transition is weak for this study, probably due to the relatively small sample of the "infected" state. On the other hand,

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% Infected birds by sample

40 35 30 25 20 15 10 5 0 Jun

Jul

Aug

Sep

Oct

Nov

Dec

Fig. 1. Apparent prevalence rate of Serin avian pox, computed as the percentage of infected birds from total number trapped by 15–day periods (n = 799 birds). Fig. 1. Tasa de prevalencia aparente de la viruela aviar en el Verdecillo, computado como el porcentaje de aves infectadas sobre el total de capturadas para periodos de 15 días (n = 799 aves).

other individual attributes, notably the state of pox infection, were clearly related to survival rates; furthermore, transition between states is clearly asymmetric, with birds surviving from pox more likely = 0.654) to move to the uninfected states than ( the reverse ( = 0.032).

Discussion The Serin avian pox outbreak mainly affected birds from September to November, which seems to be the most common period of high prevalence for this kind of disease (Davidson et al., 1980; Van Ripper

Table 2. Multistate models of state–specific survival, capture, and transition for Serins captured May– December 1996. a Factors used to model variation in 15–day survival (S), capture (p) and transition probabilities ( ): pox. Classified as uninfected (1) or infected (2) at time of release; as. Age–sex groupings (male, female, and juvenile [unsexed]); t. 15–day capture; b Quasilikelihood = 2.43. Tabla 2. Modelos de multiestado con tasas de supervivencia, captura y transición dependientes del estado, para los datos de Verdecillo capturados de mayo a diciembre de 1996. a Factores utilizados para modelar la variación en la tasa de supervivencia a 15 días (S), tasa de captura (p) y probabilidades de transición probabilities ( ): pox (viruela). Clasificada como no infectados (1) o infectados (2) en el momento de la liberación; as. Agrupación según edad–sexo (macho, hembra, y juvenil [no sexado]); t. Período entre capturas de 15 días, b Quasi razón de verosimilitud = 2,43.

Modela

QAICcb

Spox ppox+t

QAICc weights

Num. Par

pox+as

0.46837

pox+as

0.9563

0.29035

Spox+as ppox+t

pox

2.7874

0.11623

Spox+as ppox+t

pox+as

3.8465

0.06844

Spox ppox+t

5.343

0.03239

Sas ppox+t

pox

6.0193

0.02309

Spox ppox+t

12.0621

0.00113

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Table 3. Model–averaged estimates and unconditional confidence intervals for state–specific 15– day survival, capture, and transition for Serins captured May–December 1996: a Infection status determined by external examination (presence or absence of warty lesions characteristic of avian pox). "Not infected" birds may include some birds that have been previously infected and are likely immune (see text). Tabla 3. Estimas promediadas entre modelos e intervalos de confianza no condicionados para las tasas específicas de cada estado de supervivencia a los 15 días, de captura, y de transición para los Verdecillos capturados de mayo a diciembre de 1996. a Estatus de infección determinado mediante inspección externa (presencia o ausencia de lesiones ulcerosas características de la viruela aviar). La categoría de "No infectado" puede incluir a algunas aves previamente infectadas pero que han desarrollado inmunidad (ver texto). 95% CI a

Parameter

State

Lower

Upper

Survival (S)

Uninfected

0.868

0.025

0.809

0.910

Infected

0.458

0.171

0.180

0.765

0.194

0.089

0.074

0.422

0.242

0.079

0.121

0.426

0.234

0.072

0.122

0.400

0.154

0.051

0.078

0.282

0.019

0.017

0.003

0.104

0.048

0.031

0.013

0.158

0.253

0.065

0.147

0.400

0.342

0.071

0.218

0.492

0.246

0.062

0.145

0.387

0.315

0.074

0.190

0.474

0.108

0.047

0.045

0.240

0.210

0.067

0.108

0.369

0.391

0.092

0.231

0.578

0.041

0.031

0.009

0.168

0.296

0.083

0.162

0.479

0.861

0.215

0.153

0.995

0.886

0.176

0.205

0.9957*

0.882

0.181

0.198

0.996

0.832

0.254

0.123

0.994

0.373

0.444

0.014

0.961

0.602

0.424

0.045

0.980

0.890

0.167

0.221

0.996

Capture (p)

Uninfected

Infected

Transition ( )

Period

0.916

0.126

0.304

0.996

0.887

0.169

0.224

0.995

0.910

0.135

0.289

0.996

0.774

0.306

0.100

0.991

0.870

0.198

0.178

0.995

0.926

0.111

0.342

0.997

0.570

0.509

0.022

0.987

0.905

0.146

0.257

0.996

Uninfected

0.032

0.030

0.005

0.178

Infected

0.654

0.254

0.173

0.945

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Animal Biodiversity and Conservation 27.1 (2004)

Non infected 1.0

15–day survival

0.8

0.6 Infected

0.4

0.2

0 1988

Male Female Juvenile 1996–No pox 1996–pox

1990

1992

1994

1996 Year

1998

2000

2002

Fig. 2. Estimates of mean 15–day survival and empirical confidence intervals (within year) based on CJS modelling for Serins by age–sex group for 1990–2000 (excluding 1996). Superimposed estimated state–specific survival rates for 1996: Uninfected. Probability of survival for Serins uninfected by pox at time t; Infected. Probability of survival for infected birds. Fig. 2. Estimas de la tasa de supervivencia quincenal media e intervalos de confianza empíricos (dentro del año) basado en modelos para los Verdecillos de CJS, por grupo de edad–sexo para el periodo 1990–2000 (excluyendo 1996). Superpuesto se proporciona la estima de la tasa de supervivencia específica del estado para 1996: No infectado. Probabilidad de supervivencia para los Verdecillos no infectados por la viruela en el tiempo t; Infectado. Probabilidad de supervivencia para las aves infectadas.

III et al., 2002; Buenestado et al., in press). Serin avian pox seemed to be very virulent, reducing 15– day survival from 0.87 in healthy birds to 0.46 in infected birds, which means that more of 50% of sick birds did not survive to the first two weeks after infection. Nevertheless, if infected birds survive, probability to recovery from pox seemed quite high (0.65). This is in accordance with data from other species, which have reported a high percentage (15–60%) of birds with healed lesions from previous infections, which reflects that the bird survived to the disease (McClure, 1989; Van Ripper III et al., 2002). A higher recovery than infection rate has also been reported for the Mycoplasma outbreak in eastern United States (Faustino et al., in press). Models involving the effects of age and sex, and interaction with disease state, generally were not supported. This does not necessarily indicated that such effects did not occur, and may be due to the sparseness of our data. As noted earlier, two aspects of our data collection have implication for these analyses. First, because not all captured birds were subjected to histopathology, we may have incorrectly classified some birds as "not infected" that in fact had the disease; conversely, some birds exhibiting lesions may have been falsely identified as "infected" (although this is

less probable). Second, birds that have been previously infected with pox may exhibit no visible lesions, and are thus indistinguishable form birds that have not been infected. This potentially creates an indistinguishable mixture of previously exposed (and thus, presumed to be immune) birds, and birds that have not been exposed (and are therefore at risk to the disease). We consider both of these issues, and their implications for our study, below. Misclassification of states Van Riper et al. (2002) found that more than 90% (N = 22) of birds with lesions were histopathologically positive for the disease. We only histologically examined 1 bird, and so cannot compute an estimate of a "false positive" rate; however, we are confident that most if not all birds with lesions were either currently infected with pox, or had recently been infected and were recovering. We think that it is much more likely that we misclassified as uninfected, perhaps because they were mildly symptomatic (e.g., had few or no lesions; Seet et al., 2003) and these were missed during our field examinations. Classifying some infected birds as healthy should have caused our state–specific survival rates

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Table 4. Multistate models of covariate relationships to state–specific survival, capture, and transition for Serins captured May–December 1996: a Quasilikelihood = 2.43. Tabla 4. Modelos de multiestados de las relaciones de las covariantes con la tasa de supervivencia, de captura y de transición, dependientes del estado, para los Verdecillos capturados de mayo a diciembre de 1996: a Quasi razón de verosimilitud = 2,43.

Covariates included

Model

Mass

Spox ppox+t

QAICc

QAICc weights

Num. Par

0.000

0.338

pox

0.679

0.241

pox+wt

1.451

0.164

0.102

pox

Spox+wt ppox+t Spox ppox+t

Spox+wt ppox+t

pox+wt

2.390

Spox*wt ppox+t

pox

2.796

0.084

pox*wt

3.520

0.058

Spox ppox+t Spox*wt ppox+t Mass, wing length

pox*wt

6.553

0.013

Spox ppox+t

pox*wt+wng

0.000

0.414

Spox ppox+t

pox

1.406

0.205

2.571

0.114

2.573

0.114

3.038

0.091

3.948

0.058

9.109

0.004

0.000

0.662

1.552

0.305

6.480

0.026

9.083

0.007

Spox+wt+wng ppox+t Spox+wt ppox+t

pox+wt+wng

pox+wt

Spox+wng ppox+t

pox+wng

Spox+wt+wng ppox+t

pox

Spox*wt+pox*wng ppox+t Mass, wing length, tail, P3 Spox ppox+t

pox*wt+pox*wng

pox

Spox+wt+wng ppox+t Spox+tail+p3 ppox+t

pox+wt+wng pox+tail+p3

Spox+wt+wng+tail+p3 ppox+t

to be biased low, so that the actual relative survival of uninfected to infected birds would have been even greater than what we observed. On the other hand, our low estimated rates of infection (i.e., transition to infected from evidently uninfected) could have been partially an artefact of some infected birds being misclassified as uninfected. However, we think that the misclassification problem is relatively unimportant compared to the second problem related to immunity, discussed below. Additionally, if the presence of asymptomatic infected birds (see Seet et al., 2003) within Serins, had been very important, it should had reduced survival rate of "apparently" uninfected birds, but the comparison of this survival rate to that of birds in years with no pox, suggests that this is not the case. Unidentified asymptomatic (immune) birds Our statistical models, which are based on models of disease transmission (Bailey, 1975; Anderson & May, 1992; Clayton & Moore, 1997), differ from these in certain aspects that are critical to interpreting our results. Like our models, infectious disease models

pox+tail+p3

often define disease states, and model rates of transition between these states. Additionally, our models take into account imperfect and potentially heterogeneous detectability (capture), which otherwise could confound inferences. Disease models typically assume that living individuals are in 1 of 3 possible states: susceptible (X), i.e., never having been infected and thus having no immunity; infected (Y), and post–infected (Z), often (but not always) assumed immune (incapable of reinfection). We can model transition among these states by iXY, the probability that over the interval [i, i+1] a susceptible individual becomes infected iYZ, the probability that an infected animal becomes immune, and iZX, iZY the probabilities that a post–infected animal becomes either susceptible or reinfected. These last 2 probabilities are assumed zero in the case where exposure confers complete immunity; likewise it would ordinarily be assumed that iXZ = 0. i.e., an animal must first become infected before becoming "post–infected" or immune. If (as we have done in this paper) we model transition as the product survival conditioned on the state at the first occasion and movement between states, then

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Animal Biodiversity and Conservation 27.1 (2004)

Table 5. Model–averaged estimates and unconditional confidence intervals for estimates of covariate relationships for survival transition for Serins captured May–December 1996: LCI. Lower 95% confidence interval; UCI. Upper 95% confidence interval. Tabla 5. Estimas promediadas para los distintos modelos e intervalos de confianza no condicionados para las estimas de la relación de las covariantes con la tasa de supervivencia, para los Verdecillos capturados de mayo a diciembre de 1996: LCI. Intérvalo de confianza inferior del 95%; UCI. Intérvalo de confianza superior del 95%. Covariates included

Parameter

Mass

LCI

UCI

–0.149

0.638

–1.399

1.102

pox

2.060

0.698

0.694

3.430

0.080

0.260

–0.429

0.589

–0.001

0.254

–0.498

0.497

0.731

0.860

–0.954

2.416

pox

–3.744

0.898

–5.503

–1.984

–0.112

0.415

–0.925

0.701

0.019

0.329

–0.626

0.664

–0.282

0.686

–1.627

1.063

pox

–2.069

0.733

0.632

3.507

–0.071

0.173

–0.267

0.410

pox*wt 0

pox*wt

Mass, wing length

0.008

0.085

–0.157

0.174

pox*wt

–0.002

0.103

–0.204

0.200

pox*wng

–0.001

0.058

–0.114

0.113

wng

0.786

1.053

–1.278

2.851

pox

–4.223

1.273

–6.719

–1.728

–0.499

0.610

–1.695

0.698

wng pox*wt

0.289

–0.252

0.265

0.713

–1.592

1.204

pox

1.963

0.748

0.497

3.430

0.060

0.145

–0.225

0.344

wng

0.021

0.102

–0.178

0.220

tail

–0.003

0.048

–0.098

0.092

0.000

0.052

–0.101

0.102

0.929

1.014

–1.058

2.917

pox

–4.363

1.167

–6.651

–2.075

–0.248

0.460

1.150

0.654

0.199

0.354

–0.494

–0.892

–0.003

0.105

–0.208

0.202

0.014

0.127

0.235

0.262

pox*wt pox*wng

= SiX = SiX (1– i YZ = S iY i YY Y = S (1– i i

1.176

–0.291

0.132

–0.413

0.148

0.007

wng

0.405

–0.194

pox*wng

Mass, wing length, tail, P3

0.381 –0.001

i YZ i

YZ i

= SiZ (assuming immunity).

For the Serin problem, it seems to us that we cannot completely observe these 3 states. Rather,

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144

subject to misclassification, we believe that our observed state "2" corresponds to the state Y (infected), and retrospectively–i.e., when an observed capture of "2" is later followed by "1" (asymptomatic), we believe that we can infer Z, the post–infection state, but only if birds are recaptured and observed to be uninfected after having previously been observed to be infected. When birds are either observed for the first time as asymptomatic ("1"), recaptured as still asymptomatic (e.g., a capture history of 101), or never recaptured (e.g., 100), we think that these birds’ actual state is unknown (either X or Z). In principle it might also be possible to infer previous infection from the existence of healed lesions; however we observed very few of these; nearly all birds either had active lesions, or were observed to have no lesions. Antibody tests (which we did not conduct) also could reveal that the bird had previously been exposed, and was now immune (Z). Thus we conclude that our observable states consisted of Y (infected), Z (based on previous capture in the state Y), and U (unknown–free of lesions but not known based on prior capture to be either susceptible or immune). The state U is thus an (unknown) mixture of either susceptible (X) or immune (Z) NU = NX + (1 – )NZ with the mixing proportion. We can model observable transitions (conditional on recapture) transitions for these 3 states as: Pr(UU) =

XX i i

ZZ = i

+ (1 + i)

X + i

(1 – i)SiZ

ZY = i

SX i i

(assuming complete immunity, i.e.,

Pr(UY) =

XY i i

SX i i

Pr(YY) =

+ (1 + i)

= SiY (1 –

Pr(YZ) =

= Si

Pr(ZZ) =

ZZ i

YZ i

ZZ , i

XY i

= 0,

apparent survival spans for several years in which there were no observed outbreaks of pox, and we were confident that these estimates well represent background survival for non–infected (X) birds. = 0.03) Our estimate of low apparent infection ( contrasts with the high apparent prevalence rate (> 30%) found at the peak of the infection, by simple inspection of number of infected individuals from the total trapped birds. As noted above, this estimate undoubtedly underestimates true probability of infection (i.e., iXY) to the degree that < 1. However, we think that a more likely explanation for this low estimate is fact that recapture rates of symptomatic birds were consistently higher than those of asympotamic birds. Even accounting for the fact that some misclassification likely occurred, this suggests that infected birds are more prone to capture, possibility a consequence of greater dependency on an easy food source (McClure, 1989). We believe that these relatively higher recapture rates result in inflated estimates of disease prevalence rate, when such estimates rely on unadjusted capture frequencies. To illustrate this point we constructed a simple model, to predict the actual (versus apparent) incidence of pox; for the purposes of this illustration we assume that no misclassification occurs, and that confine inferences to the observable states of asymptomatic (1) and symptomatic (2). In terms of the parameters we have estimated, the probability of being in state s for a bird in the population at time i depends on 3 elements: (1) the state the bird was in (infected or not) at time i – 1, (2) the probability of survival, dependent on state at time i – 1, and (3) the probability of moving from one state to the next, given the state at time i – 1. For the state s = 1 (asymptomatic), this can be written as

= Si ,

(assuming complete immunity, i.e., ZY = 0). If immunity is incomplete, then ZY > 0, ZY < 1, complicating Pr(UY) and Pr(ZZ). We make the following conclusions regarding our inferences with CMR data: First, our estimated survival rates for negative (s = 1) birds unidentifiably mixes survival for X (susceptible) and Z (immune birds). Because the latter may be higher than the former this overall might be thought of as an overestimate. Second, the ‘infection rate’ i12 confounds the probability of being susceptible (i.e., being a member of the NX, with iXY the probability of infection for susceptible animals; because in general < 1 this should result in underestimating iXY). Third, the survival rate for infected birds, and probability of recovery, should still be unbiased, at least under the assumption that birds become immune once infected and that virulence of avian pox in symptomatic and asymptomatic infected birds is similar. With regard to the first point, our CJS estimates of

Likewise, the probability of s = 2 (symptomatic) is

Both of these event probabilities involve state– specific survival from i – 1 to i, because irs = Sir irs. The proportion of birds at time i that are infected, is therefore

Because of the recursive nature of this expression, it has no solution without imposing initial conditions. If we assume an initial period before the outbreak of pox (t = 0), in our case May or earlier, then we can specify that Pr(s 0 = 2) = 0, Pr(s0 = 1) = 1. These leads to an ability to recursively predict P(i) from the above expression. We have done so using our point estimates of survival and

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Animal Biodiversity and Conservation 27.1 (2004)

Predicted prevalence

0.05 0.04 0.03 0.02 0.01 0.00

May

Jul

Sep

Nov

Jan

Fig. 3. Predicted incidence of pox based on estimates of state–specific survival and transition (table 3) and assuming 0 incidence before May and 0 probability of transition from uninfected to infected after November. Fig. 3. Incidencia de la viruela predicha según las estimas de supervivencia y transición específicas del estado (tabla 3), asumiendo una incidencia 0 antes de mayo y una probabilidad 0 de transición del estado no infectado al infectado a partir de Noviembre.

transition parameters (table 3), using the additional assumption that of no further N to S transition following November (our last observed incidence of pox), that is i12 = 0 after this point. The predictions are displayed in figure 3, which shows a predicted asymptotic increase to a prevalence of about 4% by the end of the outbreak period, followed by a sharp decline (with the only remaining infestations being unrecovered, infected birds). This is in contrast to the apparent prevalence of pox (fig. 1), which does not properly take into account conditioning on survival, transition, and recapture rates, thus tending to overestimate prevalence. Our results indicate that caution is needed when estimating the prevalence of a disease in natural populations, either when relying on visual surveys at bird tables or on the capture of individuals (especially at baited traps). We stress the need to estimate recapture rates when estimating population dynamics parameters, a point that is repeatedly raised in all EURING conferences but that ecologists and evolutionary biologists are frequently reluctant to accept (Lebreton et al., 1993). Acknowledgements We are most grateful to Jordi Domènech, David Boné, Esther Vilamajor and Anna Serra for field assistance and Lluïsa Arroyo for assistance in the lab. We also thank Chris Jennelle for many useful comments and discussion on the paper. This work was funded by BOS 2003–09589 research project from the Spanish Research Council, Ministerio de Ciencia y Tecnología.

References

Anderson, R. & May, R., 1992. Infectious Diseases Of Humans. Oxford Univ. Press, Oxford. Arnall, L. & Keymer, I. F., 1975. Bird diseases. T. F. H. Publications, Neptune City, N.J. Bailey, N. T. J., 1975. Mathematical Theory of Infectious Diseases and its Applications. Charles Griffin, London. Benning, T. L., Lapointe, D., Atkinson, C. T. & Vitousek, P. M., 2002. Interactions of climate change with biological invasions and land use in the Hawaiian Island: Modeling the fate of indemic birds using a geographic information system. Proceedings of the National Academy of Sciences of the United States of America, 99: 14246–14249. Boch, H. & Schneidawind, H., 1994. Krankheiten des Jagdbaren Wildes. Parey, Berlin. Brawner, W. R., Hill, G. E. & Sunderman, C. A., 2000. Effects of coccidial and mycoplasmal infections on carotenoid–based plumage pigmentation in male house finches. The Auk, 117: 952–963. Buenestado, F., Gortazar, C., Millán, J., Höfle, U. & Villafuerte, R. (in press). Descriptive study of an avian pox outbreak in wild Red–Legged Partridges (Alectoris rufa) in Spain. Epidemiology and Infection. Burnham, K. P. & Anderson, D. R. 2002. Model selection and multimodel inference: a practical, information–theoretic approach. Springer Verlag, Berlin. Clayton, D. H. & Moore, J., 1997. Host–Parasite Evolution: general principles & avian models. Oxford Univ. Press, Oxford. Conroy, M. J., Senar, J. C., Hines, J. E. & Domènech,

146

J., 1999. Development and application of a mark– recapture model incorporating predicted sex and transitory behaviour. Bird Study, 46: S62–S73. Davidson, W. R., Kellogg, F. E. & Doster, G. L., 1980. An epornitic of avian pox in wild bobwhite quail. J. Wildl. Dis., 16: 293–298. Del Pino, M., 1977. Enfermedades de los pájaros de jaula. Editorial Aedos, Barcelona. Dhondt, A. A., Tessaglia, D. L. & Slothower, R. L., 1998. Epidemic mycoplasmal conjunctivitis in House Finches from eastern North America. J. Wildl. Dis., 34: 265–280. Domènech, J. & Senar, J. C., 1997. Trapping methods can bias age ratio in samples of passerine populations. Bird Study, 44: 348–354. – 1998. Trap type can bias estimates of sex ratio. Journal of Field Ornithology, 69: 380–385. Domènech, J., Senar, J. C. & Conroy, M. J., 2001. Birds captured at automatic baited traps are heavier. Butlletí Grup Català d’Anellament, 18: 1–8. Faustino, C. R., Jennelle, C. S., Connolly, V., Davis, A. K., Swarthourt, E. C., Dhondt, A. A. & Cooch, E. G. (in press). Mycoplasma gallisepticum infection dynamics in a house finch population: seasonal variation in survival, encounter and transmission rate. Journal of Animal Ecology. Fischer, J. R., Stallknecht, D. E., Luttrell, M. P., Dhondt, A. A. & Converse, K. A., 1997. Mycoplasmal conjunctivitis in wild songbirds: the spread of a new contagious disease in a mobile host population. Emerg. Infect. Dis., 3: 69–72. Gog, J., Woodroffe, R. & Swinton, J., 2002. Disease in endangered mertapopulations: the importance of alternative hosts. Proceedings of the Royal Society of London series B, 269: 671–676. Gortazar, C., Millan, J., Hofle, U., Buenestado, F. J., Villafuerte, R. & Kaleta, E. F., 2002. Pathology of Avian Pox in Wild Red–Legged Partridges (Alectoris rufa) in Spain. Annals of the New York Academy of Sciences, 969: 354–357. Hansen, W. R., 1987. Avian pox. In: Field guide to wildlife diseases. General field procedures and diseases of migratory birds: 135–141 (M. Friend, Ed.). Madison, Wisconsin. – 2004. Avian pox. In: Field manual of wildlife diseases. General field procedures and diseases of birds: 163–169 (M. Friend & J. C. Franson, Eds.). Biological Resources Division. U.S. Dept. Interior & U.S. Geological Survey, Madison, Wisconsin. Hartup, B. K., Bickal, J. M., Dhondt, A. A., Ley, D. H. & Kollias, G. V., 2001. Dynamics of conjunctivitis and Mycoplasma Gallisepticum infections house finches. The Auk, 118: 327–333. Hess, G., 2003. Disease in Metapopulation Models: Implications for Conservation. Ecology, 77: 1617– 1632. Hochachka, W. M. & Dhondt, A. A., 2000. Density– dependent decline of host abundance resulting from a new infectious disease. Proceedings of

Senar & Conroy

the National Academy of Sciences of the United States of America, 97: 5303–5306. Hudson, P., Rizzoli, A., Dobson, A., Heesterbeek, H. & Dobson, A., 2001. The Ecology of Wildlife Diseases. Oxford Univ. Press, Oxford. Lebreton, J. D., Pradel, R. & Clobert, J., 1993. The statistical analysis of survival in animal populations. Trends in Ecology and Evolution, 8: 91–95. May, R. M., 1983. Parasitic infections as regulators of animal populations. American Scientist, 71: 36–45. McClure, E., 1989. Epizootic lesions of house finches in Ventura country, California. Journal of Field Ornithology, 60: 421–430. Newton, I., 1998. Population limitation in birds. Academic Press, San Diego. Nolan, P. M., Hill, G. E. & Stoehr, A. M., 1998. Sex, size, and plumage redness predict house finch survival in an epidemic. Proceedings of the Royal Society of London series B, 265: 961–965. Ralph, C. J. & Fancy, S. G., 1995. Demography and movements of apapane and Iiwi in Hawaii. The Condor, 97: 729–742. Scott, M. E., 1988. The impact of infection and disease on animal populations: implications for conservation biology. Conservation Biology, 2: 40–56. Seet, B. T., Johnston, J. B., Brunetti, C. R., Barrett, J. W., Everett, H., Cameron, C., Sypula, J., Nazarian, S. H., Lucas, A. & McFadden, G., 2003. Poxviruses and immune evasion. Annual Review of Immunology, 21: 377–423. Senar, J. C., 1986. Gafarró (Serinus serinus). In: Historia Natural dels Països Catalans: 344– 345. (R. Folch, Ed.). Enciclopedia Catalana, Barcelona. Thompson, C. W., Hillgarth, N., Leu, M. & McClure, H. E., 1997. High parasite load in house finches (Carpodacus mexicanus) is correlated with reduced expression of a sexually selected trait. The American Naturalist, 149: 270–294. Van Ripper III, C., Van Ripper, S. G. & Hansen, W. R., 2002. Epizootiology and effect of avian pox on Hawaiian forest birds. The Auk, 119: 929–942. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46: S120–S139 Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and Management of Animal Populations: Modeling, estimation, and decision making. Academic Press, New York. Woodroffe, R., 1999. Managing disease threats to wild mammals. Animal Conservation, 2: 185–193. Yunick, R. P., 1971. A platform trap. EBBA News, 34: 122–125. Zahn, S. N. & Rothstein, S. I., 1999. Recent increase in male House Finch plumage variation and its possible relationship to avian pox disease. The Auk, 116: 35–44.

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Methodological advances J.–D. Lebreton & K. H. Pollock

Lebreton, J.–D. & Pollock, K. H., 2004. Methodological advances. Animal Biodiversity and Conservation, 27.1: 147–148. The study of population dynamics has long depended on methodological progress. Among many striking examples, continuous time models for populations structured in age (Sharpe & Lotka, 1911) were made possible by progress in the mathematics of integral equations. Therefore the relationship between population ecology and mathematical and statistical modelling in the broad sense raises a challenge in interdisciplinary research. After the impetus given in particular by Seber (1982), the regular biennial EURING conferences became a major vehicle to achieve this goal. It is thus not surprising that EURING 2003 included a session entitled "Methodological advances". Even if at risk of heterogeneity in the topics covered and of overlap with other sessions, such a session was a logical way of ensuring that recent and exciting new developments were made available for discussion, further development by biometricians and use by population biologists. The topics covered included several to which full sessions were devoted at EURING 2000 (Anderson, 2001) such as: individual covariates, Bayesian methods, and multi–state models. Some other topics (heterogeneity models, exploited populations and integrated modelling) had been addressed by contributed talks or posters. Their presence among "methodological advances", as well as in other sessions of EURING 2003, was intended as a response to their rapid development and potential relevance to biological questions. We briefly review all talks here, including those not published in the proceedings. In the plenary talk, Pradel et al. (in prep.) developed GOF tests for multi–state models. Until recently, the only goodness–of–fit procedures for multistate models were ad hoc, and non optimal, involving use of standard tests for single state models (Lebreton & Pradel, 2002). Pradel et al. (2003) proposed a general approach based in particular on mixtures of multinomial distributions. Pradel et al. (in prep.) showed how to decompose tests into interpretable components as proposed by Pollock et al. (1985) for the Cormack– Jolly–Seber model Pledger et al. (in prep.) went on in their thorough exploration of models with heterogeneity of capture (Pledger & Schwarz, 2002; Pledger et al., 2003), by considering the use of finite mixture models for the robust design. Given the level of details in demographic traits presently addressed by capture–recapture, the problem of heterogeneity, once apparently settled by fairly reassuring messages (Carothers, 1973, 1979), is becoming again a central issue, with potential disastrous consequences if improperly handled. Heterogeneity models, that bear also a relationship to "multi–event models" (Pradel, in press), will thus certainly be increasingly useful. Pollock, Norris, and Pledger (in prep.) reviewed the capture–recapture models as applied to community data (Boulinier et al., 1998) and developed general removal and capture– recapture models when multiple species are sampled to estimate community parameters. Because of unequal delectability between species, these approaches bear a clear relationship to heterogeneity models, which will be more and more a reference for comparative studies of communities and "macroecology" (Gaston & Blackburn, 2000).

Jean–Dominique Lebreton, CEFE / CNRS, 1919 Route de Mende, 34 293 Montpellier cedex 5, France. E–mail: lebreton@cefe.cnrs-mop.fr Kenneth H. Pollock, Zoology, Biomathematics and Statistics, North Carolina State Univ., Box 7617, Raleigh NC 27695–7617, U.S.A. E–mail: pollock@unity.ncsu.edu ISSN: 1578–665X

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Bonner & Schwarz (2004) proposed a capture–recapture model with continuous individual covariates changing over time more fully developed in Bonner & Schwarz (2004). The difficulty here is to set up a sub– model predicting the covariate value when an individual is not captured. While multi–state models permit an ad hoc treatment by categorizing the covariate, Bonner and Schwarz bring a sound answer by considering the covariate obeys a Markov chain with continuous state–space. Otis & White (2004) presented a thorough, simulation–based, investigation of two approaches used to test the contrasting hypotheses of additive and compensatory hunting mortality based on band recovery data. The two approaches are the usual ultra–structural model and a new one based on a random effects model. This paper can be viewed as part of a revival of studies of the dynamics of exploited populations, in the broad sense, including the study of man–induced mortality in the framework of conservation biology (Lebreton, in press). This revival is a direct consequence of the increasing impact of man on the biosphere and of continuing methodological progress (Ferson & Burgman, 2000). The use of random effects models (see also Schaub & Lebreton, 2004) directly builds upon the seminal work by Anderson and Burnham (1976). Stauffer presented a Winbugs implementation of the Cormack–Jolly–Seber model that complemented other presentations in the conference and the short course. Finally, Morgan, Besbeas, Thomas, Buckland, Harwood, Duck and Pomery, proposed a thorough and timely review of integrated modelling, i.e., in our context, of models considering simultaneously capture–recapture demographic information and census information. These methods were covered in other sessions, in relation to Bayesian methodology. Integrated modelling appears indeed to be the logical way of combining all pieces of information arising from integrated monitoring, and as one of the great methodological challenges for our community in the years to come (Besbeas et al., 2002). Methodological progress in population dynamics is apparently still on an upward trajectory and we look forward to many exciting contributions at future EURING conferences! References Anderson, D. R., 2001. Euring 2000 conference. Euring Newsletter 3, July 2001. www.euring.org/about_euring/newsletter3/anderson.htm Anderson, D. R. & Burnham, K. P., 1976. Population ecology of the Mallard. VI the effect of exploitation on survival. US DI, Fish and WIldlife Service, Washington D.C. Besbeas, P., Freema, S. N., Morgan, B. J. T. & Catchpole, E. A., 2002. Integrating mark–recapture–recovery and census data to estimate animal abundance and demographic parameters. Biometrics, 58: 540–547. Bonner, S. J. & Schwarz, C. J., 2004. Continuous time–dependent Individual covariates and the Cormack– Jolly–Seber model. Animal Biodiversity and Conservation, 27.1: 149–155. – (in revision). An extension of the Cormack–Jolly–Seber model for continuous covariates with application to Microtus pennsylvanicus. Biometrics. Boulinier, T., Nichols, J. D., Sauer, J. R., Hines, J. E. & Pollock, K. H., 1998. Estimating species richness: the importance of heterogeneity in species detectability. Ecology, 79: 1018–1028. Carothers, A. D., 1973. The effect of unequal catchability on Jolly–Seber estimates. Biometrics, 29(1): 79–100. – 1979. Quantifying unequal catchability and its effects on survival estimates in an actual population. Journal of Animal Ecology, 48: 863–869. Ferson, S. & Burgman, M. (Eds.), 2000. Quantitative methods for conservation biology. Springer, New York. Gaston, K. J. & Blackburn T. M., 2000. Pattern and process in macroecology. Blackwell Science, Oxford. Lebreton, J. D. (in press). The dynamics of exploited populations. Australian and New–Zealand journal of Statistics. Lebreton, J. D. & Pradel, R., 2002. Multistate recapture models: modeling incomplete individual histories. Journal of Applied Statistics, 29: 353–369. Otis, D. L. & White, G. C., 2004. Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations. Animal Biodiversity and Conservation, 27.1: 157–173. Pledger, S., Pollock, K. H. & Norris, J. L., 2003. Open capture–recapture models with heterogeneity. I. Cormack–Jolly–Seber model. Biometrics, 59: 786–794. Pledger, S. & Schwarz, C. J., 2002. Modelling heterogeneity of survival in band–recovery data using mixtures. Journal of Applied Statistics, 29: 315–327. Pollock, K. H., Hines, J. E. & Nichols, J. D., 1985. Goodness–of–fit tests for open capture–recapture models. Biometrics, 41: 399–410. Pradel, R. (in press). Multievent models. Biometrics. Pradel, R., Gimenez, O. & Lebreton, J.–D. (in prep.). Principles and interest of GOF tests for multistate models. Pradel, R., Wintrebert, C. & Gimenez, O., 2003. A proposal for a goodness–of–fit test to the Arnason– Schwarz multisite capture–recapture model. Biometrics, 59: 43–53. Seber, G. A., 1982. The estimation of animal abundance and related parameters. Griffin, London. Sharpe, F. R. & Lotka, A. J., 1911. A problem in Age–distribution. Philosophical magazine, series 6 21: 435–438.

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Continuous time–dependent individual covariates and the Cormack–Jolly–Seber model S. J. Bonner & C. J. Schwarz

Bonner, S. J. & Schwarz, C. J., 2004. Continuous time–dependent Individual covariates and the Cormack– Jolly–Seber model. Animal Biodiversity and Conservation, 27.1: 149–155. Abstract Continuous time–dependent individual covariates and the Cormack–Jolly–Seber model.— The Cormack– Jolly–Seber model provides the basic framework for analyzing the survival of animals in open populations using capture–recapture data. Extensions of this model have already been developed that allow the survival and capture probabilities to vary between individuals based on auxiliary variables, but none can allow for variables that are continuous, time–dependent, and vary among individuals. We summarize a new method for incorporating this type of variable into the Cormack–Jolly–Seber model by modelling the distribution of the unobserved values of the variable conditional on the observed values, given a few basic assumptions about how the variable changes over time. We begin with a hypothetical scenario as motivation for our model and also present the results of two examples used in developing the model. Key words: Capture–recapture, Cormack–Jolly–Seber model, Auxiliary variables, Continuous time–dependent individual covariates. Resumen Covariantes continuas individuales dependientes del tiempo y el modelo de Cormack–Jolly–Seber.— El modelo de Cormack–Jolly–Seber proporciona el marco básico para analizar la supervivencia de animales en poblaciones abiertas utilizando datos de captura–recaptura. Si bien se han desarrollado ampliaciones de este modelo que permiten variar las probabilidades de supervivencia y de captura entre individuos a partir de variables auxiliares, en ninguna de ellas es posible utilizar variables continuas, dependientes del tiempo y que varíen de un individuo a otro. El presente estudio analiza un nuevo método que permite la incorporación de este tipo de variable en el modelo de Cormack–Jolly–Seber mediante la modelación de la distribución de los valores no observados de la variable según los valores observados, tomando como referencia algunas asumciones básicas acerca de cómo la variable cambia con el tiempo. En primer lugar, presentamos un escenario hipotético con objeto de definir el modelo, para posteriormente indicar los resultados de dos ejemplos que utilizamos para su desarrollo. Palabras clave: Captura–recaptura, Modelo de Cormack–Jolly–Seber, Variables auxiliares, Covariantes continuas individuales dependientes del tiempo. C. J. Schwarz & S. J. Bonner, Dept. of Statistics and Actuarial Science, Simon Fraser Univ., 8888 University Drive, Burnaby BC, V5A 1S6 Canada. Corresponding author: C. J. Schwarz. E–mail: cschwarz@stat.sfu.ca

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Introduction The basic model for studying survival probabilities in open animal populations using capture–recapture data is the Cormack–Jolly–Seber (CJS) model (Cormack,1964; Jolly, 1965; Seber, 1965). In the model’s original definition, the survival and capture probabilities are assumed to be homogeneous for all animals in the population. In many applications this assumption is not justified and the capture and survival probabilities are allowed to vary as functions of auxiliary variables (Pollock, 2002). These variables may be classified as either continuous or discrete, time–dependent or constant, and individual or external (also called environmental). Here we present a new model aimed specifically at incorporating data from continuous, time–dependent, individual covariates into the CJS framework. While there are eight possible classifications of variables using the system above, five distinct cases need to be considered separately. The simplest case involves covariates which are both discrete and constant over time. For these variables, both external and individual, the captured animals can be divided into distinct groups based on the value of the variable and survival modelled separately in each group. Two examples are the effect of gender on the European Dipper (Cinclus cinclus) and the comparison of survival in two Swift (Apus apus) colonies, both from Lebreton et al. (1992). When the variable under consideration is continuous, but still constant over time, the survival and capture probabilities are both modelled using the framework of generalized linear modelling (McCullagh & Nelder, 1989). That is, the survival and capture probabilities are each considered dependent on a linear transform of the variable through a selected link function. An example is the study of the effect of agricultural pesticide use on Sage Grouse (Centrocercus urophasianus), which models survival as a function of blood cholinesterase using the log–log link to achieve proportional hazards (Skalski et al., 1993). The same method can be used for time–dependent, external covariates, both discrete and continuous, the only difference being that for such variables the linear predictor changes over time but is the same for all individuals. Early examples incorporating time–dependent, external covariates include studies of the effect of winter temperatures on the survival of the Grey Heron (Ardea cinerea) using recoveries of dead birds only (North & Morgan, 1979), and on the survival of the European Starling (Sturnus vulgaris) using recapture data (Clobert & Lebreton, 1985). The difficulty posed in the cases involving time– dependent, individual variables is that the value of the variable is unknown for some events that contribute to the model’s likelihood (i.e. capture occasions when a previously captured animal is not observed). In the case of discrete, time–dependent, individual variables, a solution to this problem is the multi–state model which incorporates a Markov chain into the CJS model to describe the movement of individuals between the different values of the

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variable (Arnason, 1973; Schwarz et al., 1993). The likelihood contribution for each individual is then adjusted by including the transition probabilities for every pair of consecutive states and summing over all possible values of the unobserved quantities. When the variable is continuous the missing values cannot be modelled by a Markov chain which takes a finite number of values. Thus, it is necessary to develop a new model predicting the distribution of the missing values of the continuous variable. This paper describes one particular model. As part of the EURING Technical Meeting’s proceedings, this paper is not intended to provide mathematical details of our model, of its derivation, or of its parameter estimation. Instead, we provide a heuristic derivation of the model, a brief discussion of the method used to estimate the model parameters, and a description of two examples that were used during the model’s development. Full mathematical details of the model and parameter estimation methods can be found in Simon Bonner’s M.Sc. report, which is available on–line (Bonner, 2003). A manuscript with full details has been submitted to Biometrics (Bonner & Schwarz, 2003). Methods In this paper we consider only the simplest case with one scalar covariate and k equally–spaced capture occasions. The basic idea behind our model is that animals living in the same study area should react similarly to the forces in their environment and so should have similar changes in the value of the covariate. For example, imagine a hypothetical capture–recapture experiment with 5 capture occasions designed to study the relationship between body mass and survival of some animal. One would expect that when food is plentiful and competition low, all animals gain mass and when conditions worsen or food becomes scarce, all animals lose mass. Possible records for four individuals are shown in figure 1. Three of the individuals are captured on all five occasions and show a definite trend in body mass, perhaps in response to environmental stresses. Between occasions two and three the animals gain mass and between occasions three and four they lose mass. The changes are similar for all animals, though small differences do occur as a result of individual variation. The fourth individual is captured only on occasions 2 and 4, and in order to make estimates of the effect of body mass on survival it is necessary to have some inference of the distribution of the individual’s mass at the other occasions. The proposed model does this by supposing that the body mass of this individual follows the same trends observed in the other 3 individuals. That is, the body mass rises between capture occasions 2 and 3, drops between capture occasions 3 and 4, and remains almost constant between capture occasions 4 and 5. Note that it is not necessary to make inference about the mass on the first occasion because the CJS model conditions on the animal’s first release.

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3 4 Capture occasion

Fig. 1. Hypothetical body mass observations for four individuals in a capture–recapture experiment with five capture occasions. Observations for each individual indicated by a distinct plotting symbol. Fig. 1. Observaciones hipotéticas de la masa corporal de cuatro individuos en un experimento de captura–recaptura que incluía cinco ocasiones de captura. Las observaciones correspondientes a cada individuo se indican por medio de una línea discontinua.

More specifically, our model imposes the assumption that for all individuals the change in the value of the variable between capture occasions t and t + 1 is normally distributed with mean t, which varies with t, and constant variance, 2. Denoting the values of the variable for individual i at capture occasion t by zit, this defines the conditional relationship: zi,t+1xzit i N(zit +

)

(1)

The values of the covariate for a single individual form a Markov chain in discrete time with continuous state–space, in some sense, the logical extension of the multi–state model to the continuous case. In accordance with our motivating example, the distribution of the change in the covariate for individual i at one capture occasion given the value at the previous occasion is normally distributed with the same mean for all individuals. Values of the continuous variable are linked to the survival and capture probabilities using the logistic link function (Lebreton et al., 1992). Denoting the capture and survival probabilities as functions of the covariate by (z) and p(z) respectively, the link functions are: (z) =

and

p(z) =

(2)

where ( 0, 1) and ( 0, 1) are the vectors of coefficients of the survival and capture curves respectively. Ultimately, these coefficients determine how the covariate affects the survival and capture probabilities. The model with a single scalar covariate contains k + 4 parameters to estimate: k – 1 mean differences ( 1,..., k–1), the variance parameter ( 2),

and the four coefficients of the logistic functions (( 0, 1) and ( 0, 1)). The likelihood function for the model is similar to the likelihood of the basic CJS model with three modifications. First, the survival and capture probabilities are replaced by the functions of the covariate in equation 2. Second, a new product of terms is added for each individual which accounts for the changes in the covariate between each pair of adjacent capture occasions. Third, for each individual it is necessary to integrate their contribution to the likelihood with respect to each unobserved covariate value in order to account for all possible configurations of the missing covariates. The second and third modifications are analogous to summing over all possible transitions for the missing states in the likelihood of the multi–state model. However, the integrals are potentially of dimension k – 1 and it is impossible to find maximum likelihood estimates of the parameters analytically. Instead, parameter estimates are computed via a Bayesian estimation scheme using the Metropolis–Hastings algorithm. As discussed in the EURING 2003 short course, Bayesian analysis is particularly well suited for estimation problems involving large proportions of missing data, including capture–recapture experiments. The primary reason for this is that Bayesian statistics does not differentiate between the unknown parameters and the missing data in the same way that frequentist statistics does. Rather, both parameters and missing data are considered as unknown random variables with some underlying distribution. In a Bayesian analysis, inference about the parameters is derived from the posterior distribution, which is proportional to the product of the prior distributions and the model’s likelihood function (Carlin & Louis, 1996). In both examples

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1.0

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0.6 0.4

0.2

0.0

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30 40 50 Body mass

Fig. 2. Estimated survival (left) and capture (right) probabilities as functions of body mass (g) for the Meadow Vole (solid lines) with point–wise 95% credible intervals (broken lines). Probabilities estimated for the multi–state model are shown as point estimates with 95% confidence intervals for each of the three states. Fig. 2. Supervivencia estimada (izquierda) y probabilidades de captura (derecha) como funciones de masa corporal (g) para el topillo de Pensilvania (líneas continuas) con intervalos puntuales creíbles al 95% (líneas discontinuas). Las probabilidades estimadas para el modelo multiestado se indican como estimaciones puntuales con intervalos de confianza al 95% para cada uno de los tres estados.

presented, the prior distributions were chosen to simplify the calculations needed to compute estimates rather than to represent true a priori information. Conjugate priors were used for the expected change on each occasion, t, and the variance, 2, and these were the normal and the inverse Gaussian, respectively. Improper flat priors with equal mass at all points of the real line were chosen for the coefficients of both the survival and capture probability functions. As is the case with most analyses involving a large number of parameters, the posterior distribution in the examples does not belong to a regular family of distributions and numerical methods were needed to generate parameter estimates. We used the Metropolis–Hastings (MH) algorithm, a specific Markov chain Monte Carlo (MCMC) technique that is common in Bayesian analysis (Chib & Greenberg, 1995). Like all MCMC methods the MH algorithm works by successively simulating values of the parameters in such a way as to generate a Markov chain whose stationary distribution is equal to the posterior distribution. Using selected starting values, the chain is iterated many times until it appears that the tail of the chain has converged in distribution. Values of the parameters from the remaining iterations of the chain are then used as if they formed a random sample from the posterior distribution. In the following examples, the algorithm was iterated 1,000,000 times in total and values from the final 200,000 iterations were used to

generate point estimates and credible intervals for each parameter (Louis & Carlin, 1996). Multiple chains were run using different starting values to check the convergence of the chain. Results Meadow Voles The primary data set used in developing the model concerned the effect of body mass on the survival of the North American Meadow Vole (Microtus pennsylvanicus). The data set contained observations of 515 voles captured at the Patuxent Wildlife Research Center, Maryland, on four capture occasions in the fall of 1981 and spring of 1982. To satisfy the assumption that all animals change weight in a similar manner, captures of juvenile animals were removed from the data set. A vole was considered juvenile if its mass was less than 22 g (Nichols et al., 1992) and for each individual, only the observations where the mass was actually less than this mark were deleted. No individual was observed with a mass less than 22 g after being captured with a mass greater than 22 g. Records for individuals captured only on the last occasion were also removed because they do not contribute to the likelihood function. The final data set comprised 450 captures of 199 voles.

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0.6 0.4

0.2

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0.0 32

36 BCI

36 38 BCI

Fig. 3. Estimated survival (left) and capture (right) probabilities as functions of body condition index (g/mm) for the Lesser Snow Goose (solid lines) with point–wise 95% credible intervals (broken lines). Probabilities estimated for the multi–state model are shown as point estimates with 95% confidence intervals for each of the two states. Fig. 3. Supervivencia estimada (izquierda) y probabilidades de captura (derecha) como funciones de condición corporal (g/mm) para el ansar nival (líneas continuas) con intervalos puntuales creíbles al 95% (líneas discontinuas). Las probabilidades estimadas para el modelo multiestado se indican como estimaciones puntuales con intervalos de confianza al 95% para cada uno de los dos estados.

The estimated capture and survival probabilities as functions of the voles’ body mass are in shown figure 2 along with point–wise 95% credible intervals. The estimated survival probability decreases slightly with the weight of the animal and the estimated capture probability increases slightly, though neither of these effects is significant at the = 0.05 level. A constant survival probability near 0.8 and constant capture probability near 0.9 appears to fit all animals. For comparison, a multi–state model using discrete mass classes was also fit to the data. Following Nichols et al. (1992), the body mass for the adult voles was divided into three states (22–33 g, 34–45 g, and > 45g) and estimates of the capture and survival probabilities for each state (fig. 2), along with the transition probabilities, were computed using Program MARK (White & Burnham, 1999). To match the assumptions of our model, transition probabilities in the multi–state model were allowed to change over time, but survival and capture probabilities were assumed constant. For both survival and capture there is considerable overlap of the 95% confidence intervals for the point estimates from the multi–state model and the point– wise 95% credible intervals for our continuous probability functions over the observed range of body mass. This suggest no discrepancies between the two methods. As with our method, the multi–state model shows no significant differences in either the survival or capture probabilities.

Snow Geese The second data set used in developing the model contained information on 31,240 Lesser Snow Geese (Chen carulescens) captured in Northern Manitoba over a 19–year period. Capture occasions occurred on an annual basis when the adult geese underwent post–breeding molt and became flightless for a short period of time. In this experiment, the researchers also weighed the birds and collected vital body measurements at each capture. Here we have examined the effect of a body condition index (BCI) defined as the ratio of body mass to culmen length (g/mm). Because the MH algorithm for computing the parameter estimates was computationally intensive, the analysis was restricted to a 6–year subset of the original data for the purpose of model development. Further, the records for several geese were missing values for either body mass or culmen length on at least one occasion and all records for these individuals were removed. As in the previous example, observations for juvenile birds, identified at the time of capture, and for birds captured only on the 6th occasion were also removed from the data set. The final data set contained a total of 474 observations for 314 geese. As in the previous example, the results show no significant effect of BCI on either the survival or capture probability of the geese (fig. 3). The 95%

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credible intervals are relatively wide and suggest a survival probability of 0.75 and capture probability of 0.3 for all birds. As above, a multi–state model was also fit for comparison with our model. In this case the geese were divided into 2 states using the median observed value of the BCI. Again, there is significant overlap between the 95% credible intervals for the continuous capture and survival probability functions and the 95% confidence intervals of the point estimates from the mulit–state model (fig. 3). However, while the estimated survival probability function passes very close to the survival estimates for the multi–state model, there appears to be some discrepancy between the estimated capture probability function and multi–state capture estimates. We believe that this might be an indication that the logistic link function is not appropriate in this case. If there actually was a significant change in the capture probability of the magnitude suggested by the estimates from the multi–state model, then the continuous capture probability function would need to be very steep near the median BCI value. For a logistic curve, this would result in very low capture probabilities for birds with low BCI and very high capture probabilities for geese with high BCI. Instead, the best fitting logistic function may be fairly flat and the true effect may not be observed. Another reason for the discrepancy may be the relatively small size of the data set caused by removing the observations with missing data. Both the 95% credible intervals for the continuous function and 95% confidence intervals for the multi–state estimates are very wide, indicating that the capture probability estimates produced by both models are highly variable. We are continuing to develop the model in order to address both of these issues. Discussion The method presented here provides a general framework for incorporating the effect of continuous, time–dependent, individual covariates into open population capture–recapture models. The model assumes that changes in the covariate are similar for all individuals in the population and constructs the distribution of the unknown values of the auxiliary variable conditional on the observed values. This information is then incorporated into the CJS model likelihood using selected link functions in order to estimate the survival and capture probabilities as continuous functions of the variable. In the specific model described, changes in the covariate are assumed to be normally distributed with time–dependent mean and constant variance, and the logistic function is used as the link for both the capture and survival probabilities. We believe that this model presents significant advantages over the current approach of categorizing continuous covariates and incorporating them into a multi– state model.

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The differences between the proposed model and the multi–state model are similar to the differences between a model using completely separate, static groups of animals and one incorporating information on a continuous, but constant, variable. The main advantage of the proposed model is that it can incorporate continuous covariates without requiring them to be categorized. Although continuous covariates may used in a multi–state model by dividing their range into discrete intervals which are treated as distinct states (see Nichols et al., 1992, for example), the divisions may be arbitrary and different categorizations may lead to different conclusions about the variable’s effect. Using a small number of divisions may impose unrealistic assumptions about the similarity of individuals in the same state and obscure the underlying changes in the capture or survival probabilities. Using a larger number of divisions may lead to problems with model identifiability. In some situations, the results of a multi–state model may also be more difficult to interpret. In particular, if a large number of divisions is used then it becomes hard to describe changes in the covariate over time based on the transition rates. In comparison, our model produces direct information about changes in the variable’s distribution. Comparing different models fit using different sets of auxiliary variables will also provide a simple way to test the effect of each variable on the animals’ survival probability. The main disadvantage of our approach is that it imposes assumptions on both the distribution of the covariate and on the relationship between the covariate and the survival and capture probabilities. If these assumptions are not satisfied then the model will fit the data poorly and provide erroneous conclusions. In particular, the use of the logistic link function in the current model may be inappropriate in some situations, e.g. if the survival or capture probabilities are not monotonic functions of the auxiliary variable. If this is the case and a suitable link function cannot be found, then the multi–state model, which makes no assumptions about the relationship between the capture and survival probabilities for the different states, will be more appropriate. As future work, we envisage several extensions of our model to accommodate different assumptions about the behaviour of the covariate and the relationship between the covariate and the survival and capture probabilities. One of the basic assumptions of our model is that the changes in the covariate have the same distribution for all individuals in the population. As in both examples described, this assumption would likely not be satisfied when considering changes in body measurements for both adult and juvenile animals. A simple extension would use two values of the expected change for each pair of capture occasion, one for adults and one for juvenile. It should also be possible to fit models in which the expected change in the auxiliary variable depends on the value of the variable itself. For example, modelling the behaviour of the variable on the

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log scale would generate a multiplicative model in which the expected change in the value of the covariate is proportional to its current value. Another constraint of the proposed model is that the survival and capture probabilities for a given value of the covariate are assumed constant over time. Time– dependent survival probabilities could be based on the Cox proportional hazards model for survival analysis, which allows for variation in survival over time under the assumption that the relative hazard for two different values of the auxiliary variable is constant (Cox, 1972). The same model could be used for the capture probability as well. Acknowledgments Most of this work was completed during Simon Bonner’s Masters thesis supported in part by a PGS–A graduate award from the Canadian National Science and Engineering Research Council, and a Graduate Fellowship from Simon Fraser University. The sample data sets used in developing the model were provided by Dr. J. Nichols of the Patuxent Wildlife Center and Dr. E. Cooch of Cornell University. Many thanks to the referees for their valuable feedback, and to Dr. J.–D. Lebreton who helped through the revision process. Thanks also to Dr. T. Swartz and Dr. K. L. Weldon at Simon Fraser University who served on Simon Bonner’s M. Sc. committee. References Arnason, A. N., 1973. The estimation of population size, migration rates, and survival in a stratified population. Research in Population Ecology, 15: 1–8. Bonner, S. J., 2003. Continuous Time–Dependent Individual Covariates in the Cormack–Jolly–Seber Model. M. Sc. Thesis, Simon Fraser Univ.. (Available online at: www.stat.sfu.ca/alumni) Bonner, S. J. & Schwarz, C. J., 2004. An Extension of the Cormack–Jolly Seber model for continuous covariates with application to Microtus pennsylvanicus. Biometrics (in press). Carlin, B. P. & Louis, T. A., 1996. Bayes and Empirical Bayes Methods for Data analysis. Chapman and Hall, London.

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Chib, S. & Greenberg, E., 1995. Understanding the Metropolis–Hastings algorithm. The American Statistician, 49: 327–335. Cormack, R. M., Estimation of survival from the sighting of marked animals. Biometrika, 51: 429–438. Cox, D. R., 1972. Regression models and life– tables. Journal of the Royal Statistical Society, 34: 187–220. Jolly, G. M., 1965. Explicit estimates from capture– recapture data with both death and immigration– stochastic model. Biometrika, 52: 225–247. Clobert, J. & Lebreton, J.–D., 1985. Dépendence de Facteurs de Milieu dans les Estimations de Taux de Survie par Capture–Recapture. Biometrics, 41: 1031–1037. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modelling survival and testing biological hypotheses using marked animals: A unified approach with case–studies. Ecological Monographs, 62: 67–118. McCullagh, P. & Nelder, J. A., 1989. Generalized linear models. Second edition. Chapman and Hall, New York, U.S.A. Nichols, J. D., Sauer, J.R., Pollock, K. H. & Hestbeck, J. B., 1992. Estimating transition probabilities for stage–based population projection matrices using capture–recapture data. Ecology, 73: 306–312. North, P. M. & Morgan, B. J. T., 1979. Modelling Heron survival using weather data. Biometrics, 35: 667–681 Pollock, K. H., 2002. The use of auxiliary variables in capture–recapture modelling: an overview. Journal of Applied Statistics, 29: 85–102. Seber, G. A. F., 1965. A note on the multiple recapture census. Biometrika, 52: 249–259. Schwarz, C. H., Schweigert, J. F. & Arnason, N. A., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Skalski, J. R., Hoffmann, A. & Smith, S.G., 1993. Testing the Significance of Individual and Cohort Level Covariates in Animal Survival Studies. In Marked Individuals in the Study of Bird Population (J.–D. Lebreton & P. M. North, Eds.). Bikhäuser Verlag, Boston. White, G. C. & Burnham, K. P., 1999. Program mark–survival estimation from populations of marked animals. Bird Study, 46 Supplement: 120–138.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations D. L. Otis & G. C. White

Otis, D. L. & White, G. C., 2004. Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations. Animal Biodiversity and Conservation, 27.1: 157–173. Abstract Evaluation of ultrastructure and random effects band recovery models for estimating relationships between survival and harvest rates in exploited populations.— Increased population survival rate after an episode of seasonal exploitation is considered a type of compensatory population response. Lack of an increase is interpreted as evidence that exploitation results in added annual mortality in the population. Despite its importance to management of exploited species, there are limited statistical techniques for comparing relative support for these two alternative models. For exploited bird species, the most common technique is to use a fixed effect, deterministic ultrastructure model incorporated into band recovery models to estimate the relationship between harvest and survival rate. We present a new likelihood–based technique within a framework that assumes that survival and harvest are random effects that covary through time. We conducted a Monte Carlo simulation study under this framework to evaluate the performance of these two techniques. The ultrastructure models performed poorly in all simulated scenarios, due mainly to pathological distributional properties. The random effects estimators and their associated estimators of precision had relatively small negative bias under most scenarios, and profile likelihood intervals achieved nominal coverage. We suggest that the random effects estimation method approach has many advantages compared to the ultrastructure models, and that evaluation of robustness and generalization to more complex population structures are topics for additional research. Key words: Compensatory mortality, Exploitation, Band recovery, Ultrastructure model, Random effects. Resumen Evaluación de los modelos ultraestructurales y de efectos aleatorios empleados en la recuperación de anillas para estimar las relaciones entre las tasas de supervivencia y las tasas de cosecha en poblaciones bajo explotación.— El aumento en la tasa de supervivencia poblacional ocurrido tras un episodio de explotación estacional se considera como un tipo de respuesta compensatoria por parte de la población. La ausencia de un aumento se interpreta como una evidencia de que la explotación se traduce en una mayor mortalidad anual de la población. Pese a su importancia para la gestión de especies bajo explotación, sólo se dispone de un número limitado de técnicas estadísticas que permiten comparar el apoyo relativo que reciben estos dos modelos alternativos. Para las especies de aves bajo explotación, la técnica más habitual consiste en utilizar un modelo de efectos fijos, de ultraestructura determinista, incorporado a los modelos de recuperación de anillas para estimar la relación entre la tasa de cosecha y la tasa de supervivencia. En el presente estudio explicamos cómo emplear una nueva técnica basada en la razón de verosimilitud, en un marco que aume que la supervivencia y la cosecha son efectos aleatorios que covarían a lo largo del tiempo. Para ello, llevamos a cabo un estudio bajo dicho marco utilizando la simulación Monte Carlo, con objeto de evaluar el rendimiento de las dos técnicas mencionadas. El rendimiento de los modelos ultraestructurales fue bastante deficiente en todos los escenarios simulados, obedeciendo, principalmente, a propiedades distribucionales patológicas. Los estimadores de efectos aleatorios y sus estimadores de precisión asociados presentaron un sesgo relativamente pequeño en la mayor parte de los escenarios, mientras que los intervalos de verosimilitud del perfil alcanzaron una cobertura nominal. Sugerimos que el planteamiento ISSN: 1578–665X

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basado en el método de estimación mediante modelos de efectos aleatorios brinda numerosas ventajas en comparación con los modelos ultraestructurales, y que la evaluación de la robustez y la generalización a estructuras poblacionales más complejas constituyen temas para una investigación adicional. Palabras clave: Mortalidad compensatoria, Explotación, Recuperación de anillas, Modelo ultraestructural, Efectos aleatorios. David L. Otis, U.S. Geological Survey, Biological Resources Division, Iowa Cooperative Fish and Wildlife Research Unit, 09 Science II, Iowa State Univ., Ames, IA 50011, U.S.A.– Gary C. White, Dept. of Fishery and Wildlife Biology, Colorado State Univ., Fort Collins, CO 80523 U.S.A. Corresponding author: D. L. Otis. E–mail: dotis@iastate.edu

Animal Biodiversity and Conservation 27.1 (2004)

Introduction Theories and models of the effects of exploitation (harvest) on vital rates of animal populations represent a fundamental and rich component of the population ecology literature. A key concept of this literature is that density dependence, effected through likely mechanisms of intraspecific competition or resource limitation, causes change in population vital rates such as survival and reproduction (Begon et al., 1996), i.e., vital rates are functions of population density. Evidence for or against hypotheses derived from this paradigm has been pursued for decades by scientists in laboratory and field studies and for all manner of taxa. Despite this effort, no general consensus or unifying principles have emerged, due to the difficulties inherent in design of critical experiments and accurate measurement of responses. Such is the case with the subject of concern in this paper, i.e., the relationship between harvest rates and vital rates of wild game bird populations. A long history of thought and work on the degree to which bird populations compensate for seasonal harvest (Errington, 1946) continues to the present. Contemporary emphasis on this topic has focused on contrasting evidence for compensatory versus additive mortality and, to a lesser extent, density– dependent versus density–independent reproduction. In this paper, we will be concerned only with mortality rates. A seminal paper on this topic was authored by Anderson & Burnham (1976), who developed models that related instantaneous competing mortality risks and annual rates of harvest (H) and natural (V) mortality, using a structural model that decomposed annual survival (S) into 2 seasonal components: Sh. Survival during the hunting season, and Sn. Survival during the non–hunting remainder of the year. They argued that the assumption of no natural mortality during the hunting season was mathematically reasonable, and under this simplifying assumption of temporal separation of mortality sources, S = (1 – H) Sn. Under the assumption of complete additivity of hunting mortality, Sn = S0, where S0 is the annual survival rate that would be realized if there were no hunting mortality (Nichols et al., 1984). Alternatively, the compensatory model is constrained by an extra parameter, usually expressed as a threshold (C). If H < C, then Sn = S0 / (1 – H) and thus S = S0. Nichols et al. (1984) pointed out that max (C) = 1 – S0 by definition. However, the functional form of S when H > C has been left unspecified in this model, other than the fact that it must be a non– increasing function of H. Burnham et al. (1984) proposed the ultrastructure model S = S0 (1 – bH) (hereafter ultra–linear) as a general statistical model that accommodates a continuous family of relationships between harvest rate and annual survival, and allows empirical estimation of the relationship. The parameter b is interpreted as an index to the degree of dependence between S

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and H. When b is close to zero there is evidence for compensation and when b is close to 1, there is evidence for additivity. Note that the threshold parameter C is not incorporated into the model. Using numerical techniques, Burnham et al. (1984) produced MLE estimates for several sets of band recovery data for adult mallards (Anas platyrhynchos). An associated small Monte Carlo simulation study of the statistical properties of the method suggested that could have significant bias, the sampling distribution of was often bimodal, and that often exhibited significant negative bias. Several authors have subsequently employed this approach using band recovery data from waterfowl species (Barker et al., 1991; Smith & Reynolds, 1992). An alternative ultrastructure model S = S0 (1 – H)b (hereafter ultra–power) was proposed by K. P. Burnham (Colorado State University, pers. comm.) and used by Rexstad (1992) in an analysis of Canada goose (Branta canadensis) band recovery data from two age classes. The author cited poor convergence rates of numerical optimization techniques in his analysis. Better performance was achieved using only the adult age class, but the author expressed reservations about the usefulness of the ultrastructure technique. Numerical instability in ultrastructure analyses of band recovery data has also been reported for black ducks (Anas rupribes; Conroy et al., 2002) and mourning doves (Zenaida macroura; Otis, 2002). Based on this review, our assessment is that researchers interested in asking questions about density–dependent and compensatory responses to harvest mortality in game bird species have few reliable analytical tools at their disposal, and little guidance about data requirements necessary to produce reliable results from ultrastructure models. More importantly, however, we deviate from considering the relationship between harvest and survival within a fixed effect, deterministic framework, and develop an alternative technique based on the assumption that both survival and recovery rates are random variables that may covary with time (Anderson & Burnham, 1976). Conceptualization of a population survival rate as a random effect with an associated process variance has more recently been used as the basis for alternative trend estimation techniques (Franklin et al., 2002), in population viability analysis (White, 2000), and in investigation of density dependence between survival and abundance (Barker et al., 2002). Schaub & Lebreton (2004) considered the situation in which band recovery information is available on two potentially competing sources of mortality. They modelled mortality rates as random effects, and considered estimation of correlation between these processes. We assume that harvest rate can also be considered as a random effect that covaries to some degree with survival because of an effect of harvest on survival. Deviations from this relationship due to a myriad of other biological and environmental mechanisms are modeled as random process error. Our objectives are: (1) to use Monte Carlo simulation

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to assess the performance characteristics of ultrastructure model estimators, and (2) to suggest and evaluate a new technique based on a random effect parameterization of band recovery data. Estimation methods We assume a traditional band recovery model for the situation in which N marked birds are released in each of k years, and recovery data are provided by hunters who report bands from i marked individuals in subsequent years. The recovery data are represented as {Ri, i = 1,...,k} where the ith vector of recoveries has elements {r ij, j = i,...,k}. The multinomial likelihood is parameterised by {Si, i = 1,...,k – 1; fi, i = 1,...,k}, where Si represents annual survival from year i to year i + 1 and fi is the recovery rate in the ith hunting season. We note that recovery rate fi is the product of the probability that a banded bird is harvested and retrieved (harvest rate; Hi) and the probability that the band is reported ( ). This model is the adult–only, time– specific Model M1 in Brownie et al. (1985). Ultrastructure estimation The estimation procedure is simply to substitute the functional form for the annual survival into the multinomial likelihood function for band recovery data and use numerical techniques to find maximum likelihood estimates. For example, in the ultra–linear model, we substitute Si = S0 (1 – bHi), where S0 is annual survival in the absence of harvest (assumed constant over time), b is the slope parameter, and Hi = fi / . We assume here that is a known constant and that there is no crippling (unretrieved) loss of banded birds. Note that this model (and the ultra–power model) is not linear in the unknown parameters, and thus closed form solutions for the estimators do not exist. PROGRAM SURVIV (White, 1983) has typically been used to accomplish the estimation, but we wrote code for PROC IML (SAS Institute, 1990) using the numerical optimisation procedures contained within IML. Estimation using a random effect model We assume a standard single age class, year– specific band recovery model with parameterisation following Brownie et al. (1985). If annual survival rate Si and recovery rate fi are assumed to be fixed parameters, then the likelihood function for estimation of S and f is (1) where Mi designates the multinomial distribution for the ith cohort of released birds, Ni = size of i, j the ith cohort release, and Rij = number of hunter recoveries from the ith cohort in the jth year, and there are k + 1 releases.

Alternatively, we assumed both survival and recovery rates were random effects. We thus conceptualised true annual survival as a process that naturally varies around some expected value, that could be modelled as a function of specific but unknown parameters. Annual recovery rate, and thereby harvest rate, was similarly considered to be a random process that varies about its expected value. Process variance for both types of parameters may be affected by biotic and abiotic factors such as weather, habitat conditions, hunting effort, or harvest regulations. Additionally, we allowed for the possibility that these two processes may not be independent, but functionally linked. In our situation, covariance between survival and recovery (harvest) processes can be considered as an index to the relationship between survival and harvest. A process covariance of zero is consistent with the hypothesis of completely compensatory hunting mortality. If covariance is moderately negative, additive mortality is suggested. Positive covariance is biologically implausible and was not considered. Thus, our emphasis in development of an alternative technique for assessment of the relationship between harvest and survival focused on estimation of process covariance or correlation. Let E{Si } = S, Var{Si } = 2S, E{fi} = f, Var{fi} = 2f, Cov{Si, fi} = Sf, and consider the vector of MLE estimates derived from the likelihood in Eq. (1). We have E{ } = X , where

and

The unconditional variance–covariance matrix of = + , where

and is the estimated sampling variance–covariance matrix of . Using a generalized least squares approach, the normal equation for estimation of produces the estimator (2) subject to minimization of (3) This estimator is a function of the unknown parameters 2S, 2f, and Sf, and thus a solution must be obtained by using iterative techniques on Eqs. (2) and (3) to estimate these process parameters and .

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Table 1. Parameter values used in simulation of ultrastructure and MVN random effects estimators. Tabla 1. Valores de los parámetros empleados en la simulación de los estimadores de los modelos ultraestructural y de efectos aleatorios MVN.

Parameter N Y

Description

Values

Number banded and released in each year

1,000; 3,000; 5,000

Number of years of banding and recovery

11; 21; 31

True mean annual survival rate

0.30; 0.45; 0.60

True mean annual recovery rate

0.05; 0.10; 0.15

True correlation between survival and recovery rate

–0.90; –0.45; 0.00

Coefficient of variation of true survival and recovery rates

0.10; 0.20; 0.30

We also considered an estimator of 2S, 2f, and Sf, based on the multivariate normal (MVN) likelihood:

where x x is the determinant of

Simulation study We investigated the statistical performance of the ultrastructure and random effects estimators using Monte Carlo simulation methods. Initial simulations of the random effects least squares estimator produced an unacceptable proportion of numerically unstable estimates, so we report only results for the MVN random effects estimator. Three values were chosen for each of the 6 necessary design parameters, resulting in 36 = 729 combinations. Parameter values were 6 chosen to span the range typical for banding studies of game birds (table 1). The correlation Sf (and hence covariance) between the survival and recovery processes were chosen to simulate a strongly additive relationship ( Sf = –0.90), a moderately additive relationship ( Sf = –0.45), and a completely compensatory relationship ( Sf = 0.00). Simulations were done in SAS (SAS Institute, 1999), using PROC IML and the nonlinear minimization algorithms available therein. Two separate SAS codes were used to simulate the two ultrastructure estimators and the MVN random effects estimator. One complete replication of the 729 scenarios (hereafter, replication) took approximately 24 hours on a 3GHz Intel processor running Windows XP for the ultrastructure simulations, and approximately 12 hours for one complete

replication of the MVN simulations. Because computer time was limited, slightly fewer replications of the ultrastructure simulations were completed (see Results). For each replication, a band recovery data set was randomly generated as follows: (1) for each year of recovery, choose a random pair of survival and recovery rates from a bivariate normal distribution parameterized by specified values of the mean vector and covariance matrix; (2) if this pair of values does not meet the multiple criteria (1 – S) > f, 0 < S < 1 and 0 < f < 1, then reject the pair and return to step 1 above; (3) for each banded cohort and recovery year, choose a random number of birds that died during the year from a binomial distribution with parameters equal to the number of individuals in the cohort that are still alive, and probability equal to 1 minus the random survival rate for that year; (4) for each banded cohort and recovery year, generate a random number of recoveries from a binomial distribution with parameters equal to the number of deaths in the cohort for that year, and probability equal to the conditional detection rate r = f / (1 – S) for that year. Given the recovery and banding data, the likelihood function for the respective methods was generated and numerical techniques used to generate point estimates of the parameters of interest. A boundary constraint restricted the ultrastructure estimator to (0,1). Variance estimators for the ultrastructure estimators were calculated from the Hessian matrix. For the MVN random effects estimator, profile likelihood intervals (Burnham & Anderson, 2002) for Sf, S, f, S, and f were derived numerically for a representative subset of the 729 scenarios based on 1,000 simulations for each scenario. Confidence interval coverage ( = 0.05) was computed for the true parameter value from which the data were simulated and also for the realized value of the five parameters for each of the simulations.

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Table 2. Simulated performance statistics of the ultrastructure power model estimator . Refer to table 1 for definition of simulation design factors. Values summarized in the table are the mean of ( ), average estimated standard error [ ], and average empirical standard deviation of [ ] over all replications of a fixed value of the process correlation of S and f ( Sf). Tabla 2. Estadísticas del rendimiento simulado del estimador del modelo de potencia ultraestructural . Para una definición de los factores del diseño de simulación, véase la tabla 1. Los valores resumidos en la tabla corresponden a la media de ( ), promedio de error estándar estimado [ ], y promedio de desviación estándar empírica de [ ] en todas las replicaciones de un valor fijo de la correlación del proceso de S y f ( Sf).

= –0.9

= –0.45

= 0.00

0.30

0.985

0.780

0.061

0.850

1.522

0.328

0.371

1.577

0.461

0.45

0.988

0.441

0.049

0.796

0.961

0.388

0.313

1.105

0.442

0.60

0.982

0.220

0.105

0.677

0.611

0.454

0.267

0.775

0.413

0.05

0.973

0.992

0.102

0.779

1.802

0.393

0.360

1.946

0.463

0.10

0.990

0.269

0.059

0.782

0.829

0.386

0.316

0.959

0.439

0.15

0.993

0.181

0.054

0.762

0.462

0.390

0.276

0.552

0.413

0.1

0.966

0.884

0.134

0.728

1.397

0.423

0.325

1.522

0.442

0.2

0.993

0.388

0.047

0.788

0.979

0.381

0.309

1.076

0.433

0.3

0.996

0.169

0.033

0.807

0.718

0.366

0.318

0.859

0.440

1,000

0.968

0.741

0.123

0.746

1.476

0.411

0.329

1.673

0.446

3,000

0.993

0.398

0.049

0.784

0.919

0.384

0.316

1.009

0.437

5,000

0.994

0.303

0.043

0.792

0.699

0.375

0.307

0.775

0.432

0.977

0.808

0.095

0.775

1.412

0.404

0.423

1.562

0.482

Y 21

0.988

0.382

0.059

0.781

0.943

0.387

0.301

1.058

0.440

0.989

0.252

0.060

0.766

0.739

0.380

0.228

0.837

0.393

We acknowledge two technical points about the simulation procedure relevant to expected values of the estimates. First, the average value of any of the simulation parameters for a given replication will deviate slightly from the value in table 1 because the parameter values were chosen randomly and independently in each replication. Comparisons are based on differences between an individual estimate and the relevant value in table 1. Secondly, paired random values of S and f were chosen with the constraint that (1 – S) > f. Although the theoretically logical constraint is (1 – S) > H, the range of values chosen for S and f made it very unlikely that (1 – S) < H, unless reporting rate was very small. Moreover, we chose to specify values of f rather than H (= f / ) because: (1) recovery rates are commonly reported in band recovery studies and thus a rea-

sonable range was easily determined; (2) recovery rates directly determine the number of recoveries; (3) the correlation between S and f is the same as that for S and H (assuming is a constant), and therefore results are invariant to choice of . This constraint logically induces some bias in the estimators, but we found this bias to be trivial in our interpretation of the results. Results Ultrastructure models We generated 200 replications of the ultrastructure models for each of the 36 = 729 parameter combinations. Estimates of the ultra–linear slope and ultra–power exponent were nearly equal in all

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225

180

200 150

140 Frequency

Frequency

175 125 100 75

120 100 80 60

f = 0.05 f = 0.10 f = 0.15

160

S = 0.30 S = 0.45 S = 0.60

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 b C 120

N = 1,000 N = 3,000 N = 5,000

100 Frequency

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 b

80 60 40 20 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 b

Fig. 1. Example frequency distributions of ultra–power model estimator for: A. on S(S); B. Sf = –0.45, conditional on f; C. Sf = 0.00, conditional on N.

= –0.90, conditional

Fig. 1. Ejemplo de distribuciones de frecuencia del estimador del modelo de ultra–potencia para = –0,90, condicionadas a S(S); B. Sf = –0.45, condicionadas a f; C. Sf = 0.00 condicionada a N. Sf

simulations, as were their estimated standard errors . We therefore report only results for the ultra–power model. We summarize in table 2 the mean, average estimated standard error [ ], and average empirical standard deviation of [ ] over all replications of a fixed value of Sf and a value of one other factor. Thus, each statistic is the average of 193*34 = 15,633 data sets. For all cases, is appropriately near 1 in strongly additive cases ( Sf = –0.9). When Sf = –0.45, was generally between 0.7 and 0.8. Because of the lack of a 1–to–1 correspondence between Sf and b, it is not possible to formally evaluate bias. However, as results are further described below, it is logical to infer that had large positive bias in the moderately additive case. For the case of total compensation, i.e., Sf = 0.00, had significant positive bias in all cases (table 2). With respect to estimates of precision, estimated sampling errors

were positively biased (relative to the empirical sampling error by 2–4 fold in most cases, and by an order of magnitude in several cases. For a fixed value of Sf , estimated sampling errors decreased as expected as simulation parameter values increase, but empirical sampling errors remain relatively constant, with few exceptions (table 2). A more detailed examination of the results reveals that average values of performance metrics are very misleading because of the pathological empirical frequency istributions of the estimator in all simulated cases. Typical examples of sets of frequency distributions of based on 193 replications each of a single combination of true parameter values are presented in fig. 1. When Sf = –0.9, the overwhelming majority of data sets resulted in = 0.999 (the numerical estimation algorithm had the constraint < 1). As previously discussed, the estimated variance of the estimator from a single replication

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was usually quite large, and clearly greatly overestimates the empirical variance of because of its degenerate distribution (fig. 1A). For Sf = –0.45, > 50%, and more typically > 80 % of the values of = 0.999; the remainder of the estimates were close to zero (fig. 1B). For Sf = 0.00, the modal value of the distribution of was approximately zero, with a second substantial mode at 0.999. In a multi–factor simulation study, standard statistical methods such as ANOVA or multiple regression analysis would be employed to gain insight into the parameters and combinations thereof that have the most influence on metrics such as bias and confidence interval coverage. However, we concluded that the distributional properties of precluded both the construction of confidence intervals and attempts to conduct formal sensitivity analyses. Informal examination of simulation results revealed that the performance of the estimator was not affected significantly by changes in values of the individual design parameters, particularly for nonzero values of Sf .

Table 3. Simulated performance statistics of the MVN random effects model estimators. Values summarized in the table are the mean , average estimated of the estimates , and average empirical standard error conditional on a standard deviation of fixed value of one of the design parameters. Each entry is based on 250 replications of 3 5 = 243 scenarios, or 60,750 data sets: P. Parameter; V. Value; * Not computed. Tabla 3. Estadísticas del rendimiento simulado de los estimadores del modelo de efectos aleatorios MVN. Los valores resumidos en la tabla corresponden a la media de las , promedio de error estándar estimaciones , y promedio de desviación estimado estándar empírica de , condicionados a un valor fijo de uno de los parámetros del diseño. Cada entrada se basa en 250 replicaciones de 35 = 243 escenarios, o 60.750 conjuntos de datos: P. Parámetro; V. Valor; * No computado.

MVN random effects models For each of the 36 = 729 scenarios shown in table 1, 250 replications were simulated. Five parameter estimates result from the estimation procedure: , , , , and . For each estimator, we summa, average estimated rize in table 3 the mean standard error , and average empirical standard deviation of conditional on a fixed value of one of the design parameters. Thus, each statistic in table 3 is generated from 250*3 = 60,750 data sets. Bias Average bias of all parameter estimates was negligible over all replications and factor combinations (table 3). We were particularly interested in estimators of Sf, S, and f, and the relative influence of the design parameters on their bias. Therefore we performed simple multi–factor ANOVAs with both bias (the average difference between the estimate and the true parameter value) and relative bias (bias / true parameter value) as the response variables, and each of the 6 design factors in the simulation study as main effects. The three factors with generally the greatest influence were number of years banded, number banded each year, and the CV of the survival and recovery rates. Bias of each of the estimators was generally small, but nearly always negative (table 4). The magnitude of the bias of was not substantially influenced by the three simulation design factors. Bias of both and decreased markedly between N = 1,000 and N = 3,000, and between Y = 11 and Y = 21 (table 4). Precision We considered the true standard error of an estimator for a specific scenario to be the empirical

P S

CVs

CVf

V 0.30

0.297

0.0153

0.0162

0.45

0.448

0.0213

0.0227

0.60

0.593

0.0265

0.0285

0.05

0.0499

0.00255

0.0027

0.10

0.0999

0.00480

0.0051

0.15

0.1500

0.00706

0.0075

0.10

0.0957

0.0189

0.20

0.1870

0.0463

0.30

0.2783

0.0120

0.10

0.0957

0.0084

0.20

0.1898

0.0055

0.30

0.2832

0.0098

–0.90

–0.879

0.216

0.130

–0.45

–0.454

0.393

0.298

0.00

–0.021

0.437

0.350

standard deviation of the 250 estimates generated from the independent data sets. Thus, for each estimator, we have 36 = 729 true standard errors, one for each scenario. Histograms of the true standard error of [ ], conditional on each value of Sf , revealed that the frequency distribution of each was heavily right–skewed. The mean true standard error [ ] of each distribution increased as Sf approached zero, i.e., = 0.130 ( SF = –0.90), = 0.300 ( Sf = –0.45), = 0.350 ( Sf = 0.00). We investigated the effects of simulation design parameters on precision of by arbitrarily designating

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6.0 5.5 5.0 4.5 4.0 3.5 Prb

3.0 2.5 2.0 1.5 1.0 0.0 –0.5 –1.0 –1.5 0.0

0.1

0.2

0.3

0.4 0.5 Sdrho

0.6

0.7

0.8

0.9

Fig. 2. Scatterplot of the percent relative bias (Prb) of the estimated average standard error of Sf, plotted against the empirical standard error of (Sdrho). Each average is taken over 250 simulated data sets. Plot symbol indicates value of Sf: A = –0.90, B = –0.45, C = 0.00. Fig. 2. Diagrama de dispersión del sesgo relativo porcentual (Prb) del promedio de error estándar estimado de Sf, representado gráficamente en comparación con el error estándar empírico de (Sdrho). Cada promedio se toma con respecto a 250 conjuntos de datos simulados. El símbolo de la representación gráfica indica el valor de Sf: A = –0,90; B = –0,45; C = 0,00.

0< < 0.20 as an acceptable range of precision. Of the 729 simulated scenarios, there were 313 scenarios that produced acceptable precision. Increasing values of S, f, and CV were associated with slightly increased proportions of cases with acceptable precision, but design parameters Sf and Y had the largest influence: 2/3 of the acceptable scenarios had Sf = –0.90, and slightly more than 1/2 had Y = 31 (table 5). We investigated the performance of the estima, i.e., , by first plotting the average tor of relative bias of from the 250 replications of a given scenario against the corresponding (fig. 2). The estimator was generally unbiased on < 0.5, although there were average when several cases of large positive bias when Sf = – 0.90. There was increasing positive bias as becomes large, and most of these situations occurred when Sf = –0.45 or Sf = 0.00. Generalizations are difficult to make about the combinations of the design factors that resulted in poor precision. It is generally true, however, that when at least 2 of the 3 following conditions held, estimation of precision was poor on the average: N = 1,000, CV = 0.10, Y = 11.

was examined in furThe performance of ther detail by looking at the 250 individual estimates for several representative scenarios. Specifically, we were interested in the co–occurrence of poor estimates of both sf and . Unreliable results occurred when was at a boundary value of –1.0 or 1.0. This occurred commonly under poor design scenarios (small values of N, Y, and CV); when Sf = –0.90, 66% of the estimates were at the lower boundary, and 4% were at the higher, and most of these estimates had extremely large estimated standard error. When Sf = 0.00, 25% of the estimates were at the lower boundary, and 18% were at the higher, with similarly large estimated standard errors. For good design scenarios, only eight of the total 500 estimates were at the negative lower bound. We suspect that these pathological individual estimates, which cause inflated average positive bias estimates, may indicate unstable performance of the numerical estimation algorithms, which we consider in more detail in the Discussion. The estimator had an average percent relative bias (Prb) of l –10% for above average values of S (fig. 3). Prb was slightly less for smaller values of S, but the estimates were more erratic. The perform-

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1.2 1.0 0.8 0.6 Prb

0.4 0.2 0.0 –0.2 –0.4 –0.6 0.02

0.04

0.06

0.08

Fig. 3. Average percent relative bias (Prb) of replications of 729 simulated scenarios.

0.10 Sds

0.12

0.14

, plotted against

0.16

0.18

0.20

(Sds). Averages are taken over 250

Fig. 3. Promedio del sesgo relativo porcentual (Prb) de , representado gráficamente en comparación con S (Sds). Los promedios se toman con respecto a 250 replicaciones de 729 escenarios simulados.

ance of the estimator , i.e., the estimated standard error of , and the similar estimator of the process recovery rate were better than the corresponding estimator for process correlation. Absolute percent relative bias of both estimators was generally < 10%. There were 44 scenarios for ) < –20%. Of these, 38 scenarios which Prb ( had Y = 31, CV = 0.10. Eight scenarios resulted in Prb( ) > 20%, and of these, 6 scenarios had Y = 11, CV = 0.10. There were 48 scenarios for which ) < –20%. Of these, 39 scenarios had Prb( Y = 31, CV = 0.10. Three scenarios resulted in Prb > 20%, and of these, 2 scenarios had Y = 11, CV = 0.10. With respect to the general influence of the simulation design factors, Prb of both and was always negative when averaged over all data sets for a fixed value of a single design factor (table 6). The average Prb taken over all 729 scenarios was –7.7% for and –8.0% for Profile likelihood confidence interval coverage Achieved confidence interval coverage on the value of Sf used to simulate the data was well below (< 75%) the nominal level of 95% for sparse data and small process variance in survival and recovery rates (table 7), i.e., N = 1000, Y = 11, and CV = 0.1. However, confidence interval coverage on Sf improved to m 92% when Y m 21 or CV = 0.3.

Confidence interval coverage on the remaining parameters was generally adequate (> 90%) with the exception of f, where coverage was < 90%. Confidence interval coverage for the realized values of each simulation improved with increasing sample sizes, reaching 100% for N = 5,000 and Y = 31. It is generally true, as was also previously noted in results for precision of the estimates, that when at least 2 of the 3 following conditions hold, coverage is < 90%: N = 1,000, CV = 0.10, Y = 11. Discussion Limited previous evaluation of the performance of the ultrastructure estimators demonstrated negative bias in the estimators of variance (Burnham et al., 1984; Barker et al., 1991), positive bias in the estimator of the slope parameter, and a bimodal distribution for the slope estimator (Burnham et al., 1984). Unlike our study, data in these earlier studies were generated from a fixed effect ultrastructure model, so performance of the estimators was evaluated using data that conform to the true underlying model. For our study, it is straightforward to show that E(Si x fi) = S [(1 – Sf) + ( Sf / f) fi]. Clearly, this model structure differs from the ultrastructure models, and unless Sf = 0, there is no 1– to–1 correspondence between b and Sf. Thus, the poor per-

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Animal Biodiversity and Conservation 27.1 (2004)

formance demonstrated in our ultrastructure estimators is enhanced by differences in the true underlying data–generating model and the assumed ultrastructure model. A second important point to consider in interpretation of ultrastructure estimator results is the constraint 0 < < 1 used in the numerical search algorithm. This constraint significantly influenced the summary performance statistics for . For data sets in which the maximum of the likelihood occurred outside the admissible interval (and for which was therefore 0 or 1), estimated variances calculated from the Hessian matrix will be positively biased. This fact accounts for the large biases reported in table 2, and the contrast with the previous results of Burnham et al. (1984) and Barker et al. (1991), who specified larger admissible intervals for . The expected values of for different simulation parameter sets are also differentially affected by the constraint on admissible values. For the case b = Sf = 0, we predict that the observed consistently large positive bias of would have been even larger with an expanded interval of admissible values, because the number of estimates at the upper boundary of 1 (fig. 1C) would have been even larger. When Sf = –0.90, 95–99% of the estimates were at the upper bound, and the relationship between the expected value and the upper bound constraint of is unknown in this case. It may be that the ultrastructure estimator and its variance were well–behaved with an increased upper boundary constraint, although bias and associated interpretation of the estimates are difficult to assess under the random effects model structure. For intermediate values of Sf, interpretation is also difficult, but the bimodal distribution of the estimator for a given data set seems symptomatic for the ultrastructure estimator. Typically, > 10% of the estimates were at the lower bound and > 75% of the estimates were at the upper bound when Sf = –0.45 (fig. 1B), and we predict this "goalpost" distribution would have been exacerbated by a larger admissible boundary interval. In summary, our results demonstrate that under a random effects model, the ultrastructure model exhibited undesirable distributional properties for all scenarios except when Sf = –0.90, in which case results were difficult to interpret because of boundary constraints. Significant positive bias in was documented when Sf = 0.00. When considered together with previous simulation evaluations and documented cases of practical application, we discourage use of the ultrastructure estimation technique in the future. Average relative bias of the random effects estimator was negligible for our simulated sceof was most narios. The true variance significantly affected by pSf, and decreased moderately with increasing values of N, CV, and Y. Values of S and f had little effect on precision. Averaged over all values of N, CV, and Y, l 0.35, 0.30, 0.13 for Sf = 0.00, –0.45, and –0.90, respectively. Thus, our random effects technique is much more sensitive in detecting a relationship between sur-

Table 4. Bias (B), i.e., the average difference between the estimate and the true parameter value, of estimators of process standard deviation of survival rate ( S), process standard deviation of recovery rate ( f), and process correlation between survival and recovery rate ( Sf), conditional on values of the most influential design parameters. Each entry is based on 250 replications of 35 = 243 scenarios, or 60,750 data sets: F. Factor; V. Value. Tabla 4. Sesgo (B), es decir, la diferencia media entre el valor estimado y el valor verdadero del parámetro, de los estimadores de la desviación estándar del proceso de la tasa de supervivencia ( S), la desviación estándar del proceso de la tasa de recuperación ( f), y la correlación del proceso entre la tasa de supervivencia y la tasa de recuperación ( Sf), condicionadas a los valores de los parámetros de diseño más influyentes. Cada entrada se basa en 250 replicaciones de 35 = 243 escenarios, o 60.750 conjuntos de datos.

B( )

1,000

–0.078

–0.049

–0.016

3,000

–0.033

–0.032

–0.006

5,000

–0.016

–0.026

–0.002

–0.104

–0.075

–0.031

–0.048

–0.033

–0.016

0.025

0.001

0.022

0.1

–0.032

–0.033

–0.007

0.2

–0.053

–0.039

–0.019

0.3

–0.042

–0.035

–0.013

)

vival and harvest rates that is consistent with an additive mortality hypothesis, as opposed to the compensatory mortality hypothesis. The performance of the variance estimator was more sensitive to simulation design parameters. Minimum values of these parameters resulted in increased chance of significant bias, which was most often positive, especially when Sf = 0.0 or –0.45. We suspect that this result may be an artifact caused by numerical optimization problems at boundary values. We constrained admissible values of to the interval (–1, 1), and when estimates were at either boundary, corresponding estimates were often very large. Investigation of improved numerical optimization techniques for the random effects procedure is a future topic of research. As would be expected, profile likelihood confidence interval coverage improved with increasing

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Table 5. Percentage (p) of n = 313 simulated cases for which the empirical standard deviation of the estimated process correlation was < 0.2, as a function of values of the simulation design parameters. Tabla 5. Porcentaje (p) de n = 313 casos simulados para los que la desviación estándar empírica de la correlación estimada del proceso fue < 0,2, expresada como una función de los valores de los parámetros de diseño de la simulación.

0.30

0.31

0.05

0.28

0.10

0.26

1,000

0.22

0.19

–0.90

0.68

0.45

0.34

0.10

0.35

0.20

0.34

3,000

0.37

0.26

–0.45

0.23

0.60

0.35

0.15

0.37

0.30

0.40

5,000

0.41

0.55

0.00

0.09

Table 6. Average percent relative bias ( ) of the , i.e., the standard error of the estimator estimator of the standard deviation of the process survival rate ( S), and the similar estimator of the process recovery rate f. Each entry is based on estimates from 250 replications of 35 = 243 scenarios, or 60,750 data sets. Tabla 6. Promedio de sesgo relativo porcentual ( ) del estimador , es decir, el error estándar del estimador de la desviación estándar de la tasa de supervivencia del proceso ( S), y el estimador de la tasa de recuperación del proceso similar .f Cada entrada se basa en estimaciones de 250 replicaciones de 35 = 243 escenarios, o 60.750 conjuntos de datos. (

)

(

Value

1,000

–3.3

–3.8

3,000

–9.4

–9.1

5,000

–10.2

–11.0

–5.4

–5.6

–3.1

–3.4

–14.5

–14.9

0.1

–14.7

–15.3

0.2

–5.3

–4.3

0.3

–3.0

–4.4

–0.90

–9.2

–9.8

–0.45

–6.7

–7.4

0.00

–7.1

–6.7

0.30

–8.3

–9.2

0.45

–8.1

–8.2

0.60

–6.4

–8.2

0.05

–10.9

–11.7

0.10

–7.4

–7.3

0.15

–4.6

–5.0

)

number of years of banding. Our results suggest that Y = 11 is not adequate to obtain useful estimates of to assess the degree of compensatory mortality Sf operating in a populationwhen there is little process variance in survival and recovery rates. However, useful results were obtained with Y = 21, or CV = 0.3. For the limited set of scenarios simulated in table 7, the process CV is partially confounded with the number of banding occasions. However, the results presented suggest that the profile likelihood confidence interval will perform adequately for most of the 729 designs simulated in this study, with the exceptions being studies with little process variation and/or small number of banding occasions. The random effects estimator had relatively small, but consistent negative bias that averaged –6.8 % over all simulations. The variance estimator of also performed relatively well, and had substantial negative bias only when the true relative variation in the survival process was smallest, i.e., CV = 0.10. Thus, can be considered an alternative to a method of moments estimator (Burnham & White, 2002) used in the technique of empirical Bayes shrinkage estimators (Burnham, unpublished manuscript), and may provide useful in applications such as minimum viable population modeling (White, 2000). The generally satisfactory performance of the random effects method is due in large measure to the fact that the data were generated from the bivariate normal distribution, which is the underlying distribution assumed in the derivation of the estimators. Although we could have chosen alternative distributions such as logit–normal or beta with attractive attributes such as asymmetry and a (0,1) domain, we opted to use the most straightforward and easily interpretable parameter structure in this initial development and evaluation of a new method. Use of the normal distribution also allowed a clean check of whether the method was producing unbiased estimators, so that evaluation was not confounded by back–transformation complications or other factors such as the (1 – S) > f truncation constraint. The robustness of the estimators to the assumption of a bivariate normal

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Animal Biodiversity and Conservation 27.1 (2004)

Table 7. Profile likelihood 95% confidence interval coverage for random effects parameters. Nominal coverage for both the true parameter value from which the recovery data were simulated and the realized value of the particular simulation are reported. Reported coverage for S, f, S and f was computed for the pooled data from the three simulated values of Sf: P. Parameter; TV. True value; R. Realized; NS. Number of simulations. Tabla 7. Cobertura del perfil de intervalos de confianza al 95% para los parámetros de efectos aleatorios. Se indica la cobertura nominal para el valor verdadero del parámetro a partir del cual se simularon los datos de recuperación y el valor obtenido de la simulación concreta. Cobertura indicada para S, , S y f se calculó para los datos combinados de los tres valores simulados de Sf: P. Parámetro; TV. Valor verdadero; R. Valor obtenido; NS. Número de simulaciones. Coverage N 1,000

0.30

0.1, 0.3

0.05

0.03

0.005

–0.90

3,000

5,000

–0.45 0.0

6,000

0.914

0.962

0.1, 0.3

6,000

0.849

0.931

0.1

3,000

0.939

0.940

0.3

3,000

0.918

0.959

0.1

3,000

0.943

0.945

0.3

3,000

0.926

0.992

0.1

1,000

0.671

0.670

0.3

1,000

0.988

0.993

0.1

1,000

0.722

0.721

0.3

1,000

0.927

0.967

0.1

1,000

0.704

0.699

0.3

1,000

0.934

0.977

0.45

0.2

3,000

0.917

0.990

0.10

0.2

3,000

0.824

0.969

0.09

0.2

3,000

0.931

0.991

0.02

0.2

3,000

0.916

0.985

–0.90

0.2

1,000

0.920

0.962

–0.45

0.2

1,000

0.932

0.998

0.0

0.2

1,000

0.949

0.999

0.45

0.3

3,000

0.918

0.979

0.15

0.3

3,000

0.884

0.976

0.135

0.3

3,000

0.941

0.997

0.045

0.3

3,000

0.942

0.995

–0.90

0.3

1,000

0.942

1.000

–0.45

0.3

1,000

0.928

1.000

0.0

0.3

1,000

0.933

1.000

distribution for (S, f) is of obvious importance in further development of the random effect method. However, we predict that, given a reasonable range of values for S and f, performance of the method will be more sensitive to the influential design parameters we identified than to the true underlying distribution of the random effect parameters.

There are many avenues for further investigation and development of the general random effect technique proposed here. Generalization of the models to multiple age classes should be straightforward and useful. We remain unsure about the poor performance of estimates based on a general least squares framework, but additional investiga-

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Table 8. Number banded and band recoveries for female mallards banded in Wisconsin (USA) 1961– 1996. Tabla 8. Número de aves anilladas y recuperaciones de anillos correspondientes a ánades reales hembras anilladas en Wisconsin (EE.UU.) 1961–1996.

Recoveries Year Banded 61

75 76 77

2,377

2,553

0 112 91

1,715

107

986

880

2,569

177

1,007

1,203

867

905

747

939

666

15 11

715

627

76 77

651

40 11

1,670

0 107

1,083

646

884

1,081

947

1,193

708

584

732

786

1,923

710

1,063

1,297

1,266

1,076

820

872

529

17 13

42 18

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Animal Biodiversity and Conservation 27.1 (2004)

Recoveries 78

0 0

95 96

121

8 10

17 11

32 11

0 39

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Example Table 9. Results from the multivariate normal model for female mallards banded in Wisconsin (U.S.A.), 1961–1996: 95% PL. 95% profile likelihood (L. Lower; U. Upper); P. Parameter; E. Estimate; SE. Standard error. Tabla 9. Resultados del modelo normal multivariante correspondientes a ánades reales hembras anilladas en Wisconsin (EE.UU.), 1961–1996: 95% PL. Probabilidad del perfil (L. Mínimo; U. Máximo); P. Parámetro; E. Estimación; SE. Error estándar.

95% PL P

Female adult mallards were banded in Wisconsin (U.S.A.) prior to the fall hunting season during 1961–1996 (Franklin et al., 2002). Band recoveries and number banded are shown in table 8. The multivariate normal model maximum likelihood estimates and profile likelihood confidence intervals for the means and process variances are estimated with good precision (table 9). The estimate of Sf and its confidence interval suggest that hunting on this population during 1961–1996 was compensatory. Note that has poor precision, which is consistent with our inference from the simulation results that precise estimation of process correlation under compensatory mortality is difficult.

0.0507

0.0024

0.0496

0.0544

0.5291

0.0147

0.4996

0.5608

0.0126

0.0018

0.0100

0.0168

0.0785

0.0163

0.0100

0.0679

–0.0197

0.2208

–0.4336

0.3934

tion may provide refinements that improve performance and result in more robust estimators. However, we suggest that a more important future study would be to investigate the performance of our technique within a density–dependent population framework. Boyce et al. (1999) characterized compensation as a demographic result of a density–dependent population mechanism, and provided a discrete–time model that incorporates seasonal harvest and density dependence that would be applicable to the situation assumed in this paper. Lebreton (unpublished manuscript) also discussed compensation as a consequence of density dependence, and argued that we should expect the quantitative relationship to be weak. He further suggested that even a weak relationship could be important to the dynamics of a population, which is consistent with our contention that the utility of our techniques or other techniques that attempt to estimate the relationship between survival and harvest should be evaluated within the context of models of the entire annual cycle of the population. Without this context, estimates of correlations or slopes cannot be practically interpreted. Finally, we acknowledge the possible applicability of modern Bayesian or empirical Bayesian techniques to the current problem. Because of computational and philosophical issues attendant to these techniques, we chose to approach the problem from a frequentist, likelihood–based perspective. Royle & Link (2002) provide an excellent description of use of Bayesian techniques within the context of survival rate estimation from capture–recapture data.

Acknowledgements We thank K. P. Burnham and J. D. Lebreton for sharing copies of unpublished manuscripts. References Anderson, D. R. & Burnham, K. P., 1976. Population ecology of the mallard: VI. The effect of exploitation on survival. U.S. Fish and Wildlife Service Resource Publication, 128. Barker, R. J., Hines, J. E. & Nichols, J. D., 1991. Effect of hunting on annual survival of grey ducks in New Zealand. Journal of Wildlife Management, 55: 260–265. Barker, R. J., Fletcher, D. & Scofield, P., 2002. Measuring density dependence in survival from capture–recapture data. Journal of Applied Statistics, 29: 305–313. Begon, M., Mortimer, M. & Thompson, D. J., 1996. Population ecology. 3rd Ed. Blackwell Science. Oxford, U.K. Boyce, M. S., Sinclair, A. R. E. & White, G. C. 1999. Seasonal compensation of predation and harvesting. Oikos, 87: 419–426. Brownie, C. A., Anderson, D. R., Burnham, K. P., & Robson, D. S., 1985. Statistical inference from band–recovery data—a handbook. Second Edition. U.S. Fish and Wildlife Service Resource Publication, 131. Burnham, K. P., White, G. C. & Anderson, D. R., 1984. Estimating the effect of hunting on annual survival rates of adult mallards. Journal of Wildlife Management 25, 48: 350–361. Burnham, K. P. & Anderson, D. R., 2002. Model selection and multimodel inference. 2nd ed. Springer–Verlag, New York. Burnham, K. P. & White, G. C., 2002. Evaluation of some random effects methodology applicable to bird ringing data. Journal of Applied Statistics, 29: 245–264. Conroy, M. J., Miller, M. W. & Hines, J. E., 2002. Identification and synthetic modelling of factors affecting American black duck populations. Wild-

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life Monograph, 150: 1–64. Errington, P. L., 1946. Predation and vertebrate populations. Quarterly Review of Biology, 21: 144–177. Franklin, A. B., Anderson, D. R. & Burnham, K. P., 2002. Estimation of long–term trends and variation in avian survival probabilities using random effects models. Journal of Applied Statistics, 29: 267–289. Nichols, J. D., Conroy, M. J., Anderson, D. R. & Burnham, K. P., 1984. Compensatory mortality in waterfowl populations: a review of the evidence and implications for research and management. Transactions of the North American Wildlife and Natural Resources Conference, 49: 535–554. Otis, D. L., 2002. Survival models for harvest management of mourning dove populations. Journal of Wildlife Management, 66: 1052–1063. Rexstad, E. A., 1992. Effect of hunting on the annual survival of Canada geese in Utah. 1992. Journal of Wildlife Management, 56: 297–305. Royle, J. A. & Link, W. A., 2002. Random effects and shrinkage estimation in capture–recapture mod-

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els. Journal of Applied Statistics, 29: 329–351. SAS Institute, 1999. SAS/STAT Users Guide. Version 8. SAS Institute. Cary, North Carolina, U.S.A. – 1990. SAS/IML Software. Version 6. SAS Institute. Cary, North Carolina, ® U.S.A. Schaub, M. & Lebreton, J., 2004. Testing the additive versus the compensatory hypothesis of mortality from ring recovery data using a random effects model. Animal Biodiversity and Conservation, 27.1: 73–85. Smith, G. W. & Reynolds, R. E., 1992. Effect of hunting on mallard survival, 1979–1988. Journal of Wildlife Management, 56: 306–316. White, G. C., 1983. Numerical estimation of survival rates from band–recovery and biotelemetry data. Journal of Wildlife Management, 47: 716–728. – 2000. Population viability analysis: data requirements and essential analysis. In: Research Techniques in Animal Ecology: Controversies and Consequences: 288–331 (L. Boitani & T. K. Fuller, Eds.). Columbia Univ. Press, New York.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Computing and software G. C. White & J. E. Hines

White, G. C. & Hines, J. E., 2004. Computing and software. Animal Biodiversity and Conservation, 27.1: 175–176. The reality is that the statistical methods used for analysis of data depend upon the availability of software. Analysis of marked animal data is no different than the rest of the statistical field. The methods used for analysis are those that are available in reliable software packages. Thus, the critical importance of having reliable, up–to–date software available to biologists is obvious. Statisticians have continued to develop more robust models, ever expanding the suite of potential analysis methods available. But without software to implement these newer methods, they will languish in the abstract, and not be applied to the problems deserving them. In the Computers and Software Session, two new software packages are described, a comparison of implementation of methods for the estimation of nest survival is provided, and a more speculative paper about how the next generation of software might be structured is presented. Rotella et al. (2004) compare nest survival estimation with different software packages: SAS logistic regression, SAS non–linear mixed models, and Program MARK. Nests are assumed to be visited at various, possibly infrequent, intervals. All of the approaches described compute nest survival with the same likelihood, and require that the age of the nest is known to account for nests that eventually hatch. However, each approach offers advantages and disadvantages, explored by Rotella et al. (2004). Efford et al. (2004) present a new software package called DENSITY. The package computes population abundance and density from trapping arrays and other detection methods with a new and unique approach. DENSITY represents the first major addition to the analysis of trapping arrays in 20 years. Barker & White (2004) discuss how existing software such as Program MARK require that each new model’s likelihood must be programmed specifically for that model. They wishfully think that future software might allow the user to combine pieces of likelihood functions together to generate estimates. The idea is interesting, and maybe some bright young statistician can work out the specifics to implement the procedure. Choquet et al. (2004) describe MSURGE, a software package that implements the multistate capture– recapture models. The unique feature of MSURGE is that the design matrix is constructed with an interpreted language called GEMACO. Because MSURGE is limited to just multistate models, the special requirements of these likelihoods can be provided. The software and methods presented in these papers gives biologists and wildlife managers an expanding range of possibilities for data analysis. Although ease–of–use is generally getting better, it does not replace the need for understanding of the requirements and structure of the models being computed. The internet provides access to many free software packages as well as user– discussion groups to share knowledge and ideas. (A starting point for wildlife–related applications is http://www.phidot.org)

Gary C. White, Dept. of Fishery and Wildlife Biology, Colorado State Univ., Fort Collins, CO 80523, U.S.A. E–mail: gwhite@cnr.colostate.edu James E. Hines, USGS, Patuxent Wildlife Research Center, 12100 Beech Forest Rd, Laurel, Md 20708, U.S.A. E–mail: jim_hines@usgs.gov ISSN: 1578–665X

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References Barker, R. J. & White, G. C., 2004. Towards the mother–of–all–models: customised construction of the mark–recapture likelihood function. Animal Biodiversity and Conservation, 27.1: 177–185. Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O., Lebreton, J.–D., 2004. M–SURGE: new software specifically designed for multistate capture–recapture models. Animal Biodiversity and Conservation, 27.1: 207–215. Efford, M. G., Dawson, D. K. & Robbins, C. S., 2004. DENSITY: software for analysing capture–recapture data from passive detector arrays. Animal Biodiversity and Conservation, 27.1: 217–228. Rotella, J. J., Dinsmore, S. J. & Shaffer, T. L., 2004. Modeling nest–survival data: a comparison of recently developed methods that can be implemented in MARK and SAS. Animal Biodiversity and Conservation, 27.1: 187–205.

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Towards the mother–of–all–models: customised construction of the mark–recapture likelihood function R. J. Barker & G. C. White

Barker, R. J. & White, G. C., 2004. Towards the mother–of–all–models: customised construction of the mark–recapture likelihood function. Animal Biodiversity and Conservation, 27.1: 177–185. Abstract Towards the mother–of–all–models: customised construction of the mark–recapture likelihood function.— With a proliferation of mark–recapture models and studies collecting mark–recapture data, software and analysis methods are being continually revised. We consider the construction of the likelihood for a general model that incorporates all the features of the recently developed models: it is a multistate robust–design mark–recapture model that includes dead recoveries and resightings of marked animals and is parameterised in terms of state–specific recruitment, survival, movement, and capture probabilities, state–specific abundances, and state–specific recovery and resighting probabilities. The construction that we outline is based on a factorisation of the likelihood function with each factor corresponding to a different component of the data. Such a construction would allow the likelihood function for a mark– recapture analysis to be customized according to the components that are actually present in the dataset. Key words: Capture–recapture, Mark–recapture, Likelihood.

Resumen Hacia el origen de todos los modelos: una construcción personalizada de la función de verosimilitud en los estudios de marcaje–recaptura.— Dada la proliferación de modelos de marcaje–recaptura y de los estudios que recopilan datos al respecto, los programas y los métodos de análisis se hallan sujetos a una continua revisión. En el presente estudio examinamos la construcción de la función de verosimilitud para un modelo general que incorpora todas las características de los modelos desarrollados recientemente. Se trata de un modelo multiestado robusto de marcaje–recaptura, que incluye las recuperaciones de individuos muertos y los reavistajes de animales marcados, parametrizándose con respecto al reclutamiento específico a un estado, la supervivencia, el movimiento y las probabilidades de captura, las abundancias específicas de un estado, las recuperaciones específicas de un estado y las probabilidades de reavistaje. La construcción que presentamos se basa en una factorización de la función de verosimilitud, de modo que cada factor corresponde a un componente distinto de los datos. Dicha construcción permitiría personalizar la función de verosimilitud en los análisis de marcaje–recaptura de acuerdo con los componentes que están realmente presentes en el conjunto de datos. Palabras clave: Captura–recaptura, Marcaje–recaptura, Probabilidad. Richard J. Barker, Dept. of Mathematics and Statistics, Univ. of Otago, P. O. Box 56, Dunedin, New Zealand.– Gary C. White, Dept. of Fishery and Wildlife Biology, Colorado State Univ., Fort Collins, Colorado, 80523 U.S.A.

ISSN: 1578–665X

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Introduction The ability of biologists to collect varied and interesting data on the re–encounter of marked animals means that existing methods of analysis are often inadequate. This has been a major driver in the development of new mark–recapture models over the past 10–20 years and emphasises the importance of the relationship between data collection, models, and software development. As software has become increasingly available and mark–recapture models increasingly understood, more diverse mark– recapture data have become available. This in turn has spurred the development of new methods. Our ability to learn from studies has always been limited by computing ability and the availability of suitable software. Early mark–recapture software such as program BROWNIE (Brownie et al., 1985) or JOLLY (Pollock et al., 1990) was relatively easy to use but offered relatively little choice in models. Three significant advances in mark–recapture software were: (1) program SURVIV (White, 1983) which allowed customized development of general multinomial models; (2) program SURGE (Lebreton & Clobert, 1986) which allowed use of a generalized linear model framework for open population mark–recapture models; and (3) program MARK (White & Burnham, 1999) which offered a "user– friendly" windows–based front–end that allows a wide choice of models and the flexibility of the generalized linear modelling approach. Program SURGE has also been generalized to MSURGE (Choquet et al., 2004) which allows multi–state models to be fitted with the model specified through a windows–type front–end. In the past 10 years in particular there has been a proliferation of mark–recapture models. As these models have developed they have been incorporated into program MARK. This has culminated in a rapidly–lengthening model list of model choices confronting a user beginning a new analysis in MARK. At the 1997 EURING meeting the new analysis menu in MARK offered a choice between 10 models. Currently, there are 47 models for the user to choose from. The models available in MARK can be categorized as open–population models that are conditional on release including the Cormack–Jolly–Seber (CJS) model (Cormack, 1964; Jolly, 1965; Seber, 1965), band recovery models (Brownie et al., 1985), multistate models (Brownie et al., 1993; Schwarz et al., 1993), joint recapture, dead recovery, and resighting models (Barker, 1997; Burnham, 1993), closed population models (Otis et al., 1978; Norris & Pollock, 1996; Pledger, 2000), robust design models (Pollock, 1981; Kendall et al., 1997) including robust–design generalizations of multistate models (Schwarz & Stobo, 1997; Kendall & Bjorkland, 2001) and of joint recapture, resighting and recovery models (Lindberg et al., 2001), and open population models that consider abundance or population growth (Pradel, 1996; Schwarz & Arnason, 1996). The correct choice of model depends on the nature of the

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data and the parameters of interest. Good knowledge of the mark–recapture literature is required for the user to correctly select the model. Many of the models in MARK represent small variations on a basic model type. For example the joint models of Burnham and Barker are closely related: Burnham’s model extends the CJS model to include data from dead recoveries of animals and Barker’s model extends Burnham’s to include resightings of animals between sampling occasions. The Pradel "survival and seniority", "survival and lambda", and "survival and recruitment" models represent three different parameterisations of the same model. The "Recoveries only" and "Brownie at al. recoveries" models in MARK also represent different parameterisations of the same model. A strength of MARK is the ability to add constraints and covariate effects to the model easily in a generalized linear modelling framework. Although the generalized linear model approach is very flexible, multiple parameterisations of the same model are included in MARK because constraints that are easily included under one parameterization may be difficult to include under another. For example, an analysis of open population data using Pradel’s (1996) model in which recruitment rate f is equal for all periods, or follows a linear trend on the log–scale, is easily implemented if the model is parameterized in terms of and f. Including this same constraint when the model is parameterized in terms of = + f is more difficult. If a logit–link is used for and a log–link for in this situation to model the effect of covariates, say, the required constraints become nonlinear and cannot be coded using the design matrix. Another example where constraints that are linear under one parametrization and nonlinear under another is the closed–population heterogeneity model of Agresti (1994) and Tjur (1982). Using a log–linear specification in which the encounter history probabilities are expressed in terms of parameters representing log– odds of capture and log–odds ratios the required constraints are linear. If the model is instead parameterised in terms of capture (p) and recapture (c) probabilities, as is done in MARK, the required constraints are nonlinear. As the choice of models and complexity of software grows it is worth reflecting on whether the software can be improved. Below are some requirements we consider necessary for good mark–recapture software: 1. A choice of models that allows full use of all relevant data should be available. The choice of model should be guided in an obvious way based on the data available and the parameters of interest. One challenge is incorporating data from several sources. For example, a mark–recapture study might generate live recaptures, dead recovery, and resighting data. Alternatively, data might be available from several simultaneous studies. Multiple sources of data are important because it is often expensive to mark large numbers of animals but follow–up information can be relatively easy to

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obtain. This is especially true as marking technology improves. Using radio–telemetry it is now possible to obtain large amounts of data, often in continuous time, from a relatively small number of tags. Making full use of these data is important so that maximum value is obtained from the study. 2. There should be flexibility to include nonlinear constraints in a simple way. It should also be possible for the user to estimate functions of parameters without having to reparameterize the model. 3. The software needs to have effective methods for dealing with large numbers of nuisance parameters. Many of the parameters in mark–recapture models are needed to describe aspects of the sampling process that are not of biologically intrinsic interest. The focus of the biologists should be on modelling interesting demographic patterns rather than the sampling processes required to obtain useful estimates of demographic parameters. 4. The models that are available should lend themselves to hierarchical extensions. Hierarchical modelling is of increasing interest as methods for fitting these models develop. Firstly, hierarchical models allow modeling of the relationship between parameters and parameter covariates; this process allows relevant biological questions to be answered. Secondly, they provide a framework for dealing with large numbers of nuisance parameters. Of the two reasons, the first is perhaps the most important. To illustrate this point, suppose it were possible for biologists to visit a study site regularly, and instead of marking and recapturing animals, they could somehow record the correct values of critical parameters. The biologists would almost certainly want to treat these repeated observations as data and fit some sort of model to the parameters. Hierarchical mark– recapture models that allow parameters to be modelled as random variables (i.e., as "data") are poorly developed. Most mark–recapture models focus on modeling the sampling process and regard the parameters as fixed constants to be estimated. In many respects, this focuses on the wrong process; it is the demographic mechanism that generated the parameters that should be of primary focus. Recently developed models such as the joint model of Burnham (1993) or the robust design model (Kendall et al., 1997) have a specific likelihood function that has been coded in program MARK. We can envisage, however, a general model that incorporates all the features of the recently developed models: it is a multi–state robust–design mark–recapture model that includes dead recoveries and resightings of marked animals and is parameterised in terms of state–specific recruitment, survival and capture probabilities, state–specific abundances, and state–specific recovery and resighting probabilities. One approach to developing software for a general model would be to write code that programs the complete likelihood function for the model. While it could be used as a basis for virtually all mark– recapture analyses it would be far too general for most studies. Irrelevant components of the model

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would need to be disabled by making appropriate parameter constraints. An alternative approach would be to compartmentalize the model in such a way that, through an interface, the user was able to construct a customized likelihood by making appropriate selections from a choice of modules. Future work would focus on developing new modules rather than completely rewriting the likelihood function to accommodate new sampling and modeling developments. In the rest of this paper we outline how such a customized likelihood function might be constructed. A general likelihood for capture–recapture models in discrete–time An underlying feature of almost every mark–recapture (MR) model is the need to model the capture process. Data from a MR experiment can be considered as repeated categorical measures with missing values. These missing values arise because animals can avoid capture and because the probability that an observation is missing depends on the survival status of the animal, it is important that the capture process is modelled correctly. Two approaches to modelling MR data have been developed. The first is based on direct modelling of the encounter histories using capture and survival probabilities. These approaches are exemplified by the sequence of models M0 through Mtbh as coded in program CAPTURE (Otis et al., 1978) for closed populations and by models such as the CJS model and the band return models of Brownie et al. (1985). These models are parameterised directly in terms of capture and survival probabilities. Model selection tends to focus on selecting a simplified description of the sampling process, and comparing different demographic parameterisations. The capture probabilities are often regarded as nuisance parameters in these analyses. More recently, loglinear models have been used for analysing mark–recapture data. Most of the work has been on closed population models, although there has been some work on modelling open population models. The log–linear approaches pioneered by Fienberg (1972), Cormack (1989) and Agresti (1994) provide a general framework for analysing mark–recapture data. In this approach the data can be thought of as contributing to a 2k contingency table with each sample generating a binary classification according to whether or not an animal is caught (0,1). The cell corresponding to the null history 00…0 is missing. The likelihood function is expressed as a linear function on the log (or multinomial logit) scale of main–effects and interactions. The most general (saturated) model contains k main–effects and all interactions up to and including the (k – 1)–way interaction. A k–way interaction cannot be fitted due to the fact that the null cell 00…0 is unobservable. The emphasis in the log–linear approach is in selecting between models with different interaction terms included in the model. For closed populations,

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model M0 corresponds to a model in which the main effects are equal, and all interactions are zero. Model Mt arises when the main effects are different but all interactions are zero. For these populations the interactions are usually regarded as nuisance parameters; the emphasis is on adopting a parsimonious model to obtain an estimate of abundance with small error. For closed populations the log–linear approach is a powerful technique that allows a full range of models to be fitted including versions of Mb, Mtb, Mh, Mbh and Mtbh in a likelihood framework. The loglinear approach has been applied to open populations by Cormack (1994) and more recently Rivest & Daigle (2004). For these populations, certain interactions must be included in the model to allow for mortality and births. The research emphasis has largely been on identifying equivalent log–linear models for standard models such as the Jolly–Seber model (Cormack, 1994) or the robust design (Rivest & Daigle, unpublished). However, there has been recognition of the potential that log–linear models have for increasing the richness of model structure in a parsimonious manner. For large studies involving many years the potential number of models is very large. For example, a 20– year single–state mark–recapture model has the potential for a model that has up to 1,048,575 parameters with a very much larger number of reduced parameter models! Clearly we are only ever likely to explore a very small fraction of these potential models, most of which will be too complicated to be useful. This large number of potential models also makes clear the importance of identifying a reasonable set of possible models be identified before model fitting and that an approach based on a philosophy of exploring all possible is fruitless. An advantage of the log–linear approach is that the fit of the model is often improved by the addition of a small number of interaction terms that allow some dependencies between samples but without the large number of parameters needed to specify full dependency. Although log–linear models have only been explored for closed populations, and for straight– forward Jolly–Seber and robust–design type models, they can in principle be extended to allow for other types of data including dead–recoveries and live–resightings, and to multiple capture/recapture states. For multi–state models, each additional state contributes another level within the capture classification. That is, instead of a 2k cross–classification it leads to a Sk cross–classification. The inclusion of dead recoveries and live resighting also contribute additional classification states. We can envisage then a general model that has multi–state captures/recaptures, dead recoveries, and live resightings. Instead of a simple 0 (not caught) and 1 (caught) we can extend the MARK LDLD…LD data classification to 00,10,20,…,S0,11,21,…,S1, 12,22,…,S2,01,02 where the "L" member of the LD pair indicates capture state 0,1,…,S and the "D" member of the LD pair for the interval beginning

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with sample i takes the value 1 if the animal is found dead in [i, i+1), 2 if resighted alive in [i, i+1), and 0 otherwise. For this model, we have a k3(S+1) cross–classification with the potential for k3(S+1)–1 identifiable parameters. For example, a 5–period model with 3 states could have up to 499 parameters increasing to 3,999 for a 10–period study. This result emphasises the need for constraints that reduce the number of parameters in order to provide relatively simple descriptions of such complicated data sets. In the log–linear approach it is relatively easy to find a parametric summary of important features of the data using standard linear modeling and model selection techniques. By incorporating the appropriate interactions for describing between–sample dependencies virtually any mark–recapture model based on a multinomial likelihood in discrete time can be accommodated. The difficulty lies in interpreting the coefficients of the fitted model. For the biologist, the parameters need to be expressed in terms of natural parameters such as survival and capture probabilities. If the log–linear approach is to develop, research needs to focus on identifying constraints on interaction terms that correspond to a reasonable set of constraints on the capture/ recovery/resighting processes such as age–dependence, temporary trap response, and simple heterogeneity (Agresti, 1994; Tjur, 1982). An alternative to the log–linear approach is to identify a useful set of constrained models by directly modelling the capture and demographic processes. This has been the traditional approach used in mark–recapture modelling. Below, we sketch out how the very general model described above could be constructed by adding factors that correspond to different components of the data. Such a construction would allow the likelihood function for a mark–recapture analysis to be customized according to the components that are actually present in the dataset. Factorization of the CJS and JS models Principal among open population capture–recapture models is the Cormack–Jolly Seber (CJS) model, first developed by Cormack (1964) and later extended by Jolly (1965) and Seber (1965). Under the assumption that all animals have the same probability of survival between sample i and i+1, and the same probability of capture in sample i, data in the CJS model can be summarized by an encounter history, with the count Xw denoting the number of animals that share the history w. The probability model used for inference is based on the joint distribution of the set of encounter histories which we denote by [{Xw}]. In the CJS model we condition on the first release of each animal; implicit in this is the idea that although the numbers of animals released in each sample are usually random variables, their distribution is uninformative about the parameters of interest: the survival and capture probabilities.

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Following Link & Barker (in press) we summarize the number of animals first caught at i by ui. Excluding the null history 0 = 00…0, we can write the joint distribution of the encounter history counts conditional on the first captures as [{Xw g 0x{ui}] which factors as: [{Xw g 0x{ui}] = [{Xw g 0x{Ri}] x [{Ri}x{ui}] The first term, [{Xw g 0x{Ri}] is the distribution of the encounter history counts conditional on the releases and so represents the CJS model. The second term [{Ri}x{ui}] represents the distribution of the releases conditional on the first captures. It is needed to complete the description of the encounter histories but is usually of little interest. If there are no losses on capture, then [{Ri}x{ui}] = 1.0 and the CJS model is [{Xw g 0x{ui}]. If there are losses on capture then the term [{Ri}x{ui}] can be factored out. In our discussion below we refer to [{Xw g 0x{ui}] as the CJS model; it is understood that if losses on capture are present, they are factored out of the model. Where interest is in all demographic processes contributing to population dynamics, the CJS model is inadequate as the only demographic parameters are the survival probabilities. Information in the data about birth and abundance processes is contained in the distribution of the {ui}. For example, the Jolly– Seber model is represented by multiplying the distribution [{Xw g 0x{ui}] by [{ui}x{Ui}] where Ui denotes the number of unmarked animals in the population at the time of sample i (Seber, 1982). Thus, the Jolly– Seber model can be constructed from three distinct factors, each representing a distinct aspect of the sampling process. An alternative extension of the CJS model developed by Schwarz & Arnason (1996) building on ideas from Crosbie & Manly (1985) is to multiply [{Xw g 0x{ui}] by [u.xN] x [{ui}x{u.}] where [u.xN] represents the distribution of the number of unmarked animals caught during the study conditional on N, number of animals that were ever available for capture during the study. The term [{ui}x{u.}] represents the distribution of captures of unmarked animals conditional on u.; that is, it models the first capture of an animal given that it was ever caught. If the survival probabilities are constrained to one, and if all animals that were ever available for capture were present in the study population at the start of the experiment, then [u.xN] x [{ui}x{u.}] x [{Xw g 0x{ui}] describes the distribution of the encounter histories for a closed population study with population size N. Although the term [{Xw g 0x{ui}] is discussed above with reference to the CJS model it can be any appropriate distribution describing a set of encounter histories conditional on the first captures. Of the models currently in MARK, the term [{Xw g 0x{ui}] is sufficiently general to describe the following models: Recaptures only, Known fates, Recoveries only, Both (Burnham), Both (Barker),

and Brownie et al. Recoveries. The Schwarz and Arnason formulation thus provides a convenient way of adding birth and population growth modelling to most of the open population models in MARK, something currently only available for "Popan" (Schwarz & Arnason, 1996) and "Pradel" (Pradel, 1996) models. With appropriate modification to the distribution [{u i }x{u.}] the model [{Xw g 0x{ui}] could be the following models currently in MARK: Robust design, Multi–Strata Recaptures only, Barker Robust Design, Multi–Strata – Live and Dead Enc. All of these models can be extended to model recruitment or growth simply by adding to the likelihood the term [{ui}xu.] describing the distribution of the first captures conditional on capture at least once during the study. A complete description of the data requires the term [u.xN] where N is the number of animals that were ever available for capture during the experiment. This term is included in the MARK "Popan" model however Link & Barker (in press) show that for open populations, this term is approximately ancillary to the estimable parameters in the model; that is, it contains virtually no information about the estimable parameters. This provides justification for omitting [u.xN], the approach taken by Pradel (1996). For closed populations the term must be included in order to estimate N. Currently, program MARK is structured so that a distinct likelihood function is written for each model. However, the ability to write the models in terms of components of the sampling and demographic process as described above for the Jolly–Seber model suggests that it may be possible to customize the likelihood function to recognize different types of data and question. Future developments would not need to re–derive existing components, but would instead improve existing components or add new ones. We envisage that a user would specify the type and structure of data and indicate whether modeling of births and abundances was of interest. The appropriate [{Xw g 0x{ui}] and [{ui}xu.] terms would then be constructed based on option choices. Construction of [{Xw g 0x{ui}] The core component of the mark–recapture model is the term [{Xw g 0x{ui}]. It is here that the key information about the capture and survival processes is obtained from the data. This component can also be factored and provides the key to adding information from dead recoveries and resightings. The distribution of [{Xw g 0x{ui}] is the product of multinomial factors with indices {ui} and probabilities {Pr (Xwxui)}. An animal with encounter history w that was first caught at i, contributes to [{Xw g 0 x{ui}] through Pr (history = wxfirst caught at i) = Pr (Xwxui). If l indexes the sampling occasion when the animal was last caught, we can factor Pr (Xwxui) in the CJS model into three parts:

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Pr (Xwxui) = Pr (survival from sample i to lxfirst caught at l) x Pr (recaptures and releasesxfirst caught at i and survival to l) x Pr (not caught againxlast caught in sample l). These terms also occur in joint live–dead models (Burnham, 1993; Barker, 1997; Barker et al., in press) with slight modification. For these models and for some animals, we have the additional information that they have survived beyond the last capture period, at least until the last capture occasion that defined the start of the interval in which they were last resighted alive or were found dead. If we index this occasion by l, and the sampling occasion when the animal was first released by j, then for the joint models we have the factorization: Pr (Xwxui) = Pr (survival from sample i to lxfirst caught at l) x Pr (recaptures and releasesxfirst caught at i and survival to l) x Pr (resightingsxfirst caught at i and survival to l, recapture and releases) x Pr (encounter history after lxalive at time of sample l). This represents the CJS model with an additional factor for the resightings but with Pr (encounter history after lxalive at time of sample l) replacing Pr (not caught againxlast caught in sample l). This term incorporates the contribution made by a dead recovery or a live resighting in [l, l + 1). The term Pr (recaptures and releasesxfirst caught at i and survival to l) is the same as the equivalent term in the CJS model if there is no or random temporary emigration, but differs if there is permanent emigration. The distinction between the Barker model and the Burnham model is that the latter does not include the resighting terms and the recaptures term has been modified in the Barker model to allow Markovian temporary emigration. We can write Pr (Xwxui) for the Barker or Burnham models as the CJS model multiplied by

survival probabilities one, and that all animal that were ever in the study population were present at the start of the study. The CJS parameterization of [{Xw g 0x{ui}] is sufficiently general to accommodate behavior effects as these can be coded using the design matrix. With an interface that allows constraints to be made on main effects and interactions for a multinomial logit link function, the heterogeneity models of Agresti (1994) and Tjur (1982) could also be fitted. Generalization to robust design To generalize the CJS core, or the Burnham & Barker joint models, to robust design versions we need to add a factor that describes captures and recaptures of animals during the sequence of samples that together define primary sampling occasion i. In the robust design, each of the capture occasions indexed by i is subdivided into secondary occasions. If we let pij denote the probability of capture in secondary occasion j (j = 1,…,J) of primary occasion i then the probability pi = 1 – (1 – pi1) (1 – pi2)...(1 – piJ) corresponds to the capture probability in the CJS model or the joint models of Burnham and Barker. We can write: [Secondary and primary recaptures] = [Secondary capturesxPrimary captures] x x [Primary captures] where [Primary captures] is the joint distribution of the primary capture events (animal is caught at least once in primary period i) and is governed by the primary capture probabilities pi. To complete the generalization we need to add the factor [{ui}xu.] which introduces the birth and growth parameters into the model if these are of interest. Kendall et al. (1995) used this approach to develop the likelihood function for the robust design extension of the CJS model except that instead of [{ui}xu.] they add the factor [{ui}x{Ui}] as in the Jolly–Seber model. Generalization to Multi–state models

Pr (resightingsxfirst caght at i and survival to l, recapture and releases)

As for the single–state case, the full multi–state likelihood can be factored into three distinct parts:

Pr (encounter history after lx alive at time of sample l) [{XMw}xN(.)] = [u.(.)xN(.)] x [{ui(s)}xu.(.)] x [{Xw g 0}x{ui(s)}]

x Pr (not caught againxlast caught in sample l) so, at least in principle, we can construct the Barker or Burnham models from a core CJS model by adding these factors. Closed population models To obtain closed population models, we need to add the factors [u.xN] and [{u i xu.}] to the [{Xw g 0x{ui}] core that forms the CJS model. Because the population is closed we also need the constraints that the birth rates are all zero, the

where XMw denotes the number of animals with the multi–state encounter history w, N(.) is the number of animals that were ever present in the population during the study, u(.) is the number of unmarked animals that were caught during the study and ui(s) is the number of unmarked animals caught in state s in sample i. The term [{XMw g 0}x{ui(s)}] is the joint distribution of the counts of the multi–state encounter histories conditional on the numbers first caught at each time and state. As such it represents any multi– state model conditional on the first captures including the Arnason–Schwarz (Schwartz et al., 1993)

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model, the memory model of Brownie et al. (1993) and any extensions that include dead recoveries or live resightings. As for the single–state model the term [{ui(s)}xu.(.)] contains the useful information about changes in abundance and recruitment and provides the means for adding these components to the multi–state model. The multi–state model written as above is not in the form of an expression involving a CJS core with a multi–state factor(s) added to the model. As an alternative we can write: [{XMw}xN(.)] = = [u.(.)xN(.)] x [{ui(.)}xu.(.)] x [{Xw g 0}x{ui(.)}] x [XMw g 0}x{Xw g 0}] Here, Xw represents the encounter history collapsed across the states and so the first three factors are in the same form as the extended CJS model discussed above. However, each of these factors represent distributions that are complicated functions of state–specific capture, movement and survival probabilities. Whether this is a productive avenue to explore depends on how tractable these expressions turn out to be. The advantage would be that the full model can be built up from a CJS–type core, however the functions at the heart of this core will need to accept multi–state arguments. Discussion The development of general software that allows biologists to focus on biological questions rather than battle with technological limitations of data analysis is important. We have outlined one approach to software development that may be useful in this context. Regardless of the approach, key issues that need to be addressed in future developments include the need for a suitable mechanism for incorporating alternative model parameterisations, multistate complications, handling large numbers of nuisance parameters, and ability to carry out hierarchical modelling. A mechanism for offering alternative model parameterisations is important because biologists are often interested in estimating functions of parameters. The invariance property of the maximum likelihood estimator (Cox & Hinckley, 1974) means that it is easy to find the MLE and its variance for a function of parameters. A second reason for reparameterizing is that some parameter scales are more natural for introducing certain kinds of constraints. The programming problem is finding a compromise between having a wide choice of parameter transformations and the ease of use of the software. For example, nonlinear constraints on parameters can be programmed using Lagrange multipliers, as in POPAN (Arnason & Schwarz, 2002). However, computer code for incorporating nonlinear constraints in this manner would require the parsing of complicated algebraic expression making the programming problem difficult.

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The multistate model is an exciting model that seems to present special difficulties (Lebreton & Pradel, 2002). A particular problem is that the likelihood function can have multiple–maxima, particularly when constraints are introduced into the model. This problem is exacerbated by the very large number of parameters for even moderately– sized multistate problems making it difficult to adequately explore the likelihood function graphically. An advantage of the approach that we have outlined of partitioning the likelihood function into components is that the likelihood function could be optimised sequentially. This is the approach taken by Schwarz & Arnason (1996) and if applied to the multi–state model might make it easier to explore the problem parts of the likelihood function. A crucial issue is the management of a large number of nuisance parameters. It is not difficult to construct a model for data that has several hundred parameters, many of which are of little interest to the researcher. This problem is exacerbated if unobservable and misclassified states are included in the model (Kendall, 2004). These nuisance parameters are needed however to correctly specify the model and to maximize the information that is extracted from the data about the demographic parameters. Because having many nuisance parameters causes reduced precision of parameter estimates it is usually desirable to decrease their number. The standard approach in mark–recapture modeling is to use model–selection to reduce the number of nuisance parameters. The main difficulty with this approach is that there are often a large number of plausible models with differing structure imposed on nuisance parameters. The need to carry out model–selection over these parameters is a distraction from what should be the core focus of model selection: exploring biologically interesting hypotheses by comparing a small set of models that have various restrictions on the demographic parameters. An alternative could be to fit a very general model with few restrictions on nuisance parameters and then consider biological hypotheses by restricting the demographic parameters. An intermediate approach is the use of hierarchical models in which nuisance parameters are modelled as random effects; currently software that allows this approach is only available for relatively simple models such as the CJS model. An alternative to reducing the number of nuisance parameters by restricting them is to eliminate them entirely from the model. The elimination of nuisance parameters is a well–known and difficult problem (see Berger et al., 1999 for a review). This can be done by integrating the nuisance parameters out of the model; this is essentially a Bayesian approach and can be done with respect to a non–infomative prior distribution for the nuisance parameters. Computationally this approach is prohibitive for anything but very simple mark–recapture models. Conditional likelihood can also be used to eliminate nuisance parameters. This approach is suitable when components of the suffi-

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cient statistics have conditional distributions that depend only on the parameters of interest. The simplest example where conditional likelihood is used in mark–recapture analysis is the 2–sample closed–population model Mt. Assuming that the number of animals with the four possible capture histories 11, 10, 01, and 00 are multinomial with index N we can write:

where the sufficient statistics are ni, the number of animals caught in sample i and m2, the number of marked animals caught in sample 2. The hypergeometric term represents the distribution [m2xN] and does not depend on the nuisance parameters hence can be used as a conditional likelihood. Conditional likelihoods have also been used for modelling heterogeneity in closed populations (Agresti, 1994; Tjur, 1982) and in the memory model of Brownie et al. (1993). Conditional likelihoods remain an interesting subject for research but at present there is little scope for their use in mark–recapture modelling. For a variety of reasons, including those outlined above, we believe that hierarchical models are going to become of increasing importance to biologists. Programs such as MARK, POPAN, SURGE and SURVIV, and the possible extensions that we have outlined above code the likelihood function for specific situations. Constructing likelihood functions for hierarchical models is prohibitive in most cases because of the multi– dimensional integrations required. An alternative approach that has recently become popular is vague prior Bayesian analysis based on McMC, for example using program WinBUGS (Spiegelhalter, 2000). In this approach a prior distribution for parameters is specified, and this distribution itself may have a so–called hyper– prior distribution. Inference is made by summarizing the posterior distribution which is approximated by Monte Carlo simulation. Simple models such as the CJS model are easily coded in WinBUGS, but it becomes more difficult for some of the more complicated models. An alternative to WinBUGS is to develop specific mark–recapture code. For example, a HyperMARK module could be developed that clips on a user–specified prior distribution to the distributions used for expressing the mark–recapture likelihood function. McMC could then be used to sample from to sample from the posterior distribution. The key issues here are (1) writing the HyerMARK code in an efficient and user–friendly manner that makes specification of biologically relevant models easy and (2) the user would need to accept the logic of Bayesian inference. If we can accept a vague

prior as a statement of knowledge about parameters before the experiment then the Bayesian logic is impeccable, however the appropriate description of near–ignorance before an experiment is carried out is controversial. References Agresti, A., 1994. Simple capture–recapture models permitting unequal catchability and variable sampling effort. Biometrics, 50: 494–500. Arnason, A. N. & Schwarz, C. J., 2002. POPAN–6: exploring convergence and estimate properties with SIMULATE. Journal of Applied Staitistics, 29: 649–668. Barker, R. J., 1997. Joint modelling of live–recapture, tag–resight, and tag–recovery data. Biometrics, 53: 666–677. Barker, R. J., Burnham, K. P. & White, G. C., in press. Encounter history modelling of joint mark– recapture, tag–resighting and tag–recovery data under temporary emigration. Statistica Sinica. Berger, J. O., Brunero, L. & Wolpert, R. L., 1999 Integrated Likelihood Methods for Eliminating Nuisance Parameters. Statistical Science, 14: 1–28. Brownie, C., Anderson, D. R., Burnham, K. P. & Robson, D. S., 1985. Statistical Inference from Band–Recovery Data: A handbook, 2nd edition. U.S. Fish and Wildlife Service, Resource Publication 156. Washington D.C., U.S. Dep. of the Interior. Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H. & Hestbeck, J. B., 1993. Capture–recapture studies for multiple strata including non–Markovian transitions. Biometrics, 49: 1173–1187. Burnham, K. P., 1993. A theory for combined analysis of ring recovery and recapture data. In: Marked Individuals in Bird Population Studies: 199–213 (J.–D. Lebreton & P. North, Eds.). Birkhauser Verlag, Basel. Choquet R., Reboulet, A.–M., Pradel, R., Gimenez, O. & Lebreton, J. D., 2004. M–SURGE New software for multistate recapture models. Animal Biodiversity and Conservation, 27.1: Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438. – 1989. Log–linear models for capture–recapture. Biometrics, 45: 395–413. – 1994. Unification of mark–recapture analyses by loglinear modelling. In: Statistics in Ecology and Environmental Monitoring (D. J. Fletcher & B. F. J. Manly, Eds.). Otago Conference Series 2, Univ. of Otago Press, Dunedin. Cox, D. R. & Hinkley, D. V., 1974. Theoretical Statistics. Chapman and Hall, London, U.K. Crosbie, S. F. & Manly, B. F. J., 1985. Parsimonious modelling of capture–mark–recapture studies. Biometrics, 41: 385–398. Fienberg, S. E., 1972. The multiple–recapture census for closed populations and incomplete contingency tables. Biometrika, 59: 591–603.

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Jolly, G. M., 1965. Explicit estimates from capture– recapture data with both death and immigration stochastic model. Biometrika, 52: 225–247. Kendall, W. L., 2004. Coping with unobservable and misclassified states in capture–recapture studies. Animal Biodiversity and Conservation, 27.1: 97–107. Kendall, W. L. & Bjorkland, R., 2001. Using open robust design models to estimate temporary emigration from capture–recapture data. Biometrics, 57: 1113–1122. Kendall, W. L., Nichols, J. D. & Hines, J. E., 1997. Estimating temporary emigration and breeding proportions using capture–recapture data with Pollock’s robust design. Ecology, 78: 563–578. Kendall, W. L., Pollock, K. H. & Brownie, C., 1995. A Likelihood–based approach to capture–recapture estimations of demographic parameters under the robust design. Biometrics, 51: 293–308. Lebreton, J.–D. & Clobert, J., 1986. Users manual for program SURGE. Version 2.0. C.E.F.E, C.N.R.S., Montpellier, France. Lebreton, J.–D. & Pradel, R., 2002. Multistate recapture models: modelling incomplete individual histories. Journal of Applied Statistics, 29: 353–369. Lindberg, M. S., Kendall, W. L., Hines, J. E. & Anderson, M. G., 2001. Combining band recovery data and Pollock’s robust design to model temporary and permanent emigration. Biometrics, 57: 273–281. Link, W. A. & Barker, R. J. (in press) Hierarchical models for open population capture–recapture data. Biometrics. Norris, J. L. & Pollock, K. H., 1996. Non–parametric MLE under two closed capture–recapture models with heterogeneity. Biometrics, 52: 639–649. Otis, D. L., Burnham, K. P., White, G. C. & Anderson, D. R., 1978. Statistical inference from capture– recapture data in closed animal populations. Wildlife Monographs, 62: Pollock, K. H., 1981. Capture–recapture models allowing for age–dependent survival and capture

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rates. Biometrics, 37: 521–529. Pollock, K. H., Nichols, J. D., Brownie, C. & Hines, J. E., 1990. Statistical inference for capture– recapture experiments. Wildlife Monographs, 107: 1–97. Pledger, S., 2000. Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics, 56: 434–442. Pradel, R., 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Rivest, L.–P. & Daigle, G., 2004. Loglinear models for the robust design in mark–recapture experiments. Biometrics, 60: 100–107. Schwarz, C. J. & Arnason, A. N., 1996. A general methodology for the analysis of capture–recapture experiments in open populations. Biometrics, 52: 860–873. Schwarz, C. J. Schweigert, J. F. & Arnason, A. N., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Schwarz, C. J. & Stobo, W. T., 1997. Estimating temporary migration using the robust design. Biometrics, 53: 178–194 Seber, G. A. F., 1965. A note on the multiple recapture census. Biometrika, 52: 249–259. – 1982. The Estimation of Animal Abundance and Related Parameter. Charles Griffin and Co., London. Spiegelhalter, D., Thomas, A. & Best, N. G., 2000. WinBUGS Version 1.3 user manual. Medical Research Council Biostatistics Unit, Cambridge. Tjur, T., 1982. A connection between Rasch’s Item Analysis Model and a Multiplicative Poisson Model. Scandinavian Journal of Statistics, 9: 23–30. White, G. C., 1983. Numerical estimation of survival rates from band–recovery and biotelemetry data. Journal of Wildlife Management, 47: 716– 728. White, G. C. & P. Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46(supplement): 120–139.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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Modeling nest–survival data: a comparison of recently developed methods that can be implemented in MARK and SAS J. J. Rotella, S. J. Dinsmore & T. L. Shaffer

Rotella, J. J., Dinsmore, S. J. & Shaffer, T. L., 2004. Modeling nest–survival data: a comparison of recently developed methods that can be implemented in MARK and SAS. Animal Biodiversity and Conservation, 27.1: 187–205. Abstract Modeling nest–survival data: a comparison of recently developed methods that can be implemented in MARK and SAS.— Estimating nest success and evaluating factors potentially related to the survival rates of nests are key aspects of many studies of avian populations. A strong interest in nest success has led to a rich literature detailing a variety of estimation methods for this vital rate. In recent years, modeling approaches have undergone especially rapid development. Despite these advances, most researchers still employ Mayfield’s ad–hoc method (Mayfield, 1961) or, in some cases, the maximum–likelihood estimator of Johnson (1979) and Bart & Robson (1982). Such methods permit analyses of stratified data but do not allow for more complex and realistic models of nest survival rate that include covariates that vary by individual, nest age, time, etc. and that may be continuous or categorical. Methods that allow researchers to rigorously assess the importance of a variety of biological factors that might affect nest survival rates can now be readily implemented in Program MARK and in SAS’s Proc GENMOD and Proc NLMIXED. Accordingly, use of Mayfield’s estimator without first evaluating the need for more complex models of nest survival rate cannot be justified. With the goal of increasing the use of more flexible methods, we first describe the likelihood used for these models and then consider the question of what the effective sample size is for computation of AICc. Next, we consider the advantages and disadvantages of these different programs in terms of ease of data input and model construction; utility/flexibility of generated estimates and predictions; ease of model selection; and ability to estimate variance components. An example data set is then analyzed using both MARK and SAS to demonstrate implementation of the methods with various models that contain nest–, group– (or block–), and time–specific covariates. Finally, we discuss improvements that would, if they became available, promote a better general understanding of nest survival rates. Key words: Nest success, Nest survival, Generalized linear models, Avian demography, Mayfield. Resumen Modelización de datos de supervivencia en nidos: estudio comparativo de varios métodos desarrollados recientemente que pueden implementarse en MARK y SAS.— La estimación del éxito de nidificación y la evaluación de los factores potencialmente relacionados con las tasas de supervivencia de los mismos son aspectos clave de numerosos estudios sobre poblaciones de aves. El gran interés por el éxito de nidificación se ha traducido en una rica literatura que detalla varios métodos de estimación de esta tasa vital. En los últimos años, los enfoques de modelización han experimentado un rápido desarrollo. No obstante, pese a estos avances, la mayoría de los investigadores siguen empleando el método ad–hoc de Mayfield (Mayfield, 1961) o, en algunos casos, el estimador de probabilidad máxima de Johnson (1979) y Bart & Robson (1982). Tales métodos permiten el análisis de datos estratificados, pero, en cambio, no permiten modelos más complejos y realistas de la tasa de supervivencia en nidos cuando se incluyen covariantes que cambian según el individuo, la edad del nido, el tiempo, etc., y que pueden ser continuas o categóricas. Actualmente, con la ayuda de Program MARK, así como de Proc GENMOD y Proc NLMIXED de SAS, es posible implementar métodos que permiten a los investigadores evaluar rigurosamente la importancia de varios factores biológicos susceptibles de incidir en las tasas de ISSN: 1578–665X

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supervivencia en nidos. Por consiguiente, no está justificada la utilización del estimador de Mayfield sin antes evaluar la necesidad de emplear modelos más complejos para determinar la tasa de supervivencia en nidos. Con objeto de incrementar el empleo de métodos más flexibles, primero describimos la probabilidad empleada para estos modelos, para posteriormente tomar en consideración cuál es el tamaño de muestra eficaz para el cálculo de AICc. Seguidamente, tomamos en consideración las ventajas y desventajas de estos programas por lo que respecta a la facilidad de introducción de datos y de construcción de modelos, la utilidad/flexibilidad de las estimaciones y predicciones generadas, la facilidad de la selección de modelos y la capacidad para estimar los componentes de la varianza. A continuación, analizamos un conjunto de datos de ejemplo utilizando los programas MARK y SAS con objeto de demostrar la implementación de los métodos con varios modelos que contienen nido–, grupo– (o bloque–), y covariantes específicas al tiempo. Por último, comentamos varias mejoras que, si estuvieran disponibles, fomentarían una mejor comprensión general de las tasas de supervivencia en nidos. Palabras clave: Éxito de nidificación, Supervivencia en nidos, Modelos lineales generalizados, Demografía en aves, Mayfield. Jay J. Rotella, Ecology Dept., Montana State Univ., Bozeman, Montana 59717, U.S.A.– Stephen J. Dinsmore, Dept of Wildlife and Fisheries, Box 9690, Mississippi State, Mississippi 39762, U.S.A.– Terry L. Shaffer, Northern Prairie Wildlife Research Center, U S Geological Survey, 8711 37th Street Southeast, Jamestown, North Dakota 58401, U.S.A.

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Introduction Nest success is a key component of reproductive rate for many species of birds, which can be defined as the probability that a nest survives from initiation to completion and has at least one offspring leave the nest. Accordingly, estimates of nest success are crucial to many studies of avian populations, and methods for estimating this vital rate have received considerable attention (e.g., Mayfield, 1961; Johnson, 1979; Hensler & Nichols, 1981; Bart & Robson, 1982; Pollock & Cornelius, 1988; Bromaghin & McDonald, 1993; Aebischer, 1999; Natarajan & McCulloch, 1999; Rotella et al., 2000; Dinsmore et al., 2002). Williams et al. (2002) provide a useful review of historical development, available approaches, and estimation programs. In most studies, nests are found at various (and perhaps unknown) ages, i.e., days since the first egg was deposited in the nest, and estimation is done on a nest’s daily survival rate during the period of time it is under observation. The estimated probabilities of surviving each day in the entire nesting cycle are then multiplied together to estimate nest success. The Mayfield method, either in its original form or as expanded by Johnson (1979) and Bart & Robson (1982), requires the assumption of a constant daily nest survival rate for all nests in a sample over the time period being considered. Thus, nest success data are frequently divided into groups for analysis with the Mayfield method, e.g., stratification by stage of the nesting cycle, season, and habitat conditions. But, stratification can lead to small samples for many strata if multiple covariates are used to classify data, which is common because most nesting studies investigate how daily survival rates of nests vary in relation to multiple explanatory variables, many of which are measured on continuous scales. Despite these limitations, the Mayfield method is still frequently used to analyze nest success data, e.g., Chase (2002), Liebezeit & George (2002), Moorman et al. (2002), Tarvin & Garvin (2002). Accordingly, many studies fail to fully explore their interesting biological questions regarding spatial and temporal variation in daily nest survival. Yet, researchers clearly appear to be interested in such questions. For example, Chase (2002) and Liebezeit & George (2002) both used multiple–logistic regression to compare features of successful and unsuccessful nests. Given that these authors employed the Mayfield method to estimate nest success, it seems that they were aware of the bias inherent in conducting logistic regression analyses on what is essentially apparent nest success, i.e., the proportion of nests in the sample that are successful. Such an analysis is only valid if inactive nests can be found with the same probability as active ones, a condition rarely met in studies of real nests (see Shaffer [2004] for a detailed explanation). Such problematic analyses are no longer necessary, however, given recent advances in analysis methods. Concurrently, Dinsmore et al. (2002),

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Stephens (2003), and Shaffer (2004) developed methods for modeling daily survival rates of nests as functions of hypothesized nest–, group– and time– specific covariates. The likelihood–based methods allow visitation intervals to vary among observations and make no assumptions about when nest failure occurs within an interval. Values for time–specific explanatory variables, such as nest age, date, disturbance, and weather, are allowed to vary daily. Models can contain variables that are measured on categorical or continuous scales. Thus, these methods provide a highly–flexible and powerful alternative to traditional constant–nest–survival methods and allow a wide variety of competing models to be assessed via likelihood–based information–theoretic methods (see Burnham & Anderson 2002). Although the modeling methods presented by Dinsmore et al. (2002), Stephens (2003), and Shaffer (2004) all use the likelihood presented by Dinsmore et al. (2002: pp 3478), there are important differences among the approaches and how they are implemented. For example, methods of Dinsmore et al. (2002) are implemented in program MARK (White & Burnham, 1999), whereas those of Stephens (2003) and Shaffer (2004) are executed in SAS (SAS Institute, 2000). Key differences also exist in terms of input data, how covariates are handled, ease of model specification, the need to provide starting values for parameters, the potential to model random effects, simulation capabilities, and how desired model output is obtained. To facilitate use of these methods, this paper describes the general approach, points out the advantages of implementing the method in different programs, and, where possible, provides methods for re–coding data such that future projects can easily switch from one method to another. In this way, researchers can take advantage of the best features of each method based on project goals and investigator experience with the relevant software packages. A generalized linear models approach for nest survival The approaches to modeling nest–survival described by Dinsmore et al. (2002), Stephens (2003), and Shaffer (2004) extend the model described by Bart & Robson (1982). Each employs a generalized linear modeling approach (McCullagh & Nelder, 1989) based on the same binomial likelihood, where daily survival rates are modeled as a function of nest–, group, and/or time–specific covariates. Daily survival rates can then be estimated from the resulting model and multiplied together, as appropriate, to estimate nest success. To illustrate the model likelihood, let Si denote the probability that a nest survives from day i to day i + 1 (i.e., Si is a daily survival rate). Consider a nest that was found on day k, was active when revisited on day l, and was last checked on day m (k < l < m). Because the nest is known to have survived the

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first interval, its contribution to the likelihood for that interval is Sk Sk+1...Sl–1. During the second interval, the nest either survives with probability Sl Sl+1...Sm–1 or fails with probability (1 – Sl Sl+1...Sm–1). The likelihood is thus proportional to the product of probabilities of observed events for all nests in the sample (see Dinsmore et al., 2002). In the nest survival models discussed here, a link function is used to characterize the relationship between daily survival rate and the covariates of interest. A variety of link functions can be used (White & Burnham, 1999; Williams et al., 2002). Here, focus will be on use of the logit link (and its inverse) as it is the natural link for the binomial distribution (McCullagh & Nelder, 1989). The logit link is frequently used in mark–resight modeling, provides a flexible form, bounds estimates of survival in the (0,1) interval, and is available in MARK and SAS. Dinsmore et al. (2002), Stephens (2003), and Shaffer (2004) all used the logit link in their work, and Lebreton et al. (1992) presented methods for estimating confidence intervals and back– transforming to model parameters and estimates of their variances and covariances when the logit link is used. Until very recently, the log link was most commonly used in likelihood–based extensions of the Mayfield method, e.g., Johnson (1979), Bart & Robson (1982), and Rotella et al. (2000). The log link is convenient because, unlike for the logit link, covariate values can be summed across a visitation interval to obtain the appropriate covariate value for the interval being modeled (Dinsmore et al., 2002). However, convergence problems can arise when the log link is used, and the log link does not constrain survival to the interval (0,1). Further, the methods described herein provide alternative methods of obtaining the appropriate covariate value for the interval being modeled such that there is no loss of utility when using the logit link instead of the log link. Thus, use of the log link will not be considered here. With the logit link, daily survival rate of a nest on day i is modeled as

where the xji (j = 1,2,…,J) are values for J covariates on day i and the { j} are coefficients to be estimated from the data. Logit transformation of the above expression yields 0 + j j xji. Thus, it can be seen that the relationship between the logit of Si, i.e., ln (Si / (1 – Si)), and the covariates is linear, whereas the relationship between S i and the covariates is logistic or S–shaped. Once the { j} have been estimated, the {Si} for given values of {xij} can be estimated from the inverse–link function. Note that the above formulation allows daily survival rates to vary among groups of nests (i.e., group–specific covariates), among individual nests (i.e., nest–specific covariates), and among days (i.e., time–specific covariates).

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The model likelihood provides insight into the effective sample size for nest–survival data collected from periodic nest visits. A nest that survives an interval of t days is modeled as t Bernoulli trials, whereas a nest that fails the interval is modeled as a single Bernoulli trial. The effective sample size is thus the sum of (1) the total number of days that all nests were known to have survived (each day that a nest was known to survive contributes 1 to effective sample size, i.e., the outcome of each Bernoulli trial is known) and (2) the number of intervals that ended in failure (each interval in which a nest was known to fail contributes 1 to effective sample size, i.e., the exact day of failure was not known but it is certain that the nest failed during the interval). For example, a nest that survived 45–day intervals and subsequently failed would contribute 21 to the study’s effective sample size, regardless of the length of the interval during which the failure occurred. The parameters { j} of competing models are estimated iteratively by the method of maximum likelihood in software designed for generalized linear models. Accordingly, a variety of likelihood– based methods are available for obtaining parameter estimates and evaluating competing models. In the rare case of control–treatment experiments where nests are randomly allocated to treatment groups, likelihood ratio tests can be used to formally test hypotheses about whether specific covariates are associated with variation in nest survival. If an a priori set of candidate models is posed, then information–theoretic measures such as Akaike’s information criterion (AIC) can be used to select which model or models to use for inference (Burnham & Anderson, 2002). Assumptions of the daily nest–survival models described here are: (1) homogeneity of daily survival rates; (2) nest fates are correctly determined; (3) nest discovery and subsequent nest checks do not influence survival; (4) nest fates are independent; (5) all visits to nests are recorded; and (6) nest checks are conducted independently of nest fate. If data are available for > 1 interval length, an extension of the model presented by Rotella et al. (2000) can be used to evaluate and possibly relax assumption 3 as shown below. Assumption 1, by virtue of the fact that daily survival rates can be modeled as a function of group–, nest–, and time– specific covariates, is far less restrictive than is necessary for Mayfield’s method. If nest age is to be considered in models of daily survival rate then it is also assumed that nests can be correctly aged when they are first found (Dinsmore et al., 2002). Three approaches to modeling nest survival Nest survival model in program MARK (approach 1) The nest survival model of Dinsmore et al. (2002) is implemented in Program MARK (White & Burnham, 1999). Minimally, five pieces of information are re-

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quired for each nest: (1) the day the nest was found; (2) the last day the nest was checked when alive; (3) the last day the nest was checked; (4) the fate of the nest (0 = successful, 1 = depredated), and (5) the number (frequency) of nests that had this history. In MARK, these pieces of information are used to generate an encounter history for each nest in live/dead (LDLD…) format. Encounter histories, which are assigned to groups, can also include individual and group– and time–specific covariates. Group covariates can also be incorporated through the design matrix (White & Burnham, 1999). For example, to model daily nest–survival rates for Mountain Plovers (Charadrius montanus), Dinsmore et al. (2002) standardized 19 May as day 1 of the nesting season and numbered all nest–check dates sequentially thereafter. They assigned each encounter history to one of 12 groups (two sex groups in each of six years). For each nest, they included 78 individual covariates. Seventy six of the covariates accounted for the daily age of the nest on each of the 76 days of the nesting season as a continuous covariate. Beginning on the day the nest was found, nest age (in days) was entered sequentially until hatching age (29 d); all other values were zero. An example encounter history (see tables 1 and 2) for a single nest was: 53 59 63 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27; where the first five numbers represent the critical information for the nest, and the remaining values indicate daily values of nest age (i.e., the xij values for the i th nest). [Information about the last two covariates, i.e., sex and year, was provided by group assignment (see above).] The use of 76 individual covariates for nest age is a straightforward way of providing information to Program MARK about the age of each nest on each day of the nesting season; however, other options exist for handling nest age if one is proficient with the special functions (i.e., programming statements) that are allowed as entries in the design matrix of Program MARK. For example, one could simply enter the nest’s age on the first day of the nesting season (a single value instead of 76 values) and subsequently use this value and special functions in the design matrix to create the ages on other days (see the program’s help files for details). In Program MARK, the design matrix allows additional constraints to be placed on parameter estimates. In the Mountain Plover example, daily values for maximum daily temperature and daily precipitation were entered in the design matrix. The design matrix can also be used to specify individual covariates to be included in the model. If one were interested in estimating the effects of observer visits to nests on nest survival (Rotella et al., 2000), an additional 76 covariates could be added to the encounter history to indicate whether the nest was visited (1 = visited) or not (0 = not visited) on each day. As was done for nest age by

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Dinsmore et al. (2002), observer effects could be modeled as a single parameter. More complex models of observer effects could be developed if additional covariates contained information on the nature of the visit, e.g., how closely the nest was approached, how long the visit lasted, and whether or not nest contents were handled. Clearly, as shown by Dinsmore et al. (2002), a diverse collection of models can be considered with this method. Simpler models that have been commonly employed in past studies of nest survival can also be evaluated. A model that includes a single value of Si (i.e., constant for all groups, nest ages, dates, and weather conditions) is similar to that of Johnson (1979) and Bart & Robson (1982). Evaluating a model that allows {Si} to vary among groups is analogous to (but more analytically efficient than) conducting a stratified analysis with methods of Johnson (1979) and Bart & Robson (1982) and testing for homogeneity among group–specific survival rates with methods of Sauer & Williams (1989). Nest survival models in SAS (approaches 2 and 3) Stephens (2003) and Shaffer (2004) each presented methods for analyzing nest survival data containing multiple interval lengths in SAS (SAS Institute, 2000). These two different approaches have many similarities and several key differences. As was the case for Dinsmore et al.’s (2002) approach, these methods require no assumptions about when nest losses occur during an observation interval for which a nest failure is recorded and can handle diverse types of covariates. The data input for SAS is different from that used in MARK but ultimately provides the same information. For SAS, each row of input typically contains information for one observation interval for an individual nest. An observation interval is the length of time (t; an integer, typically measured in days) between any two successive nest visits. Note that for a given nest, different observation intervals do not need to be of the same length. The minimum data that must be provided are the length of the interval (t) and the nest’s fate for the interval (IFate; 1 = successful, 0 = unsuccessful). In addition, individual and group– and time–specific covariates can be included. For example, the date (StartDate) and age of the nest (StartAge) at the start of the interval might be recorded. Other individual covariates such as habitat measures associated with the nest site could be included. Group covariates such as habitat type or year could also be included. The example Mountain Plover nest encounter history presented earlier consists of data from two observation intervals and thus, would be entered as two SAS observations (observation one: 6 1 53 4; and observation two: 4 0 59 10; where the variables are t, IFate, StartDate, and StartAge). Here, the nest (1) survived the first interval, which was 6 days long and began on day 53 of the

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nesting season when the nest was 4 days old and (2) failed during the second interval, which was 4 days long and began on day 59 of the nesting season when the nest was 10 days old. The group assignments used by Dinsmore et al. (2002) can be achieved by adding two additional covariates for sex and year. Because interval lengths typically are > 1 d, it is necessary to use SAS procedures that allow logistic regression to be done iteratively for each day in an interval. Stephens (2003) and Shaffer (2004) achieved this in slightly different ways with similar end results. Proc GENMOD – approach 2 Shaffer (2004) used Proc GENMOD (SAS Institute, 2000) to achieve this by (1) specifying use of a binomial probability distribution, (2) defining the inverse link function between an interval’s fate and the covariates of interest as

and (3) specifying the response as the ratio of the outcome (IFate = 1 or 0) to the number of trials (n = 1) for each interval. For covariates that varied across an interval, e.g., nest age or weather conditions, Shaffer (2004) averaged daily values within each observation interval and used the averages as explanatory variables. For nest age and date, the average value across the interval can easily be obtained using the values for these variables at the start of the interval, the interval length, and SAS’s DATA Step. Thus, a variety of models can then be readily evaluated using various combinations of covariates and, if desired, transformed values of covariates, e.g., squared values for evaluating quadratic terms in models. Proc NLMIXED – approach 3 Stephens (2003) implemented nest–survival models in Proc NLMIXED of SAS (SAS Institute, 2000) using the same data input format as Shaffer (2004) in approach 2. However, this method uses programming statements from within NLMIXED to iteratively do logistic regression for each of the days in an interval (see below). Through programming statements, covariates that vary across an interval in a predictable fashion, e.g., date and age, can be calculated for each day of the interval thus, avoiding the need to work with values that are averaged across an interval. Consider a model that includes (1) a covariate x1 that does not vary by time, (2) nest age, and (3) date. This method models the probability that a nest survives a given interval (i.e., probability that IFatei = 1) as:

Applying this model to a 2–d observation interval that started on the 20th day of the nesting season for a nest that was 15 days old at the start of the interval and whose value for covariate x1 was 8 would yield:

Because Stephens’(2003) method allows covariates to vary for different days within an interval, observer effects on nest survival can be modeled in a straightforward manner. Specifically, an index variable (visit) is created with programming statements such that it takes on a value of 1 for the first day of an interval (day the nest was visited) and 0 otherwise. This variable can then be used to evaluate whether variation in daily survival rates was associated with observer visits. If additional covariates contain information on the nature of a nest visit, these covariates can be allowed to interact with the visit variable to test for their potential influence on survival rate. To illustrate, consider a 2–d interval and a model that includes the effect of (1) an observer visit and (2) a single covariate (x1) on daily survival rate. With Stephens’ (2003) method, IFatei =

Procedures in SAS allow for examination of a rich collection of models for nest survival and provide an alternative to Program MARK. The NLMIXED procedure also allows models to include random effects as well as fixed effects, i.e., mixed models (SAS Institute, 2000). Mixed models are appropriate if levels of some covariates (i.e., fixed effects) represent all possible levels, or at least the levels for which inferences are desired, i.e., fixed factors, whereas for others covariates (i.e., random effects), the levels observed are only a random sample of a larger set of potential levels of interest (Breslow & Clayton, 1993; Littell et al., 1996; Pinheiro & Bates, 2000). Examples of covariates that might be treated as random effects are study site, year, or individual. This is true because it will often be the case that the sites, years, or individuals studied, i.e., the particular experimental units, are selected at random from the population of interest, i.e., the population of sites, years, or individuals. As stated by Pinheiro & Bates (2000: 8), "they are effects because they represent a deviation from an overall mean". Thus, the effect of choosing a particular site, year, or individual may be a shift in the expected response value for observations made on that experimental unit relative to those made on other experimental units experiencing the same levels for the fixed effects. In other words, multiple observations made on the same site, year, or individual may be correlated, and, if so, this should be accounted for in the

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analysis. In a broad discussion of data analysis, Littell et al. (1996: vii) stated that, "we firmly believe that valid statistical analysis of most data sets requires mixed model methodology." Proc NLMIXED allows inclusion of a single random factor, which can then be modeled as an influence on daily survival rate (SAS Institute, 2000). The random effects are assumed to follow normal distributions, typically with zero mean and unknown variances. Examples of the utility of mixed models are provided by Stephens (2003), who reported strong support for models that considered study site as a random factor in a multi– site data set, and by Shaffer (2004) who illustrated use of a random site effect to analyze data from a randomized block study design. The NLMIXED procedure can be used regardless of whether Stephens (2003) or Shaffer’s (2004) approach to handling values of time–varying covariates is used. Strengths and weaknesses of each approach It is clear that each of the methods described above can consider a wide variety of models and allow for substantial improvements over analyses that have been typically employed in nesting studies. All methods will yield the same results for models that do not consider covariates that vary during an observation interval (Shaffer, 2004). For models that do consider time–varying covariates such as nest age, differences in results obtained from Shaffer’s (2004) method and those of Dinsmore et al. (2002) or Stephens (2003) will be slight, especially if intervals are short and effects of time–varying covariates are modest (Shaffer, 2004). Regardless of these similarities, there are advantages and disadvantages to each approach that may influence which method is most appropriate for use in a specific study. Program MARK is readily available at no cost at the following URL: http://www.cnr.colostate.edu/ ~gwhite/mark/mark.htm. Also, its use is well documented (e.g., White & Burnham, 1999; Dinsmore et al., 2002), and formal coursework and internet teaching materials are readily available to those interested in using the software. Software support is available through an electronic analysis forum at: http://www.phidot.org/forum/index.php. Further, Program MARK provides a consistent approach to implementing a broad variety of mark–resight analyses. Once the software is learned, MARK allows users to build a variety of models and to easily employ different link functions. AIC model–selection and model–averaging capabilities are built into the software. In addition, because Program MARK has a module specifically designed for analyzing nest–survival data, effective sample size is automatically calculated and used in AICc calculations. One drawback to use of Program MARK is the method in which nest–specific, time–varying

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covariates are entered. If nesting seasons comprise many days and a number of individual and time–varying covariates are to be considered, input files can be cumbersome. Further complications with the handling of covariates can be encountered during the modeling process if one is interested in evaluating various transformations of the covariates, including interactions or polynomial terms. The difficulty can be mitigated if all covariates of interest, including transformed values and products of multiple variables (for interaction terms) are included in the input file. Also, the tools available for transforming covariates and creating interaction terms within Program MARK are continually being improved. Monte Carlo simulation studies are useful for evaluating various properties of estimation and model–selection methods under different modeling scenarios. Program MARK does not currently have simulation capabilities for nest–survival data. Thus, Monte Carlo methods cannot yet be employed in this software. Because SAS is commonly used for a many types of statistical analyses, many researchers have ready access to the software and are familiar with its use. In SAS, the formatting of input data is flexible, and powerful tools are available for subsequent data manipulation through the DATA Step. In addition, programming capabilities are available within the relevant procedures. Thus, link functions other than the logit can be implemented (though not as easily as in Program MARK), and model specification is straightforward. Specification of models involving categorical covariates can be somewhat tedious in Proc NLMIXED, as categorical covariates must be recoded as indicator variables in a DATA Step. Proc GENMOD allows the use of a CLASS statement for specifying categorical covariates. This can save time and frustration if there are multiple categorical covariates, especially if those covariates take on numerous values, and models with and without interactions are being considered. As noted above, Proc NLMIXED allows random effects to be incorporated, which may be of interest in some studies. A variety of output data sets can be created in SAS that can be useful for examining predicted values across ranges of covariate values. AIC model–selection and model–averaging capabilities are not built into the software, but macros for performing the computations are available for Proc GENMOD at http://www.npwrc.usgs.gov/resource/ tools/nestsurv/nestsurv.htm. A macro that generates a table of AIC model–selection results from PROC NLMIXED is located at http://www.montana.edu/rotella/ nestsurv/. Because SAS procedures are not specifically designed for analyzing nest–survival data, the effective sample size is not automatically calculated and used in AICc calculations. Rather, the number of observations (intervals) present in the input file is considered to be the sample size. Fortunately, the effective sample size is easily calculated in a DATA Step, and the value can then be used to correctly

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calculate AICc (see code at http://www.npwrc.usgs.gov/ resource/tools/nestsurv/nestsurv.htm or http:// www.montana.edu/rotella/nestsurv/). Monte Carlo simulations are possible in SAS using various programming statements and macros (Fan et al., 2003). Simulations can be useful during the planning stages of a study by allowing the researcher to understand how results may vary as a function of the generating model, estimating models, sampling designs, and sampling effort. SAS programs for conducting simulations and summarizing results of analyses with Proc NLMIXED can be obtained at http://www.montana.edu/rotella/nestsurv/ Program MARK and Proc NLMIXED both allow one to input starting values for each parameter in the model. By default, when the logit link is used and one does not specify starting values for the parameters, Program MARK starts the optimization of all parameter estimates at 0.01; and NLMIXED uses a default value of 1. Model convergence can be affected by the choice of starting values, and researchers may have to experiment with different starting values to achieve convergence. This is usually not an issue unless sample sizes are small, or models are fairly complex or involve a random effect. Non–convergence can also occur when strong collinearity exists among covariates or with erroneously coded data (e.g., interval lengths < 1). Useful strategies for researchers who are experiencing convergence problems with MARK or NLMIXED is to obtain starting values by first fitting a simpler model that is nested within the model you are currently building. In our experience, this approach is critical when working in NLMIXED with models containing random effects. If convergence problems still exist, then we suggest obtaining starting values for the model’s parameters using Proc GENMOD. In most cases, starting values obtained in this manner will be very similar to or identical to final estimates produced by MARK or Proc NLMIXED (see Shaffer, 2004). A limitation of this approach is that models involving observer effects or random effects cannot be fitted in Proc GENMOD. However, Proc GENMOD can still be used to estimate starting values for other covariates in the model. Given that each of the methods described above has slight advantages and disadvantages, it is worth noting that it is not particularly difficult to convert a dataset prepared for analysis in SAS into an input file appropriate for use in Program MARK. This can be accomplished using SAS’ DATA Step and programming statements to re–configure a SAS input file that consists of multiple data records per nest into a MARK input file that contains a single record for each nest. Such a conversion, which is implemented in the example below, allows users to readily analyze data in either program MARK or SAS and thus, make the best use of each program’s advantages. Converting a dataset prepared for analysis in Program MARK into an input file appropriate for use in SAS can also be accomplished (see below). However, the

input file for Program MARK will not necessarily include the information about the dates of all nest visits, and thus, in such cases, estimation of observer effects on daily nest survival rate will not be possible unless the input data are modified to include such information. An example To demonstrate the implementation of the methods described above in both SAS and MARK, an analysis of data for Mallard (Anas platyrhynchos) nests that were monitored during 2000 in the Coteau region of North Dakota as part of a larger study (Stephens, 2003) is presented for various models that contain nest–, group– and time–specific covariates. The data set contains information from a total of 1,585 observation intervals made on 565 nests that were monitored on 18 sites during a 90–d nesting season. Interval lengths were typically 4, 5, or 6 d (average = 4.66 d, SD = 1.41 d). Here, the following subset of the covariates measured by Stephens (2003) was considered: (1) nest age (Age; 1 to 35 d), (2) date (Date; 1 to 90), (3) vegetative visual obstruction at the nest site (Robel; Robel et al., 1970), (4) the proportion of grassland cover (PpnGr) on the 10.4–km2 study site that contained the nest, (5–7) the habitat type in which the nest was located (3 indicator variables, each coded as 0 or 1, that were used to distinguish among nests found in native grassland [NatGr], planted nesting cover [PlCov], wetland vegetation [Wetl], and roadside right–of–ways [Road]), (9) study site (Site), and (10) nest–visitation status (Ob–an indicator variable coded as 1 on the day a nest was visited and 0 otherwise). Data were originally recorded in interval–specific form, i.e., each row of data contained information for one observation interval for an individual nest (table 1). Data in this format were appropriate for analysis by approaches 2 and 3 in SAS but needed to be re–formatted before being analyzed in Program MARK. Accordingly, SAS DATA Steps and programming statements (appendix 1) were used to re–configure the original data file into a MARK input file with a single record per nest (table 2). Likewise, the MARK input file can be re–configured appropriately for analysis in SAS (appendix 2). Competing models of nest survival were first evaluated using Program MARK. Because these analyses were done for illustrative purposes, the model list was kept brief, and justification of the models is not provided. No model in the list included study site because this was considered to be a random effect and so was beyond the current capabilities of nest–survival analyses in MARK. Although effects of observer visits on nest survival could have been considered in MARK, they were not because we considered it best to estimate observer effects in models that considered as many other sources of heterogeneity as possible, i.e., in models that contained the full suite of both fixed and random effects.

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Table 1. Input format for interval–specific nest–survival data: ID. Nest number; Species. Species code; Site. Study site; Hab. Habitat code; Int. Observation interval; t. Interval length (d); IFate. Nest fate for the interval; SDate. Date at the start of the interval; SAge. Nest age at the start of the interval; Robel. Vegetative visual obstruction at nest site; PpnGr. Proportion of grassland cover on the 10.4 km2 study site. (Also, see table 2.) Tabla 1. Formato para la introducción de datos específicos a un intervalo, relacionados con la supervivencia en: ID. Número del nido; Species. Código de la especie; Site. Lugar del estudio; Hab. Código del hábitat; Int. Intervalo de observación; t. Duración del intervalo (d); IFate. Destino del nido para el intervalo; SDate. Fecha al inicio del intervalo; SAge. Edad del nido al inicio del intervalo; Robel. Obstrucción vegetativa visual en el lugar del nido; PpnGr. Proporción de cobertura de pasto en los 10,4 km2 que ocupa el lugar del estudio. (Véase también la tabla 2.) ID

Species

Site

Hab

Int

IFate SDate

SAge

Robel

PpnGr

Mall

PlCov

Mall

PlCov

4.50

0.96

4.50

Mall

PlCov

0.96

4.50

0.96

Mall

PlCov

Mall

PlCov

4.50

0.96

4.50

0.96

Mall

PlCov

Mall

PlCov

4.50

0.96

4.50

0.96

Mall

PlCov

0.88

0.96

PlCov

0.88

Mall

0.96

PlCov

0.88

0.96

Mall

PlCov

0.88

0.96

2,206

Mall

Road

6.00

0.80

2,206

Mall

Road

6.00

0.80

2,206

Mall

Road

6.00

0.80

2,206

Mall

Road

6.00

0.80

…

The most parsimonious fixed–effects model of nest survival included Age and PpnGr (table 3). This model was 1.06 AICc units better than the second–best model, which included Age but not PpnGr, and was ≥ 2.90 AICc units better than all other models evaluated. The best model indicated that daily survival rate increased with nest age ( = 0.0188, SE = 0.007) and grassland extent ( = 0.369, SE = 0.211) (fig. 1). Models that held daily survival rate constant or simply allowed it to vary by habitat type, i.e., the only model types that have been used in many recent publications on nest survival (see above), received little support ( AICc ≥ 6.11). Shaffer (2004) showed that analyses using his methods in Proc GENMOD yield (1) the same results as those obtained in MARK for fixed–effects models with time–invariant covariates and (2) similar answers to those provided by MARK for models with time– varying covariates, e.g., nest age or date. Results from the GENMOD approach are not presented here (see appendix 3 for example code), but they were indeed nearly identical to those obtained with MARK.

Proc NLMIXED was used to evaluate the same list of fixed–effects models; results were virtually identical to those obtained from MARK. For exam= 0.3689, ple, t to at least 4 decimal places ( SE = 0.2105). To evaluate the importance of considering more complex models, three additional models were evaluated using Proc NLMIXED (appendix 4). These models were created by adding observer effects, a random effect of site, or both effects to the most parsimonious fixed–effects model. Of the 12 models considered, the two most parsimonious models both included a random effect of site (table 4). Spatial process variance was estimated as 0.089 (SE = 0.052) by the better of these two models. The second–most parsimonious model ( AICc = 0.33) provided some evidence of a negative effect of observer visits on daily survival rate for the day immediately following a nest visit ( = –0.844, SE = 0.629). The point estimate indicates that the effect was potentially of a size that is of interest, but the lack of precision makes inference difficult. For example, on a site with 50% grassland cover, daily survival

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Table 2. Input format for nest–survival data to be analyzed in Program MARK (White & Burnham, 1999; Dinsmore et al., 2002) where the numeric variables are: 1. Nest identification number; 2. The date the nest was found; 3. The last date the nest was known to be alive; 4. The date that the nest’s final fate was determined; 5. The nest’s fate (0 = successful, 1 = depredated); 6. The number of nests with this history; 7–95. The age of the nest on the first 89 days of the 90–day nesting season (a 0 is used for all dates preceding the date the nest was found and following age 35, i.e., the maximum age); 96. Vegetative visual obstruction at the nest site; 97. Proportion of grassland cover at nest site (10.4 km2); 98–101. Indicator variables indicating the habitat type a nest was in. (See table 1 for alternative format.) Tabla 2. Formato para la introducción de datos relativos a la supervivencia en nidos que se analizarán con Program MARK (White & Burnham, 1999; Dinsmore et al., 2002), donde las variables numéricas son: 1. El número de identificación del nido; 2. La fecha en que se encontró el nido; 3. La última fecha en que se tenía conocimiento de que el nido estaba vivo; 4. La fecha en que se determinó el destino final del nido; 5. El destino del nido (0 = satisfactorio, 1 = depredado); 6. El número de nidos con este historial; 7–95. La edad del nido durante los primeros 89 días de la estación de anidamiento de 90 días (para todas las fechas anteriores a la fecha en que se encontró el nido, se emplea un 0, seguido de la edad 35; es decir, la edad máxima); 96. Obstrucción vegetativa visual en el lugar del nido; 97. Proporción de cobertura de hierba en el lugar del nido (10,4 km2); 98–101. Variables indicadoras del tipo de hábitat en que se encontraba el nido. (Para un formato alternativo, ver tabla 1.)

Nest Survival Group = 1; /* 1 */ 1 35 35 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.5 0.9616 0 1 0 1; /* 2 */ 1 15 21 1 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.875 0.9616 0 1 0 1; … /* 2206 */ 73 89 89 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 6 0.8002 0 0 0 1;

for a 15–d old nest would be predicted as 0.911 (SE = 0.033, 95% CI = 0.842 to 0.981) if it were visited and 0.960 (SE = 0.010, 95% CI = 0.939 to 0.981) otherwise, where the estimates were obtained using the ESTIMATE statement (1 statement for each of the 2 scenarios) of Proc NLMIXED (see appendix 4). When the random effect of site was added to various fixed–effects models, estimates of coefficients for fixed effects were quite stable with one notable exception. The coefficient for grassland extent was reduced from 0.369 to 0.086, while the estimated standard error increased slightly (0.211 to 0.233), which alters the inferences that can be drawn from this data set in important ways, i.e., the importance of grassland extent is called into question. Because our emphasis here is on illustrating the various analysis methods, these changes are not discussed further, and other models that might be suggested by the data are not explored. However, Stephens (2003) conducted a detailed analysis of both a priori and exploratory models for the larger, multi–year dataset that included the data analyzed here.

Recommendations The methods presented by Dinsmore et al. (2002), Stephens (2003) and Shaffer (2004), which are elaborated on here, allow a variety of competing models to be assessed via likelihood–based information–theoretic methods. Thus, they provide excellent alternatives to traditional constant–survival methods, and these three approaches can be used interchangeably as best suits a particular problem. The methods presented here: (1) can be used to conduct analyses of stratified data (appropriate if the simplifying assumptions of constant survival apply) and provide estimates that are almost identical to Mayfield estimates (or various refinements), (2) perform comparisons of survival rates among groups, (3) allow a much broader variety of covariates and competing models to be evaluated, and (4) should be employed in most nest–survival studies. It is worth noting that these methods can also be used for analyzing survival data collected from radiomarked individuals using ragged (uneven) intervals among animals and over time.

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1.00 DSR – Age 30 DSR – Age 15 DSR – Age 1

Daily survival rate (DSR)

0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91

0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of study area in grassland

Fig. 1. Estimated relationship between daily survival rate (S) and the proportion of a study site consisting of grassland (PpnGr) for Mallard nests of different ages. Estimates from the best fixedeffects model where log (S / (1 – S)) = 2.43 + 0.019 · Age + 0.369. PpnGr. Fig. 1. Relación estimada entre la tasa de supervivencia diaria (S) y la proporción de un lugar de estudio formado por pastos (PpnGr) en nidos de distintas edades de Mallard. Las estimaciones de los mejores modelos de efectos fijos fueron log (S / (1 – S)) = 2.43 + 0.019 · Age + 0.369. PpnGr.

Table 3. Summary of model–selection results obtained in Program MARK (White & Burnham, 1999; Dinsmore et al., 2002) for fixed–effects models of daily survival rate for Mallard nests studied by Stephens (2003) in North Dakota. K is the number of parameters in the model, and wi is the model weight. Tabla 3. Resumen de los resultados sobre la selección de modelos obtenidos con Program MARK (White & Burnham, 1999; Dinsmore et al., 2002) para los modelos de efectos fijos de la tasa de supervivencia diaria en nidos de Mallard estudiados por Stephens (2003) en Dakota del Norte. K es el número de parámetros que contiene el modelo, y wi es el peso del modelo.

Model 0

+ 0 0

*Age +

*PpnGr

*Age *Age +

*Age + 1 1

*Robel

NatGr + 2

*CRP + 3

*PpnGr

0 0

+ 0 0

*Robel

*Date 1 1

*NatGr +

*CRP +

*Wetl

*Wetl 4

AICc

1563.010

0.000

0.465

1564.066

1.056

0.274

1565.906

2.896

0.109

1567.344

4.334

0.053

1567.368

4.358

0.053

1569.117

6.107

0.022

1570.775

7.765

0.010

1570.826

7.817

0.009

1571.957

8.948

0.005

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Table 4. Summary of model–selection results obtained in Proc NLMIXED (SAS Institute, 2000) for fixed–effects and mixed models of daily survival rate for Mallard nests studied by Stephens (2003) in North Dakota. Tabla 4. Resumen de los resultados sobre la selección de modelos obtenidos con Proc NLMIXED (SAS Institute, 2000) para modelos mixtos y de efectos fijos de la tasa de supervivencia diaria correspondiente a los nidos de Mallard estudiados por Stephens (2003) en Dakota del Norte.

Model 0

+ 0

AICc

*PpnGr + b1*site

*PpnGr +

*Age +

*Age + 1

+ 0

K *Ob + b1*site

*Ob

*PpnGr 2

*Age *Age +

*Age +

*Robel

NatGr +

*CRP +

*Wetl

*PpnGr 1

0 0

+ 0 0

*Robel

*Date 1 1

*NatGr +

*CRP +

*Wetl

Despite these advances, further analysis improvements would be useful. The methods presented here do not consider age–specific encounter probabilities, where "age" refers to the age of a newly encountered nest, as do some methods for survival analysis (Pollock & Cornelius, 1988; Williams et al., 2002). Information on age–specific nest encounter probabilities can provide information about survival probabilities prior to encounter. The utility of such information is presented by Williams et al. (2002). Improved methods of estimating goodness–of– fit and for detecting and estimating overdispersion, or extra–binomial variation, would be useful given that a variety of factors may cause overdispersion. Nest–success data are commonly collected according to multilevel designs that result in grouped data, e.g., multiple observations on at least some nests, multiple nests per site, and multiple sites within each year. Thus, undetermined random effects of individuals, sites, and years could cause overdispersion or within–group correlations in daily survival rates, e.g., nest fates from multiple nests from within a colony or from a given study plot may not be independent. In addition, the spatial clustering of covariate levels could generate spatial correlation in nest survival rates and thus cause overdispersion. The random–effects model implemented in Proc NLMIXED offers an improvement as it can estimate random effects due to one source, e.g., site. However, current methods in NLMIXED do

AICc

1554.013

0.000

0.529

1554.340

0.327

0.449

1562.265

8.252

0.009

1563.010

8.996

0.006

1564.066

10.053

0.003

1565.906

11.892

0.001

1567.344

13.330

0.001

1567.368

13.355

0.001

1569.117

15.103

0.000

1570.775

16.762

0.000

1570.826

16.813

0.000

1571.957

17.944

0.000

not accommodate multi–level nonlinear mixed models (e.g., some random effects associated with site, some associated with year, and others associated with individuals), although, as mentioned above, they will be of interest in some studies. Recently, Bayesian techniques (Cam et al., 2002; Link et al., 2002; Williams et al., 2002) were used to address individual heterogeneity in mark–resight analysis, and to estimate age–specific nest–survival rates (He et al., 2001; He, 2003). Thus, Bayesian techniques may hold potential for improving future modeling of nest survival data. A composite likelihood approach (Lele & Taper, 2002) has been used successfully for nest–success data (M. Taper, S. Lele, & J. J. Rotella) and should allow more thorough treatment of multiple random effects in nest– survival data in the future, e.g., through simultaneous consideration of factors such as individuals, sites, and years. In some studies, uncertainty will exist about nest ages and when transitions among nest stages occur (Williams et al., 2002). This problem has been addressed for stratified data (Stanley, 2000) but not yet for data sets containing more complex sets of covariates. References Aebischer, N. J., 1999. Multi–way comparisons and generalized linear models of nest success: extensions of the Mayfield method. Bird Study, 46

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(supplement): S22–S31. Bart, J. & Robson, D. S., 1982. Estimating survivorship when the subjects are visited periodically. Ecology, 63: 1078–1090. Breslow, N. E. & Clayton, D. G., 1993. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88: 9–25. Bromaghin, J. F. & McDonald, L. L., 1993. Weighted nest survival models. Biometrics, 49: 1164–1172. Burnham, K. P. & Anderson, D. R., 2002. Model Selection and Multi–model Inference: a Practical Information–Theoretic Approach, 2nd ed. Springer–Verlag, New York. Cam, E., Link, W. A., Cooch, E. G., Monnat, Y.–A., & Danchin, E., 2002. Individual covariation in life– istory traits: seeing the trees despite the forest. American Naturalist, 159. Chase, M. K., 2002. Nest site selection and nest success in a song sparrow population: the significance of spatial variation. Condor, 104: 103–116. Dinsmore, S. J., White, G. C. & Knopf, F. L., 2002. Advanced techniques for modeling avian nest survival. Ecology, 83: 3476–3488. Fan, X., Felsovalyi, A., Sivo, S. A. & Keenan, S. C., 2003. SAS for Monte Carlo studies: a guide for quantitative researchers. SAS Institute, Inc., Cary, North Carolina. He, C. Z., 2003. Bayesian modeling of age–specific survival in bird nesting studies under irregular visits. Biometrics, 59: 962–973. He, C. Z., Sun, D. & Tra, Y., 2001. Bayesian modeling age–specific survival in nesting studies under Dirichlet priors. Biometrics, 57: 281–288. Hensler, G. L. & Nichols, J. D., 1981. The Mayfield method of estimating nest success: A model, estimators and simulation results. Wilson Bulletin, 93: 42–53. Johnson, D. H., 1979. Estimating nest success: the Mayfield method and an alternative. Auk, 96: 651–661. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies. Ecological Monographs, 62: 67–118. Lele, S. & Taper, M. L., 2002. A composite likelihood approach to estimation of (co)variance components. Journal of Statistical Planning and Inference, 103: 117–135. Liebezeit, J. R., & George, T. L., 2002. Nest predators, nest–site selections, and nesting success of the dusky flycatcher in a managed ponderosa pine forest. Condor, 104: 507–517. Link, W. A., Cam, E., Nichols, J. D. & Cooch, E., 2002. Of BUGS and birds: an introduction to Markov chain Monte Carlo. Journal of Wildlife

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Management, 66: 277–291. Littell, R. C., Milliken, G. A., Stroup, W. W. & Wolfinger, R. D., 1996. SAS system for mixed models. SAS Institute Inc., Cary, North Carolina. Mayfield, H. F., 1961. Nesting success calculated from exposure. Wilson Bulletin, 73: 255–261. McCullagh, P. & Nelder, J. A., 1989. Generalized linear models, 2nd ed. Chapman and Hall, New York. Moorman, C. E., Guynn Jr., D. C. & Kilgo, J. C., 2002. Hooded warbler nesting success adjacent to group–selection and clearcut edges in a southeastern bottomland forest. Condor, 104: 366–377. Natarajan, R. & McCulloch, P. C. E., 1999. Modeling heterogeneity in nest survival data. Biometrics, 55: 553–559. Pinheiro, J. C. & Bates, D. M., 2000. Mixed–effects models in S and S–plus. Springer–Verlag, New York. Pollock, K. H. & Cornelius, W. L., 1988. A distribution–free nest survival model. Biometrics, 44: 397–404. Robel, R. J., Briggs, J. N., Dayton, A. D. & Hulbert, L. C., 1970. Relationships between visual obstruction measurements and weights of grassland vegetation. Journal of Range Management, 23: 295–298. Rotella, J. J., Taper, M. L. & Hansen, A. J., 2000. Correcting nesting success estimates for possible observer effects: maximum–likelihood estimates of daily survival rates with reduced bias. Auk, 117: 92–109. SAS Institute, 2000. SAS/STAT user’s guide, Version 8. SAS Institute, Inc., Cary, North Carolina. Sauer, J. R. & Williams, B. K., 1989. Generalized procedures for testing hypotheses about survival or recovery rates. Journal of Wildlife Management, 53: 137–142. Shaffer, T. L., 2004. A unified approach to analyzing nest success. Auk, 121: 526–540. Stanley, T. R., 2000. Modeling and estimation of stage–specific daily survival probabilities of nests. Ecology, 81: 2048–2053. Stephens, S. E., 2003. The influence of landscape characteristics on duck nesting success in the Missouri Coteau Region of North Dakota. Ph. D. Dissertation, Montana State Univ. Tarvin, K. A. & Garvin, M. C., 2002. Habitat and nesting success of blue jays (Cyanocitta cristata): importance of scale. Auk, 119: 971–983. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 Supplement: 120–138. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations: modeling, estimation, and decision making. Academic Press, New York.

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Appendix 1. Code for converting an example dataset prepared for analysis in SAS (SAS Institute, 2000) into an input file for Program MARK (White & Burnham, 1999). Apéndice 1. Código para convertir un conjunto de datos de un ejemplo preparado para ser analizado en SAS (SAS Institute, 2000) a un archivo de datos para Program MARK (White & Burnham, 1999).

proc sort data = sasuser.mall2000nd; by id int; run; data mark1; set Sasuser.mall2000nd; retain firstday; retain firstage; by id; if first. id then do; firstday= sdate; /*date at start of 1st interval = date found*/ firstage=sage; /*age at start of 1st interval = age when found*/ end; if last. id then do; lastdaylive= sdate; /*date at start of last interval for nest*/ lastday=sdate + t; /*date at end of last interval for nest*/ end; if firstday=. then delete; if lastday=. then delete; if lastdaylive=. then delete; /* create indicator variables for different nesting habitats*/ if hab=1 then NatGr=1; else NatGr=0; /*Native Grassland*/ if hab=2 or hab=3 or hab=9 then PlCov=1; else PlCov=0; /*Planted Cover*/ if hab=7 or hab=22 then Wetl=1; else Wetl=0; /*Wetland sites*/ if hab=20 or hab=8 then Road=1; else Road=0; /*Roadside sites*/ drop sdate sage; run; data mark2; set mark1; array age{90}; /*where 90 is the number of days in the study’s nesting season*/ do i=1 to 90; if i<firstday then age{i}=0; else age{i}=firstage+i-firstday; end; do i=1 to 90; if age{i}>35 then age{i}=0; /*where 35 is the species’ maximum nest*/ end; run; data markinp; set mark2; /*Create a text file with the necessary output for MARK*/ /*The directory used in the statement below must exist on the computer being used*/ file ‘C:\My Documents\nest success\MallMARK.inp’ ; /*Use ID number for first nest in the data set to put header line on file for MARK*/ /*the next line can be deleted if ID numbers aren’t in the dataset. But the header line*/ /*for MARK must then be put in by hand before using it in MARK.*/ if id=1 then put «Nest Survival Group=1 ;» ; /*Code below assumes that ifate is 1 for a nest that survives an interval and*/ /*that ifate is 0 for nests that fail during an interval between 2 visits.*/ if ifate = 1 then put «/* « id « */ « firstday lastday lastday « 0 1 « age1-age89 robel PpnGr NatGr PlCov Wetl Road «;»; else put «/* « id « */ « firstday lastdaylive lastday « 1 1 « age1-age89 robel PpnGr NatGr PlCov Wetl Road «;»; run;

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Appendix 2. Code for converting a particular dataset prepared for analysis in Program MARK (White & Burnham, 1999) into an input file for SAS (SAS Institute, 2000). Apéndice 2. Código para convertir un conjunto de datos concreto preparado para ser analizado con Programa MARK (White & Burnham, 1999) a un archivo de datos para SAS (SAS Institute, 2000).

* * * *

This program reads the MARK input file shown in table 3 and creates a SAS data set that can be analyzed with Proc Genmod or Proc Nlmixed. Because the MARK format does not contain information on all intermediate visits to a nest, a maximum of 2 records is generated for each nest.;

data mark2sas; array age(89) age1-age89; Infile ‘C:\My Documents\nest success\MallMARK.inp’ firstobs=2 lrecl=750; Input junk $ id junk $ firstday lastdaylive lastday markfate freq age1-age89 Robel PpnGr NatGr PlCov Wetl Road; If markfate=0 then do; /* successful nest - generate 1 interval with ifate=1 */ ifate=1; sdate=firstday; t = lastday - firstday; sage = age{sdate}; output; end; If markfate=1 then do; /* unsuccessful nest - generate 1 interval with ifate=1 and 1 interval with ifate=0 */ ifate=1; sdate=firstday; t = lastdaylive - firstday; sage = age{sdate}; output; ifate=0; sdate=lastdaylive; t = lastday -lastdaylive; sage = age{sdate}; output; end; keep id Robel PpnGr NatGr PlCov Wetl Road ifate sdate t sage; run; proc print; run;

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Appendix 3. Example code for analyzing fixed–effects models of nest–survival data from periodic nest visits with Proc GENMOD in SAS (SAS Institute, 2000). Note: code will have to be modified accordingly for other datasets, e.g., variable names will need to be adjusted. Apéndice 3. Código de ejemplo para analizar con Proc GENMOD en SAS (SAS Institute, 2000) modelos de efectos fijos correspondientes a los datos de supervivencia en nidos a partir de visitas periódicas a los mismos. Nota: el código deberá modificarse en función de los otros conjuntos de datos; es decir, se deberán ajustar los nombres de las variables.

* Sample code for fitting logistic–exposure models (Shaffer, 2004) to * interval nest–visit data. * * Variables are as follows: * * Ifate = 0 if nest fails the interval, and 1 if it survives. * Trials = 1 in all cases. * Avgage = age (d) of the nest at interval midpoint. * PctGr = proportion of grassland cover. * t = interval length (d) * * Macros for generating AIC analyses and model-averaged parameter * estimates are available from * http://www.npwrc.usgs.gov/resource/tools/nestsurv/nestsurv.htm *; Proc Genmod Data=Mall; Fwdlink link = log((_mean_**(1/t))/(1-_mean_**(1/t))); Invlink ilink = (exp(_xbeta_)/(1+exp(_xbeta_)))**t; Model Ifate/Trials = Avgage PpnGr / Dist=bin; Ods output modelfit=modelfit; Ods output modelinfo=modelinfo; Ods output ParameterEstimates=ParameterEstimates; Title ‘b0 + b1*avgage + b2*pctgr4’; Run;

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Appendix 4. Code for analyzing fixed– and random–effects models of nest–survival data from periodic nest visits with Proc NLMIXED in SAS (SAS Institute, 2000). Note: code will have to be modified accordingly for other datasets, e.g., variable names will need to be adjusted. Apéndice 4. Código para analizar con Proc NLMIXED en SAS (SAS Institute, 2000) modelos de efectos fijos y aleatorios correspondientes a los datos de supervivencia en nidos a partir de visitas periódicas. Nota: el código deberá modificarse en función de los otros conjuntos de datos; es decir, se deberán ajustar los nombres de las variables.

* This file: * 1. inputs a dataset containing nest survival information, * 2. calculates the effective sample size for the dataset, * 3. runs a variety of models of NEST SURVIVAL in NLMIXED, * 4. creates an AICc table for model selection, & * 5. outputs the AICc table to HTML, RTF, and pdf files. * Step 1: calculate the effective sample size according to the methods * of Dinsmore et al. (2002). Here, n–ess is incremented by 1 for * each day a nest was under observation and survived and by 1 for * each interval for which a nest was under observation and failed.; * This step: * 1. calculates the contribution to n–ess for each observation * interval and adds that contribution to the sum of ness, i.e., * ness column is a running total * 2. creates dummy/indicator variables for each of 4 habitat types; data Mall; set Sasuser.mall2000nd; if ifate=0 then ness+1; else if ifate=1 then ness+t; /* create indicator variables for different nesting habitats */ /* Native Grassland */ if hab=1 then NatGr=1; else NatGr=0; /* CRP & similar */ if hab=2 or hab=3 or hab=9 then CRP=1; else CRP=0; /* Wetland sites */ if hab=7 or hab=22 then Wetl=1; else Wetl=0; /* Roadside sites */ if hab=20 or hab=8 then Road=1; else Road=0; run; * This step finds the actual n-ess for the dataset, * which is the maximum value in the ness column.; Proc Univariate data=Mall; var ness; output out=ness max=ness; run; * This step sorts the data by site, which is used as a random factor in * some models below. PROC NLMIXED assumes that a new realization * occurs whenever the SUBJECT= variable changes from the previous * observation, so your input data set should be clustered according * to this variable. You can accomplish this by running PROC SORT * prior to calling PROC NLMIXED using the SUBJECT=variable as the * BY variable. ; Proc Sort data=Mall; by site; run; * This step reformats the Fit Statistics table of NLMIXED * so it displays more decimal places in the created tables; proc template; define table Stat.Nlm.FitStatistics; notes «Fit statistics»;

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Appendix 4. (Cont.)

column Descr Value; header H1; define H1; text «Fit Statistics»; space = 1; end; define Descr; header = «Description»; width = 30; print_headers = OFF; flow; end; define Value; header = «Value»; format = 12.4; print_headers = OFF; end; end; run; * Run the most parsimonious fixed-effects model for this datset; Proc Nlmixed data=Mall tech=quanew method=gauss maxiter=1000; parms B0=0, B1=0, B2=0; p=1; do i=0 TO t-1; logit=B0+B1*(SAge+i)+B2*PpnGr; p=p*(exp(logit)/(1+exp(logit))); end; model ifate~binomial(1,p); Run; * Run the most parsimonious fixed-effects model for this datset * with the addition of an observer effect on dsr for day 1 * of each interval (done with a dummy variable called ‘Ob’); Proc Nlmixed data=Mall tech=quanew method=gauss maxiter=1000; parms B0=0, B2=0, B4=0, B8=0; p=1; do i=0 TO t-1; if i=0 then Ob=1; else Ob=0; logit=(B0+(B8*Ob))+B2*(SAge+i)+B4*PpnGr; p=p*(exp(logit)/(1+exp(logit))); end; model ifate~binomial(1,p); Run; * Run the most parsimonious fixed-effects model for this datset * with the addition of a random effect of site, i.e., run a * mixed model where the random effect influences the intercept.; Proc Nlmixed data=Mall tech=quanew method=gauss maxiter=1000; parms B0=2.42, B2=0.019, B4=0.38, vsite=0.5; p=1; do i=0 TO t-1; if i=0 then Ob=1; else Ob=0; logit=(B0+u)+B2*(sage+i)+B4*PctGr4; p=p*(exp(logit)/(1+exp(logit))); end;

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Appendix 4. (Cont.)

model ifate~binomial(1,p); random u~normal(0,vsite) subject=site; Run; * Run the most parsimonious fixed-effects model for this datset * with the addition of an observer effect on dsr for day 1 * and the addition of a random effect of site.; Proc Nlmixed data=Mall tech=quanew method=gauss maxiter=1000; parms B0=2.42, B2=0.019, B4=0.38, B8=-1, vsite=0.5; p=1; do i=0 TO t-1; if i=0 then Ob=1; else Ob=0; logit=(B0+u+B8*Ob)+B2*(sage+i)+B4*PctGr4; p=p*(exp(logit)/(1+exp(logit))); end; model ifate~binomial(1,p); random u~normal(0,vsite) subject=site; * The following lines of code estimate the daily survival rate * (DSR) for a 15-day old nest on a site with a grassland * proportion of 0.5 on (1) the day of a nest visit and * (2) a day without a nest visit; estimate ‘dsr-visited’ exp(B0+B2*(15)+B4*0.5+B8*1)/(1+exp(B0+B2*(15)+B4*0.5+B8*1)); estimate ‘dsr-not visited’ exp(B0+B2*(15)+B4*0.5+B8*0)/(1+exp(B0+B2*(15)+B4*0.5+B8*0)); Run;

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Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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M–SURGE: new software specifically designed for multistate capture–recapture models R. Choquet, A.–M. Reboulet, R. Pradel, O. Gimenez, J.–D. Lebreton

Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O., Lebreton J.–D., 2004. M–SURGE: new software specifically designed for multistate capture–recapture models. Animal Biodiversity and Conservation, 27.1: 207–215. Abstract M–SURGE: new software specifically designed for multistate capture–recapture models.— M–SURGE (along with its companion program U–CARE) has been written specifically to handle multistate capture–recapture models and to alleviate their inherent difficulties (model specification, quality of convergence, flexibility of parameterization, assessment of fit). In its domain, M–SURGE covers a broader range of models than a general program like MARK (White & Burnham, 1999), while being more user–friendly than MS–SURVIV (Hines, 1994). Among the main features of M–SURGE is a wide class of models and a variety of parameterizations: (1) M–SURGE covers conditional models with probability of recapture depending on the current state (Arnason–Schwarz type models) as well as on the current and previous state (Jolly–movement type models). In both cases, age and/or time–dependence and multiple groups can be considered. (2) Combined survival–transition probabilities can be represented as such or decomposed into transition and survival probabilities. (3) Among the transition probabilities with the same state of departure, the one to be computed by subtraction can be freely picked by the user. User–friendliness is enhanced by the easiness with which constrained models are built, using an interpreted language called GEMACO. Examples of various types of multistate models are developed and presented. Key words: Constraints, Language, Linear Model, Population dynamics. Resumen M–SURGE: el nuevo programa diseñado específicamente para los modelos multiestado de captura–recaptura.— M–SURGE, al igual que su compañero, el programa U–CARE, se ha escrito con el propósito específico de manejar modelos multiestado de captura–recaptura, lo que a su vez permite mitigar las dificultades inherentes a los mismos (especificación de los modelos, calidad de la convergencia, flexibilidad de parametrización, evaluación del ajuste). En su terreno, M–SURGE abarca una gama de modelos más extensa que un programa general, como el MARK (White & Burnham, 1999), al tiempo que resulta más accesible para el usuario que el MS–SURVIV (Hines, 1994). De entre las principales características del M–SURGE, cabe destacar una amplia gama de modelos y varias parametrizaciones: (1) M–SURGE abarca los modelos condicionales con probabilidad de recaptura según el estado actual (modelos tipo Arnason–Schwarz), y según el estado actual y previo (modelos tipo Jolly–movement). En ambos casos, es posible examinar los efectos dependientes de la edad y/ o del tiempo, así como grupos múltiples. (2) Las probabilidades combinadas de supervivencia–transición pueden representarse como tales, o descomponerse en probabilidades de transición y supervivencia. (3) Por lo que respecta a las probabilidades de transición con el mismo estado de partida, el usuario puede elegir libremente la probabilidad que deberá calcularse por sustracción. Además de ser un programa muy accesible para el usuario, también debe subrayarse la facilidad con que permite construir modelos constreñidos utilizando un lenguaje interpretado denominado GEMACO. En este estudio desarrollamos y presentamos varios tipos de modelos multiestado. Palabras clave: Limitaciones, Lenguaje, Modelo lineal, Dinámica poblacional. Remí Choquet, Anne–Marie Reboulet, Roger Pradel, Olivier Gimenez, Jean–Dominique Lebreton, CEFE, UMR 5175, CNRS, 1919 route de Mende, 34293 Montpellier Cedex 5, France. Corresponding author: R. Choquet, E–mail: remi.choquet@cefe.cnrs.fr ISSN: 1578–665X

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Introduction Adequate statistical models play a central role in the ability to answer biological questions, and, as a consequence, to the development of specific fields of biology (e.g., the Generalized Linear Models (GLM), McCullagh & Nelder, 1989, and more recently generalized additive models, Hastie & Tibshirani 1990, and generalized linear mixed models, McCulloch & Searle, 2001). In practice the availability of software to fit models belonging to a well–defined class of models in a flexible way is crucial. For instance, the development of GLIM (Payne, 1986) greatly helped disseminate the use of GLM (Crawley, 1982) and promoted many different uses of categorical data analyses relevant to a variety of biological fields. In animal population ecology, capture–recapture methods (see the overview by Seber, 1982, and the recent book by Williams et al., 2002) have been key tools over the last fifty years to estimate population size and demographic parameters such as survival, recruitment and dispersal. In the case of survival estimation, for instance, the development of family of models in the spirit of GLM (Lebreton et al., 1992) and the availability of flexible software such as SURGE (Pradel & Lebreton, 1991; Reboulet et al., 1998) and MARK (White & Burnham, 1999) "have given rise to literally hundred of papers exploiting this very powerful methodology" (Seber & Schwarz, 2002). Recent developments in capture–recapture methodology and an increasing emphasis on spatial aspects of population dynamics and variation between individuals give a central role to multistate capture–recapture models (Lebreton & Pradel, 2002, and references therein). In multistate models, individuals are sampled on discrete occasions, at which they may be captured or not, and can die or move within a finite set of sites or states, between occasions. By considering states based on various competing events in an individual’s life, one can analyze complex individual histories, dealing with accession to reproduction over several sites (Lebreton et al., 2003) or with mixtures of information such as live recaptures and dead recoveries (Lebreton et al., 1999). We present here a general computer program, called M–SURGE, for "Multistate Survival Generalized Estimation", to fit a wide class of multistate capture–recapture models in a flexible way. M– SURGE, along with its companion program U– CARE, has been written specifically to handle multistate capture–recapture models with the ultimate concern to alleviate their inherent difficulties (model specification, quality of convergence, flexibility of parameterization, number of estimable parameters,…). In its present version, M–SURGE covers a broader range of models conditional on numbers released, than a general program like MARK (White & Burnham, 1999), while being more user–friendly than MS–SURVIV (Hines, 1994). The philosophy to define a model is to build the associ-

ated constraint matrix with a simple language (i.e. without any manipulation inside the matrix and/or any programming). The purpose of this paper is to provide an overview of M–SURGE abilities and key features, to present the class of models covered and the language used to define models, then to present a few illustrative examples and discuss perspectives of development. An overview of M–Surge: abilities and key features M–SURGE covers a wide class of models and a variety of parameterizations: (1) M–SURGE covers models conditional on numbers released with probability of recapture depending on the current state (Arnason–Schwarz type models) as well as on the current and previous state (Jolly–MoVement type models) (Brownie et al., 1993). In both cases, age and/or time–dependence and multiple groups can be considered; (2) combined Survival–Transition probabilities can be represented as such or decomposed into transition and survival probabilities (Hestbeck et al., 1991); (3) among the transition probabilities with the same state of departure, the one to be computed by subtraction can be freely picked by the user. The second main feature is that user–friendliness is enhanced by the easiness with which constrained models are built, using a language called GEMACO. This language is like those in general statistical software such as SAS or GLIM (i.e., a formula such as t+g generates a model with additive effects of time and group), thus avoiding tedious, time consuming and error–prone matrix manipulations using an editor or a spread–sheet. Class of models covered As SURGE and more recent programs for capture– recapture analysis, M–SURGE consider constrained models obtained by linear constraints among parameters, in the spirit of generalized linear models (Lebreton et al., 1992). The vector, , of parameters of direct interest to the biologist, or, for short, "biological parameters" is expressed as a linear transformation of a vector, , of "mathematical parameters". To keep the biological parameters, which are probabilities, in–the interval [0,1], a link function f is used: f( ) = X

= f-1(X )

The matrix, X, is a "matrix of constraints", which can be a genuine design matrix in the case of a designed experiment. In M–SURGE, it will be built by GEMACO using the model definition language described below. Often, X will be based on a mix of 0/1 variables (for equality constraints on survival probabilities e.g.) and of external covariates (e.g.,

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capture effort or weather covariates). An overview of linear constraints in single state capture–recapture models is provided by Lebreton et al. (1992). Our presentation of multistate models will use the following general notation: s is the number of states; K is the number of occasions; k = 2,…,K is the current occasion for a recapture; i = 1,…,s is the index of the departure state; j = 1,…,s is the index of the arrival state. All matrices of parameters are written with i as row index and j as column index. The most general model in M–SURGE is an age and time dependent version of the JollyMoVe model (JMV) (Brownie et al., 1993). For the sake of clarity, we do not specify the occasion, the cohort and the group in this section. In JMV models, capture probabilities depend on both previous and current states. Thus, the generic set of recapture parameters is a matrix of probabilities of dimension s x s. Models with recapture probabilities depending only on the current states are also considered, with age and/or time dependence. The generic set of recapture parameters is then a vector of probabilities P = (pi) or, using the diagonal operator D, a s x s diagonal matrix D(P). The time–dependent version of this model corresponds to the Arnason–Schwarz model (Arnason, 1972, 1973; Schwarz et al., 1993). However models presently considered in M–SURGE work conditional on releases (i.e., without estimation of population size). The time–dependent model bears then to the original Arnason–Schwarz model the same relationship as the Cormack–Jolly–Seber (CJS) model does to the full Jolly–Seber model. This class of models will then be called "Conditional Arnason– Schwarz" models (CAS). They are obtained as particular cases of JMV models obtained by dropping the dependence of probabilities of capture on the previous state. JMV and CAS models reduce to Cormack– Jolly–Seber models (CJS) when there is a single state (s = 1), since the matrix of survival transi-

tion–probabilities reduces to a scalar, , and the matrix or vector of recapture probabilities reduces to a scalar, p. In both JMV and CAS models, survival–transition probabilities can be represented in M–SURGE in two ways: 1. Combined survival–transition probabilities. The generic set of survival–transition parameters is a matrix = ( i,j) where i,j is the probability that an animal in state i at time k–1 is in state j at time k. The row sums, Fi = are survival probabilities conditional on the state of departure, i, and are [ 1 (= 1 if and only if survival is 1). 2. Separate movement and fidelity–survival probabilities. The parameters considered are s fidelity– survival probabilities, Fi arranged in a s x s diagonal matrix F, and the transition probabilities conditional on survival, arranged in a s x s matrix

Once (s – 1)*s parameters are known, the remaining s are obtained by difference. The separate and combined parameterizations are related by the matrix relationship = F* . They are further commented by Hestbeck et al. (1991). By combining the JMV or CAS recapture probability parameterizations with the separate/combined survival–transition parameterizations, one obtains four parameterizations (table 1). M–SURGE provides thus a great amount of flexibility since only the CAS —separate survival–

Table 1. Parameterizations of multistate capture–recapture models used in M–SURGE. See text for notation and further explanations. Tabla 1. Parametrizaciones de los modelos multiestado de captura–recaptura utilizados en el M– SURGE. Para anotaciones y más detalles al respecto, consultar el texto.

Survival–transition parameters Recapture parameters

Combined

Separate

JMV

Matrix P = (pi,j): depends on previous and current state

Matrix = ( i,j): matrix of survival–transition probabilities

Matrix = F* F = D(Fi) : diagonal matrix of fidelity–survival = ( i,j): matrix of transition probabilities

CAS

Vector P = (pj): depends on current state only

Matrix

= (

i,j

)

Matrix

= F*

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Fig. 1. Selection of time dependent models in the menu "Models". Fig. 1. Selección de los modelos dependientes del tiempo en el menú "Models".

transition models were until now easily fitted, using MARK. The preferable parameterization will depend on the biological context. The JMV model appears as a natural step in testing the fit of the CAS model (Pradel et al., 2003). In M–SURGE, the starting point for any model is an umbrella model with pre–programmed variation in parameters, later submitted to constraints. The generic sets of parameters, and P or F, and P, according to the parameterization chosen, may vary in different ways: (1) by group, if there are several permanent groups of individuals, such as sexes or species, or disconnected study sites; (2) between cohorts, cohort being taken in the sense of individuals first released with a mark on a same occasion; (3) over time (i.e., between occasions). The user has to choose variation by time only, or by cohort and time as a first step in the analysis. The former models are called "time–dependent models", the latter, "time– and cohort–dependent models" (denoted "time.cohort", in relation with the Model Description Language implemented in GEMACO, see next section). The parameterization and the choice of the umbrella model (time or time.cohort) are selected by simple menus as in figure 1.

(Generator of Matrices of Constraints). For the sake of simplicity, one constraint matrix has to be defined for each type of parameter (survival, capture and transition or capture and combined survival–transition depending on the particular kind of parameterization previously chosen). M–SURGE assembles the overall matrix. For instance, with matrices X1 and X2 for survival–transition and recapture probabilities, respectively, the overall matrix is, in block matrix notation:

The model definition language in GEMACO

Key words for main effects

The constraint matrix, X, associated with the model, f( ) = X , is obtained by describing it using a Model Definition Language (MDL). This language is interpreted in a section of M–SURGE called GEMACO

In capture–recapture modelling, several classical effects, such as time, age and group, have been widely used to explain variability in the data (Lebreton et al., 1992). In the model definition language of M–

The main step in defining a matrix of constraints for one type of parameter consists of typing a phrase in the MDL that will be interpreted by GEMACO to build X automatically. The language is based on main keywords for various effects such as time (t) or group (g) and operators such as the dot (.). This language expands the tensor notation for analysis of variance models (Wilkinson & Rogers, 1973; McCullagh & Nelder, 1989) already advocated by Lebreton et al. (1992) for CJS models. GEMACO and the MDL offers very wide possibilities that make the building of nearly any biologically meaningful model a fairly easy task.

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Fig. 2. Window structure of the GEMACO interface. As shown here, the recapture structure of the CJS model has just been built. The notation, "t", means that recapture probabilities vary over time as it must for the CJS model. The corresponding design matrix, automatically computed by GEMACO, has popped up in top left area. In this example, the numbers of occasions and groups are 7 and 2, respectively. Fig. 2. Estructura de las ventanas de la interfaz de GEMACO. Tal y como puede observarse en la figura, la estructura de recaptura del modelo CJS acaba de construirse. La anotación "t" significa que las probabilidades de recaptura varían a lo largo del tiempo, según lo exigido por el modelo CJS. La correspondiente matriz de diseño, calculada automáticamente por GEMACO, se muestra en el área superior izquierda. En este ejemplo, los números de ocasiones y de grupos son 7 y 2, respectivamente.

SURGE, these effects are represented by reserved keywords, with synonyms to facilitate the writing of models. The effects and their associated keywords are described in table 2. In what follows, these effects are first considered by themselves (i.e., as main effects in the meaning of analysis of variance) then combined (i.e. with interaction). As a first example of the capabilities of GEMACO, let us assume we want to run a CJS–type model with survival constant over time and varying over groups, and recapture probability varying with time only. One has only to define the structure for survival and recapture probabilities to be "g" and "t" respectively, exactly as in the tensor notation of this model ( g, pt). The time–dependent CJS model in the strict sense is written as ( t, pt). The phrases for model definition are typed in a specific window called "GEMACO interface" (fig. 2). As an example, let us consider cohort and time variation in survival probability over a single state and 4 occasions (i.e., 3 intervals). The vector of parameters is written as (with the first index for cohort and the second for time):

= (

)' = X1

(where ’ stands for the transpose of ). Defining the model as "t" creates a matrix X with as many rows as components in , in the same order, with columns corresponding to the time index (second subscript):

Defining the model as "c" (for cohort) creates a matrix X1 with K – 1 columns:

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Effects "from" and "to" take their meaning only when there are several (s > 1) states. When “from” is applied for instance to survival–transition probabilities with 3 states, all matrices F will be equal to (rows = previous state, columns = current state):

. If the structure "to" is used, the transition probability matrices will be equal to:

. Further keywords are described in Choquet et al. (2003).

with 1 = 1 – 4 – 7, 5 = 1 – 2 – 8, 9 = 1 – 3 – 6. Several effects can be combined using these operators since in a . b and a + b, a and b can themselves be model formulae. As a consequence of these rules, the JMV model with separate transition–survival representation is generated by "from.t" for fidelity–survival F, "from.to.t" for transition and capture P. The Conditional Arnason–Schwarz time–dependent model is gene r a t e d b y "f r o m . t " f o r f i d e l i t y – s u r v i v a l F, "from.to.t" for transition , and "to.t" for capture P. The default priority order of operations is (+ < .). This order can be changed using brackets (e.g., "[a + t] . g").

Combining effects with operators External covariates Two operators can be used to combine effects to generate more complex models. Let a and b be two factors with ma and mb categories, respectively: 1. Dot product (.): a . b is the product column by column of a by b (i.e. the set of all combinations of categories of the factors a and b, or a model with interaction). The result, a.b, is a factor with ma x mb categories. This dot product is the "crossing operator" of McCullagh & Nelder (1989). 2. Sum (+): a + b joins the columns of a and b. If the intercept (constant column equal to one) is obtained as linear combination of the variables in a and in b, the first column of b is suppressed to avoid linear redundancy. The result, a + b, has then ma + mb – 1 columns. Otherwise, all the columns of a and b are kept. In the previous example, one obtains for t.g and t+g the respective matrices for survival:

and

Let us assume in the previous example that a time– dependent covariate x is available as a column vector,

The X matrix corresponding to a linear effect of this time–dependent covariate is:

which reduces to i + t * x. Hence the matrix product of a factor by an external covariate replaces this factor by the linear effect of the covariate. The default priority order of operations is (+ < . < *), and, as above, can be changed using brackets (e.g., "[a . t] * x"). Several covariates related to different effects can be used simultaneously provided they are prepared in a file with a specific format. They are then used as x(1), x(2),etc. Aggregation of parameters: lists

The CJS model with several groups will be denoted as ( t.g, Pt.g). The dot operator is very useful with the "from" and "to" effects when there is more than one state (s > 1): "from.to" applied to a transition probability matrix induces a variation by rows and columns (i.e.; a matrix with all elements different):

M–SURGE offers several possibilities of "grouping parameters" in the broad sense. First, one often needs to build effects less fine than time, age or any given factor. This is obtained by lumping categories. For instance in an analysis of data concerning the dipper, Cinclus cinclus, over 6 year intervals, Lebreton et al. (1992) described how floods impacted the survival probability in years 2

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Table 2. Effects and keywords used in the Model Definition Language of M–SURGE. The sentences in this language are interpreted in GEMACO to build the matrices of constraints X. Tabla 2. Efectos y palabras clave utilizados en el lenguaje para la definición de modelos del M–SURGE. Las frases escritas en este lenguaje se interpretan en GEMACO para construir las matrices de limitaciones X.

Effects

Keywords and abbreviations/synonyms

Comments

Constant or Intercept

intercept, i

To obtain constant parameters

Time

time, t

Categorical variation over time ("factor" with K – 1 levels)

Age

age, a

Categorical variation over age (time elapsed since first capture) ("factor" with K – 1 levels). More refined age variations are introduced later

Cohort

cohort, c

Categorical variation between cohorts (batches of individuals released for the first time with a mark on a same occasion) ("factor" with K – 1 levels)

Group

group, g

Categorical variation between groups

Departure state ("from") or previous state (capture)

from, f previous, p

Forces rows in j or y matrices to differ. In JMV models, forces rows in p matrices to differ Not active on p in CAS models

Arrival state ("to") or current or next state (capture)

to, next, n, current

Forces columns in j or y matrices to differ Not active on F parametersIn JMV models, forces rows in p matrices to differ In CAS models forces terms in p vector to differ

and 3. The resulting X matrix is obtained by lumping years 2 and 3 on the one hand, and years 1, 4, 5, and 6 on the other. This is done in M–SURGE using lists of parameter categories, each list corresponding to a set of categories lumped together. In the dipper example, the model formula to reduce the variation over time to two levels would be "t(1 4 5 6, 2 3)". Similarly, over 7 occasions, to distinguish the first year after capture from the other ones, as two age–classes, one would use "a(1, 2 3 4 5 6 7)" or "a(1, 2:7)". The user can also define shortcuts for complex expressions (e.g. "sex" for "g(1 3, 2 4)" if groups 1 and 3 are females and groups 2 and 4 are males). Complex models using repeatedly such expressions (e.g., as "sex.t") are then written easily. Aggregation of parameters: the aggregation operator The lists make it possible only to aggregate parameters within a same main effect. The aggregation operator & makes it possible to aggregate parameters corresponding to categories of different effects (i.e. that cannot be handled within a same list). The syntax f(1).to(1)&to(2) applied to the combined survival transition for two states builds the following constraint 11 = 12 = 22.

Redundancy in the transition matrix The transition matrix is associated to a Markov chain (i.e., the sum of each row is equal to one). Thus, only s x (s – 1) parameters out of the s x s parameters should be estimated. One redundant parameter has to be chosen for each row. This is open to the user’s choice, based on a transition pattern matrix. This matrix is a s x s matrix made of 0’s and 1’s, with rows corresponding to the previous state and columns to the current state. The elements, tij, are equal to 1 except for one element per row, set equal to 0, defining the position of the redundant parameter for this row. Hence, there must be exactly one zero per row. With three states, for instance, if instead of using the parameters { 12 13 21 23 31 32} we want to use the parameters { 11 12 21 22 32 33}, we have to change the transition pattern matrix from

which is the default, to .

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Table 3. Correspondence between models as defined in the notation proposed by Lebreton et al. (1992) and the Model Definition Language in GEMACO: ML. Model in Lebreton et al. (1992, p. 88, 91 and 95); MGs. Model used in GEMACO for survival; MGc. Model used in GEMACO for capture.

Table 4. Correspondence between models used in Brownie et al. (1993) in the notation proposed by Lebreton et al. (1992) and the Model Definition Language in GEMACO: MB Model in Brownie et al. (1993) (with s = site); MGt. Model used in GEMACO for transitions; MGc. Model used in GEMACO for capture.

Tabla 3. Correspondencia entre modelos, según lo definido en la anotación propuesta por Lebreton et al. (1992) y el Lenguaje para la definición de modelos en GEMACO: ML. Modelo en Lebreton et al. (1992, p. 88, 91 y 95); MGs. Modelo utilizado en GEMACO para la supervivencia; MGc. Modelo utilizado en GEMACO para la captura..

Tabla 4. Correspondencia entre los modelos utilizados en Brownie et al. (1993) en la anotación propuesta por Lebreton et al. (1992) y el lenguaje para la definición de modelos en GEMACO: Mb. Modelo en Brownie et al. (1993) (con s = localidad); MGt. Modelo utilizado en GEMACO para las transiciones; MGc. Modelo utilizado en GEMACO para la captura.

MGt

MGc

, pt t

,P (JMV) s*t s*t

f.to.t

, p

s*t

,Ps*t (CAS)

f.to.t

to.t

t(1 4 5 6, 2 3)

,P s*t

f.to

to.t

,Ps

f.to

, pc*t c*t

t.g

t+g

MGs

, p

, pc+t

ac*t*s

, pperiod*s a(1,2,3:10).t.g

ac*t*s

, pperiod

a(1,2,3:10).t.g

MGc

t(1:7,8:10).g t(1:7,8:10)

So, GEMACO provides an unambiguous notation to describe multistate models (table 4).

Illustrative examples

Discussion and perspectives

Case studies in Lebreton et al. (1992)

In this paper, we have presented M–SURGE and the set of models which are presently available. The set of models considered is based on: (1) the central role of JMV–type models, a large set of models for which GOF tests are now available (Pradel et al., 2003); (2) combined and separated fidelity transitions formulations; (3) a language which allows to build easily constraint matrices for multistate models. Optimized algorithms (in particular, use of the explicit gradient of the likelihood, profile likelihood calculations) are used to fit the model and will be described in another paper. We are continuously exploring new features with working versions. For instance, Gimenez (2003) introduced efficient starting point for quasi–Newton algorithms. We hope to use this work to improve the reliability of the results. We only described here basic models. However GEMACO is able to generate in a straightforward fashion very complex models, such as models mixing recapture and recoveries (Catchpole et al., 1998; Lebreton et al., 1999) and multistate recruitment models (Lebreton et al., 2003). Thus, we hope that the language developed for building constraint matrices and the fast and reliable computa-

Pradel & Lebreton (1991) introduced program SURGE to fit single state models. Three case studies described in Lebreton et al. (1992) were analyzed with this program using the PIM (Parameter Index Matrix) approach. The main models used in these three studies are described in table 3. While some models are quite classical, some other ones are not (e.g., flood effect on survival, for which aggregation of categories was used to describe a particular variation over time). Wintering Canada Geese Brownie et al. (1993) introduced a routine to be used with MS–SURVIV that allows easy implementation of the JMV model and of constrained versions of this model. We show here how models considered in Brownie et al. (1993) can be implemented simply with M–SURGE (table 4). In the notation used by Brownie et al. (1993), the first two models (JMV) and (CAS) are written in the same way, although capture parameterizations differ. In GEMACO, the sentences for recapture parameters in the two models do differ.

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tions in M–SURGE will help to put the emphasis on biological questions. References Arnason, A. N., 1972. Parameter estimates from mark–recapture experiments on two populations subject to migration and death. Researches on Population Ecology, 13: 97–113. – 1973. The estimation of population size, migration rates and survival in a stratified population. Researches on Population Ecology, 15: 1–8. Brownie, C., Hines J. E., Nichols J. D., Pollock K. H. & Hestbeck J. B., 1993. Capture–recapture studies for multiple strata including non– Markovian transitions. Biometrics, 49: 1173–1187. Catchpole, E. A., Freeman S. N. & Morgan B. J. T., 1998. Integrated recovery/recapture data analysis. Biometrics, 54: 33–46. Choquet, R., Reboulet A. M., Pradel R., Gimenez O. & Lebreton, J.–D., 2003. User’s manual for program M–SURGE 1.0. Montpellier, CEFE/ CNRS. Crawley, M. J., 1982. GLIM for Ecologists. Blackwell Science, U.K. Gimenez, O., 2003. Estimation et Tests d’Adéquation pour les Modèles de Capture–Recapture Multiétats. Thèse de doctorat, Univ. Montpellier II. Hastie, T. & Tibshirani R., 1990. Generalized Additive Models. Chapman & Hall, New York. Hestbeck, J. B., Nichols J. D. & Malecki, R.A., 1991. Estimates of movement and site fidelity using mark–resight data of wintering Canada Geese. Ecology, 72: 523–533. Hines, J. E., 1994. MSSURVIV User’s Manual. National Biological Survey, Laurel, MD. Lebreton, J. D., Almeras T. & Pradel R., 1999. Competing events, mixtures of information and multistratum recapture models. Bird Study, 46: 39–46. Lebreton, J.–D., Burnham, K. P., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecological Monographs, 62: 67–118.

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Lebreton, J. D., Hines, J. E., Pradel, R., Nichols, J.D. & Spendelow, J. A., 2003. Estimation by capture–recapture of recruitment and dispersal over several sites. Oïkos, 101: 253–264. Lebreton, J. D. & Pradel, R., 2002. Multistate recapture models: modelling incomplete individual histories. Journal of Applied Statistics, 29(1–4): 353–369. McCullagh, P. & Nelder, J. A., 1989. Generalized Linear Models. Chapman & Hall, New York. McCulloch, C. E. & Searle, S. R., 2001. Generalized, Linear, and Mixed Models. Chapman & Hall, New York. Payne, C. D., 1986. The Generalized Interactive Modelling system. Royal Statistical Society and Numerical Algorithm Group, Oxford. Pradel, R. & Lebreton, J. D., 1991. User’s manual for program SURGE (version 4.2). CEFE/CNRS et Praxème, Montpellier. Pradel, R., Wintrebert, C. & Gimenez, O., 2003. A Proposal for a Goodness–of–Fit Test to the Arnason–Schwarz Multisite Capture–Recapture Model. Biometrics, 59: 43–53. Reboulet, A. M., Viallefont, A., Pradel, R. & Lebreton, J.D., 1998. Selection of survival and recruitment models with Surge 5.0. Bird Study, 46 (suppl.): 148–156. Schwarz, C. J., Schweigert, J. F. & Arnason, A.N., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Seber, G. A. 1982. The estimation of animal abundance and related parameters. Ch. Griffin & Company Ltd., London. Seber, G. A. F. & Schwarz, C. J., 2002. Capture– recapture : before and after EURING 2000. Journal of Applied Statistics, 29: 5–18. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 (suppl.): 120–139. Wilkinson, G. N. & Rogers, C. E., 1973. Symbolic description of factorial models for analysis of variance. Applied Statistics, 22: 392–399. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, San Diego.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

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DENSITY: software for analysing capture–recapture data from passive detector arrays M. G. Efford, D. K. Dawson & C. S. Robbins

Efford, M. G., Dawson, D. K. & Robbins, C. S., 2004. DENSITY: software for analysing capture–recapture data from passive detector arrays. Animal Biodiversity and Conservation, 27.1: 217–228. Abstract DENSITY: software for analysing capture–recapture data from passive detector arrays.— A general computer–intensive method is described for fitting spatial detection functions to capture–recapture data from arrays of passive detectors such as live traps and mist nets. The method is used to estimate the population density of 10 species of breeding birds sampled by mist–netting in deciduous forest at Patuxent Research Refuge, Laurel, Maryland, U.S.A., from 1961 to 1972. Total density (9.9 ± 0.6 ha–1 mean ± SE) appeared to decline over time (slope –0.41 ± 0.15 ha–1y–1). The mean precision of annual estimates for = 14%). Spatial analysis of closed–population capture– all 10 species pooled was acceptable ( recapture data highlighted deficiencies in non–spatial methodologies. For example, effective trapping area cannot be assumed constant when detection probability is variable. Simulation may be used to evaluate alternative designs for mist net arrays where density estimation is a study goal. Key words: Passive detector arrays, Density estimation, Capture–recapture, Mist–netting, Birds. Resumen DENSITY: programa empleado para el análisis de datos de captura–recaptura procedentes de matrices de detectores pasivos.— En este estudio se describe un método general de cómputo intensivo que permite ajustar las funciones de detección espacial a datos de captura–recaptura procedentes de baterías de trampas pasivas, como las trampas de cebo y las redes japonesas. Este método es utilizado para estimar la densidad de población de 10 especies de aves reproductoras, muestreadas mediante la colocación de redes japonesas en un bosque de árboles de hoja caduca del Centro de Investigación Patuxent, en Laurel, Maryland, Estados Unidos, desde 1961 hasta 1972. La densidad total (9,9 ± 0,6 ha–1 promedio ± EE) parecía disminuir con el tiempo (gradiente –0,41 ± 0,15 ha–1y–1). La precisión media de las estimaciones anuales correspondientes a la totalidad de las 10 especies recogidas fue aceptable ( = 14%). El análisis espacial de los datos de captura–recaptura de la población cerrada revelaron deficiencias en las metodologías no espaciales. Así, por ejemplo, no puede suponerse que el área efectiva de colocación de trampas sea constante cuando la probabilidad de detección es variable. En los casos en que la estimación de la densidad sea objeto de estudio, la simulación permite evaluar diseños alternativos para baterías de redes japonesas. Palabras clave: Baterías de trampas pasivas, Estimación de la densidad, Captura–recaptura, Redes japonesas, Aves. Murray G. Efford, Landcare Research, Private Bag 1930, Dunedin, New Zealand.– Deanna K. Dawson & Chandler S. Robbins, USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, U.S.A. Corresponding author: Dr M. G. Efford. E–mail: effordm@landcareresearch.co.nz

ISSN: 1578–665X

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Introduction Rigorous sampling of animal populations to estimate or index density raises the problem of incomplete detection (e.g. Burnham, 1981; MacKenzie & Kendall, 2002; Pollock et al., 2002; Rosenstock et al., 2002; Thompson, 2002). Detection probability generally has been described by a single parameter p. An estimate of p may be used to obtain a population estimate N from a count C: (1) Estimation of detection probability protects the population estimate or index from the confounding effects of season, time of day, observer, weather, habitat, etc. If detections relate to a known area A then population density may be calculated as (2) This strategy applies to "limited–area" counts when a stationary observer makes instantaneous observations of bird locations and includes only those within a known area (e.g. the double–observer approach of Nichols et al., 2000 and the removal method of Farnsworth et al., 2002). These methods are called "active" because they require continuous attention and discrimination by the observer. "Passive" counts are obtained when a detector (e.g. trap, mist net, or camera) records individuals at a point. Individuals are included in the count only when they encounter and interact with a detector. Passive detectors are commonly deployed in arrays of varying geometry and size. The general term "passive detector array" (PDA) is suggested to emphasize the common features of spatial capture data from diverse field studies (table 1). Passive detection combined with mark and release on a series of occasions spaced closely in time is a common source of data for closed–population experiments (e.g. Otis et al., 1978). Equations (1) and (2) and their capture–recapture equivalents do not provide an adequate framework for estimating animal density from capture data from PDAs. The area A is unknown and difficult to define. It follows that both N and p are ill–defined for these data except in an operational and probably circular sense (e.g. "N is the number of animals potentially exposed to the PDA"). Thus N varies both with animal behaviour and with the configuration of the PDA. The widespread use of the term "abundance" for N acknowledges its vagueness in this context. Further complications arise because the component detectors (traps, mist nets) of a PDA may interact. Interaction commonly occurs when an animal detained in one trap is not immediately available for capture in a different trap. An alternative framework is advocated for estimating density from closed population data from

arrays of passive detectors. The framework is conceptually consistent with that of distance sampling (Buckland et al., 1993; Rosenstock et al., 2002), but it offers major advantages for passive count data. Here we introduce the spatial detection model for PDAs and a numerical method for model fitting (Efford, 2004), along with software designed to make the method generally accessible. We assess the potential of the method for estimating the population density of birds captured in mist nets, using a dataset collected in Maryland, U.S.A., by CSR. Spatial model for the detection process Assume that animals occupy stationary home ranges whose centres are a realization of a homogeneous random spatial point process with intensity (density) D. Populations that have a natural boundary are explicitly excluded. Passive sampling uses detectors in a known spatial configuration to sample the unknown distribution of animals. An individual–based model is proposed for the detection process. The core of the model is a spatial detection function g(r) for the simplest possible case: one animal and one detector. The probability of detecting animal i is assumed to be a decreasing function of the distance r between its range centre and the detector. The simplest useful detection function has two parameters. In the formulation discussed here, these correspond to measures of home range size ( ) and susceptibility to capture (g(0)). This definition of g is more useful than a global one at the level of the entire array, as parameter estimates are "portable" to other detector configurations (i.e. different arrays). Given some ancillary information, the three parameters D, g(0) and define the detection process. The required ancillary information is: (i) the configuration of the detector array (i.e. x–y coordinates of detectors), (ii) the nature of the spatial point process (here assumed to be Poisson), (iii) a model for resolving conflicts between incompatible detection events (e.g. animal caught in two traps at once), and (iv) the shape of the detection function (assumed here to be half–normal). Writing a computer algorithm to simulate capture data from this model is straightforward except for (iii), which is addressed later. Fitting the spatial detection model Our formulation of closed population sampling in terms of D, g(0) and is useful only if there is a practical method of estimation. An expression for the likelihood is currently lacking, and therefore maximum likelihood estimators cannot be derived. Instead, D, g(0) and are estimated by simulation and inverse prediction (Carothers, 1979; Pledger & Efford, 1998). Briefly, this method uses Monte Carlo sampling of populations with known D, g(0)

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and to generate data that may be "matched" to the field data. "Matching" uses statistics from the data as surrogates for D, g(0) and . Each statistic is chosen for its conditional monotonic relation to a parameter. The statistics used here are the closed population estimate , the corresponding estimate of mean detection probability , and the mean distance between successive detections of the same individual . Inverse prediction (Brown, 1982; Pledger & Efford, 1998) provides a formal framework for estimating D, g(0) and from , and , complete with prediction standard errors. The parameter vector x and simulated observations y may be used to fit the multivariate multiple regression y=

+ Bx + E

where λ is a 3 x 1 vector of intercepts, B is a 3 x 3 matrix of coefficients, and E is a 3 x 1 vector of error terms with multivariate normal MVN(0, V) distribution. With sufficient replications, the elements of λ, B and E are estimated virtually without sampling error. Given a single observation yP the point estimates of D, g(0), and (together, the vector xP) are given by xP = B–1(yP – )

Table 1. Examples of passive detectors used in animal population studies. Detectors in which animals are detained provide the further option of selective non–release: D. Detections; Eb. Effect on behaviour; * Individual natural marks are required for capture–recapture analysis. Tabla 1. Ejemplos de métodos de trampeo utilizados en estudios de poblaciones animales. Los métodos de trampeo en los que los animales quedan retenidos proporcionan la opción más avanzada de no liberación selectiva: D. Detecciones; Eb. Efecto en el comportamiento; * Para el análisis de captura– recaptura se precisan marcas naturales individuales.

Type Sherman trap

single

detained

Mist net

multiple

detained

Pitfall trap

multiple

detained

Crab pot

multiple

detained

Fixed camera*

multiple

not detained

(3)

The model equation rearranges to: x = B–1(y – ) + B–1E where the random error B –1E has distribution MVN(0, ), where = B–1 V B–1T. Detection conflicts As noted previously, simulations with a one– animal, one–detector detection function may lead to conflicts when multiple animals interact with multiple detectors. The problem is severe when either animals or detectors are at high density (e.g. when traps become "saturated"). This issue was addressed by a discrete–event simulation of the trapping process (see also Efford, 2004). Potential detection events (animal i at detector j) were treated as competing Poisson processes. The time to first occurrence of each potential event then follows an exponential distribution with rate parameter ij = –ln(1–g(rij)) where rij is the distance between animal i and detector j and g(r) = g(0)exp(–rij2/(2 2)). Independent pseudorandom exponential variates, one for each potential event, were sorted by magnitude to establish priority among events. Simulated events were discarded (i.e. did not occur) if they were inconsistent with previous events or they occurred after one unit of elapsed detection time. Mist nets allow multiple simultaneous detections; for this detector type the only consistency constraint was that an individual could appear at no more than one detector per sample.

Software The Windows ® program DENSITY (www.landcare research.co.nz/services/software/density) analyses closed population capture–recapture data from arrays of passive detectors. Two input text files are required. One file contains the locations (x–y coordinates) of the detectors. The second file records detection events (individual ID, sample number and detector ID). Detection events may for convenience be stratified by "session". Each session is analysed separately. A graphic interface enables the visualization of spatial detection data (fig. 1). Program usage is described in an online help file. DENSITY implements the proposed method for simulation and inverse prediction. Starting values of the parameters are determined automatically (see the Appendix for a description of how these are calculated). Model fitting proceeds by Monte Carlo simulation of detection samples from random populations with known parameter vectors. In each simulation a new set of animal locations is generated for a rectangular area that includes the PDA and a buffer zone. The width of the buffer zone should be at least 3 in order to include all individuals with a reasonable chance of detection. Parameter values follow a full factorial design, i.e. they lie at the vertices of a "box" in parameter space. The dimensions of the parameter "box" are fixed as a percentage (e.g. ± 10%) of the current best esti-

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mate of the parameter values. Statistics ( , and ) from the multiple simulations conducted at each vertex are averaged to remove sampling variance. A sample size of 20–100 simulations appears adequate, but this requires further investigation. Estimation follows equation (3). If the estimated vector (D, g(0), ) lies outside the initial "box" then the estimate becomes the starting point for another simulation cycle. This avoids extrapolation of the linear approximating function. Once a satisfactory prediction is obtained (i.e. the estimate lies inside the box), further simulations are conducted to estimate the variance–covariance matrix in statistic space V, and its equivalent in parameter space (see above). The automatic algorithm sometimes fails to find a parameter "box" that includes the fitted parameter values, in which case a degree of supervision may be required to fit the model. It is usually sufficient to provide better starting values, either manually or by applying a constant scalar adjustment (e.g. x 0.5) to the "automatic" initial value for g(0). The adjustment may be stored for further use with the same detector configuration. It may also be necessary to increase the size of the "box" or the number of replicate simulations at each vertex. Outputs of DENSITY include both the "N–p" analyses of conventional closed population models (Otis et al., 1978; Chao & Huggins, in press a, in press b) and numerical estimates of D, g(0) and by inverse prediction. The user may also simulate sampling with novel detector arrays to identify efficient ways to allocate sampling effort and to predict precision and bias. This meta–functionality is described as "power analysis" to distinguish it from the simulations embedded in the estimation of D, g(0) and by inverse prediction. Test of assumptions Goodness–of–fit tests for the present spatial detection model have yet to be developed. However, a test has been developed for one key assumption. This is the assumption that capture locations are sampled from a stationary distribution subject only to the modelled effects of competition for and among detectors and, in particular, that capture location is not affected by previous capture. A suitable test statistic is the t–value for a comparison of the mean distance between first and second capture when captures are in consecutive samples versus the mean distance between first and second capture when captures are separated by more than one sample. Values of this statistic may be compared to its bootstrap sampling distribution from simulated realizations of spatial sampling with appropriate D, g(0) and (see DENSITY online help). Methods Birds were mist–netted on a forested site on the Patuxent Research Refuge, Maryland, U.S.A.

Efford et al.

(39° 3’ N, 76° 48’ W) in early summer from 1959 to 1972. The initial study design used 21 12–m nets at 61–m (200–foot) intervals on the arms of a cross (Stamm et al., 1960). This was changed in 1960 to a 4 x 11 grid, with net spacing 100 m between rows and 61 m along rows. On six nonconsecutive days in late May and early June, nets were open between about 0600 hours and 1800 hours Eastern Daylight Time and were checked every 2 hours. On initial capture birds were ringed with uniquely numbered aluminium rings, identified to species and, when possible, to age and sex, measured, and released. The ring number and the date and location of capture were also recorded for previously marked birds. Noon & Sauer (1990) presented some analyses of survival and recruitment from these data. Here we analyse only closed–population data from the grid layout (1961 to 1972). Thirty–seven bird species netted over 1961–1972 were likely to have been breeding in the forest. We focus on the 10 species with at least 200 captures each (Baeolophus bicolor, Cardinalis cardinalis, Empidonax virescens, Hylocichla mustelina, Oporornis formosus, Piranga olivacea, Seiurus aurocapillus, Setophaga ruticilla, Vireo olivaceus, Wilsonia citrina). Analysis of mist–netting data in DENSITY Captures were pooled by day (i.e. within–day recaptures were ignored). Most species were captured and recaptured in only small numbers in each sampling session (year), and it was necessary to group data for analysis. Analyses were conducted by species for data pooled over three consecutive years (1961–1963, 1964–1966, etc.), considering only recaptures within a year. In addition both annual and 3–year–pooled analyses were conducted on the pooled sample from the 10 species most commonly caught. This enabled us to assess empirically the effect of pooling species with different detection functions. Half–normal spatial detection functions were fitted by simulation and inverse prediction. Bird range centres were assumed to follow a Poisson distribution. For closed–population estimation ( , ), Chao’s second coverage estimator for model Mth was used (Otis et al., 1978; Lee & Chao, 1994). This estimator is a sensible and conservative option given the likely presence of non–spatial heterogeneity. However, some study designs do not allow formal probabilistic estimation of N (e.g. when the locations of detectors only partly overlap between samples). In these situations it is still possible to estimate density by the method presented here, but it is necessary to use an ad hoc surrogate for such as the number of individuals caught (Mt+1). To evaluate the effect of this substitution on the estimates, the density of all 10 species pooled was also estimated using Mt+1 for and setting

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Fig. 1. Graphic interface to DENSITY. Fig. 1. Interfaz gráfica del DENSITY.

Initial parameter values were determined as described in the Appendix. The factorial design in parameter space spanned ± 10% of the initial values; statistics were averaged from 100 simulations with each combination of parameter values. Nets were modelled as multi–catch detectors with marking and live release. The variance–covariance matrix was estimated by conducting 200 further simulations at the fitted values. The precision of density estimates was expressed as .

120 100 80 60 40

Results 20 The detection model was fitted successfully to the 3–year grouped data for all species. Vireo olivaceus maintained the highest population density throughout the study (table 2). Precision depended strongly on the number of recaptures in the sample (fig. 2). Relative number of captures was a poor measure of relative density; for example, captures of Seiurus aurocapillus outnumbered those of Empidonax virescens, but the estimated density of Empidonax virescens was always more than twice that of Seiurus aurocapillus (table 2). This is consistent with large species differences in the fitted detection functions (table 2; fig. 3). Three–year estimates for the 10 species pooled (table 2) were close to the sums of the individual species estimates for each interval (1961–1963 12.9 ha–1; 1964–1966 11.0 ha–1; 1967–1969 10.3 ha–1; 1970–1972 7.6 ha –1). Thus the pooled data appear to provide usable estimates of total density despite species differences in detection. The number of within–year recaptures ( m) for all 10 species pooled ranged from 52 to 129 (97.4 ± 7.9

0 0

20 40 60 80 Number of recaptures

100

Fig. 2. Precision of density estimated by inverse prediction ( %) for 10 bird species mist–netted at Patuxent Research Refuge, Maryland, U.S.A., 1961–1972, as a function of the number of recaptures. Each point represents one 3–year pooled estimate for one species. Fig. 2. Precisión de la densidad estimada por predicción inversa ( %) para 10 especies de aves capturadas con redes japonesas en el Centro de Investigación Patuxent, Maryland, Estados Unidos, 1961–1972, en función del número de recapturas. Cada punto representa una estimación combinada de tres años para una especie.

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Table 2. Density ( ha-1) and spatial detection parameters ( , ) of breeding bird species at Patuxent Research Refuge, Maryland, U.S.A., in 3–year intervals 1961–1972. Estimates by inverse prediction (SE). Also shown is the number of captures, including recaptures, over the entire study (NC). , ) de especies de aves Tabla 2. Densidad ( ha-1) y parámetros de detección espacial ( reproductoras en el Centro de Investigación Patuxent, Maryland, Estados Unidos, en intervalos de tres años: 1961–1972. Estimaciones por predicción inversa (EE). También se indica el número de capturas, incluyendo las recapturas, a lo largo de todo el estudio (NC).

Year interval Species

1961–63

1964–66

1967–69

1970–72

NC Vireo olivaceus 1,015

5.50

0.034

(0.92) (0.007) (5)

3.35

0.038 61

(0.56) (0.008) (6)

3.55

0.041 66

(0.51) (0.006) (5)

2.85

0.027

(0.80) (0.009) (8)

Hylocichla mustelina 743

1.68

0.041

(0.24) (0.007) (8)

1.43

0.019 85

(0.34) (0.009) (16)

1.11

0.028 103

(0.22) (0.006) (12)

1.36

0.035

(0.25) (0.008) (9)

Seiurus aurocapillus 436

0.42

0.076

(0.09)

(0.019) (10)

0.33

0.070 111

(0.08) (0.018) (13)

0.37

0.050 114

(0.09) (0.015) (16)

0.30

0.035 118

(0.11) (0.014) (22)

Empidonax virescens 385

1.28

0.036

(0.27) (0.011) (10)

1.06

0.039 68

(0.29) (0.012) (10)

0.88

0.043 64

(0.22) (0.016) (10)

0.97

0.020

(0.45) (0.011) (20)

Oporornis formosus 366

0.53

0.072

(0.11) (0.019) (8)

0.42

0.023 124

(0.16) (0.009) (24)

0.66

0.038 87

(0.20) (0.012) (12)

0.40

0.055

(0.10) (0.020) (13)

Wilsonia citrina 288

0.36

0.058

(0.09) (0.018) (12)

0.50

0.023 93

(0.25) (0.012) (24)

0.19

0.037 155

(0.06) (0.013) (28)

0.08

0.043 180

(0.04) (0.018) (55)

Baeolophus bicolor 262

0.51

0.027

(0.16) (0.014) (18)

0.52

0.013 115

(0.26) (0.007) (37)

0.61

0.057 80

(0.12) (0.016) (10)

0.44

0.008 111

(0.35) (0.008) (64)

Setophaga ruticilla 230

1.19

0.037

(0.90) (0.023) (11)

1.68

0.037 55

(0.50) (0.018) (10)

1.40

0.010 69

(1.19) (0.010) (36)

0.38

0.029

(0.24) (0.029) (29)

Cardinalis cardinalis 214

0.35

0.027

(0.14) (0.015) (27)

0.60

0.010 114

(0.62) (0.008) (73)

0.87

0.015 92

(0.36) (0.008) (21)

0.30

0.017 132

(0.13) (0.010) (43)

Piranga olivacea 200

1.04

0.012

1.09

0.015 71

0.64

0.008 145

(0.60) (0.009) (43)

(0.74) (0.009) (22)

(0.35) (0.005) (46)

11.39

11.05 0.024 84

10.06 0.029 89

(0.81) (0.003) (3)

(0.94) (0.003) (4)

(0.77) (0.003) (4)

0.52

0.004 142

(0.45) (0.005) (87)

Pooled 4,139

0.035

7.11

0.022

(0.78) (0.003) (6)

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Density ha–1

Detection function g

0.05 0.04 c

0.03

15 10 5

0.02 0

1962

0.01 a

0.00 0

1964 1966 1968 1970 1972 Year

50 100 150 200 250 300 Distance from range centre (m)

Fig. 3. Fitted detection functions for the five species most often caught in mist nets at Patuxent Research Refuge, Maryland, U.S.A., 1961–1972: a. Vireo olivaceus; b. Hylocichla mustelina; c. Seiurus aurocapillus; d. Empidonax virescens; e. Oporornis formosus. Parameters were obtained by averaging 3–year pooled estimates for each species (table 2). Fig. 3. Funciones de detección ajustadas para las cinco especies más frecuentemente capturadas con redes japonesas en el Centro de Investigación Patuxent, Maryland, Estados Unidos, 1961–1972 (a. Vireo olivaceus; b. Hylocichla mustelina; c. Seiurus aurocapillus; d. Empidonax virescens; e. Oporornis formosus). Los parámetros se obtuvieron calculando el promedio de las estimaciones combinadas de tres años para cada especie (tabla 2).

mean ± SE), sufficient to allow us to compute annual estimates of density. The estimated density of common breeding birds ranged between 4.8 ha–1 (1971) and 12.2 ha –1 (1966) (9.93 ± 0.64 ha –1 mean ± SE) and appeared to decline over time (slope of linear trend –0.41 ± 0.15 ha–1y–1; fig. 4). Density estimates using Mt+1 for (8.16 ± 0.54 ha–1) were lower by 17 ± 3% than those using Mth, but this discrepancy was much less than that between the alternative estimates of N (51 ± 2%). Estimates of g(0) and were inversely correlated between years (r = –0.83, P < 0.001), which may be due in part to their inverse sampling covariance. The precision of annual density estimates was high ( = 0.144 ± 0.010) and only slightly worse than that of population size estimates ( = 0.123 ± 0.008).

Fig. 4. Temporal variation in annual population density of common breeding bird species at Patuxent Research Refuge, Maryland, U.S.A., 1961–1972, estimated by inverse prediction ± 1 SE. (See text for details.) Fig. 4. Variación temporal en la densidad anual de la población de especies de aves comunes reproductoras en el Centro de Investigación Patuxent, Maryland, Estados Unidos, 1961–1972, estimada mediante predicción inversa ± 1 EE (Para más detalles ver el texto.)

Discussion The conventional parameterization of closed–population models in terms of N and p is incomplete because it neglects space. Conversely, a spatial parameterization (D, g(0), ) has major benefits where the underlying dispersion model (localized detection) fits the biology of the study animal. More often than not, ecologists want to measure population density D rather than N. Detection functions g(r) are fundamental to distance analysis, which uses a sample of detection distances (Buckland et al., 1993). Our approach does not model detection distances as such, but estimates parameters of the detection function from the pattern of recaptures. By our definition, the parameters are independent of a particular detector configuration. This means that simulations may be conducted to compare the efficiency of alternative, novel configurations using values of g(0) and estimated from the field. is also a convenient measure of home range size. There may be pathological detector configurations for which is a biased estimate of , but this remains to be investigated. Closed–population estimator selection Our method uses an empirical as an input to inverse prediction. Analyses were presented using

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both (1) Chao’s sample coverage estimators that are believed to be robust to temporal and individual heterogeneity (Mth), and (2) the number of distinct individuals caught, which is almost certainly less than the number that would be caught over a longer period. This raises the issue of how to select an appropriate estimator of N from among the many available (e.g. Otis et al., 1978; Chao & Huggins, in press a, in press b). More work needs to be done on this topic, but the outcome is not critical for the adoption of the method. is used here only in the context of a particular PDA and detection function. Biases inherent in the estimator and context will also arise during simulation and be automatically down–weighted or removed. This applies specifically to spatial heterogeneity in access by animals to detectors and the general negative bias of incomplete counts from a small number of samples. In some situations (e.g. detection on a continuously shifting array), Mt+1 may be the only available measure of N, and an estimate of density using Mt+1 and inverse prediction may be acceptable. Nevertheless, density estimates of common breeding birds at Patuxent did depend on the estimator for N, and similar effects have been observed with field data for other species (e.g. Efford, 2004). Our provisional interpretation is that such field datasets include unmodelled heterogeneity that causes modest negative bias in , particularly with N–estimators that are not robust to individual heterogeneity. The available tests for heterogeneity in the closed-population capture histories (e.g. Otis et al., 1978) do not distinguish heterogeneity caused by spatial location (see below) from other heterogeneity. They are therefore inappropriate for model selection in this context. Until new methods are developed the use of robust estimators such as those for Mth and Mh is recommended. Simulations reported elsewhere (Efford, 2004) suggest that is robust to the arbitrary choice of 2– D distribution (Poisson vs even) and detection function (half–normal vs uniform). Conventional capture probability and the spatial detection function

given time interval. pxy is a spatial variable that may be contoured for given g and detector locations (example in fig. 5). This provides a useful perspective on the functioning of the entire PDA as a sampling device, discussed in the next section. Constant–area assumption The conventional N–p parameterization relies on constant area A. Our fully spatial description of the detection process allows us to consider this assumption in more detail. Clearly the assumption does not hold when there is variation in the scale of movement by individuals, indexed by the parameter of the detection function. In our experience is often a decreasing function of population density. However, there is also reason to believe that A depends on the detection function parameter g(0). A may be defined operationally as the area within which every animal is close enough to a detector that it is "counted" by the population (N) estimator. For animals whose detection function declines gradually towards the edge of their range, this implies a fixed threshold pT of individual detection probability for inclusion in N. Consider the effect of increasing the non–spatial parameter of the spatial detection function g(0) to say g(0)’ while keeping constant. Animals outside A will now be counted (included in N) because p’xy > pT. A will correspondingly increase to A’ defined by the locus of points at which p’xy = pT. Our postulate that N estimators may be characterized by a threshold pT is unproven. Nevertheless, the argument provides strong reason for doubting the common, if implicit, assumption that A is sensitive only to the scale of movement (home range size). "Abundance" as conventionally measured with passive detector arrays is confounded with variation in both the spatial and non–spatial components of detection, and with the size and configuration of detector arrays. The force of arguments against index methods (Anderson, 2003) should lead us also to reject as a surrogate for population density. The method described here allows researchers to overcome the problem by estimating density itself. Pollock’s robust design

Detection in conventional N–p frameworks is described at the level of the detector array. In other words, p is the probability that an average individual of the target population is detected somewhere in the array. When density is low or detectors may register multiple individuals (e.g. mist nets) competition between animals for detectors may be ignored. Then the cumulative array–level probability of detection p xy of an individual at x,y can be predicted from its pairwise interactions with detectors:

Here gxy(j) represents the probability of detecting an animal with range centred at x,y in detector j over a

It is conceptually simple to substitute for in the robust design of Pollock (1982). The appropriate unit for recruitment Bt is then animals per unit area per unit time. However, it is untidy to use a spatial parameterization (g(0), ) for detection probability in the closed population model and a non–spatial parameterization (p) in the open population model. Further work is needed to determine whether it is beneficial and practical to incorporate a spatial detection function in the open population model used to estimate apparent survival ( t). We envisage modelling between– session home–range shifts in the open model, which would require at least one additional time– specific parameter.

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Fig. 5. Contours of daily array–level detection probability pxy for common breeding birds at Patuxent Research Refuge, Maryland, U.S.A. Detection was assumed to follow a half–normal function with parameters equal to the estimated means for 1961–1972 (g(0) = 0.029, = 87.3 m). Contour levels range upwards in increments of 0.02 from 0.02 at the edge. Stars on the plot represent the locations of mist nets (vertical spacing 100 m). Fig. 5. Contornos de probabilidad de detección diaria pxy de los distintos niveles de la batería de captura para aves comunes reproductoras en el Centro de Investigación Patuxent, Maryland, Estados Unidos. Se asumió que la detección seguía una función seminormal con parámetros iguales a los promedios estimados para 1961–1972 (g(0) = 0,029, = 87,3 m). Los niveles de contornos varían en sentido ascendente en incrementos de 0,02, desde 0,02 en el borde. Las estrellas que aparecen en la representación gráfica representan los lugares donde se colocaron las redes de niebla (distancia vertical de 100 m).

Mist–netting to estimate the density of bird populations The application of spatial detection functions in the analysis of capture–recapture data from mist nets is now briefly discussed. The main requirement for such analysis is that birds occupy equal–sized home ranges that are more or less stationary for the duration of a detection session. The effect of transients and of heterogeneous home range sizes on spatial detection estimates remains to be investigated. A related requirement is that recaptures within a detection session provide information on the spatial scale of movements. This can be met if the PDA is large enough to span several home ranges and either capture rates are high or netting continues for many occasions. "Net shyness" is sometimes invoked to explain a declining trend in capture numbers during a mist– netting session (e.g. Swinebroad, 1964; MacArthur & MacArthur, 1974; Karr, 1981). Avoidance by birds of nets at which they have been captured previously has the potential to bias estimates of density by inverse prediction. The general occurrence of net shyness appears controversial, and alternatively may be explained as avoidance of areas of human activity (Murray, 1997). If the learned response is to a site–specific hazard rather than to the device itself, then a solution is to shift the location of nets part way through a detection session (e.g. Stamm

et al., 1960). Spatial data from such a design may be analysed in DENSITY by specifying the occasions on which detectors were operated at each location. Finally it is noted that the fitted density and detection function for breeding birds at Patuxent (in round terms D = 10 ha-1, g(0) = 0.03, = 90 m) is likely to be a good basis for simulations to optimize detector configurations in future field studies of similar species. Acknowledgements Our use of inverse prediction draws on previous work with Shirley Pledger, Andrew Tokeley and Dave Ramsey, for whose help we are most grateful. We thank Paul Lukacs, Gary White, and an anonymous referee for comments on a draft, David Fletcher for mathematical advice, and Christine Bezar for editing. Funding for MGE was provided by Landcare Research and the New Zealand Animal Health Board. References Anderson, D. R., 2003. Index values rarely constitute reliable information. Wildlife Society Bulletin, 31: 288–291.

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Brown, P. J., 1982. Multivariate calibration. Journal of the Royal Statistical Society, Series B, 44: 287–321. Buckland, S. T., Burnham, K. P., Anderson, D. R. & Laake, J. L., 1993. Density estimation using distance sampling. Chapman and Hall, London. Burnham, K. P., 1981. Summarizing remarks: environmental influences. Studies in Avian Biology, 6: 324–325. Carothers, A. D., 1979. Quantifying unequal catchability and its effect on survival estimates in an actual population. Journal of Animal Ecology, 48: 863–869. Chao, A. & Huggins, R. M. (in press a). Classical closed population models. In: The handbook of capture–recapture methods (B. Manly, T. McDonald & S. Amstrup, Eds.). Princeton Univ. Press, Princeton. – (in press b). Modern closed population models. In: The handbook of capture–recapture methods (B. Manly, T. McDonald & S. Amstrup, Eds.). Princeton Univ. Press, Princeton. Efford, M. G., 2004. Density estimation in live– trapping studies. Oikos, 106: 598-610. Farnsworth, G. L., Pollock, K. H., Nichols, J. D., Simons, T. R., Hines, J. D. & Sauer, J. R., 2002. A removal model for estimating detection probabilities from point–count surveys. Auk, 119: 414–425. Karr, J. R., 1981. Surveying birds with mist nets. Studies in Avian Biology, 6: 62–67. Lee, S.–M. & Chao, A., 1994. Estimating population size via sample coverage for closed capturerecapture models. Biometrics, 50: 88–97. MacArthur, R. H. & MacArthur, A. T., 1974. On the use of mist nets for population studies of birds. Proceedings of the National Academy of Sciences, 71: 3230–3233. MacKenzie, D. I. & Kendall, W. L., 2002. How should detection probability be incorporated into estimates of relative abundance? Ecology, 83: 2387–2393. Murray, B. G. Jr., 1997. Net shyness in the Wood Thrush. Journal of Field Ornithology, 68: 348–357.

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Nichols, J. D., Hines, J. E., Sauer, J. R., Fallon, F. W., Fallon, J. E., & Heglund, P. J., 2000. A double observer approach for estimating detection probability and abundance from point counts. Auk, 117: 393–408. Noon, B. R. & Sauer, J. R., 1990. Population models for passerine birds: structure, parameterization, and analysis. In: Wildlife 2001: populations: 441–464 (D. R. McCullough & R. H. Barrett, Eds.). Elsevier Applied Science, London. Otis, D. L., Burnham, K. P., White, G. C. & Anderson, D. R., 1978. Statistical inference from capture data on closed animal populations. Wildlife Monographs, No. 62. Pledger, S. & Efford, M., 1998. Correction of bias due to heterogeneous capture probability in capture–recapture studies of open populations. Biometrics, 54: 888–898. Pollock, K. H., 1982. A capture–recapture design robust to unequal probability of capture. Journal of Wildlife Management, 46: 752–757. Pollock, K. H., Nichols, J. D., Simons, T. R., Farnsworth, G. L., Bailey, L. L. & Sauer, J. R., 2002. Large scale wildlife monitoring studies: statistical methods for design and analysis. Environmetrics, 13: 105–119. Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T., 1989. Numerical recipes in Pascal: the art of scientific computing. Cambridge Univ. Press, Cambridge. Rosenstock, S. S., Anderson, D. R., Giesen, K. M., Leukering, T. & Carter, M. F., 2002. Landbird counting techniques: current practices and an alternative. Auk, 119: 46–53. Stamm, D. D., Davis, D. E. & Robbins, C. S., 1960. A method of studying wild bird populations by mist–netting and banding. Bird–Banding, 31: 115– 130. Swinebroad, J., 1964. Net–shyness and Wood Thrush populations. Bird–Banding, 35: 196–202. Thompson, W. L., 2002. Towards reliable bird surveys: accounting for individuals present but not detected. Auk, 119: 18–25.

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Appendix. Automatic calculation of initial values for inverse prediction search algorithm. Apéndice. Cálculo automático de los valores iniciales para buscar el algoritmo de predicción inversa.

Initial values of the three parameters D, g(0), required for inverse prediction are here denoted by subscript S. Good initial values often make the difference between a speedy, fruitful search and failure. Calculations in DENSITY use the simplifying assumption of negligible competition among animals for detectors. Calculation of S and g(0)S uses Monte Carlo integration; accuracy depends on the number of random points sampled within the nominal detection area A. The user may vary the sampling intensity as an option in DENSITY. Initial detection scale

The expected distance between recaptures may be inferred from

and the detector configuration:

(1)

where the indices i and j refer to traps, pi is the "naïve" probability of an animal located somewhere within area A being caught in trap i, and Rij is the distance between traps i and j. With a half–normal detection model (2) where ri is the distance between an animal’s range centre at x,y and trap i. The integrals are evaluated by sampling points x,y within an area A. The area A is limited to locations where animals are "detectable" by some criterion (e.g. P > 0.01, see below). A factor of g(0)2 appears in both numerator and denominator of equation 1, and cancels out. Numerical minimization (the "golden" routine of Press et al., 1989) is used to find the value of matches the observed for the given detector configuration. Evaluation of equation (1) for which is time–consuming (O(T2) where T is the number of traps). Only the lower triangle and diagonal of the symmetric Rij pi pj matrix need be evaluated. Initial core detection probability g(0)S A similar but faster approach is used to obtain g(0)S. Given a value for equation (2) may be used to estimate the "naïve" probability that an animal is caught somewhere in the PDA:

To what observable quantity should we relate g(0)? We propose the mean number of captures within a detection session, conditional on an animal having been caught once:

This may be an unreliable indicator of g(0) when there is a large "learned trap response" (i.e. c ^ p or c p p in the notation of Otis et al., 1978). However, it has the advantage of not requiring an estimate of N or D. (3)

where t is the number of capture occasions and A is the area within which P > 0. The procedure is again to sample P from A and use numerical minimization to find the value of g(0) equals the observed , given S, t and the detector configuration. for which

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Appendix. (Cont.)

Initial DENSITY DS When D has been estimated by inverse prediction it is possible retrospectively to infer the boundary strip width W (e.g. Otis et al., 1978) that, if applied as a buffer around the PDA, would have yielded the "correct" density. Inferred values of W typically show a quadratic relationship to (unpubl. results). We base our initial estimate of density on this relationship as follows:

where AW is the effective area corresponding to W. A quadratic is used to predict W: W = a

2 S

+ c

The default polynomial coefficients in DENSITY are a = 0, b = 2, c = 0 (i.e. W = 2 S). Coefficients may be changed by the user if experience with a particular species and detector configuration provides more information. The relationship between and W appears to vary with the properties of the estimator for , and also with the detector configuration and possibly the duration of the study. Estimators that are robust to individual heterogeneity (Mh) tend to yield larger , and therefore require a smaller W to yield the same .

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Population dynamics and monitoring applied to decision–making M. J. Conroy & D. C. Lee

Conroy, M. J. & Lee, D. C., 2004. Population dynamics and monitoring applied to decision–making. Animal Biodiversity and Conservation, 27.1: 229–230. Research in wildlife conservation and management often affects decisions made by managers. Improving understanding through applied research is key to advancing the ability to manage birds and other organisms efficiently. Indeed, many papers from EURING 2003 and previous EURING meetings describe research on problems of pressing management concern. In this session, we focus on a subset of studies in which modeling and statistical estimation is explicitly connected to management decision-making. In these decision–centric studies, data are gathered and models are constructed with the explicit intention of using the resulting information to inform decisions about conservation. Whereas ecological models often produce information of value to decision makers, decision models explicitly include two additional features. First, management options are modeled via decision variables that link to system attributes that are directly responsive to management actions, such as harvest and habitat management. Second, certain outcomes are assigned value, via an objective or utility function. Both of these features involve factors beyond the usual consideration of ecological modeling; the first implies the presence of one or more "decision makers", and the second characterizes the societal preferences of each possible outcome resulting from a prospective decision. Our plenary paper, by Tim Haas (Haas, 2004), ventures the furthest into the realm of human behavior and societal processes by modeling the political context for conservation of the endangered cheetah (Acinonyx jubatus) in Africa. Haas shows that scientific information (e.g., population monitoring and population viability analyses) reaches decision makers through multiple pathways, each of which can modify or reinterpret the information signal. A predictive understanding of the country’s political as well as ecological processes is essential. Hass uses a system of interacting ecological and political influence diagrams to capture the stochastic, temporal processes of managing cheetah population in Kenya. The model predicts likely management decisions made by various actors within these countries, (e.g., the President, the Environmental Protection Agency, and rural residents) and the resulting probability of cheetah extinction following these decisions. By approaching the problem in both its political and ecological contexts one avoids consideration of decisions that, while beneficial from a purely conservation point of view, are unlikely to be implemented because of conflicting political objectives. Haas’s analysis demonstrates both the promise and challenges of this type of modeling, and he offers suggestions for overcoming inherent technical difficulties such as model calibration. The second paper, by Simon Hoyle and Mark Maunder (Hoyle & Maunder, 2004), uses a Bayesian approach to model population dynamics and the effects of commercial fishing bycatch for the eastern Pacific Ocean spotted dolphin (Stenella attenuata). Their paper provides a good example of why Bayesian analysis is particularly suited to many management problems. Namely, because it allows the integration of disparate pieces of monitoring data in the simultaneous estimation of population parameters; allows for

Michael J. Conroy, Cooperative Fish and Wildlife Research Unit, D.B. Warnell School of Forest Resources, Univ. of Georgia, Athens, GA 30602 U.S.A. E–mail: mconroy@uga.edu Danny C. Lee, USDA Forest Service, Pacific Southwest Research Station, 1700 Bayview Drive, Arcata, CA 95521 U.S.A. E–mail: dclee@fs.fed.us ISSN: 1578–665X

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incorporation of expert judgment and data from other systems and species; and provides for explicit consideration of uncertainty in decision–making. Alternative management scenarios can then be explored via forward simulations. In the third paper, Chris Fonnesbeck and Mike Conroy (Fonnesbeck & Conroy, 2004) present an integrated approach for estimating parameters and predicting abundance of American black duck (Anas rubripes) populations. They also employ a Bayesian approach and overcome some of the computational challenges by using Markov chain–Monte Carlo methods. Ring–recovery and harvest data are used to estimate fall age ratios under alternative reproductive models. These in turn are used to predict abundance of black ducks in each of 3 breeding areas. Finally, calibration of model parameters is obtained by comparing predicted with observed abundance. Although not currently implemented, the authors discuss how a Bayesian approach can be integrated into decision–making procedures using conditional modeling and application of reinforcement or machine learning. The next paper, by Martin Drechler and Franz Wätzold (Drechler & Wätzold, 2004), considers the problem of optimally allocating a conservation budget over time to maximize the survival probability of an endangered species. This must be done in the presence of uncertainty both about the biological system (e.g., probability of extinction under alternative plans), as well as about the availability of future funding. On the one hand, it would be undesirable to imprudently spend money now that might be needed for future conservation effort, when funds may be limited. On the other hand, failure to take action (and thus spend funds) sooner might lead to a higher probability of extinction. Provided estimates of uncertainty in funding, a model for trend in funding, and a model relating funding levels to viability, stochastic dynamic programming can be used to solve for an optimal amount of expenditure during any budget period. The final paper, by Clint Moore and Bill Kendall (Moore & Kendall, 2004), examines the costs incurred when uncalibrated indices to abundance are used in lieu of unbiased abundance estimates to make management decisions. Indices are often used instead of abundance estimates in the belief that the latter are too difficult and expensive. Moore and Kendall analyzed the impacts of using indices when making silvicultural decisions for the joint benefit of two bird populations, an endangered woodpecker and a shrub– nesting neotropical migrant. They computed the expected cost of uncertainty in the relationship between the monitoring index and population size, in currency units of the composite objective for both species. The authors found that substantial degradation of decision value can occur, depending on how uncertain the relationship between the index and true abundance. The results have important implications for managers, who may endeavor to cut costs by using index–based methods, while in the process incur these hidden costs of loss of decision utility. The five papers summarized above provide a good sampling of applications and methodological approaches, but are not a comprehensive coverage of the topic. For useful introductions to decision theory and methods, we suggest that readers consult Lindley (1985) or Clemen & Reilly (2001). A more detailed coverage of optimal decision–making, decision–making under uncertainty, and adaptive resource management is provided in Williams et al. (2002: chapters 21–25). References Clemen, R. T. & Reilly, T., 2001. Making Hard Decisions with Decision Tools. Pacific Grove, California, U.S.A. Drechler, M. & Wätzold, F., 2004. A decision model for the efficient management of a conservation fund over time. Animal Biodiversity and Conservation, 27.1: 283–285. Fonnesbeck, C. J. & Conroy, M. J., 2004. Application of integrated Bayesian modeling and Markov chain Monte Carlo methods to the conservation of a harvested species. Animal Biodiversity and Conservation, 27.1: 267–281. Haas, T. C., 2004. Ecosystem management via interacting models of political and ecological processes. Animal Biodiversity and Conservation, 27.1: 231–245. Hoyle, S. D. & Maunder, M. N., 2004. A Bayesian integrated population dynamics model to analyze data for protected species. Animal Biodiversity and Conservation, 27.1: 247–266. Lindley, D. V. , 1985. Making Decisions. John Wiley & Sons, New York, U.S.A. Moore, C. T. & Kendall, W. L., 2004. Costs of detection bias in index–based population monitoring. Animal Biodiversity and Conservation, 27.1: 287–296. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and Management of Animal Populations. Academic Press, San Diego, CA, U.S.A.

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Ecosystem management via interacting models of political and ecological processes T. C. Haas

Haas, T. C., 2004. Ecosystem management via interacting models of political and ecological processes. Animal Biodiversity and Conservation, 27.1: 231–245. Abstract Ecosystem management via interacting models of political and ecological processes.— The decision to implement environmental protection options is a political one. Political realities may cause a country to not heed the most persuasive scientific analysis of an ecosystem’s future health. A predictive understanding of the political processes that result in ecosystem management decisions may help guide ecosystem management policymaking. To this end, this article develops a stochastic, temporal model of how political processes influence and are influenced by ecosystem processes. This model is realized in a system of interacting influence diagrams that model the decision making of a country’s political bodies. These decisions interact with a model of the ecosystem enclosed by the country. As an example, a model for Cheetah (Acinonyx jubatus) management in Kenya is constructed and fitted to decision and ecological data. Key words: Social systems, Ecological systems, Influence diagrams. Resumen Gestión de ecosistemas mediante modelos interactivos de procesos políticos y ecológicos.— La decisión de implementar opciones de protección medioambiental es de carácter político. Las realidades políticas de un país pueden permitir ignorar los análisis científicos más rotundos acerca de la futura salud de un ecosistema. Una comprensión predictiva de los procesos políticos que conducen a la toma de decisiones sobre la gestión de los ecosistemas puede contribuir a orientar las políticas relativas a dichas áreas. Con este objetivo, el presente artículo desarrolla un modelo estocástico temporal acerca de cómo los procesos políticos influyen y son influidos por los procesos de los ecosistemas. Dicho modelo se ha estructurado a partir de un sistema de diagramas de influencia interactivos que configuran la toma de decisiones de las instituciones políticas de un país. Dichas decisiones interactúan con un modelo del ecosistema presente en el país. Así, a modo de ejemplo, se elabora un modelo para la gestión del guepardo (Acinonyx jubatus) en Kenia, ajustándose a los datos ecológicos y de toma de decisiones. Palabras clave: Sistemas sociales, Sistemas ecológicos, Diagramas de influencia. Timothy C. Haas, School of Business Administration, Univ. of Wisconsin at Milwaukee, P. O. Box 742, Milwaukee, WI 53201, U.S.A.

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Introduction Ultimately, the decision to implement ecosystem protection policies is a political one. Currently, the majority of ecosystem management research is concerned with ecological and/or physical processes. A management option that is suggested by examining the output of these models and/or data analyses may not be implemented unless the option addresses the goals of each involved social group (hereafter, group). For example, Francis & Regier (1995) describe efforts to sustain the Great Lakes ecosystem. These authors identify the following major barriers to the sustainable management of this ecosystem: 1. Social science research and model building is restricted by research funding decisions to "safe" projects – typically the economic benefits of Great Lakes resource utilization. These authors see a strong need for social science research to understand the goals and restrictions that drive the many groups that advise, regulate, pollute, and advocate for the Great Lakes ecosystem. 2. Great Lakes physical and biological science is University department compartmentalized and hence ecosystem models that integrate limnological and terrestrial subsystems are under–developed. 3. Because of (2), science–based management policies are lacking in their reliability and hence are either ignored, corrupted or, at best have limited impact during the political process of negotiating treaties between Canada and the U.S. for the regulation of pollution, fishing, and recreation on the Great Lakes. As a step towards meeting these needs, an Ecosystem Management System (EMS) is described herein that links political processes and goals to ecosystem processes and ecosystem health goals. This system is used to identify first the set of ecosystem management policies that have a realistic chance of being accepted by all involved groups, and then, within this set, those policies that are most beneficial to the ecosystem. Haas (2001) gives one way of defining the main components, workings, and delivery of an EMS. The central component of this EMS is a quantitative, stochastic, and causal model of the ecosystem being managed (hereafter, the EMS model). The other components are (2) links to data streams, (3) freely–available software for performing all ecosystem management computations and displays, and (4) a web–based archive and delivery system for items 1–3. This article focuses on ecosystem management in developing countries. One of the first questions then, is what theoretical framework should be used to model political groups in developing countries? The "new institutionalists" (see Gibson 1999, pp. 9– 14, 163, 169–171; Brewer & De Leon, 1983; Lindblom, 1980) draw on political economy theory to stress the following: (a) decision makers are pursuing their own personal goals, e.g. increasing their influence and protecting their job; and (b) decision makers work to modify institutions to help

Haas

them achieve these goals. This view of the policy making process is particularly relevant for studying wildlife management in developing countries: as Gibson (1999, pp. 9–10) states "New institutionalists provide tools useful to the study of African wildlife policy by placing individuals, their preferences, and institutions at the center of analysis. They begin with the assumption that individuals are rational, self–interested actors who attempt to secure the outcome they most prefer. Yet, as these actors search for gains in a highly uncertain world, their strategic interactions may generate suboptimal outcomes for society as a whole. Thus, rational individuals can take actions that lead to irrational social outcomes." New institutionalism is not limited to explaining ecosystem management policymaking in developing countries. Healy & Ascher (1995) document the effect that individual actor goal seeking behavior had on how analytical ecosystem health models were used to manage national forests in the United States during the 1970s, ’80s, and ’90s. Another paradigm for political decision making is the descriptive model (see Vertzberger, 1990). This approach emphasizes that humans can only reach decisions based on their internal, perceived models of other actors in the decision making situation. These internal models may in fact be inaccurate portrayals of the capabilities and intentions of these other actors. The ecosystem model component of the EMS described in Haas (2001) can be extended to synthesize these two policy making paradigms. In Haas (2001), the ecosystem model is expressed as an influence diagram (ID) (see the online tutorial, Haas (2003b) for an introduction to IDs). To incorporate the interaction between groups and the ecosystem, a set of IDs are constructed, one for each group, and one for the ecosystem. Then, optimal decisions computed by each of these group IDs through time are allowed to interact with the solution history of the ecosystem ID. The model that emerges from the interactions of the group IDs and the ecosystem ID is called an interacting influence diagrams (IntIDs) model. In this model, each group makes decisions that they perceive will further their individual goals. Each of these groups however, has a perceived, possibly inaccurate internal model of the ecosystem and the other groups. The IntIDs model approach then, synthesizes insights from political economics (groups acting to maximize their own utility functions) and descriptive decision making theory (groups using —possibly— distorted internal models of other groups to reach decisions). By choosing from a pre–determined repertoire of options, each group implements the option that maximizes a multiobjective (multiple goals) utility function. This is accomplished by having each group’s ID contain a decision option node representing the different actions that the group can take. Each group has an overall goal satisfaction node (hereafter, a utility node) that is influenced by the group’s goals. Each group implements an op-

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tion that maximizes the expected value of its utility node. This is called evaluating the ID (see Nilsson & Lauritzen, 2000). A schematic of the architecture of an IntIDs model is given in figure 1. As new types of actions are observed, the fixed repertoire of actions is periodically enlarged and the EMS model is re–fitted to the entire set of actions observations —see the Parameter Estimation with CA section, below. Ecosystem management emerges as each group implements management actions that best satisfy its goals conditional on the actions of the other groups and the ecosystem’s status. Conditional on these implemented management options, the marginal distributions of all ecosystem status variables are updated. By simulating these between–group and group–to–ecosystem interactions many years into the future, predictions of future ecosystem status can be computed. Ecosystem status, state, or health (hereafter status) is a multidimensional concept and has been defined differently depending on those ecosystem characteristics of most interest to the analyst. This article focuses on the status of an endangered species within an ecosystem. One way to quantify this characteristic is with the number of animals of such a species in the ecosystem at a particular spatio–temporal point. Kelly & Durant (2000) identify the cheetah in East Africa as an endangered species. The example given below of cheetah survivability in Kenya contains count variables for cheetah, and cheetah prey (herbivores having a biomass less than 35 kg). Future extensions of this model will have variables for individual cheetah prey herbivores such as Thomson’s gazelle. An output of the EMS model is the probability distribution on each of these counts by region and time point. These distributions are used to compute a typical measure of species survivability: the probability of extinction (POE) defined as the chance of a non–sustainable cheetah count at a specified spatial location and future point in time. The uses and benefits of an EMS that combines both political and ecological processes are: (1) a (possibly empty) set of ecosystem management policies can be found that are both politically acceptable and effective at protecting the ecosystem; (2) the most likely sequence of future management activities can be identified so that more plausible predictions of future ecosystem status can be computed such as extinction probabilities; and (3) international audiences can predict which countries have any chance of reaching ecosystem status goals such as averting the extinction of an endangered species. Because this modeling effort draws on several disciplines, the goals that are driving the model’s development need to be clearly stated. They are (in order of priority): 1. Usability: develop a model that, because of its predictive and construct validity, contributes to the ecosystem management debate by delivering reliable insight into how groups reach ecosystem man-

President ID

Ecosystem ID

Rural residents ID

Environmental Protection Agency ID

Legislature ID

Action Message Bulletin Board

Pastoralists ID

Ranchers ID

Fig. 1. Schematic of the IntIDs model of interacting political and ecological processes. Fig. 1. Diagrama esquemático del modelo de IntIDs (diagramas de influencia interactivos) para los procesos interactivos de carácter ecológico y político.

agement decisions, what strategies are effective in influencing these decisions, and how ecosystems respond to management actions. 2. Clarity and accessibility: develop a model that can be exercised and understood by as wide a range of users as possible. Such users will minimally need to be literate and have either direct access to the EMS website or access to a printed copy of the EMS report. For the cheetah viability example below, all groups except rural residents and pastoralists meet these minimum requirements. A major challenge will be to bring the contents of the EMS report to groups that are illiterate and/or lack web access. One idea is to deliver the EMS report to any literate members of such groups, e.g. schoolteachers. There is a tension between predictive and construct validity in that the development of a model rich enough in structure to represent theories of group decision making and ecosystem dynamics can easily become overparameterized which in–turn can reduce its predictive performance. The approach taken here is to develop as simple a model as is faithful to theories of group decision making and ecosystem dynamics —followed by a fit of this model to data so as to maximize its predictive performance. Specifically, success in the model building effort presented herein will be measured along the following two dimensions: (1) the model’s one–

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step–ahead prediction error rate wherein at every step, the model is refitted with all available data up to that step; and (2) the degree to which the model’s internal structure (variables and inter–variable relationships) agrees with theories of group behavior and ecology/population dynamics theories. The first dimension measures predictive validity and the second, construct validity. One step ahead predictive validity is seen as essential to establishing the reliability of the EMS model. Such predictive validity is however, not without its challenges. For example, it is possible that once an EMS model becomes known to the groups it is modeling, these groups may alter their behavior in response to model predictions. This Heisenberg Principle effect would invalidate model predictions. The influence of groups attempting to game EMS model predictions could be represented in the EMS model by adding another group called "the modelers." This group would post EMS model predictions to the bulletin board at every time step for all other groups to read. Once evidence on how model predictions are gamed is observed, such gaming behavior could be included in the group submodels of the EMS model and one step ahead prediction error rates computed as described in the Results section, below. Of course a second level of gaming is possible wherein groups attempt to manipulate an EMS model that includes groups attempting to game EMS model predictions. This second level of a Heisenberg Principle effect would be difficult to correct for and no solution is offered at this time. This article proceeds as follows. The Materials and methods section gives the architecture of a group ID. The Results section applies this framework to the management of cheetah in Kenya and describes how the model can be statistically fitted to observations on political and ecosystem variables represented in the model. Conclusions are drawn in the Discussion section along with brief comparisons with related efforts. Materials and methods Group ID architecture Overview A group’s ID is partitioned into subsets of connected nodes called the Situation, and Scenario subIDs (see figure 2 in Haas 2003a, Section 2). The Situation subID is the group’s internal representation of the state of the decision situation and contains Situation state nodes. Conditional on what decision option is chosen, the Scenario subID is the group’s internal representation of what the future situation (the Scenario) will be like after a proposed option is implemented. See Haas (1992, 2003a) for the cognitive theory that supports this decision making model architecture.

Actors, actions, and the time node A decision option will hereafter be referred to as an action. Groups interact with each other and the ecosystem by executing actions. The decision making group, referred to as the DM_group receives an input action that is executed by an actor referred to as the input–action–actor group or InAc_group. The subject of this action is the input–action–subject group or InS_group (which may or may not be the DM_group). The DM_group implements an output action whose subject is the target group or T_group. Actions are either verbal (message) or physical events that include all inter– and intra–country interactions. Each ID is a dynamic model and therefore has a deterministic root node Time. Time starts at t0 and increments discretely through t1, ..., tT in steps of t. Ecosystem status perceptions nodes Quantities that represent ecosystem status can be input nodes to a group ID. These nodes influence a node that represents how sensitive the group is to the value of the corresponding ecosystem status node. The idea is that a group is affected by the ecosystem but is only conscious of it through filtered, perceptual functions of the underlying ecosystem status nodes. For example, a group ID is sensitive to the presence of a land animal such as the cheetah through the animal’s density (number per hectare). This sensitivity is modeled by having the animal’s density node influence a perceived animal prevalence node that takes on the values none, few, and many. An ecosystem status node is stochastic due to it being a component of a stochastic ecosystem model. Sources of noise within this model include climate, unmodeled or mismodeled ecological functions, and inaccurate specification of model parameters. Even if an ecosystem status node was deterministic, how values on this node would affect the representation of the ecosystem quantity in a group’s perception of the ecosystem has its own set of uncertainties and noise sources. For example, consider the size of a minority population in a city. A demographic model allows for the stochasticity in birth rates, death rates, migration, and emigration. Now consider an elderly member of this city who’s only source of information is TV news and newsletters from politically conservative groups. The perceived size of the minority population of the city by this elderly person may be only minimally affected by the probability distribution of this quantity computed from the demographic model. Further, this elderly person may reason about the size of this minority population in categorical terms (small, moderate, hordes). The credence this individual will give to these different values will be determined in–part by random encounters with members of this minority group (captured by the demographic model) and by some noisy function of images viewed on TV and statements made in newsletters ("...members of minority group X are over–running this city!").

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Therefore, a separate node within each group ID is seen as necessary to capture both the more coarse resolution of perceptual models of continuously–valued quantities and the unique sources of stochasticity characteristic of perceptual processing. Image nodes A set of dimensions that defines the DM_group’s image of another group is needed. Two such dimensions, Affect and Relative Power appear in many studies of political belief systems. Affect varies over the enemy–neutral–ally–self dimension (Murray & Cowden, 1999; Hudson, 1983, chs. 2–4). Relative Power varies over the weaker–parity–stronger dimension. The Affect dimension’s self category is needed because the subject of an InAc_group’s action may be the DM_group itself. This self category includes the DM_group’s audiences (see below). Economic, militaristic and institutional goals This model is based on the cognitive–theoretic assumption that a group evaluates an input action directly on its perceived immediate and future impacts on economic, militaristic, and institutional goals. The two militaristic goals of Defend Country (inter– country), and Maintain Domestic Order (intra–country) are lumped into one Militaristic goal node. Economic and militaristic goal status is computed with a two–step process: first an assessment is made of how the input or output action changes economic or military resource amounts; then, an assessment is made of how these new resource levels affect the associated economic or militaristic goal. Here, only one institutional goal is modeled: Maintain Political Power. This goal is solely dependent on maintaining the contentment of several important audiences, discussed below. A goal node is a binary–valued random variable with values not–satisfied and satisfied. A goal node is similar to a utility variable in political economics. The nodes that affect these utility judgements are the DM_group’s InAc_group and InS_group image nodes, and the nodes representing the input action’s immediate and future impact on the DM_group’s resources. Scenario goals are influenced by Situation goals —if an output action does not cause a resource or audience node change, the Scenario goal’s distribution is the same as the corresponding Situation goal’s distribution. Audience effects The influence of audiences on a decision maker is described by research that suggests perceptions of present and future reactions of important audiences have effects on decision making and bargaining, see Asch (1951), Festinger (1957), Milgram (1974), Rubin & Brown (1975), and Partell & Palmer (1999). An input action’s impact on an audience is modeled as a function of the action’s perceived effect on

audience demands. For example, an important audience for (former) President Moi of Kenya was his ethnic group, the Kalenjin (Throup & Hornsby, 1998, p. 8). President Moi knew that only actions that brought benefits to that group would be favorably received by them. The effects of perceived audience reactions to input actions is modeled by having input action characteristics influence Audience Demands Satisfaction Change nodes which in turn, influence Audience Contentment nodes. What demands an audience has and the effect of different input actions on the satisfaction of those demands are both represented by the conditional distributions of the Audience Demands Satisfaction Change nodes. Say that the DM_group ID has k important audiences and consider the ith such audience. Let the node CAi(St) denote the perceived change in audience i’s demands satisfaction level due to an input action. CAi(St) takes on the values decreased, no change, and increased. Let the node Ai(St) denote the perceived contentment level of audience i. Ai(St) takes on the values discontented, and contented. Likewise, in the Scenario subID, output action characteristics influence Audience Contentment nodes through the Audience Demands Satisfaction Change nodes. These nodes are also influenced by input action characteristics. Situation contentment level influences Scenario contentment: if there is no change to an audience’s contentment level due to the output action, Scenario contentment level inherits the Situation’s contentment level. The only goal node influenced by audience contentment nodes is the Maintain Political Power goal, GMPP(St) —there is no goal of directly satisfying audiences because the decision maker has no concern for these audiences other than how they affect the decision maker’s hold on political power. Overall goal satisfaction Goal prioritization is modeled by a single node representing the DM_group’s overall sense of well– being. This node, denoted by U(St) (Situation) or U(Sc) (Scenario) is a deterministic function of the goal nodes wherein the coefficients in this function are interpreted as goal–importance weights and hence are assigned from knowledge of the group’s goal priorities. Group actions To avoid creating a system that can only process a historical sequence of ecosystem management actions, a group output action classification system is needed that characterizes actions along dimensions that are not situation–specific. The idea is to map an exhaustive list of possible actions onto a set of dimensions that collectively completely describe an action. Several action taxonomies or classification systems have been developed in the political science literature (see Schrodt, 1995). These

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taxonomies however, lack a set of situation–independent dimensions for characterizing an action. The approach taken here is to base a set of action characteristics or dimensions on an existing action classification system. The Behavioral Correlates of War (BCOW) classification system is chosen for this extension for two reasons. First, BCOW is designed to support a variety of theoretical viewpoints (Leng, 1999) and hence can be used to code data that will be used to estimate a model of group decision making that synthesizes realist and cognitive processing paradigms of political decision making —and BCOW is at least as rich a classification system as other systems in the literature. Second, BCOW has coding slots for recording: (i) a detailed description of an action, (ii) inter– and intra–country groups, and (iii) a short history of group interactions. This last coding category allows causal relationships to be identified and tracked through time. The BCOW coding scheme consists of a nearly exhaustive list of actions grouped into Military, Diplomatic, Economic, Unofficial (intra–country actor), and Verbal categories. The BCOW classification system exhaustively and uniquely characterizes a verbal action into either a comment on an action (Verbal: Action Comment), a statement that an action is intended (Verbal: Action Intent), or a request for an action (Verbal: Action Request). Modifications have been made to the BCOW scheme for purposes of categorizing ecosystem management actions. These modifications are as follows. First, the Unofficial Actions category of the BCOW coding system is not needed since groups internal to a country are modeled as having nearly the same range of output actions as a country– level group. Hence, all BCOW Unofficial Actions have been absorbed into one of the other action categories. For example, Hostage Taking, BCOW code 14153 is coded as a Military dimension action. Second, all BCOW Verbal Actions have been inserted into the BCOW Diplomatic Action category. Third, because BCOW does not include many actions that are peculiar to ecosystem management, such actions have been added to the BCOW taxonomy at the end of each action category listing (see Appendix 1, tables A1–A3 at www.uwm.edu/-haas/ems-cheetah/bcow.pdf). See Haas (2003a) for descriptions of nodes that determine realistic actor–input action combinations, and realistic target–output action combinations. A proposed target and output action combination influences target image and action characteristic nodes. These nodes along with Situation goal nodes, influence Scenario goal nodes. Finally, Scenario goal nodes influence the Scenario Overall Goal Satisfaction node. Each target and output action combination is used to compute the expected value of the Overall Goal Satisfaction node. At time t, the output action that maximizes this expected value is designated by coptimal(t). After determining coptimal(t), the DM_group posts to a bulletin board an action–message consisting of the time, the DM_group’s name, the target’s name,

and the BCOW action code. At the next time value, all other groups read this message. Each group assigns the values on the action characteristics associated with the BCOW action code and assigns values to the InAc_group image and InS_group image nodes. Using these values, each group computes an optimal output action and posts it to the bulletin board. When all groups have posted an output action and the ecosystem ID has posted updated distributions on its status nodes, the time variable is incremented by the value of t and the process is repeated (see fig. 2). Note that this protocol allows for feedback loops through time to emerge without need for additional model structure. There are groups that directly affect the ecosystem and groups that only indirectly affect the ecosystem. Actions by direct–affect groups always have the ecosystem as one of the targets of an output action. When such an action message is read by the ecosystem ID, its effect on the ecosystem is computed. If the action does not affect the ecosystem, e.g. a riot by the rural residents of Kenya, then the ecosystem model computes no effect on the ecosystem due to this action. Target, output action pair effectiveness The militaristic or economic effectiveness of an output action is determined in–part by its target. To represent this interaction, Scenario nodes are needed to represent the DM_group’s perception of the militaristic effectiveness of a target, output action pair given an input actor, input action pair. The nodes MilEf and EconEf take on the values negative effect, no effect, and positive effect and are influenced by Input Actor, Input Action, Target, and Output Action nodes. MilEf influences the Scenario Maintain Order goal, and EconEf influences Scenario Immediate Economic Resources Change.

Group ID hypothesis value assignment In the Results section, below, Consistency Analysis (CA) is used to fit each ID’s parameters to data. CA requires that each parameter in an ID be assigned an a–priori point value derived from expert opinion and/or subject matter theory. Let H(j) be such a value assigned to an ID’s jth parameter. Collect all of these hypothesis parameter values into the hypothesis parameter vector, H. See the Results section, below and Haas (2001, Appendix) for further discussion of CA. Because of the complexity of each group’s ID, it is difficult to directly assign hypothesis parameter values. For this reason, two optimization steps are performed to find hypothesis values that reflect the information contained in two types of hypothetical data sets. The first of these data sets is a collection of pairs of input and output node values on the group ID. Call this nontemporal data set an action– reaction data set. The second data set consists of a history of actions by all group IDs in the EMS

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t0 action messages t0 action messages

t1 action messages t1 action messages

t2 t1 action messages t2 action messages

ID1

ID2

Fig. 2. Sequential updating scheme of an IntIDs model consisting of two IDs. Bulletin board states are indicated by the boxes in the top row. A dashed arrow indicates messages are read but not removed. A solid arrow indicates message addition. Fig. 2. Esquema secuencial de actualización de un modelo de IntIDs (diagramas de influencia interactivos) que consta de dos IDs (diagramas de influencia). Los diferentes estados del tablón de anuncios se indican mediante los recuadros situados en la fila superior. Una flecha discontinua indica que los mensajes se han leído, pero no han sido eliminados. Una flecha continua indica la adición de mensajes.

model. Call this temporally–indexed data set an actions history data set. The action–reaction data set is used in an optimization procedure to find an initial H. The actions history data set is used by a second optimization procedure to refine these initial values. These two optimization steps are described in Haas (2003a, Section 2). Results Example: cheetah in Kenya

survival is reduced by lion predation inside reserves because these reserves are not big enough for cheetah to find areas uninhabited by lions. Over crowding of reserves in Africa is widespread (see O’Connell–Rodwell et al., 2000) and cheetah do not compete well for space with other carnivores (Kelly & Durant, 2000). Gibson (1999, p. 122) finds that the three reasons for poaching are the need for meat, the need for cash from selling animal "trophies", and the need to protect livestock. All input and output files for this example along with the EMS JavaTM based software is available at www.uwm.edu/-haas/ems-cheetah/.

Background ID descriptions Cheetah preservation is a prominent example of the difficulties surrounding the preservation of a large land mammal whose range extends over several countries. The main threats to cheetah preservation are loss of habitat, cub predation by other carnivores, and poaching (Gros, 1998; Kelly & Durant, 2000). Kelly & Durant (2000) note that juvenile

According to Gros (1998) and Gibson (1999, p. 164), the groups that directly affect the cheetah population are ranchers, rural residents, and pastoralists. Presidents, Environmental Protection Agencies (EPAs), legislatures, and courts indirectly affect the cheetah through their influence on these

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direct–affect groups. In this EMS model, group IDs are constructed to represent the president of Kenya, the Kenyan EPA, Kenyan rural residents, and Kenyan pastoralists. These group IDs interact with each other and an ID of the cheetah–supporting ecosystem contained within Kenya. Hypothesis parameter values for each ID in the EMS model are available at the aforementioned cheetah EMS website under the Current EMS Report link. Abbreviations used below are: Pres. president; EPA. Environmental Protection Agency; RR. Rural residents; Pas. Pastoralists; and Eco. Ecosystem. The ecosystem is directly affected only by poaching activities and land clearing. Anti–poaching enforcement is directed against either the rural residents or pastoralists and may or may not be effective at reducing poaching activity. Likewise, the creation of a preserve or the opening of an existing preserve to settlement are actions directed against the rural residents and/or pastoralists. In what follows, each group ID is described and hypothetical action–reaction data sets (tables 1–4) are given that are used to compute each group’s initial H vector (Haa 2003a, Section 2). The heuristics listed in www.uwm.edu/- haas/ems-cheetah/ heuristics.pdf are used to represent subject matter theory during the first step of this computation. Gibson (1999, pp. 155–156) argues in his case studies of Kenya, Zambia, and Zimbabwe that the president in each of these countries has a different personal priority for protecting ecosystems. Further, presidents of politically unstable countries typically place a high priority on protecting their power and staying in office (Gibson 1999, p. 7). These insights have motivated the following president ID (see www.uwm.edu/-haas/ems-cheetah/kenpres.pdf). The president has direct knowledge of rural resident and pastoralist actions. The president receives ecosystem status information exclusively from the EPA. The president’s audiences are campaign donors and the military. The president’s goals are to maintain political power and domestic order. The president’s action repertoire is: no changes, create a preserve, request increased antipoaching enforcement, open a preserve to settlement, and suppress a riot. EPA perceptions of the ecosystem’s status are represented by the cheetah prevalence and herbivore prevalence nodes. These nodes are influenced by the values of cheetah density, herbivore density, and poaching rate in the ecosystem ID. The EPA’s sole audience is the president. The EPA’s goals are to protect the environment, and to increase the agency’s staff and budget. The latter goal is motivated by an examination of the literature on bureaucracies. The main postulates of this literature are concisely stated by Ott (1981): "Managers of public enterprises —e.g. municipal fire departments, public hospitals, the Department of the Interior— have incentive structures much different from their private counterparts. In a private firm the owners create incentives for managers to

Haas

maximize the difference between revenues and private costs. Since the private manager has some contingent property rights in the revenue–minus– cost residual, he makes choices that tend to maximize the firm’s and its owners’net worth. Conversely, in the public sector there is no residual claimant: The public agency’s budget must be exhausted by approved expenditures. If there is a surplus, it is remanded to the general fund and will usually result in a reduction of the agency’s subsequent budgets. Since a surplus cannot benefit the agency, there can be no direct benefit to the agency of increasing a benefit–cost difference or of reducing the cost of achieving a given benefit level. Thus, broadly speaking, bureaucrats have strong incentives to increase costs, as these will, up to a point, increase the size of the bureau’s budget. This budget augmentation can be accomplished in one or both of two ways: (1) by under–stating the marginal cost of the bureau’s output; (2) by price discrimination. If we assume that managers of public agencies are wealth maximizers to the same extent as managers of private firms, then their behavior —i.e., their budgeting decisions, their planning, and their production— can be understood in terms of the reward structure under which they function. The pecuniary compensation of civil service managers is determined, somewhat rigidly and quite uniformly, by the number and grade of people whom they supervise; thus there is a strong incentive for bureaucrats at each level in an agency to increase the number of employees in their sections. By so doing, their operating budgets and salaries will be enlarged. The bureaucrat’s decision problem is, therefore, to present the largest budget that his political executive —the mayor, the governor, or the cabinet secretary— would approve. This entails knowing the executive’s demand for the agency’s output as well as knowing the agency’s own cost function. Knowledge of the latter is a qualification for management and comes from the seniority characteristic of civil servants who head agencies. Knowledge of the former is obtained as a result of the political process. A political candidate reveals his preferences both explicitly in his campaign platform and implicitly by embodying the preferences of those voter and special–interest groups who support him. Since department heads and cabinet secretaries are appointees of the elected politician, these political executives may, in turn, be presumed to reflect the preferences of the politician''. See also Niskanen (1971). For example, Healy & Ascher (1995) note that during the 1970’s and 1980’s the USDA Forest Service, using FORPLAN output, consistently proposed forest management plans that required large increases in Forest Service budget and staff (see also Gibson 1999, pp. 85, 115–116). Possible actions by the EPA are: decrease antipoaching enforcement, maintain antipoaching enforcement, and increase antipoaching enforcement (see www.uwm.edu/ - haas/ems-cheetah/ kenepa.pdf).

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Table 1. Hypothetical action–reaction data for the Kenya president ID: inac. Input actor; c(in). Input action; s(in). Input subject; c(ou). Output action; Target. Output action subject; Pas. Pastoralist; RR. Rural residents; EPA. Environmental Protection Agency; Eco. Ecosystem; Self. DM_group Tabla 1. Datos hipotéticos de acción–reacción con respecto a los diagramas de influencia del presidente de Kenia ID: inac. Actor de la acción estímulo; c(in). Acción estímulo; s(in). Sujeto estímulo; c(ou). Acción resultado; Target. Sujeto que recibe el resultado (el objetivo); Pas. Ganaderos; RR. Residentes rurales; EPA. Agencia de Protección del Medio Ambiente; Eco. Ecosistema; Self. Grupo DM.

Input vector inac

(in)

Output vector s

(in)

(ou)

Target

Pas

little poaching

Eco

do nothing

Pas

moderate poaching

Eco

request increased antipoaching enforcement

EPA

Pas

heavy poaching

Self

request increased antipoaching enforcement

EPA

little poaching

Eco

do nothing

moderate poaching

Eco

request increased antipoaching enforcement

EPA

heavy poaching

Self

open preserve

clear new land

Eco

do nothing

riot

Self

suppress riot

EPA

negative ecoreport

Self

create preserve

Pas

EPA

positive ecoreport

Self

do nothing

EPA

The development of both the rural resident, and pastoralist IDs, below is derived from the study of these two groups by Gibson (1999, pp. 121–123, 143–147) and is an attempt to represent quantitatively the goals, audiences, and action repertoire of these two groups as described by that author. Herbivore prevalence as influenced by the herbivore density node is the single ecosystem status node for the rural resident ID. A rural resident is pursuing the two goals supporting his/her family, and avoiding prosecution for poaching herbivores and/or cheetahs. Possible rural resident actions are: little poaching, moderate poaching, heavy poaching, clear new land, and riot (see www.uwm.edu/-haas/ems-cheetah/kenrr.pdf). The action little poaching includes the action of no poaching. This version of the rural resident ID does not distinguish between poaching herbivores versus cheetahs. There is evidence that poaching activity tends to include both herbivores and carnivores (Gibson 1999, pp. 143–145). In the ecosystem ID, a poaching action modifies the herbivore count stochastic differential equation (SDE) and the cheetah death rate SDE (see below). A change in the area of protected regions affects the herbivore SDE and the cheetah birth rate SDE. Hence, a poaching action’s affect on the ecosystem model is interpreted as the poaching

of both herbivores and cheetahs. Future versions of this EMS model will have separate group actions for the frequency of poaching herbivores for meat, and the frequency of poaching cheetahs for either trophies or to protect livestock. Such differentiation will also allow the indirect effect on the ecosystem of herbivore poaching causing a reduction in cheetah carrying capacity. Hunting big cats for trophies is market–driven and this world–wide market is not represented in either the rural resident or pastoralist IDs. The effect of this omission is that the model assumes a constant demand or constant market price for trophies. One way to model this demand–side effect on the motivation of rural residents and/or pastoralists to poach cheetahs is to develop a group ID of the buyers of such trophies. As a result of world–wide efforts to reduce the demand for trophies, this group would post lower market prices for trophies to the IntIDs bulletin board. These posted prices would, in–turn affect the perceived profit by rural residents and/or pastoralists from poaching cheetahs. This approach will be experimented with in future versions of this cheetah management EMS. Cheetah prevalence as influenced by cheetah density is the single ecosystem status node in the pastoralist ID. Pastoralists have the three goals of supporting their family, protecting their livestock,

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Table 2. Hypothetical action–reaction data for the Kenya EPA ID. (For abbreviations see table 1.) Tabla 2. Datos hipotéticos de acción–reacción con respecto a los diagramas de influencia de la Agencia de Protección del Medioambiente de Kenia. (Para las abreviaturas ver tabla 1.)

Input vector inac

(in)

Output vector s

(in)

(ou)

Target

Pas

heavy poaching

Eco

increase antipoach

Pas

clear new land

Eco

negative ecoreport

Pres

request increased antipoaching enforcement

Self

increase antipoach

Pres

Table 3. Hypothetical action–reaction data for the Kenya rural resident ID. (For abbreviations see table 1.) Tabla 3. Datos hipotéticos de acción–reacción con respecto a los diagramas de influencia de los residentes rurales de Kenia. (Para las abreviaturas ver tabla 1.)

Input vector inac

(in)

Output vector s

(in)

(ou)

Target

EPA

increase antipoach

Self

moderate poaching

Eco

Pres

open preserve

Self

clear new land

Eco

Pres

create preserve

Self

heavy poaching

Eco

and avoiding prosecution for poaching. Possible pastoralist actions are: little poaching, moderate poaching, and heavy poaching (see www.uwm.edu/ -haas/ems-cheetah/kenpas.pdf). As with the rural residents, a poaching action does not differentiate between the taking of herbivores versus cheetahs. The ecosystem ID is a modified version of the cheetah population dynamics ID of Haas (2001) and consists of four subIDs: management, habitat, direct effects on population dynamics, and population dynamics (see www.uwm.edu/- haas/ems-cheetah/ ecosys.pdf). Management nodes represent time (t), region (q), and management options (m). Cheetah habitat is characterized by chance nodes for the region’s climate (CL), unprotected land use (U), and the proportion of a region’s area that is protected (Rt). A single direct effect chance node follows: within–region poaching pressure (Pt). The node U takes on the values nomad_camel, nomad_cattle, ranching, and farming. Cheetah population dynamics is modeled with a system of SDEs consisting of the within–region nodes of birth rate (ft), death rate (rt), number of herbivores (Ht), cheetah carrying capacity (Kt), and cheetah count (Nt).

The SDE for Ht is (1) where H0 is the initial count, 0 (10,000) is the carrying capacity of the habitat (influenced by CL), is the difference between herbivore birth and 1 death rates, (= 0.01) is the diffusion parameter, and Wt is a Wiener process. The initial value, H0 is set to 0.6 0. This model is a simplified version of the relationship given in Wells et al. (1998) wherein the probability of offspring upon the meeting of a male and female is assumed to be 1.0. Poaching affects the value of 1: minor poaching, moderate poaching, and severe poaching cause 1 to take on the values 0.1, –0.1, and –0.3, respectively. If E[Ht] < 2,000, the rural resident ID’s Herbivores node is set to none, if 2000< E[Ht] < 10,000, this node is set to few, and if 10,000 < E[Ht], this node is set to many. As described in Haas (2001), the distribution of cheetah birth rate, ft is the solution of the SDE (2)

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Table 4. Hypothetical action–reaction data for the Kenya pastoralist ID. (For abbreviations see table 1.) Tabla 4. Datos hipotéticos de acción–reacción con respecto a los diagramas de influencia de los ganaderos de Kenia. (Para las abreviaturas ver tabla 1.)

Input vector inac

(in)

Output vector s

(in)

(ou)

Target

EPA

increase antipoach

Self

moderate poaching

Eco, EPA

Pres

create preserve

Self

heavy poaching

Pres, Eco

EPA

decrease antipoaching

Self

moderate poaching

Eco, EPA

where ft’ = 2ft – 1. This SDE was chosen because its solution is bounded between 0 and 1 making ft a well–defined birth rate ft c ( 0, 1). A similar development for cheetah death rate gives (3) where rt’ = 2rt – 1. Note that the birth rate decreases as f becomes increasingly positive, and the death rate decreases as r becomes increasingly positive. The tendency of more females to have litters within protected areas (see Gros, 1998) is represented by having the parameter f be conditional on the region’s status. Similarly, to represent the effect of poaching and pest hunting on r t , r is conditional on poaching pressure. The variability of the sample paths of f t and r t are controlled by the parameters f and , respectively. r All other unmodeled effects (such as migration, emigration, or age–dependent parameter values) that could influence the within–region cheetah count differential (dNt) are represented by the noise term in the cheetah count SDE: (4) were P, c, N0, and N are fixed parameters, and Kt is a deterministic function of the H t temporal stochastic process. Future versions of this cheetah count model will include terms to represent cheetah migration and emigration between adjacent regions including regions that are within the neighboring countries of Tanzania and Uganda. As mentioned above, the effect of climate change on a region is represented by the ecosystem’s climate node (CL) that influences herbivore carrying capacity. Ecosystem status output nodes are herbivore and cheetah densities. Because the ecosystem ID is conditional on region, computed herbivore and cheetah densities are region–specific. Since the group IDs are not regionally–indexed, these re-

gion–specific ecosystem ID outputs need to be aggregated across regions. Here, this aggregation is accomplished by computing at each time step, a weighted average of the expected values of ecosystem output nodes with region area as the weighting variable. These weighted averages are written to the bulletin board. Hypothesis parameter values for this ecosystem ID are taken from Haas (2001). Example model output As an example of EMS model output, figure 3 gives the event history over a three year period computed by the IntIDs EMS model using each ID’s H values. Three months is the unit of time (expressed in units of years, i.e., groups read the bulletin board every 0.25 time units). The initializing action is RRs clearing new land. This action prompts a negative ecosystem status report by the EPA. Upon receipt of this report, the president calls for increased antipoaching enforcement (Time = 2000.5) and so forth. The figure indicates a steady decline in both herbivore and cheetah density across Kenya. Say that preservation measures were enacted in 2001 and maintained through 2002. What changes in parameter values would be needed to reverse these declines? Through trial and error it has been found that the difference in herbivore birth and death rates would need to be maintained at 0.5, the cheetah birth rate parameter, f at –3.0, and the cheetah death rate parameter, r at 3.0. Because a fixed time step is used, the EMS model may produce a frequency of actions from a group that may be higher than observed. For the case of an action being repeated —such as the president’s call for increased antipoaching enforcement in the example, the repeated action should be interpreted as the group’s continued preferred response which in reality may not be made public at time points following the first time that the action is posted on the bulletin board.

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Parameter estimation with CA CA overview CA is used to fit the EMS model to data. Let U be an IntID’s r–dimensional vector of chance nodes. Let gS( ) be a goodness–of–fit statistic that measures the agreement of this distribution (referred to here as the Ux distribution) and the (possibly) incomplete sample, S. Let gH( ) be the agreement between this distribution identified by the values of H (referred to here as the hypothesis distribution) and the U* distribution. Let gsmax be the unconstrained maximum value of gS( ) over all . Let ghmax be the unconstrained maximum value of gH( ) over all . Up to errors in the approximation of gH( ), ghmax = gH( H). The CA parameter estimator maximizes gCA( ) h (1 – cH) gs( )/*gsmax*+ cHgH( ) /*ghmax* were c H c (0, 1) is the analyst’s priority of having the consistent distribution agree with the hypothesis distribution as opposed to agreeing with the empirical (data–based) distribution. Let h argmax {gCA( )} be the CA estimate of . See Haas (2001, Appendix) for further details and a comparison with other parameter estimators, and Haas (2003a, Section 5) for mathematical definitions of all CA agreement functions.

have also been developed for fitting the EMS model to a history of group actions and ecosystem observations. This group actions and ecosystem observations data set can be augmented with actions directed towards similar environmental metrics. For example, management decisions concerning any large land carnivore such as lions can be included in the data set used to estimate the parameters of the example’s cheetah management EMS model. Modeling across multiple scales Group behavior across a range of spatial scales is captured in the IntIDs EMS model structure by using a separate suite of group IDs for each country. Different temporal scales are modeled with selection of values for the time step between bulletin board updates in relation to the values chosen for the wildlife population dynamics model’s diffusion rate parameters. Ecosystem behavior across a range of spatial scales is captured thru the use of an ecosystem model at the level of a homogeneous region – similar to an ecoregion. A current shortcoming of the cheetah management model is that the population dynamics model’s diffusion rate parameters are too fast. This was done to illustrate how the ecosystem model could interact with several group models.

CA example Descriptions of related approaches The actions history–ecosystem status output (fig. 3) is used to illustrate CA. A smaller number of Monte Carlo realizations per ID causes the IntIDs model output to deviate slightly from the output of figure 3 and hence can be used as a data set that is different than the EMS model output under the IntID’s hypothesis distributions. The parameters estimated with CA are those defining the president’s Overall Goal Satisfaction node, and the ecosystem ID’s cheetah count node —resulting in 12 parameters to be estimated. For cH set to 0.5, starting and ending values of each CA agreement function are in table 5. Values of gsmax and ghmax are 23.6962 and 1.2671, respectively. The CA optimization was limited to 200 function evaluations per step and hence did not achieve convergence on either step. This run required four hours on a 500 mhz PC. Table 5 indicates that significant improvements in model fit to a data set can be achieved after only a modest exploration of the parameter space. One–step–ahead prediction error rates are given in Haas (2003a, Section 5). Also, a parameter sensitivity analysis of this model shows no highly unstable parameters (see Haas, 2003a, Section 4). Discussion A general purpose EMS has been developed that can help decision makers manage an ecosystem while taking into account political realities. Methods

Post–normal science In their development of post–normal science, Funtowicz & Ravetz (1993) argue that: (a) models as normally understood by scientists are not going to be successful in capturing the behavior of complex environmental systems, (b) diverse groups have

Table 5. CA agreement function values using artificial data: Am. Agreement measure; Pi. Percent improvement. Table 5. Valores de la función de concordancia del análisis de consistencia utilizando datos artificiales: Am. Medida de concordancia; Pi. Porcentaje de mejora.

gSEco( )

–1.9146

1.5273

179

gSGrp( )

18.049

20.2800

12 100

Eco

( )

–1.2657

–.00035

Grp

( )

–.0014

–.00164

–17

–.3191

.4594

244

gH gH

gCA ( )

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Model–Computed Actions History Kenpres

suppress riot open preserve req incr antip create preserve no changes

Kenepa

positive ecoreport negative ecoreport increase antipoach decrease antipoach no changes

Kenrr

riot clear new land heavy poaching moderate poaching litte poaching abandon settlement

Kenpas

heavy poaching moderate poaching little poaching

Cheetah density

0.0426 0.0322 0.0218 0.0114 0.0010

Herbivore density

0.5977 0.4548 0.3118 0.1688 0.0258 2000.0

2000.2

2000.5

2000.7

2001.0 Time

2001.2

2001.5

2001.7

2002.0

Fig. 3. Output action time series under H values. An arrow’s tail locates a group’s action and the arrow’s head indicates the reaction of either a group or the ecosystem. Each line on an ecosystem variable plot is the mean for one of the eleven regions in Kenya. Fig. 3. Serie temporal de los resultados de las acciones según los valores H. La cola de una flecha sitúa la acción de un grupo, mientras que su cabeza indica la reacción de un grupo o del ecosistema. Cada línea que figura en la representación gráfica variable del ecosistema corresponde al promedio de una de las once regiones de Kenia.

stakes in the outcomes of these systems and attach different values to such outcomes, (c) many of these groups are not members of established policymaking or scientific elites but nonetheless are demanding and receiving a significant role in the management of such complex environmental systems, so that (d) future management decisions should be made on "partial" scientific analysis and shared decision making that respects the values of groups that have been historically marginalized in policymaking debates.

The ID–based, combined political and ecological processes EMS model proposed here is similar to the post–normal science agenda in that it explicitly models the values and decision making processes of all groups affecting the environmental system. System complexity however, contrary to the post–normal science view, is not seen as hopeless to model but rather, stochastic models are proposed that, after being fitted to data, can have their out–of–sample or predictive validity

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demonstrated. Such demonstrations can lead to greater impact of the output of such models on the policymaking debate. Multiagent models Janssen (2002) describes a multi–agent simulation model of forest harvesting decisions of landowners in southern Indiana (U.S.A.), and in the Brazilian Amazon. This model employs finite–difference equations to represent farmers using a simple learning algorithm and a simple maximum expected utility decision making algorithm to reach harvesting decisions. For example, a decision to harvest trees is made if current economic conditions result in the utility of a harvesting proposal to be greater than that of not harvesting. This approach to a model–based EMS is different from the approach described in this article in that: (a) a procedure has not been given for fitting the model to landowner behavior observations, (b) the group behavior model is relatively simple, and (c) there is no separate ecosystem model. Differential and finite difference equation models Costanza et al. (2001) has developed a simulation model of the dynamic characteristics of humans interacting with periodically harvested fish stocks. This model accounts for different spatial and temporal scales of social and ecological processes. For example, mis–perceived spatial scale of fish populations can lead to extinction because the regulatory region scale and the natural population scale are different. For the case of northern Atlantic fisheries, the U.S. claim to regulatory control for up to 200 miles offshore results in a large–area fishing quota being set —but population spawning grounds are small and separated areas. Implementation of such quotas then, can lead to local population depletion. In this model, groups obey simple rules of behavior such as harvesting to maintain a maximum sustained yield, or unlimited fishing. Fish stocks are affected by: (a) harvesting and value– addition by humans before human consumption, (b) fishing regulation limits, (c) cheating (catches are over regulation limits), and (d) spatially heterogeneous area (three subregions). This approach to a model–based EMS is different from the approach described in this article in that: (a) there are no stochastic terms, (b) a procedure has not been given for fitting the model to fisheries observations, (c) the group behavior model is relatively simple, and (d) there is no separate ecosystem model. Carpenter et al. (1999) describes a model of multiple agents affecting a lake’s nutrient loading. A stochastic finite difference equation model of a lake’s phosphorous load along with a soils equation makes up the environmental model. Simple, deterministic, utility maximizing equations are used to represent the decision making of scientists,

economists, regulators, and farmers. Two computations are made at each time step. First, each agent decides how much phosphorous to allow into the lake. This is done by modeling these agents as utility maximizers having only partial information access. Then, the soils and lake models are updated. This approach to a model–based EMS is different from the approach described in this article in that: (a) no procedure has been given for fitting the model to observations on soils and lake status, (b) group behavior models assume high education levels and the ability to make fairly precise economic calculations, and (c) multiple spatial scales are not represented. References Asch, S. E., 1951. Effects of group pressure on the modification and distortion of judgments. In: Groups, leadership and men: 177–190 (H. Geutzkow, Ed.), Carnegie Institute of Technology Press, Pittsburgh, Pennsylvania. Brewer, G. & De Leon, P., 1983. Foundations of policy analysis. Dorsey Press, Homewood, IL. Carpenter, S., Brock, W. & Hanson, P., 1999. Ecological and social dynamics in simple models of ecosystem management. Conservation Ecology, 3(2): 4. URL: www.consecol.org/vol3/iss2/art4 Costanza, R., Low, B. S., Ostrom, E. & Wilson, J., 2001. Institutions, ecosystems, and sustainability. Lewis Publishers, Boca Raton, Florida. Festinger, L., 1957. A theory of cognitive dissonance. Stanford Univ. Press, Stanford. Francis, G. R. & Regier, H. A., 1995. Barriers and bridges to the restoration of the great lakes. In: Barriers and bridges to the renewal of ecosystems and institutions: 239–291 (L. H. Gunderson, C. S. Holling & S. S. Light, Eds.) Columbia Univ. Press, New York. Funtowicz, S. O. & Ravetz, J. R., 1993, Science for the post–normal age. Futures, 25(7): 739–755. Gibson, C. C., 1999. Politicians and poachers. Cambridge Univ. Press, Cambridge, UK. Gros, P. M., 1998. Status of the Cheetah Acinonyx jubatus in Kenya: a field–interview assessment. Biological Conservation, 85: 137–149. Haas, T. C., 1992. A Bayes network model of district ranger decision making. Artificial Intelligence Applications, 6(3): 72–88. – 2001. A web–based system for public–private sector collaborative ecosystem management. Stochastic Environmental Research and Risk Assessment, 15(2): 101–131. – 2003a. Ecosystem management via interacting models of political and ecological processes. Technical Report, School of Business Administration, Univ. of Wisconsin–Milwaukee, Milwaukee, Wisconsin. www.uwm.edu/-haas/ems-cheetah/poleco.pdf – 2003b. Tutorial on Influence Diagrams.

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URL: www.uwm.edu/-haas/idtutrl.pdf Healy, R. G. & Ascher, W., 1995. Knowledge in the policy process: incorporating new environmental information in natural resources policy making. Policy Sciences, 28: 1–19. Hudson, V. M., 1983. The external predisposition component of a model of foreign policy behavior. Ph. D. Dissertation, Ohio State Univ., Ohio. Janssen, M. A., 2002, Complexity and ecosystem management: the theory and practice of multi–agent systems. Edward Elgar Publishing, New York. Kelly, M. J. & Durant, S. M., 2000. Viability of the Serengeti Cheetah population. Conservation Biology, 14(3): 786–797. Leng, R. J., 1999. Behavioral correlates of war, 1816–1979 (Computer file), 3rd release, Middlebury College 1993. Available for download as Study Number 8606 from the Inter–University Consortium for Political and Social Research (ICPSR), Ann Arbor, Michigan. URL: www.icpsr.umich.edu Lindblom, C., 1980. The policymaking process. Prentice–Hall, New York. Milgram, S., 1974. Obedience to authority. Harper and Row, New York. Murray, S. K. & Cowden, J. A., 1999. The role of "Enemy Images" and ideology in elite belief systems. International Studies Quarterly, 43: 455–481. Nilsson, D. & Lauritzen, S. L., 2000. Evaluating influence diagrams using LIMIDs. In: Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence: 436–445 (C.

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Boutilier & M. Goldszmidt, Eds.). Morgan Kaufmann Publishers, San Francisco. Niskanen, W. A., 1971. Bureaucracy and representative government. Aldine, Atherton, Chicago. O’Connell–Rodwell, C. E., Rodwell, T., Matthew, R. & Hart, L. A., 2000. Living with the modern conservation paradigm: can agricultural communities co–exist with elephants? A five–year case study in east Caprivi, Namibia. Biological Conservation, 93: 381–391. Ott, M., 1981. Bureaucratic incentives, social efficiency, and the conflict in federal land policy. The Cato Journal, 1(2): 585–607. www.cato.org/pubs/journal/cjln2-2.html. Partell, P. J. & Palmer, G., 1999. Audience costs and interstate crises: an empirical assessment of Fearon’s model of dispute outcomes. International Studies Quarterly, 43: 389–405. Rubin, J. Z. & Brown, B. R., 1975. The social psychology of bargaining and negotiation. Academic Press, New York. Schrodt, P. A., 1995. Event data in foreign policy analysis. In: Foreign policy analysis: 145–166 (L. Neack, J. A. K. Hey & P. J. Haney, Eds.). Prentice– Hall, Englewood Cliffs, N.J. Throup, D. & Hornsby, C., 1998. Multi–party politics in Kenya. Ohio Univ. Press, Athens, Ohio. Vertzberger, Y. Y. I., 1990. The world in their minds. Stanford Univ. Press, Stanford. Wells, H., Strauss, E. G., Rutter, M. A. & Wells, P. H., 1998. Mate location, population growth and species extinction. Biological Conservation, 86: 317–324.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

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A Bayesian integrated population dynamics model to analyze data for protected species S. D. Hoyle & M. N. Maunder

Hoyle, S. D. & Maunder, M. N., 2004. A Bayesian integrated population dynamics model to analyze data for protected species. Animal Biodiversity and Conservation, 27.1: 247–266. Abstract A Bayesian integrated population dynamics model to analyze data for protected species.— Managing wildlife–human interactions demands reliable information about the likely consequences of management actions. This requirement is a general one, whatever the taxonomic group. We describe a method for estimating population dynamics and decision analysis that is generally applicable, extremely flexible, uses data efficiently, and gives answers in a useful format. Our case study involves bycatch of a protected species, the Northeastern Offshore Spotted Dolphin (Stenella attenuata), in the tuna fishery of the eastern Pacific Ocean. Informed decision–making requires quantitative analyses taking all relevant information into account, assessing how bycatch affects these species and how regulations affect the fisheries, and describing the uncertainty in analyses. Bayesian analysis is an ideal framework for delivering information on uncertainty to the decision–making process. It also allows information from other populations or species or expert judgment to be included in the analysis, if appropriate. Integrated analysis attempts to include all relevant data for a population into one analysis by combining analyses, sharing parameters, and simultaneously estimating all parameters, using a combined objective function. It ensures that model assumptions and parameter estimates are consistent throughout the analysis, that uncertainty is propagated through the analysis, and that the correlations among parameters are preserved. Perhaps the most important aspect of integrated analysis is the way it both enables and forces consideration of the system as a whole, so that inconsistencies can be observed and resolved. Key words: Bayesian analysis, Dolphin, Stenella attenuata, Model, Population dynamics, Yellowfin tuna. Resumen Modelo bayesiano integrado de dinámica de poblaciones para el análisis de datos de especies protegidas.— La gestión de las interacciones que se producen entre la flora y fauna y los seres humanos requiere disponer de información fiable acerca de las consecuencias probables que generarán las acciones de gestión. Este requisito es de carácter general, con independencia del grupo taxonómico. En el presente artículo describimos un método para estimar parámetros de dinámica de poblaciones y de toma de decisiones, de aplicación general, extremadamente flexible, que utiliza datos de un modo eficiente y proporciona respuestas en un formato útil. Nuestro ejemplo está relacionado con la captura accidental de una especie protegida, el delfín moteado (Stenella attenuata), en las pesquerías de atún de la costa este del océano Pacífico. Una toma de decisiones bien fundamentada requiere disponer de análisis cuantitativos que tomen en consideración toda la información relevante, permitiendo evaluar cómo afecta la captura accidental a estas especies y cómo afecta la normativa a las pesquerías, además de describir la incertidumbre en los análisis. El análisis bayesiano constituye un marco idóneo para proporcionar información sobre la incertidumbre que acompaña al proceso de toma de decisiones. Además de ser adecuado, permite incluir información acerca de otras poblaciones o especies en los análisis, así como un criterio experto. El análisis integrado pretende incluir en un único proceso todos los datos relevantes de una población, combinando análisis, compartiendo parámetros y estimando el conjunto de dichos parámetros de forma simultánea mediante el empleo de una función objetiva combinada. También garantiza que las presuposiciones del modelo y las estimaciones de parámetros sean coherentes a lo largo de todo el ISSN: 1578–665X

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análisis, que la incertidumbre se transmita a través del mismo, y que se mantengan las correlaciones entre los distintos parámetros. Quizá el aspecto más relevante del análisis integrado sea el modo en que permite y obliga a considerar el sistema como un todo, de forma que es posible observar y resolver las posibles contradicciones. Palabras clave: Análisis bayesiano, Delfín, Stenella attenuata, Modelo, Dinámica de poblaciones, Rabil. Simon D. Hoyle & Mark N. Maunder, Inter–American Tropical Tuna Commission, 8604 La Jolla Shores Drive, La Jolla, California 92037–1508, U.S.A.

Animal Biodiversity and Conservation 27.1 (2004)

Introduction Managing wildlife–human interactions is increasingly important as human influence on natural habitats grows. Effective management requires defined objectives and reliable information about the likely consequences of management actions, or lack of such actions. These requirements are general across all taxonomic groups and management issues. We describe a method for estimating population dynamics and decision analysis that is generally applicable, extremely flexible, uses data efficiently, and gives answers in a format that can be directly measured against management objectives. Our case study involves bycatch of dolphins in the fishery for tunas in the eastern Pacific Ocean (EPO). Impact on non–target species is a major concern in many fisheries worldwide (Hall et al., 2000), and is also a concern in areas such as harvest of waterfowl (Barbosa, 2001; Caswell et al., 2003), forestry (Noon & McKelvey, 1996), and pest control (Davidson & Armstrong, 2002). Many of these species are protected by governments. Unfortunately, it is often difficult or impossible to capture the target species without also affecting these protected species. This can result in restrictions on the harvest, leading to social, economic, and political problems. Such restrictions are often precautionary, based on inadequate information about both the target and the protected species. To make appropriate decisions, management must be able to predict the outcomes and estimate the associated uncertainty for alternative management actions. This will allow management to make informed decisions that consider the effects on both the protected and target species, while taking into consideration the uncertainty in the estimates. Bayesian analysis has become one of the most common methods for describing uncertainty in fisheries stock assessment (McAllister et al., 1994; Punt & Hilborn, 1997; McAllister & Kirkwood, 1998; Maunder & Starr, 2001) and marine mammal management (Raftery et al., 1995; Punt & Butterworth, 1999; Maunder et al., 2000; Wade, 2002; Breen et al., 2003). It is also becoming popular in wildlife management (Taylor et al., 1996; Link et al., 2002; Brooks et al., 2002). In addition to describing uncertainty, Bayesian analysis facilitates the inclusion of additional information in the form of prior distributions. The prior distributions can be developed from previous studies on the same population, studies on different populations of the same species, studies of similar species, meta–analyses (Hilborn & Liermann, 1998), or expert judgment. Bayesian analysis is a convenient way to include additional information into the analysis. However, development of priors from previous analyses has several disadvantages: it may be difficult to represent the distributional form of the prior, particularly if there is more than one parameter and the correlation among the parameters is important; also there may be information in the data about param-

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eters in the current model that were not included in the previous analysis. For example, a constant survival rate estimate from a mark–recapture analysis could be used as a prior (replacing data) in a subsequent analysis with different animals from the same population. However, the prior gives no information about annual survival variation that may be in the data, since annual survival parameters were not included in the previous analysis; the combined information in both data sets may support annual survival variation, rather than constant survival. These problems are resolved by integrating the analysis used to generate the prior with the current analysis (Maunder, 2003). This directly involves the data used to generate the prior in the current analysis. The integration of multiple sources of data in fisheries stock assessment models has been common for several decades (e.g. Fournier & Archibald, 1982), and is becoming popular in wildlife management (e.g. White & Lubow, 2002; Besbeas et al., 2002). However, even if an integrated approach is used, there is still a need to represent the uncertainty. Using an integrated model in a Bayesian framework allows for the most efficient use of information in the data and description of uncertainty (Maunder, in press), and is the most rigorous method for forward projection that incorporates both parameter and future demographic uncertainty. Integrated analysis involves fitting a single model to data from multiple sources. This integration is achieved by measuring relative model fit to each data set in a common currency, the likelihood, and then combining the likelihoods by addition after transformation to a negative log likelihood. Likelihood is not an absolute measure of fit, but can be used to compare one model with another. Information integration can be extended beyond data alone, with integration of parameter distributions from similar species, results from other analyses, prior knowledge, and expert opinion. These are incorporated by expressing them in terms of likelihood with respect to the model, and are termed penalties (likelihoodist) or priors (Bayesian). This paper switches back and forth between likelihoodist (Tanner, 1993) (rather than frequentist) and Bayesian philosophies, which involve making different kinds of inferential statements (see Wade, 1999). The likelihoodist philosophy is useful since it considers only the relative probability of alternative models, hypotheses, or parameter values, so does not require prior distributions on all parameters (Edwards, 1972). Choice of prior distributions can affect results (e.g. "non–informative" priors differ on a log or linear scale), and it can be convenient to avoid this problem when comparing models, for example (Wade, 1999; Maunder, 2003). Likelihoodist inference also has practical advantages when Bayesian Markov chain Monte Carlo (MCMC) will not converge without fixing parameters. However, some objectives, such as inference about a derived parameter independent of

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the value of model parameters (i.e. integrating over parameter space to remove nuisance parameters and estimate marginal probability distributions), and forward–projection under uncertainty, are best carried out in a Bayesian context (although MCMC convergence sometimes requires fixing parameters). This is especially true in data– poor situations, or when considering process error in a non–linear model —both common circumstances in natural resource modeling. Bayesian posterior distributions permit probability statements about parameter values. However, these posteriors are still effectively conditional probabilities, since: (1) they are conditional on the priors, which must be selected for all model components, and (2) one cannot consider all possible model structures —unconsidered structures are effectively given a prior probability of zero. Thus probability statements are valid only relative to scenarios that have been modeled. For the purposes of estimation, the combined likelihood behaves in the same way as an individual likelihood. A numerical search can be used to find the parameter values that maximize the combined likelihood. Model structures can be compared using information–theoretic approaches (Burnham & Anderson, 1998). Alternatively, Bayesian methods such as Markov chain Monte Carlo (MCMC) or sampling–importance–resampling (SIR) can be used to integrate across parameter values and model structures and estimate posterior distributions. Description of uncertainty about the current status of a population is important for management advice. However, management is also concerned with the consequences of different management actions. Therefore, it is important to extend the Bayesian analysis to estimate the outcomes and associated uncertainty for different management actions. It is possible that some actions may be more robust to uncertainty than others; therefore both expected outcome and uncertainty can be used to choose the appropriate management strategy. Analyses should also estimate the effect of the management actions on both the protected species and the fishery (Maunder et al., 2000). One high–profile example of interactions between a protected species and a commercial fishery is the tuna–dolphin interaction in the EPO (Joseph, 1994). The tuna purse–seine fishery in the EPO is of great economic importance, producing between 100 and 400 thousand metric tons of yellowfin tuna per year. The schools of yellowfin caught by purse seiners can be grouped into three categories: (1) those associated with floating objects; (2) those associated with dolphins; and (3) free–swimming schools. The majority of the catch comes from dolphin–associated schools, and only a small proportion comes from yellowfin associated with floating objects. Historically, large numbers of dolphins were killed in the sets on yellowfin associated with dolphins, which significantly reduced dolphin populations (Hall, 1997) such that

two are recognized as depleted by the U.S. government under the Marine Mammal Protection Act of 1972. There was substantial pressure to reduce the dolphin mortality, a challenge met by improved gear and dolphin–release procedures, international dolphin stock mortality limits, and individual vessel limits that together resulted in dramatic declines in mortality. With the later focus on "dolphin–safe" labeling and embargos by the United States on tuna caught in association with dolphins, pressure was exerted to reduce the fishing effort on tunas associated with dolphins, though these contentious policies have had little or no long–term effect on effort directed at yellowfin tuna associated with dolphins. These decisions were taken in a context of uncertainty about such important aspects of dolphin ecology as population sizes, reproductive rates, and natural mortality rates. Better information about these factors would facilitate decisions that take into account costs, benefits, and risks of alternative actions. The aim of this research was to demonstrate the Bayesian and integrated approaches by applying them to data for a protected species. The objectives of the particular application were to examine the effect of incidental mortality on the population dynamics of Northeastern Offshore Spotted Dolphin (Stenella attenuata), estimate population parameters, and look at likely future population trajectories. First, an age– and stage– structured population dynamics model was developed for the Spotted Dolphin population of the northeastern Pacific Ocean. Then models were developed to predict the observed data from the population dynamics model. Statistical assumptions were developed to create likelihood functions to provide fitting criteria. Priors were developed for the relevant parameters of the model. An MCMC algorithm was used to estimate the Bayesian posterior distribution. Finally, samples from the posterior distribution were used to examine different management scenarios. This analysis differed from previous analyses by Wade (1994) and Wade et al. (2002) in a number of structural assumptions (e.g. the use of recruitment deviates, modeling of stage structure, and different prior distributions), and in the integration of age and color phase data into the analysis. Methods Data Three main sources of data were used to fit the model. Absolute estimates of abundance were based on surveys carried out in the EPO in 12 years between 1979 and 2000 (Gerrodette & Forcada, 2002). These estimates, which were based on modified line–transect methods, update earlier estimates (Wade & Gerrodette, 1992, 1993; Gerrodette, 1999, 2000). An additional source of line–transect abundance estimates from observers on tuna vessels

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was not used to fit the model due to doubts about its reliability (Lennert–Cody et al., 2001), particularly concerning biases that vary through time. Proportional incidental–mortality–at–age and color–phase data for 1973 to 1978 were provided by the U.S. National Marine Fisheries Service (NMFS) (Susan Chivers, NMFS, pers. comm.). These data were collected by NMFS observers aboard tuna purse seiners in the EPO. Sample sizes over the six years were 224, 322, 46, 98, 181, and 58 individuals sampled from the dead dolphins brought aboard tuna vessels. Ageing was carried out using growth–layer groups observed in sectioned teeth (Myrick et al., 1983; Barlow & Hohn, 1984). Because Spotted Dolphins undergo changes in color pattern as they mature, five color phases (Neonate, Two–tone, Speckled, Mottled, and Fused) have been identified (Perrin, 1969) and used as stages in the model. Annual data on proportional incidental–mortality–at–color–phase from 1971 to 1990, and from 1996 to 2000 came from a summary of observer data (Archer & Chivers, 2002). Observers from the NMFS (1971 to 1990) and Inter–American Tropical Tuna Commission (IATTC) (1979 to 2000) recorded the sex and color phase of all dolphins killed during purse–seine operations. Sample sizes ranged from a high of 6,866 in 1975 to a low of 198 in 2000, and averaged 1,760 per annum over the period. Incidental–mortality–at–color–phase data were used only for years for which ageing data were not available. Information on incidental mortality (henceforth also termed "catch") due to the fishery had three sources: total catch estimates for 1959 to 1972 from Wade (1995); for 1973 to 1978 from the IATTC (1994); and for 1979 to 2000 from the IATTC (2002). The estimates for. 1959 to 1972 have "little or no statistical value" (National Research Council, 1992) because few vessels were monitored by observers before 1973, but are nevertheless used without error. The implications are considered in the discussion Model structure The Northeastern Spotted Dolphin population was modeled using a color–phase and age–structured projection model with an annual time step. Model structure was dictated by the available data and the modeling objectives. The population was structured by age and color phase (also referred to by the modeling term "stage"), which permitted use of two important data sources: long–term annual incidental mortality observed at color–phase, and a shorter time series of annual incidental mortality observed by color–phase and age, which gave better information about growth and color–phase transition rates. The model was implemented using a discrete formulation that assumed all incidental mortality occurred at the start of the year. As usual in marine mammal models, density– dependence was included, both to represent a

likely feature of population dynamics (Taylor & DeMaster, 1993; Taylor et al., 2000) and to permit an equilibrium–based MSY–like approach to be taken (see also Breiwick et al., 1984; Wade, 1993; Wade, 2002). This approach is similar to those needed to address the requirements of the Marine Mammal Protection Act. The population was assumed to be closed to migration. Our notation represents time, stages, and ages, using the subscripts t, s, and a, with T, S, and A representing the maximum possible value in each category. The model is represented diagrammatically in figure 1. Dynamics The model had an annual time step, representing annual transitions through the year classes. Transitions through the two dimensions of the population (age and stage) were modeled differently, since every year each individual moves to the next year class, but may not move to the next stage. These transitions were combined with the mortality processes in two phases: the within–year process (equation 1) involving mortality, and the between–year process (equations 2 and 3) involving transitions between ages and stages. Equation 1 gives the abundance in stage s and age class a at year end, as affected by incidental mortality and natural mortality, which are described later. Stage and age transitions occurred to start the following year (equations 2 and 3); these transition rates are also described later. (1) (2) for s > 1 (3) Unlike other year classes, from which all individuals moved up a class each year, the final year class A was modeled using a self–loop (Caswell, 2001) or "plus group" (equation 4), which individuals leave only by dying. This method avoided assuming 100% mortality of individuals after reaching age A. (4) Incidental mortality was modeled using the product of a temporally–varying harvest rate parameter ut and a stage and age class–dependent vulnerability parameter vs,a. This assumed that the vulnerability of different stage and age classes did not vary through time. Harvest rate at time t (equation 5) was a function of the incidental catch Ct and the vulnerable population Vt, which was a product of Nt,s,a and vs,a (equation 6). (5)

(6)

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Natural mortality Ms,a was constant for all ages and stages. Recruitment was modeled as a density–dependent process involving the number of sexually– mature individuals St (equation 7), population size as a proportion of estimated carrying capacity (Nddt /Nddeq), and an annual random variation term. The individual fecundity parameters (equation 8) were f eq, the average recruitment rate in an unexploited population, which was calculated as described later in the section on initial conditions; and fmax, the maximum rate of recruitment at low population size, which was an estimated parameter. The shape parameter z adjusted the shape of the density–dependence relationship, with a value of one suggesting a linear relationship between population size and recruitment. Values greater than one, which are expected in cetacean populations (Taylor & DeMaster, 1993) imply that density–dependent reductions in fecundity rates occur more at high population sizes. Since recruitment is to age 0 the important density is in the year of conception, though given high dolphin survival rates density will not change much from one year to the next. This is a common functional form for density–dependence in marine mammal populations (e.g. Breiwick et al., 1984; Wade, 1994). The annual recruitment deviate t was applied on a lognormal scale as per the usual practice, due to the multiplicative nature of mortality (Hilborn & Walters, 1992) and to avoid negative recruitment. Lognormal bias correction (–0.5"2R) was applied to recruitment deviates to ensure that the average stochastic recruitment was equal to its deterministic equivalent. Proportion mature at stage and age was ms,a, set to 0 for animals below the age of sexual maturity, and 1 above.

class dolphins had the Two–tone color pattern, presumably because the color phase change can occur during the first year of life. This phenomenon was accommodated with the parameter tr, representing the proportion of recruitment to the Two– tone color phase. Nt,1,0 = Rt (1 – tr) Nt,2,0 = Rt (tr) Vulnerability and stage transition parameters Vulnerability at age and stage represents the relative vulnerability of different groups within the population. It was modeled using a separate age–specific vulnerability curve for each stage. Vulnerability was assumed to be invariant through time and asymptotic with increasing age, and was modeled using the logistic with maximum vulnerability asy vs (equation 10). The most vulnerable group had asymptotic vulnerability of 1. The parameterization was designed to be informative, with L50 vs the age of 50% of maximum vulnerability for stage s, and sr vs the slope of the vulnerability curve. (10) where , , . Parameters asyvs, L50vs and srvs were estimated within the model. An alternative parameterization was also used, with vs,a = vs: reducing the vulnerability model to four parameters. The stage transition parameters )trs,a were modeled in the same way (equation 11).

(7) (11) Rt+1 m 0 (8) The size of the population as a proportion of carrying capacity, Nddt /Nddeq, was modeled using a flexible formulation whereby individuals of different ages and stages can make different contributions to density dependence. Thus population size and carrying capacity were expressed in terms of density–dependent individual equivalents, or Nddt and Nddeq. Nddt was calculated using equation 9. This approach was motivated by the contrast between a biomass–based model, where an individual’s contribution to density–dependence would increase as it grew, and an individual–based model. The term dds,a allows the contribution of each stage and age class to be adjusted. (9) Recruitment was always to age 0, but not always to the Neonate color phase: a proportion of 0 age

where , , . Parameters asytrs, L50trs and srtrs were estimated within the model. Initial conditions The unexploited equilibrium condition of the model was estimated at carrying capacity, with number of recruits to the 0 age class (the only location of recruitment in the model) balancing natural mortality. The following section describes the method used to estimate this state, and subsequently the initial population state. Calculations were on a "per–recruit" basis, which means that recruitment to age 0 was set to 1, and the numbers in all subsequent life history stages set relative to this single annual recruit. The pre–recruit calculations represent a generalized unexploited equilibrium state, so the time subscript is omitted. Individuals per recruit at the start of the year were represented by ns,a, and the number at the

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Stages

1–) )2,a I

)1,a

)2,a

Rt (1–tr)

1–) )3,a III

)3,a

1– )4,a IV

)4,a

1 V

Rt (tr)

Ages 0

1 1–z0,t

11 1–z1,t

A–1 1–z11,t

A 1–zA–1,t

Rt R

Fig. 1. Diagrammatic representation of the two dimensions on which individuals flow through the model: through color phases (I. Neonate, II. Two–tone, III. Speckled, IV. Mottled, and V. Fused) and through age classes (0 to A, with sexual maturity at age 11). Rt is recruitment at time t, zt,a is total mortality [1 – z = (1 – M)(1 – u)], )s,a is stage transition probability, and tr is the proportion of recruitment to the Two–tone stage. Fig. 1. Representación diagramática de las dos dimensiones en las que los individuos fluyen a través del modelo: a través de fases de colores (I. Neonato, II. Dos tonos, III. Moteado, IV. Manchado, y V. Fusionado) y a través de clases de edades (de 0 a A, con madurez sexual a los 11 años). Rt es el reclutamiento en el tiempo t, zt,a es la mortalidad total [1 – z = (1 – M)(1 – u)], )s,a es la probabilidad de transición de fase, y tr es la proporción de reclutamiento en la fase dos tonos.

end of the year by n's,a. The key parameter to estimate in this section was feq, the fecundity rate per mature individual at equilibrium. All relationships and parameter values were as in the dynamics section above, replacing N s,a with n s,a and N' s,a with n' s,a, but with two main differences. Firstly, and obviously, the stochastic term in the recruitment equation 8 was omitted. Secondly, the self–loop or plus–group was modeled with n' S,A at the end of the year, rather than n S,A at the start. (12) Fecundity rate per individual at equilibrium, feq, was calculated as the inverse of the number of breeders per recruit at carrying capacity, since together they produce one recruit (equation 13).

(13) Recruitment at carrying capacity was estimated in two stages. First, the population size produced from a single annual recruit, or number per recruit (NPR) was calculated (equation 14). NPR was not strictly population size, but population size adjusted for the flexible contribution of individuals at different life stages to density–dependence. It was then straightforward to divide carrying capacity by NPR to obtain recruitment at carrying capacity, Req (equation 15). (14)

(15)

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The starting year of the model was 1935, 20 years before significant fishing on dolphin–associated tuna began, to allow the 1955 population age distribution to differ from that for pre–exploitation equilibrium to the extent that the data suggested. Initial conditions in the starting year of N1935,s,a and N’1935,s,a for all s and a were then calculated by multiplying ns,a and n’s,a by Req. R1935 was set to Req. Parameter estimation Parameters were estimated initially in a penalized likelihood context by estimating the mode of the joint posterior. The likelihood components included data from three sources: observed mortality at age and stage; observed mortality at stage; and line– transect estimates of absolute abundance. The objective function was the sum of the negative logarithms of the likelihood function components and the penalties. Adding the negative log likelihoods relies on the components being independent of one another. The data component’s influence can be removed by removing its likelihood from the objective function. The abundance estimate likelihoods (equation 16) were assumed to be lognormally distributed (Gerrodette & Forcada, 2002). The model was made more flexible by using scalars to scale the coefficient of variation (cv) (approximately the standard deviation of the log abundance indices) and q to scale abundance (see Maunder & Starr, 2003). For the base case both q and were set to 1.

(16) Observed mortality–at–stage data collected by observers were modeled using a multinomial likelihood (equation 17). Effective sample size values were used to scale the influence of data for different years, because each dolphin aged and/or staged was not an independent sample from the population. In most years a number dolphins had been sampled from each school or sub–region, and there is a certain amount of homogeneity in age and stage within schools and regions (Kasuya et al., 1974; Hohn & Scott, 1983). Lack of independence reduces effective sample size, as do processes, not modeled, that cause the age distributions to change over time (e.g. temporal variation in M). In addition, reported sample sizes were slightly inflated, since the data were preprocessed and adjusted for dolphins without a color phase recorded. The square root of the actual sample size for the year was used as the effective sample size, which is arguably superior to the alternatives of using the reported sample size (problems as above), or the common fisheries approach (e.g. Butterworth et al., 2003) of using a fixed arbitrary sample size for each year. The latter approach has the drawback of giving equal weight to years with large and small sample sizes.

(17)

Observed mortality–at–stage–and–age data were modeled similarly, replacing equation 17’s obst,s with obst,s,a, the number of animals observed at time t in stage s at age a, and with Nt,s,a. Effective sample size was equal to the square root of the true sample size. Automatic differentiation software was used to estimate the many parameters in this model. Automatic differentiation provides very efficient fitting of statistical models (Greiwank & Corliss, 1991). The model implementation language AD Model Builder (ADMB, Otter Research, http://otter-rsch.com/ admodel.htm) gave access to the automatic differentiation routines, and estimated the variance– covariance matrix for all parameters of interest. For key parameters the software was also used to provide likelihood profiles, and Bayesian posterior probability distributions using MCMC methods with the Metropolis–Hastings algorithm (Hastings, 1970). ADMB combines the Hessian associated with the maximum likelihood estimate with any bounds on parameters to produce a bounded multivariate normal distribution, and uses this in the proposal function. A run of 10,000,000 iterations was analyzed using the Bayesian Output Analysis Program (Smith, 2001) to assess convergence using the Heidelberger & Welch (1983) stationarity and half–width tests, Geweke (1992) convergence diagnostic, and the Raftery & Lewis (1992) convergence diagnostic. The burn–in recommended by the Raftery & Lewis (1992) diagnostic was discarded, and the remaining values were used to generate the posterior. Prior distributions Prior distributions were specified for all the parameters estimated in the model, as required for Bayesian analysis. Normal priors were implemented as penalties on the negative log likelihood, equal to , where is the parameter estimate, and p and "2p are the mean and variance of the prior. Priors without penalties were uniform (on a nominal or log scale), and bounded to make them proper. Prior distributions are given in table 1. Natural mortality M was given a vague prior with mean 0.04 and standard deviation of 0.2. To improve convergence it was bounded at 0.1, which was well outside the 95% profile likelihood interval. The prior for the recruitment shape parameter z implied maximum net productivity level between 0.5 and 0.85 of equilibrium population size (Taylor & DeMaster, 1993), and was bounded at 0.2 and 15. A uniform prior was given to fmax. A normal prior could have been imposed, based on observed pregnancy rates of 0.167 females per female and inferred calving intervals (Myrick et al., 1986). However, natural mortality was parameterized as the

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Table 1. Descriptions and prior distributions for model parameters. Justifications for priors are given in the text. Tabla 1. Descripciones y distribuciones a priori para los parámetros del modelo. En el texto se detallan las justificaciones de tales distribuciones.

Parameter

Description

Distribution

Mean (SD)

M z

Natural mortality

Normal

0.04 (0.2)

0, 0.1

Shape parameter

Normal

2 (1)

0.2, 15

Nddeq

Equilibrium population size– density–dependent equivalent

Uniform

f max

Maximum fecundity

Uniform

Recruitment deviates

Normal

Two–tone recruitment

Uniform

0, 1

Stage transition slope

Uniform

0.001, 10

Age at 50% of max stage transition

Uniform

–1, 30

Asymptotic stage transition rate

Uniform

0, 1

Vulnerability slope

Uniform

0.1, 10

Age at 50% of max vulnerability

Uniform

–5, 35

Asymptotic vulnerability

Uniform

tr srs

L50

tr s

asytrs sr

v s

L50 asy

0, 108 0, 0.5 0 ("R)

n/a, n/a

0, 1

Standard deviation of recruitment variation

Fixed

0.15

Asm

Age at sexual maturity

Fixed

Maximum age

Fixed

same for all ages, which, given the higher mortality expected for Neonates, would bias fmax downward. Also, realized calving rates are likely to be lower than observed pregnancy rates, which would also bias the appropriate fmax downward. A uniform uninformative prior bounded at 0 and 0.5 was used, with the constraint that fmax must be greater than feq. Vulnerability and stage transition parameters were bounded to increase the stability of the model. For the 15–parameter vulnerability model, asymptotic vulnerability of the final stage asy5v was fixed at 1. For the five– and two–parameter models all the L50v parameters were fixed at –3 and the srv parameters at 0.1, to give constant vulnerability within a color phase. In the five–parameter model asyv5 was fixed at 1 and asyv1, asyv2, asyv3, and asyv4 were estimated, while in the two–parameter model asyv2,5 was fixed at 1 and asyv1,3,4 was estimated. To constrain recruitment deviates, a penalty term was added to the negative log–likelihood function

Bounds

likelihood analysis, unless there was evidence to the contrary. The standard deviation on the prior for individual recruitment deviates, "R, was fixed at the relatively low level of 0.15. This was justified biologically by the relatively low potential for recruitment to vary annually, given dolphins’ life history strategy. Age at sexual maturity was fixed at 11 years. Myrick et al. (1986) report two estimates of age at sexual maturity (10.7 and 12.2), based on the ages estimated by two different readers. The maximum age class implemented in the model affected dynamics only when a plus group was not used. The imposition of a maximum age class without a plus group implies senescence, about which there is little information for Northeastern Spotted Dolphins. This assumption is particularly influential if age at senescence is set too low in populations with low mortality rates. This element of the structure was investigated in the sensitivity analysis. Forward projection

This was equivalent to a lognormal prior on each individual annual recruitment deviate with a mean of 0 and standard deviation of ". Each deviate was therefore estimated to be zero in the maximum

The population was projected forward 5 years from the final year estimated, using samples from the joint posterior of the Bayesian MCMC. Recruitment

256

deviates were assumed to be zero, as recruitment variability is substantially less than parameter uncertainty. The bias adjustment factor was therefore not applied. As part of the forward projections, the effect of changes to the level of setting on dolphins was investigated by altering the mortality rate (observed mortalities/total population) due to setting on dolphins to 0, 0.5, 1, and 2 times the average level prevailing over the three years to 2000. Outcomes recorded were the relative change in total number of dolphins after 5 years. Sensitivity analysis and hypothesis tests The sensitivity of parameter estimates and management conclusions to different components of the data (line transect, stage data, and age and stage structure data) was tested by multiplying the standard deviation of the line transect data by 4 to reduce its influence, and by removing from the objective function the likelihood component associated with each of the other data components, one at a time. The sensitivity of the model to several alternative structures (hypotheses about the state of nature) was examined. Models of long–lived species are sensitive to the maximum age in the model. This sensitivity was investigated by running the model with the last age in the model with and without a self–loop (plus–group). Using a plus–group is equivalent to assuming that senescence does not occur early enough to affect population dynamics. Sensitivity to variation in vulnerability between color phases was also examined. Vulnerability was modeled using three different approaches: the 15–parameter method described earlier with three parameters per stage; a five–parameter method with constant vulnerability within a stage; and a two–parameter model with Two–tone and adult stages fully selected, and common vulnerability for the Neonate, Speckled, and Mottled stages. The support for alternative structures was investigated by comparing the likelihoods using the Akaike Information Criterion (AIC) (–2 log L + 2v); and the posterior probabilities using the Bayes factor , where L, L1, and L2 are negative log likelihoods, and v, v1, and v2 represent the number of parameters. Results Parameter estimates are presented in table 2 for the base case model, with standard deviations, profile likelihood confidence intervals, and MCMC intervals. To allow the MCMC chain to converge, z was fixed at the posterior mode. The fit of the model to the color– phase and age–structure data is presented in figure 2 and figure 3. The estimated population trajectory, with the line transect population size estimates for comparison, is presented in figure 4, with the recruitment deviates in fig. 5.

Hoyle & Maunder

The modal estimate of fmax —fecundity rate at low population size— of 0.125, although less than the pregnancy rate given by Myrick et al. (1986), the difference is not statistically significant. The value suggested by recent re–analyses of pregnancy completion rates was lower than our estimate, significantly according to likelihood profile, but not MCMC intervals. Our estimate was significantly less than Wade’s (2002) estimate according to the MCMC, but not the profile likelihood intervals. Natural mortality at 0.039 was little different from the prior mode of 0.04, but the standard deviation was far smaller. The posterior mode of the shape parameter z, at 1.69, was less than the prior mode of 2. Two–tone recruitment was estimated to be about 20%. Credibility intervals generated using MCMC were mostly narrower than the profile intervals, mainly due to fixing z to make the MCMC runs converge. When z was not fixed the chain was occasionally trapped for a period at very low values of z, near the lower boundary set at 0.2. Sensitivity analysis and hypothesis tests Fitting the model to the data with increased coefficient of variation on the abundance estimates and without each of the other two data components demonstrated the relative influence of each on parameter estimates and variance (table 3). Reducing the influence of abundance data affected primarily the variance on estimates of natural mortality and Nddeq. Removing the age by color phase by year data made the model unstable, with excessively high Nddeq and zero natural mortality, illustrating the strong influence of these data. Removing the color phase by year data resulted in slightly increased variance and slightly changed modal estimates of natural mortality, Nddeq, fmax, Two–tone recruitment, and vulnerability parameters, suggesting that it had only a small influence on results. The investigation of the plus–group could be seen as exploring the alternative hypotheses that (1) senescence (the right side of the U–shaped mammalian mortality curve) occurs sufficiently late that it affects too few animals to matter, or (2) senescence generally occurs around the age of 40, and so it is important for population dynamics. The latter hypothesis carried more weight with a Bayes factor of 48, suggesting strong evidence (Kass & Raftery, 1995) for senescence (or ageing error causing the appearance of senescence). Variation in vulnerability among color phases was apparent when all 15 vulnerability parameters were estimated separately. However, this model was over–parameterized, and had AIC of 1892.69. A five–parameter model with uniform vulnerability within each color phase had a lower AIC of 1872.95. A single parameter model where all animals had the same vulnerability had less support with AIC of 1879.97. Following the suggestion of Barlow & Hohn (1984) that age classes 5 to 15 and Neonates are under–represented, a parameter was added to differentiate the under–represented color phases (Neonates, Speckled, and Mottled) from the other

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Table 2. Parameter estimates with standard deviations from the Hessian matrix, 95% likelihood profile intervals, and Bayesian posterior distributions from MCMC. Tabla 2. Estimaciones de parámetros con desviaciones estándar de la matriz hessiana, intervalos de confianza de verosimilitud del 95%, y distribuciones bayesianas a posteriori a partir de las cadenas de Markov Monte Carlo (MCMC).

95% Likelihood profile intervals Parameter

Estimation

0.039

Nddeq fmax Z

Lower

MCMC

Upper

2.5%

97.5%

0.011

0.017

0.060

0.020

0.057

3,406

206

3,068

3,857

3,086

3,791

0.125

0.018

0.090

0.166

0.093

0.162

1.69

1.04

0.220

3.532

5.25

3.13

5.73

0.61

srtr4

4.54

0.44

–0.05

0.21

2.31

0.94

0.043

0.373

0.068

0.379

L50

1 tr

L50

L50tr4 v

asy

1,3,4 v 2,5

9.99

0.65

16.00

0.76

0.49

0.08

1.00

0.00

0.196

0.080

two color phases (Two–tone and Fused). This two– parameter model was most strongly supported with AIC of 1867.68. Depletion and implications for recovery The model estimated a mean recovery from 1995 to 2000 of 7.4%, with 95% confidence interval from 0.00% to 17.3%. Current depletion level (number of individuals as a proportion of equilibrium) was estimated as 19% (15% to 24%). Forward projections were carried out using MCMC with four scenarios of setting on dolphins. With the levels of mortality from setting on dolphins prevailing between 1998 and 2000, average recovery over the 5 years to 2005 was 4.1% (–2.1% to 12.7%). With no mortality from setting on dolphins, average recovery was 6.9% (0.5% to 15.9%). With 0.5 and 2 times the 1998–2000 mortality rates, recoveries were 5.5% (–0.8% to 14.2%) and 1.3% (–4.7 and 9.7%) respectively. Discussion The integrated approach used in this analysis is a very general methodology that can provide useful answers to some of the questions most relevant to

managers. Bayesian forward projections, such as those given above, can be used as statements of belief about the probability of future events, or relative probability given models that have not been considered. Measuring such forward projections against management objectives gives a direct way to compare the utility of management options. Decision analysis based on projecting the population forward in time under different management strategies is a major component of many Bayesian analyses (e.g. Maunder et al., 2000; Breen et al., 2003). Several components of uncertainty, such as: (1) parameter uncertainty, (2) model structure uncertainty, and (3) demographic uncertainty can be included in forward projections. Parameter uncertainty is inherent in statistical estimation of model parameters. Many different values of the model parameters may adequately represent the data, and they must all be considered as possible true states of nature. Similarly, alternative model structures may represent different possible states of nature. Demographic uncertainty describes how model parameters change over time. In general, scientific knowledge reduces parameter uncertainty and model structure uncertainty, while demographic uncertainty in the future cannot be reduced. In many situations both model structure uncertainty

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0.8

Observed Expected

1972

0.6 0.4 0.2 0

0.8

III

1982

0.6 0.4 0.2 0 I 0.8

1987

0.6 0.4 0.2 0

0.8 0.6

1996

0.4 0.2 0 I

0.8

2000

0.6 0.4 0.2 0 I

Fig. 2. Observed and estimated proportions of dolphins captured by color phase (Neonate–I, Two– Tone–II, Speckled–III, Mottled–IV, and Fused–V) for five of the years for which data were available. Sample sizes for each year are given in the text. Fig. 2. Proporciones observadas y estimadas de delfines capturados por fase de color (Neonato–I, Dos tonos–II, Moteado–III, Manchado–IV, y Fusionado–V) para cinco de los años en que se disponía de datos. Los tamaños de las muestras para cada año se facilitan en el texto.

and parameter uncertainty are more important than demographic uncertainty in the future. This is particularly true for long–lived species such as dolphins, which have very low productivity

rates. For these populations, catastrophes can be a more important component of uncertainty than annual variation in model parameters (Breen et al., 2003).

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Proportion

1973 Observed Observed Expected Expected

0.1

0.05

Proportion

1974 0.1

0.05

Proportion

19 Age

1975 0.1

0.05

Proportion

7 1976

0.1

0.05

Proportion

7 1977

0.1 0.05 0

Proportion

7 1978

0.1 0.05 0

Fig. 3. Observed and estimated proportions of dolphins captured at age from 1973 to 1978. Data are pooled across color phase. Sample sizes are given in the text. Fig. 3. Proporciones observadas y estimadas de delfines capturados segĂşn edad desde 1973 hasta 1978. Los datos se agrupan segĂşn la fase de color. Los tamaĂąos de las muestras se facilitan en el texto.

Integrating several data sources requires more complexity in the model. For example, a model based on numbers alone can be used to estimate population trajectories based on removals and abundance estimates. Adding age structure data gave information about variability in recruitment and about

total mortality rates, but required that age structure be modeled and required assumptions about ageing error. Adding the longer time series of size structure data gave more information about total mortality and recruitment, but required that stage structure be modeled.

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Abundance (thousands of animals)

4000

Line transect estimates Model estimates

3500 3000 2500 2000 1500 1000 500 0 1955

1965

1975 Year

1985

1995

Fig. 4. Research line transect estimates of abundance, with 95% confidence intervals, compared with estimated population trajectory from the model. Fig. 4. Estimaciones de abundancia a lo largo de los transectos investigados, con intervalos de confianza del 95%, en comparación con una trayectoria poblacional estimada a partir del modelo.

0.04 0.03 0.02 0.01 0.00 –0.01 –0.02 –0.03 1930

1940

1950

1960 1970 Year

1980

1990

2000

Fig. 5. Estimated recruitment deviates for all years. Fig. 5. Desviaciones del reclutamiento estimado para todos los años de estudio.

In many applications of fitting population dynamics models to data, results are more sensitive to model structure than to parameter uncertainty within a single model structure, and even small structural changes can have significant effects. For example, in the dolphin model a simple model of senescence significantly affected estimates of depletion level and recovery rate. More complex models of age–specific survival, with the Gompertz (Wilson, 1993) and Siler functions (Siler, 1979; Barlow & Boveng, 1991), may further improve the fit of the model and alter parameter estimates. Different model structures represent different hypotheses about population dynamics. It is often possible to formulate the

model so that one or more parameters can be changed to represent the different model structures. In this case, the model structural uncertainty can be included in the analysis by estimating these parameters. In other situations, it is not possible to represent different model structures by model parameters, so other techniques such as Bayesian model averaging (Hoeting et al., 1999) must be used. For example, McAllister & Kirchner (2002) included uncertainty about the structure of the stock–recruitment relationship for a Bayesian stock assessment of Namibian orange roughy. Because only a few of the possible model structures can be considered, models tend to underestimate uncertainty.

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Table 3. Parameter estimates for the model fitted without a plus group, and without each of the three fitted data components in turn. Tabla 3. Estimaciones de parámetros para el modelo ajustado sin un grupo adicional, y sin cada uno de los tres componentes de datos ajustados sucesivamente.

Without plus group

Abundance cv x 4

Without age

Without stage

Mean

0.056

0.008

0.044

0.022

0.000

0.037

0.014

Nddeq

3,535

229

3,549

552

79,821

1,679

3,377

222

f max

0.147

0.019

0.125

0.020

0.466

0.011

0.117

0.028

1.73

1.04

1.82

1.11

4.90

0.64

2.09

0.98

srtr2

5.91

2.43

5.25

3.13

0.02

70.10

5.64

2.52

5.77

0.60

5.74

0.61

6.69

0.52

5.51

0.62

4.48

0.43

4.54

0.44

3.10

0.40

4.54

0.44

–0.01

0.19

–0.05

0.21

2.93

0.15

–0.26

0.39

1.17

1.14

2.29

0.99

–0.22

729.65

2.05

1.26

9.92

0.65

9.96

0.65

9.54

0.49

10.66

0.91

16.40

0.83

15.97

0.78

9.83

0.28

16.19

0.92

3 4

L50

tr 1

L50tr2 L50

L50

asy

3 4

0.40

0.07

0.49

0.09

0.16

0.01

0.35

0.08

asyv2,5

1.00

0.01

1.00

0.00

1.00

0.01

1.00

0.01

0.189

0.079

0.196

0.080

0.881

0.016

0.217

0.086

1,3,4

The form of integrated analysis presented in this study is simple, and more advanced methods could be applied. The model uses estimates of abundance from the surveys with the associated confidence intervals. More advanced forms of integrated analyses combine two analyses that are usually carried out independently. For example, Maunder (2001a) combined a generalized linear model standardization of catch–per–unit–of–effort data with a population dynamics model. In this analysis it would be possible to combine the analysis of the sighting surveys with the population dynamics model. This would ensure that model assumptions and parameter estimates were consistent throughout the analysis, that uncertainty was propagated through the analysis, and that the correlations among parameters (between population size estimates for example) were preserved. Integrating the sighting survey analysis with the population dynamics model would also allow some parameters of the survey analyses to be shared among years. Other data such as mark–recapture data could also be integrated into the population dynamics model (Hampton & Fournier, 2001; Maunder, 2001b, Besbeas et al., 2002). It is common for analyses that use multiple data sets to have conflicting information in the data, and such conflicts must be resolved. How-

ever, the data are usually correct, if seen in the right context (barring falsification or transcription errors), and the conflict comes from inadequacy of either the population model itself, or the model used to pre–process the raw data and provide summary statistics for the analysis. Our analysis did not use the estimates of abundance based on data collected by observers on tuna vessels, as they are thought to be unreliable (Lennert–Cody et al., 2001) and contradict the other data. One reason the estimates of abundance collected by observers are thought to be unreliable is that more searching was carried out by helicopters in the later part of the time series. Including information about the methods of searching may help eliminate the contradiction between the observer data and the other data used in the model. Schnute & Hilborn (1993) describe a method that can be used to represent the uncertainty in the conflicting data sets when it is not known which data set is prone to model misspecification. Data quality must always be considered as a source of uncertainty. It is common for historic data to be of doubtful quality, and the uncertainty this implies can be included in the analysis. For example, incidental dolphin mortality before 1973 is very uncertain, with some correlation in the uncertainty between years, but was treated as

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accurate in this analysis. Preliminary investigation suggests bias in early incidental mortality, with implications for depletion and recovery estimates. Further analyses could investigate this uncertainty both by further investigation of the data themselves, and by estimating the value of a bias parameter, since there is unused information about early catches in the age structure and stage structure data. This method assumes that the uncertainty in historic catch–mortality is perfectly correlated among years. Alternatively, constant mortality rate (proportion of the population killed per year) could be assumed from 1959 to 1972; or the analysis could be started in 1973 at an exploited population size (e.g. Maunder & Starr, 2001). Other approaches that could be used to include this uncertainty include: (1) the method of Fournier et al. (1998) that fits to the catch data conditioned on effort rather than assuming it is known; or (2) sample the catch from the appropriate distribution each time the objective function is evaluated in the Bayesian analysis (e.g. Wade et al., 2002). The correlation in errors among years, due to methods used to estimate the catch, should be taken into consideration as much as possible, since correlation will increase bias. There may also be data quality issues with the age structure and color phase data, due to, for example: (1) changes through time in sampling methods, and (2) biases associated with higher probability of setting on larger groups of dolphins. Further analyses in these areas are planned. Prior distributions are always informative in some sense, which may be important if there are few data about the parameter. This fact is illustrated by differences between this analysis and previous analyses (Wade, 1994; Wade et al., 2002). The prior implemented for adult natural mortality differed from Wade et al.’s (2002) 0.009±0.02 (SE) bounded at 0.002, which implied (at the mode) that 76% of dolphins reaching adult mortality rates would survive to age 40 and die through senescence. Wade et al. (2002) sought a uniform prior on rmax (a derived parameter), implied by the combined priors on their estimated parameters. For the current model such low natural mortality and high senescence in a wild population were not considered the prior "belief" about parameter distribution, particularly given the influence of a fairly tight distribution on results. Adult natural mortality is often the parameter with most influence on population growth rate for long–lived animals (Heppell et al., 2000). The posterior mode of the natural mortality estimate from the current analysis was 0.039, and the 95% credibility intervals did not include 0.009. The prior for maximum fecundity at low population sizes also differed from Wade et al.’s (2002) uniform prior with observed fecundity (Myrick et al., 1986) of 0.167 as a lower bound and 0.333 as an upper bound. Density–dependence occurs at high population sizes in cetacean populations (Taylor & DeMaster, 1993), the population was

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depleted when pregnancy rates were observed, and pregnancy rate must be higher than fecundity rate, so 0.167 may well be close to or higher than fmax. Recent analyses that consider pregnancy completion rate suggest that realized fecundity rate during this observation period was considerably less (Susan Chivers, NMFS, pers. comm.). The posterior estimate from the current analysis was significantly below 0.167, but was lowered by some Neonate natural mortality and parameterization of survival constant at age. Truncation of fmax at 0.167 would be informative, since it permits no values below 0.167 in the posterior. This issue, and the related issues of (1) the causes of pregnancy failure, and (2) age, spatial, and school size effects on fecundity, could be investigated further by integrating the pregnancy observation data into an expanded analysis. Catastrophes and environmental transition shifts (Fiedler & Reilly, 1994; Reilly & Fiedler, 1994) were not modeled, either for parameter estimation or for forward projection. Such discontinuities magnify the number of possible fits to the data and make parameters difficult to estimate, and it is not easy to determine the risk of catastrophic events affecting the population in the future (but see Gerber & Hilborn, 2001). For the primary objective of this model —examining how the tuna fishery affects dolphins— these considerations are not directly relevant, as they might be for a population viability analysis. Models with vulnerability varying among color phases had more support than those without variation, and the best model included lower vulnerabilities for stages 1, 3, and 4 (color phases Neonate, Speckled, and Mottled). This difference in vulnerability explained the observed pattern of relatively fewer dolphins sampled at age 0 and between the ages of 5 and 15. Such a dip in catchability could result from a difference in behavior between immature and mature individuals, such as the formation of immature schools that for some reason (perhaps school size, Perkins & Edwards, 1999) are less likely to be targeted by purse seiners; immatures swimming on the edge of the school, where they are less likely to be encircled by nets; or immatures being more likely to split off from the school during the chase (Barlow & Hohn, 1984). There is some evidence for segregation into juvenile schools among Spotted Dolphins (Kasuya et al., 1974; Hohn & Scott, 1983), and for other species in the genus Stenella (Myazaki & Nishiwaki, 1978; Chivers & Hohn, 1985; Perryman & Lynn, 1994). Alternative explanations for a difference in vulnerability are also possible (Barlow & Hohn, 1984), but have not been included in the model. For example, this life stage may tend to lay down additional non– annual growth layer groups. This would result in fewer individuals being captured with each number of growth rings, and would also imply overestimation of age for older dolphins.

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Further possible changes to model structure include modeling the sexes separately, since the age structure data also include sex information and there may be differences in vital rates. It may be useful to consider bias and uncertainty in ageing, which are likely to be significant, given the quality of the growth layer groups used to age Spotted Dolphins. The age of the oldest age classes may have been underestimated (Susan Chivers, NMFS, pers. comm.). Assuming that ages were underestimated would result in lower estimates of natural mortality, and perhaps lower productivity. In summary, the methods presented here are very flexible and generally applicable to wide variety of taxa and problems, and easily extended to comparing management options and predicting future consequences. Uncertainty is an integral part of the analysis, and prior knowledge and model assumptions are handled consistently throughout. Perhaps the most important aspect of integrated analysis is the way it both enables and forces consideration of the system as a whole, so that inconsistencies can be observed and resolved. Acknowledgements This project was funded by Cooperative Agreement NA17RJ1230 between the Joint Institute for Marine and Atmospheric Research (JIMAR) and the National Ocean and Atmospheric Administration (NOAA). The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA or any of its subdivisions. The U.S. National Marine Fisheries Service provided data. Jay Barlow, Susan Chivers, Tim Gerrodette, and Steve Reilly of the NMFS helped with data and advice. Thanks to Cleridy Lennert–Cody and Michael Scott for help with the data, and Robin Allen, Bill Bayliff, Rick Deriso, Shelton Harley, Cleridy Lennert–Cody, Michael Scott, and three anonymous reviewers for reviewing the manuscript. References Archer, F. & Chivers, S. J., 2002. Age structure of the northeastern spotted dolphin incidental kill by year for 1971 to 1990 and 1996 to 2000. Administrative Report No. LJ–02–12, NOAA Fisheries, Southwest Fisheries Science Center, La Jolla. Barbosa, A., 2001. Hunting impact on waders in Spain: effects of species protection measures. Biodiversity and Conservation, 10: 1703–1709. Barlow, J. & Boveng, P., 1991. Modeling age– specific mortality for marine mammal populations. Marine Mammal Science, 7: 50–65. Barlow, J. & Hohn, A. A., 1984. Interpreting spotted dolphin age distributions. NOAA Technical Memorandum, NMFS–SWFC–48. Besbeas, P., Freeman, S. N., Morgan, B. J. T. & Catchpole, E. A., 2002. Integrating mark–recap-

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ture–recovery and census data to estimate animal abundance and demographic parameters. Biometrics, 58: 540–547. Breen, P. A., Hilborn, R., Maunder, M. N. & Kim, S. W., 2003. Effects of alternative control rules on the conflict between a fishery and a threatened sea lion (Phocarctos hookeri). Canadian Journal of Fisheries and Aquatic Sciences, 60: 527–541. Breiwick, J. M., Eberhardt, L. L. & Braham, H. W., 1984. Population dynamics of western Arctic bowhead whales (Balaena mysticetus). Canadian Journal of Fisheries and Aquatic Sciences, 41: 484–496. Brooks, S. P., Catchpole, E. A., Morgan, B. J. T. & Harris, M. P., 2002. Bayesian methods for analyzing ringing data. Journal of Applied Statistics, 29: 187–206. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference: a practical information– theoretic approach. Springer–Verlag, New York. Butterworth, D. S., Ianelli, J. N. & Hilborn, R., 2003. A statistical model for stock assessment of southern bluefin tuna with temporal changes in selectivity. African Journal of Marine Science, 25: 331–361. Caswell, H., 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates, Inc. Caswell, J. H., Afton, A. D. & Caswell, F. D., 2003. Vulnerability of nontarget goose species to hunting with electronic snow goose calls. Wildlife Society Bulletin, 31: 1117–1125. Chivers, S. J. & Hohn, A. A., 1985. Segregation based on maturity state and sex in schools of spinner dolphins in the eastern Pacific. Abstract, Sixth Biennial Conference on the Biology of Marine Mammals, Vancouver. Davidson, R. S. & Armstrong, D. P., 2002. Estimating impacts of poison operations on non–target species using mark–recapture analysis and simulation modelling: an example with saddlebacks. Biological Conservation, 105: 375–381. Edwards, A., 1972. Likelihood, Cambridge Univ. Press, Cambridge. Fiedler, P. C. & Reilly, S. B., 1994. Interannual Variability of Dolphin Habitats in the Eastern Tropical Pacific .2. Effects on Abundances Estimated from Tuna Vessel Sightings, 1975–1990. Fishery Bulletin, 92: 451–463. Fournier, D. & Archibald, C. P., 1982. A General– Theory for Analyzing Catch at Age Data. Canadian Journal of Fisheries and Aquatic Sciences, 39: 1195–1207. Fournier, D. A., Hampton, J. & Sibert, J. R., 1998. MULTIFAN–CL: a length–based, age–structured model for fisheries stock assessment, with application to South Pacific albacore, Thunnus alalunga. Canadian Journal of Fisheries and Aquatic Science, 55: 2105–2116. Gerber, L. R. & Hilborn, R., 2001. Estimating the frequency of catastrophic events and recovery from low densities: examples from populations

264

of otariids. Mammal Review, 31: 131–150. Gerrodette, T., 1999. Preliminary estimates of 1998 abundance of four dolphin stocks in the eastern tropical Pacific. Southwest Fisheries Science Center Administrative Report, LJ–99–04. – 2000. Preliminary estimates of 1999 abundance of four dolphin sticks in the eastern tropical Pacific. Southwest Fisheries Science Center Administrative Report, LJ–00–12. Gerrodette, T. & Forcada, J., 2002. Estimates of abundance of northeastern offshore spotted, coastal spotted, and eastern spinner dolphins in the eastern tropical Pacific Ocean. Administrative Report No. LJ–02–06. NOAA Fisheries, Southwest Fisheries Science Center, La Jolla. Geweke, J., 1992. Evaluating the accuracy of sampling–based approaches to calculating posterior moments. In Bayesian Statistics 4: 169– 193 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds.). Oxford Univ. Press, Oxford. Greiwank, A. & Corliss, G., Eds., 1991. Automatic Differentiation of Algorithms: Theory, Implementation and Application. SIAM, Philadelphia. Hall, M. A., 1997. Dolphin and other bycatch in the eastern Pacific Ocean tuna purse seine fishery. In: Fisheries Bycatch: Consequences and Management: 35–38. Alaska Sea Grant College Program Report No. AK–SG–97–02, Univ. of Alaska, Fairbanks. – 1998. An ecological view of the tuna–dolphin problem: impacts and trade–off. Reviews in Fish Biology and Fisheries, 8: 1–34. Hall, M. A., Alverson, D. L. & Metuzals, K. I., 2000. By–catch: problems and solutions. Marine Pollution Bulletin, 41: 204–219. Hampton, J. & Fournier, D. A., 2001. A spatially disaggregated, length–based, age–structured population model of yellowfin tuna (Thunnus albacares) in the western and central Pacific Ocean. Marine and Freshwater Research, 52: 937–963. Hastings, W. K., 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57: 97–109. Heidelberger, P. & Welch, P., 1983. Simulation run length control in the presence of an initial transient. Operations Research, 31: 1109–1144. Heppell, S. S., Caswell, H. & Crowder, L. B., 2000. Life histories and elasticity patterns: Perturbation analysis for species with minimal demographic data. Ecology, 81: 654–665. Hilborn, R. & Liermann, M., 1998. Standing on the shoulders of giants: learning from experience in fisheries. Reviews in Fish Biology and Fisheries, 8: 273–283. Hilborn, R. & Walters, C. J., 1992. Quantitative Fisheries Stock Assessment: Choice Dynamics and Uncertainty. Chapman and Hall, New York. Hoeting, J. A., Madigan, D., Raftery, A. E. & Volinsky, C. T., 1999. Bayesian model averaging. Statistical Science, 14: 382–417. Hohn, A. A. & Scott, M. D., 1983. Segregation by age

Hoyle & Maunder

in schools of spotted dolphins in the eastern tropical Pacific. Abstract, Fifth Biennial Conference on the Biology of Marine Mammals, Boston. Inter–American Tropical Tuna Commission, 1994. Letter to Dr. Michael Tillman, Southwest Fisheries Science Centre Director, September 6, 1994. – 2002. Annual Report of the Inter–American Tropical Tuna Commission: 2001. La Jolla, California. Joseph, J., 1994. The tuna–dolphin controversy in the eastern Pacific Ocean: biological, economic, and political impacts. Ocean Development and International Law, 25: 1–30. Kass, R. E. & Raftery, A. E., 1995. Bayes Factors. Journal of the American Statistical Association, 90: 773–795. Kasuya, T., Miyazaki, N. & Dawbin, W. H., 1974. Growth and reproduction of Stenella attenuata in the Pacific coast of Japan. The Scientific Reports of the Whales Research Institute, 26: 157–226. Lennert–Cody, C. E., Buckland, S. T. & Marques, F. F. C., 2001. Trends in dolphin abundance estimated from fisheries data: a cautionary note. Journal of Cetacean Research and Management, 3: 305–319. Link, W. A., Cam, E., Nichols, J. D. & Cooch, E. G., 2002. Of BUGS and birds: Markov Chain Monte Carlo for hierarchical modeling in wildlife research. Journal of Wildlife Management, 66: 277–291. Maunder M. N., 2001a. A general framework for integrating the standardization of catch–per–unit– of–effort into stock assessment models. Canadian Journal of Fisheries and Aquatic Science, 58: 795–803. – 2001b. Integrated Tagging and Catch–at–Age ANalysis (ITCAAN). In: Spatial Processes and Management of Fish Populations: 123–146 (G. H. Kruse, N. Bez, A. Booth, M. W. Dorn, S. Hills, R. N. Lipcius, D. Pelletier, C. Roy, S. J. Smith & D. Witherell, Eds.). Alaska Sea Grant College Program Report No. AK–SG–01–02, Univ. of Alaska, Fairbanks. – 2003. Paradigm shifts in fisheries stock assessment: from integrated analysis to Bayesian analysis and back again. Natural Resource Modeling, 16: 465–475. – 2004. Population Viability Analysis, Based on Combining, Integrated, Bayesian, and Hierarchical Analyses. Acta Oecologica, 26: 85–94. Maunder, M. N. & Starr, P. J., 2001. Bayesian assessment of the SNA1 snapper (Pagrus auratus) stock on the north–east coast of New Zealand. New Zealand Journal of Marine and Freshwater Research, 35: 87–110. – 2003. Fitting fisheries models to standardised CPUE abundance indices. Fisheries Research, 63: 43–50. Maunder, M. N., Starr, P. J. & Hilborn, R., 2000. A Bayesian analysis to estimate loss in squid catch due to the implementation of a sea lion population management plan. Marine Mammal Science, 16: 413–426. McAllister, M. & Kirchner, C., 2002. Accounting for

Animal Biodiversity and Conservation 27.1 (2004)

structural uncertainty to facilitate precautionary fishery management: illustration with Namibian orange roughy. Bulletin of Marine Science, 70: 499–540. McAllister, M. K. & Kirkwood, G. P., 1998. Bayesian stock assessment and policy evaluation: a review and example application using the logistic model. ICES Journal of Marine Science, 55: 1031–1060. McAllister, M. K., Pikitch, E. K., Punt, A. E. & Hilborn, R., 1994. A Bayesian approach to stock assessment and harvest decisions using the sampling/importance resampling algorithm. Canadian Journal of Fisheries and Aquatic Sciences, 51: 2673–2687. Myazaki, N. & Nishiwaki, M., 1978. School structure of the striped dolphin off the Pacific coast of Japan. Scientific Report of the Whale Research Institute, 30: 65–115. Myrick, A. C., Hohn, A. A., Barlow, J. & Sloan, P. A., 1986. Reproductive–Biology of Female Spotted Dolphins, Stenella–Attenuata, from the Eastern Tropical Pacific. Fishery Bulletin, 84: 247–259. Myrick, A. C. Jr., Hohn, A. A., Sloan, P. A., Kimura, M. & Stanley, D. D., 1983. Estimating age of spotted and spinner dolphins (Stenella attenuata and Stenella longirostris) from teeth. NOAA–Technical Memorandum–NMFS–SWFC–30. National Research Council, 1992. Dolphins and the Tuna Industry. Committee on Reducing Porpoise Mortality from Tuna Fishing. National Academy Press, Washington, DC. Noon, B. R. & McKelvey, K. S., 1996. Management of the spotted owl: A case history in conservation biology. Annual Review of Ecology and Systematics, 27: 135–162. Perkins, P. C. & Edwards, E. F., 1999. Capture rate as a function of school size in pantropical spotted dolphins, Stenella attenuata, in the eastern tropical Pacific Ocean. Fishery Bulletin, 97: 542–554. Perrin, W. F., 1969. Color pattern of the eastern Pacific spotted porpoise Stenella graffmani Lonnberg (Cetacea, Delphinidae). Zoological Journal, New York Zoological Society, 54: 135–149. Perryman, W. L. & Lynn, M. S., 1994. Examination of stock and school structure of striped dolphins (Stenella coeruleoalba) in the eastern Pacific from aerial photogrammetry. Fishery Bulletin, 92: 122–131. Punt, A. E. & Hilborn, R., 1997. Fisheries stock assessment and decision analysis: the Bayesian approach. Reviews in Fish Biology and Fisheries, 7: 35–63. Punt, A. E. & Butterworth, D. S., 1999. On assessment of the Bering–Chukchi–Beaufort Seas stock of bowhead whales (Balaena mysticetus) using a Bayesian approach. Journal of Cetacean Research and Management, 1: 53–71. Raftery, A. E., Givens, G. H. & Zeh, J. H., 1995. Inference from a deterministic population dynamics model for bowhead whales. Journal of the American Statistical Association, 90: 402–416. Raftery, A. L. & Lewis, S., 1992. How many itera-

265

tions in the Gibbs sampler? In: Bayesian Statistics 4: 763–773 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds.). Oxford Univ. Press, Oxford. Reilly, S. B. & Fiedler, P. C., 1994. Interannual Variability of Dolphin Habitats in the Eastern Tropical Pacific .1. Research Vessel Surveys, 1986–1990. Fishery Bulletin, 92: 434–450. Schnute, J. T. & Hilborn, R., 1993. Analysis of contradictory data sources in fish stock assessment. Canadian Journal of Fisheries and Aquatic Sciences, 50: 1916–1923. Siler, W., 1979. A competing–risk model for animal mortality. Ecology, 60: 750–757. Smith, B. J., 2001. Baysian Output Analysis Program (BOA) . Version 1.0.1 for Windows R. URL: http://www.public-health.uiowa.edu/boa/ boa_prog.html Tanner, M. A., 1993. Tools for statistical inference: methods for the exploration of posterior distributions and likelihood functions. Springer–Verlag, New York. Taylor, B. L. & DeMaster, D. P., 1993. Implications of Nonlinear Density–Dependence. Marine Mammal Science, 9: 360–371. Taylor, B. L., Wade, P. R., DeMaster, D. P. & Barlow, J., 2000. Incorporating uncertainty into management models for marine mammals. Conservation Biology, 14: 1243–1252. Taylor, B., Wade, P. R., Stehn, R. & Cochrane, J., 1996. A Bayesian approach to classification criteria for spectacled eiders. Ecological Applications, 6: 1077–1089. Wade, P. R., 1993. Estimation of Historical Population–Size of the Eastern Spinner Dolphin (Stenella–Longirostris–Orientalis). Fishery Bulletin, 91: 775–787. – 1994. Abundance and Population Dynamics of Two Eastern Pacific Dolphins, Stenella attenuata and Stenella longirostris orientalis. Ph. D. Thesis, Univ. of California, San Diego. – 1995. Revised estimates of dolphin kill in the eastern tropical Pacific, 1959–1972. Fishery Bulletin, 93: 345–354. – 1999. A comparison of statistical methods for fitting population models to data. In: Marine mammal survey and assessment methods: 249– 270 (A. A. Balkema, Ed.). Proceedings of the symposium on marine mammal survey and assessment methods, Seattle, Washington, U.S.A. – 2002. A Bayesian stock assessment of the eastern Pacific gray whale using abundance and harvest data from 1967–1996. Journal of cetacean research and management, 4: 85–98. Wade, P. R. & Gerrodette, T., 1992. Estimates of dolphin abundance in the eastern tropical Pacific: Preliminary analysis of five years of data (IWC SC/43/SM–13). Report of the International Whaling Commission, 42: 533–539. – 1993. Estimates of cetacean abundance and distribution in the eastern tropical Pacific. Report of the International Whaling Commission, 43: 477–493.

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Wade, P. R., Reilly, S. B. & Gerrodette, T., 2002. Assessment of the population dynamics of the northeastern offshore spotted and the eastern spinner dolphin populations through 2002. Administrative Report No. LJ–02–13, NOAA Fisheries, Southwest Fisheries Science Center, La Jolla.

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White, G. C. & Lubow, B. C., 2002. Fitting population models to multiple sources of observed data. Journal of Wildlife Management, 66: 300–309. Wilson, D. L., 1993. A comparison of methods for estimating mortality parameters from survival data. Mechanisms of Ageing and Development, 66: 269–281.

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Application of integrated Bayesian modeling and Markov chain Monte Carlo methods to the conservation of a harvested species C. J. Fonnesbeck & M. J. Conroy

Fonnesbeck, C. J. & Conroy, M. J., 2004. Application of integrated Bayesian modeling and Markov chain Monte Carlo methods to the conservation of a harvested species. Animal Biodiversity and Conservation, 27.1: 267–281. Abstract Application of integrated Bayesian modeling and Markov chain Monte Carlo methods to the conservation of a harvested species.— When endeavoring to make informed decisions, conservation biologists must frequently contend with disparate sources of data and competing hypotheses about the likely impacts of proposed decisions on the resource status. Frequently, statistical analyses, modeling (e.g., for population projection) and optimization or simulation are conducted as separate exercises. For example, a population model might be constructed, whose parameters are then estimated from data (e.g., ringing studies, population surveys). This model might then be used to predict future population states, from current population estimates, under a particular management regime. Finally, the parameterized model might also be used to evaluate alternative candidate management decisions, via simulation, optimization, or both. This approach, while effective, does not take full advantage of the integration of data and model components for prediction and updating; we propose a hierarchical Bayesian context for this integration. In the case of American black ducks (Anas rubripes), managers are simultaneously faced with trying to extract a sustainable harvest from the species, while maintaining individual stocks above acceptable thresholds. The problem is complicated by spatial heterogeneity in the growth rates and carrying capacity of black ducks stocks, movement between stocks, regional differences in the intensity of harvest pressure, and heterogeneity in the degree of competition from a close congener, mallards (Anas platyrynchos) among stocks. We have constructed a population life cycle model that takes these components into account and simultaneously performs parameter estimation and population prediction in a Bayesian framework. Ringing data are used to develop posterior predictive distributions for harvest mortality rates, given as input decisions about harvest regulations. Population surveys of black ducks and mallards are used to obtain stock–specific estimates of population size for both species, for inputs into the population life–cycle model. These estimates are combined with the posterior distributions for harvest mortality, to obtain posterior predictive distributions of future population status for candidate sets of regional harvest regulations, under alternative biological hypotheses for black duck population dynamics. These distributions might then be used for both the exploration of optimal harvest policies and for sequential updating of model posteriors, via comparison of predictive distributions to future survey estimates of stock–specific abundance. Our approach illustrates advantages of MCMC for integrating disparate data sources into a common predictive framework, for use in conservation decision making. Key words: Bayesian analysis, Integrated model, Hierarchical model, Harvest, MCMC, Waterfowl. Resumen Aplicación de la modelación integrada bayesiana y de los métodos Monte Carlo basados en cadenas de Markov para la conservación de una especie recolectada.— En el momento de tomar decisiones bien fundamentadas, es habitual que los biólogos conservacionistas deban enfrentarse a fuentes de datos dispares e hipótesis alternativas acerca de los impactos probables que tendrán las decisiones propuestas en el estado del recurso. A menudo, tanto los análisis estadísticos, como la modelación (para la proyección poblacional, por ejemplo) y la optimización o simulación, se llevan a cabo como ejercicios independientes. ISSN: 1578–665X

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Así, es posible que se construya un modelo poblacional, cuyos parámetros se estimen a partir de datos (como estudios de anillamiento y estudios poblacionales). Posteriormente, cabe la posibilidad de que este mismo modelo se emplee para predecir situaciones demográficas futuras a partir de las estimaciones de población actuales, utilizando para ello un sistema de gestión determinado. Por último, el modelo parametrizado también puede emplearse para evaluar posibles decisiones de gestión alternativas, a través de la simulación, la optimización, o ambos procedimientos. Si bien este enfoque resulta eficaz, no aprovecha al máximo la integración de datos y los componentes de los modelos para la predicción y actualización. En este estudio proponemos un contexto bayesiano jerárquico que permite efectuar dicha integración. En el caso del ánade sombrío americano (Anas rubripes), los gestores deben enfrentarse a la labor de intentar extraer una recolección sostenible de la especie, al tiempo que mantienen los stocks de individuos por encima de umbrales aceptables. El problema se ve agravado por la heterogeneidad espacial que presentan las tasas de crecimiento y la carga cinegética de los stocks de ánades sombríos, el movimiento entre los stocks, las diferencias regionales en la intensidad de la presión recolectora y la heterogeneidad en el grado de competencia por parte de un congénere cercano —el ánade real (Anas platyrynchos)— entre los stocks. Hemos formulado un modelo del ciclo vital de la población que toma en consideración estos componentes, al tiempo que permite llevar a cabo una estimación de los parámetros y una predicción de la población en un marco bayesiano. Los datos de anillamiento se emplean para desarrollar distribuciones predictivas posteriores para las tasas de mortalidad durante la recolección, expresadas como decisiones de entrada acerca de la normativa sobre recolecciones. Los estudios poblacionales del ánade sombrío y del ánade real se emplean para obtener estimaciones sobre el tamaño poblacional específicas de los stocks de ambas especies, que se emplearán como entradas para el modelo del ciclo vital de la población. Dichas estimaciones se combinan con las distribuciones posteriores para la mortalidad durante la recolección, con el propósito de obtener distribuciones predictivas posteriores de la situación demográfica futura para posibles conjuntos de normativas regionales acerca de la recolección, de acuerdo con hipótesis biológicas alternativas relativas a la dinámica poblacional del ánade sombrío. En una fase posterior, tales distribuciones pueden utilizarse tanto para la investigación de políticas óptimas en materia de recolección, como para la actualización secuencial de distribuciones posteriores del modelo mediante la comparación de distribuciones predictivas para estimaciones en estudios futuros acerca de la abundancia poblacional presente de forma específica en los stocks. Nuestro enfoque ilustra las ventajas que presentan las técnicas de Montecarlo basadas en cadenas de Markov (MCMC) para integrar fuentes de datos dispares en un marco predictivo común, con vistas a su utilización en la toma de decisiones sobre conservación. Palabras clave: Análisis bayesiano, Modelo integrado, Modelo jerárquico, Recolección, MCMC, Aves acuáticas. Christopher J. Fonnesbeck & Michael J. Conroy, Cooperative Fish and Wildlife Research Unit, D.B. Warnell School of Forest Resources, Univ. of Georgia, Athens, GA 30602 U.S.A.

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Introduction Dynamic models frequently are used in conservation biology to aid in the evaluation of alternative conservation decisions, with respect to obtaining some desired outcome. Such approaches ordinarily employ, whether explicitly or not, several elements: First, a process model is used to describe how observed system states, such as population size and composition, change through time. Second, system states and relevant parameters such as survival and recruitment rates must be estimated using data, via one or more statistical models. Third, one or more control variables must be defined; these are thought to influence the system in such a way as to lead to gains in the management objective. Examples of decisions that are frequently made, and to which decision modeling may be applied, include determination of optimal harvest regulations, setting of forest cutting policies, the timing and intensity of restocking efforts as components of endangered species conservation, and decisions about land acquisition for conservation. Fourth, we need an explicit way of describing the relative value or utility of potential management outcomes. That is, either explicitly or implicitly, there is some overarching resource goal and quantifiable resource objective in any decision–making process. For harvest decisions the objective typically is the maximization of long–term harvest yield; for forest cutting the objective may be the gain of revenue, perhaps subject to constraints on the avoidance of loss of biodiversity; for restocking efforts, perhaps the maximization of the expected time to extinction for some species; for land acquisition, perhaps the maximization of biodiversity conservation under budgetary constraints. Finally, we need a procedure that seeks some optimum combination of decisions and system conditions. Formal procedures exist for all these elements; in ecology, there is a particularly rich literature focused on dynamic modeling and statistical estimation methods, much of which is summarized in Williams et al. (2002). Likewise, there exists an extensive literature on decision theory and, dynamic optimization methods and optimal control theory, much of it also summarized by Williams et al. (2002). However, in our experience, process modeling, statistical estimation, and decision analysis are often considered as distinct enterprises. Thus, statistical models are frequently used to estimate population states and other parameters; these results may subsequently be incorporated into an existing or newly constructed population model. The parameterized model then may be applied to a decision problem, for instance, by exploratory simulation or formal optimization procedures. Although the sequence of events differs from cases to case (e.g., model constructed first, followed by parameter estimation and optimization), the idea is the same: component elements are treated separately, and integration (to the extent it occurs at all) is usually post–hoc, and often ad–hoc. Philosophically, there are close link-

ages among these elements; pragmatically, there are also strong arguments for integration. For instance, dynamic optimization models include both state dynamics and an objective function. In turn, data are required to estimate system states and state dynamics, and to assess model comportment to reality, which in turn should influence decision making. Because these elements typically depend upon a common data structure, and involve modeling, there is practical motivation for an integrated approach that leverages shared information. In this paper, we first construct a conceptual framework for integrating process modeling, parameter estimation, and model prediction, based on principles of conditional hierarchical modeling. We then demonstrate the approach using a real decision problem, involving optimal harvest management of multiple stocks of American black ducks. Methods A generic decision model We clarify these concepts by means of a generic decision model. To begin with, consider a dynamic system in which the state Xt (possibly vector– valued) evolves through time according to a specified process model f (Xt*Zt), which includes both the endogenous effect of the state, as exogenous factors Zt, such as weather; the latter are frequently modeled as random variables (fig. 1). Add to this model inputs from the vector of decisions dt, which potentially affect both transitions to future states and the utility gained from present and future states (possibly altered under management). Before proceeding further, we wish to use this generic model to reinforce our earlier points. First, figure 2 makes clear that the modeling of decision influences (both on the system itself and our objective gain from the system) are inextricably linked to the process model. In addition, any algorithm that seeks to find decisions which optimize the objective functional is constrained by system dynamics. That is, it is impossible to obtain the maximum of a dynamic decision problem without taking into account system dynamics. Finally, dynamic decisions are often subject to a finite time horizon. In natural resource management it is usually appropriate that this horizon is relatively distant. Decisions are made, usually with feedback from the current system state, in order to seek an optimal result over an appropriately long time horizon. Although simple conceptually, this integration of dynamic modeling with decision making is complex in practice. Further, decisions ordinarily cannot be based directly on the system states and a model, but rather on statistical estimates of the states, and of the parameters of the model. Thus, the actual system state being modeled evolves through time according to some (assumed) model. The observed system state is related to the actual system according to a statis-

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Random effects Z(t+1)

Random effects Z(t)

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Fig. 1. Generical model of a dynamic process with random effects. Fig. 1. Modelo genérico de un proceso dinámico con efectos aleatorios.

Random effects Z(t)

Action d(t)

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Action d(t+1)

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State X(t)

Reward u[d(t+1),X(t+1)]

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Fig. 2. Incorporation of decision variables and objective function into the stochastic process, showing feedback of future state upon the objective. Fig. 2. Incorporación al proceso estocástico de variables de decisiones y de la función objetiva, indicando el feedback del estado futuro con respecto al objetivo.

tical sampling model. Parameters of the process and decision models are now themselves based on estimates from one or more statistical sampling models, providing a prediction given the current (observed) state (fig. 3): (1) Predictions about the future state of the system now inherit statistical uncertainties due these statistical models, as do predictions about utility under each candidate management decision. The development so far has assumed that the mathematical form and parametric structure of the process model are known. Usually this will not be

the case, therefore it will be important to consider alternative process models. In the context of decision making, these alternative models become important to the extent that the utility of decisions is dependent on belief in the alternate models. Suppose we entertain a single alternate model, denoted as Model 2 (fig. 4). Identical observed system states and candidate decisions induce two sets of values for the predicted utility and predicted system state, one set under each model. Given these alternate predictions, the decision maker must first reconcile the fact that different models may lead to different utility for each decision; thus, the optimal decision may be different for each model. One approach is to form an

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Actual Reward u[d(t),X(t)]

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Observed state Y(t+1) Estimated state Y(t+1)

Estimated reward u[d(t),X(t)]

Predicted

Fig. 3. Relationship between actual and observed/estimated processes in a dyanmic system. Fig. 3. Relación entre los procesos reales y los observados/estimados en un sistema dinámico.

expectation of utility across the models, which in turn depends on model probabilities quantifying relative belief in each model. Initially, model probabilities may be impartial, attributing equal weight to each candidate model. Finally, following decision making, monitoring data are used to compare the predictions under each model for observed future states (fig. 3). These predictions and observations have at least two potential uses: (1) they provide an obvious means of evaluating the degree to which any of the predictive models performs (i.e., validation), as well as a means of discriminating among competing models. As we will describe, this leads to a natural way to update relative belief in the alternate models, using Bayes factors; (2) as noted above, optimal decisions and their utility will generally differ among alternate process models. Thus, to the degree that uncertainty exists with respect to which model best describes and predicts the process, prescribed decisions will be suboptimal; conversely, reduction of process uncertainty through time will result in improved decision making at future decision–making epochs.

Sequential conditioning as a tool for integration Our approach to this problem exploits well–known and related principles of probability, Bayesian inference and conditional modeling. In Bayesian inference, parameter values and observations Y are both modeled with probability density functions, so it makes sense to consider their joint probability Pr ( , Y) (2) Because for any sample outcome the probability of the data is a constant, Bayes’ theorem states that the posterior (that is, following sampling) probability of is proportional to the product of the sampling distribution of the data assuming and the prior (unconditional) probability of : (3) This formulation is readily generalized to alternative model forms. Conditional modeling has increasingly been recognized as a powerful tool for

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Model 1 Predicted state X1(t+1) Action d(t)

Predicted reward u1[d(t+1),X(t+1)]

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Model 2

Observed state Y(t+1) Observed reward u[d(t+1),X(t+1)]

Predicted state X2(t+1)

Predicted reward u2[d(t+1),X(t+1)]

Fig. 4. Alternative process models and predictions. Following decision making, monitoring data are obtained for comparison of the predictions under each model to observed future states. These predictions may then be used to compute Bayes factors for updating the relative belief in the alternate models, for use in the next decision–making epoch. Fig. 4. Modelos de procesos alternativos y predicciones. Tras la toma de decisiones, se obtienen datos de control para comparar las predicciones facilitadas por cada modelo con los estados futuros observados. Estas predicciones pueden utilizarse para calcular factores de Bayes y actualizar la creencia relativa en los modelos alternativos, con vistas a su utilización en el siguiente período de toma de decisiones.

modeling complex ecological relationships by decomposition as simpler element that are related in a conditional, often hierarchical, manner (e.g., Wikle, 2003). In addition, hierarchical processes naturally lend themselves to conditional sequencing. These ideas can be effectively combined to solve our generic decision problem in an adaptive, sequential updating model, by expressing the problem as:

this sequence is the probability of model M, and finally the likelihood of the data, given the model, parameter values, and predictions. We use Markov chain Monte Carlo (MCMC) methods to sample from the conditional posterior distributions of the quantities we seek (e.g., models, predictions). Although not explicitly explored in this paper, conditional decomposition is readily extended to decision making problems, by incorporating a utility function.

(4) where M represents model structure, are parameter values, is the predicted and Y the observed system state. By sequential conditioning we can see that: (5) This decomposition provides a natural, sequential way of dealing with the complexity of simultaneously modeling uncertainty in data, models, and processes. Beginning with the right–most term in (5), conditioned on a distribution of parameter values Pr( ), we have predicted values . Next in

Integrated modeling with Markov chain Monte Carlo methods Markov chain Monte Carlo is a class of general simulation techniques used primarily to solve problems of Bayesian inference (Gamerman, 1997). Specifically, these methods are used to generate samples that are distributed according to some target posterior form ( ) without having to directly sample from the posterior itself. This is most useful when ( ) is extremely complex, or otherwise difficult to analyze. For example, the joint posterior characterizing our problem in (5) can be analytical intractable for many biological problems

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All MCMC algorithms produce samples from a set of densities (hence, Monte Carlo simulation); these distributions are derived from ( ), according to the conditional probability of each i. The current set of parameter values is dependent on the previous values , thereby generating a Markov chain. The Markov chain is constructed in such a way that its limiting distribution, ( ), is the distribution of interest. The Metropolis–Hastings algorithm (Metropolis et al., 1953; Hastings, 1970) is the most general MCMC procedure, and therefore the most widely applicable. The Metropolis–Hastings algorithm estimates the posterior density using a form of rejection sampling (see Robert & Casella, 1999). The proposal function q( * ) generates candidate values for ( ) which are accepted or rejected according to each value’s probability under the target . Provided that the full support of ( ) may be sampled, the choice of q can otherwise be arbitrary. In any case, the Metropolis–Hastings algorithm estimates the posterior form directly from a subset of filtered samples, rather than relying on Bayes’ rule and conjugate distributions. Case study: integrated estimation and prediction for American black ducks To illustrate our approach we use an example of an adaptive decision–making model for American black ducks. The black duck problem involves a process of observation, predictive modeling, and optimization, with the following major elements: (1) historical data have been used to fit empirical relationships between population parameters and key hypothesized factors, under alternative models; (2) a population projection model incorporates key relationships into a discrete– time projection model; (3) population surveys are used each year to infer the state of the system; (4) based on the surveys, parameter estimates, and the projection model, a forecast of system state and of expected utility is obtained; (5) the forecast of system state is compared to observed system state at the next time epoch in order to evaluate the relative predictive ability of the alternative models and compute relative weights for each model; and (6) the parameter estimates, models, and model weights are used in dynamic optimization procedures (e.g. Lubow, 1995) to obtain optimal, state–specific decisions for maximizing expected utility. The models and data structures used are an extension of those described in Conroy et al. (2002) to multiple stocks, and is illustrated in schematic form in figure 5. The elements of this model are summarized as follows: (1) the observed system state is the number of black ducks (Ni) and mallards (Mi) in each of three geographic strata (western, central, and eastern portions of the range in Canada), as estimated from surveys conducted by the Canadian Wildlife Service; (2) historical data involving band recoveries, hunter surveys of wings collected from shot ducks, and population surveys, are used to estimate the relationship be-

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tween stratum–specific fall age ratios (Ai) and black duck and mallard abundance, under alternative models of density–dependence and density– dependence with competition (Zimpfer, 2004; Conroy et al., 2002); (3) historical data involving band recoveries and population surveys are used to estimate the relationship between non–harvest survival (Sbj) and black duck abundance, under alternative models of density–dependent (compensatory) and density–independent (additive) mortality (Conroy et al., 2002); (4) the fall age ratios calculated above, together with spring–to–summer survival(Sb, assumed constant) are used to project fall abundance in each breeding area prior to migration (Fnj); (5) historical data involving band releases stratified by breeding and harvest areas, and recoveries and recaptures stratified by harvest and breeding areas, are used to estimate rates of movement from breeding to harvest areas ( ij), and of return (fidelity) to breeding areas ( ji; Zimpfer, 2004); (6) historical data involving band recoveries, harvest regulations, and hunter numbers stratified by harvest regions, are used to develop predictions of harvest rates conditional on harvest regulations and hunter numbers. For the purposes of this study, we assumed fixed values for the parameter estimates of non– harvest survival (Sa ,Sb) and movement ( ij, ji) rates, and perfect ability to control harvest rates at specified values via harvest regulations. We further conditioned on the observed system state Xt = [N1, N2, N3, M1, M2, M3]t. Thus, focus is on uncertainty in stratum–specific estimates of age ratio, which, in turn, induce uncertainty in the projection model, from two sources: statistical uncertainty in the parameter values, conditioned on an assumed model structure Pr( *M), and uncertainty in model structure Pr(M). Predictions under these alternative, estimated models are then compared to observations of the consequent states of system Xt. Stratum–specific age ratios are estimated under a joint likelihood of wing survey and band recovery data (fig. 6). This consists of the following components:

(6)

where Wy is the number of juvenile wings in the harvest survey for a specific reproduction area, W is total (adult + juvenile) wings, {mi, Ri; i = y, a} are number of recoveries and bands, respectively, for juveniles and adults, {hi; i = y, a} are band recovery rates, is relative vulnerability of adult to juvenile harvest, and is proportion of young in the harvest (adjusted from population age ratio A by . These likelihoods are, in turn, embedded in a model describing the relationship between age ratio and black duck and mallard abundance:

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S(a)

Ai = exp (

Fi(n) = NiS(1)(1 + Ai) S(b)

Ni +

Mi)

Fj(s)

Nj(w)

Fig. 5. Flow diagram of black duck state dynamics model. Stratum–specific numbers of black ducks (Ni, i = 1,2,3) survive from spring to fall at the constant rate Sa. Stratum–specific black duck and mallard (Mi) abundance influence fall age ratios Ai according to a production model (see (6) for details; age ratios are applied to the surviving adult population to determine fall abundance prior to migration and harvest (F(s)i). Surviving birds migrate to southern harvest/wintering areas j = 1,2,3 according to fixed rates ij, with post–harvest (Hi) abundance leaving Nj, j = 1,2,3 in each wintering area. Survival over winter (Sbj) is determined by a density–dependent model as a function of area–specific abundance N(w)j, with fidelity to breeding areas at a fixed rate ij. In present study movement and survival parameters are fixed, with only the parameters of the reproduction estimation and prediction model (6) estimated from data. Fig. 5. Diagrama de flujo del modelo de la dinámica del estado del ánade sombrío. Los números de ánades sombríos específicos al estrato (Ni, i =1,2,3) sobreviven de primavera a otoño a una razón constante de Sa. La abundancia de ánades sombríos y ánades azulones específica al estrato (Mi) influye en las tasas de edad durante el otoño Ai según un modelo de producción (para detalles, ver (6); las tasas de edad se aplican a la población adulta superviviente para determinar la abundancia otoñal antes de la migración y la recolección (F(s)i). Las aves supervivientes migran hacia áreas de recolección/hibernación meridionales j = 1,2,3 según tasas fijas ij, de manera que la abundancia posterior a la recolección (Hj) deja Nj, j = 1,2,3 en cada área de hibernación. La supervivencia durante el invierno (Sbj) viene determinada por un modelo dependiente de la densidad como una función de la abundancia específica a un área N(w)j, con fidelidad a las áreas de reproducción a una tasa fija ij. En el presente estudio, los parámetros de movimiento y de supervivencia son fijos, de manera que los parámetros del modelo de predicción y de estimación de reproducción (6) son los únicos que se estiman a partir de los datos.

(7) where { i} are coefficients to be estimated. Harvest rates could not be directly estimated during the period of our study, because of problems induced by the conversion to toll–free solicitation of bands (see Conroy et al., 2002). We instead used estimates from hunter surveys in the U.S. and Canada to obtain an estimate of total annual harvest per harvest region Ht, which was assigned to age and sex categories in our model

according to differential vulnerability estimates . These estimates were used together with the projected pre–harvest population in each region (according to our process model) to estimate harvest rates and harvest mortality, with the constraint that the latter could not exceed unity. Conditioned on observed Yt = (Nt, Mt, m gt, m st, Wgt), we used our process model to generate posterior predictions of stratum–specific abundance for black ducks at the next survey period by the relationship

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Wy - Bin( ,W)

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A = exp (

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A A+

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hy = ha Age ratios (production)

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A A+

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N - MN(N,T)

ln(S') = ln(S)+

Survival bias Production bias

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Fig. 6. Model for black duck reproduction estimation and prediction using non–integrated (top) and integrated (bottom) frameworks. Binomial likelihoods (ellipses) are calculated for proportion of juveniles in the harvest ( ) from wing recovery data (W), and for stage–specific harvest rates (h) based on bandings (R) and dead recoveries (X). These are, in turn, used to calculate age ratios (A) that form the basis for the reproduction model, which includes density–dependent terms for black ducks (B) and mallards (M). This, along with a fixed survival function, is then used to predict the next year’s population size. The integrated form is identical to the non–integrated, except that the predicted population is now part of a multivariate log–normal likelihood, as the expected value of the observed population size. Each integrated model incorporates zero or more bias parameters ( ). Fig. 6. Modelo para la predicción y estimación de reproducción del ánade sombrío utilizando marcos no integrados (parte superior) y marcos integrados (parte inferior). Las probabilidades binomiales (elipses) se calculan para la proporción de individuos jóvenes en la recolección ( ) a partir de los datos de recuperación de alas (W), y para las tasas de recolección específicas a una etapa (h) basándose en anillamientos (R) y en la recuperación de aves muertas (X). Tales probabilidades se utilizan, a su vez, para calcular las tasas de edad (A) que forman la base para el modelo de reproducción, incluyendo términos dependientes de la densidad para el ánade sombrío (B) y el ánade real (M). Esto, junto con una función de supervivencia fija, se utiliza despues para predecir el tamaño que tendrá la población el próximo año. La forma integrada es idéntica a la no integrada, salvo que ahora la población prevista forma parte de una probabilidad logarítmica normal multivariante, expresada como el valor previsto del tamaño de la población observada. Cada modelo integrado incorpora cero o más parámetros de sesgo ( ).

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where

(8)

Here, are parameters fixed as constants for this analysis (e.g., movement and survival rates), and f(.) is the functional form specified under our process model. Finally, we modeled observed abundance Nt+1 via a multivariate log–normal distribution, centered at the predicted abundance Ñt+1:

(9) where subscripts are suppressed for notational simplicity. Previous approaches have involved independent estimation of , then using these estimates to predict future states Nt+1. Experience with the black duck and other duck harvest models has uncovered apparent, systematic over–prediction from this approach. An integrated, hierarchical approach, outlined above, endeavors to remediate this problem by establishing feedback between prediction and estimation (fig. 6). In our case, this would be achieved via adjustments of the coefficients of the age ratio process models At = (Nt, Mt). However, it is not known whether over– prediction is due to the reproduction model, or the survival process model (which, for the purposes of this study, we have assumed has fixed coefficients) (Conroy et al., 2002). Indeed, it is possible that neither component induces the bias, but rather, some aspect of the sampling process itself is flawed. Therefore, in addition to the (implicit) adjustment contained in the integrated age ratio process model, we explicitly modeled systematic biases in predicted survival and age ratio:

(10) where St, At are the values of survival and age ratio, respectively, predicted from the model conditioned on current states and data, and s, a are log–scale biasing factors which are estimated; the values S't, A't are then used in prediction. MCMC implementation We implemented the black duck parameter estimation and prediction models using Python (http:// python.org), an open source, object–oriented programming language. Python is a modular development environment, with a wide selection of third– party scientific and numerical tools suitable for biometric applications. We developed a Python module, PyMC (http://pymc.sourceforge.net), that implements an adaptive random walk MetropolisHastings algorithm for MCMC sampling. At each iteration of the algorithm, new parameter values are proposed according to a random walk. The

increment ct is generated using a N(0, ) density. The advantage of the random walk approach is that no problem–specific restrictions regarding the form of the proposal distribution need to be considered. Proposal distributions for Metropolis–Hastings sampling must be enveloping, such that q(x) m (x) . Therefore, most proposal distributions must be chosen manually for each variable in each problem. In contrast, the random walk algorithm functions independently of the form of the target ( ). The disadvantage of the random walk is that when steps are relatively large (i.e., large ), proposed values may frequently fall in the tails of the target distribution, resulting in unacceptably low acceptance rates; similarly, if increments are relatively small, acceptance rates may be very high, but the rate of mixing (exploration of the support of ) will be correspondingly low. Either extreme is inefficient, and therefore undesirable. The adaptive random walk implemented in PyMC addresses this inefficiency by adapting the scale parameter of the proposal distribution according to the recent acceptance probability for each parameter during the simulation. Every k iterations (k = 100 is default), the variance is decreased for acceptance rates below 20%, and increased for those above 50% (arbitrarily chosen), thereby balancing proposal acceptance and mixing. Adaptation occurs in the burn–in phase of the algorithm, and continues until all parameter acceptance rates fall within the aforementioned interval. The joint likelihood for the integrated model included three components related to estimation and prediction. Binomial likelihoods were calculated for the adult and juvenile harvest rates, based on banding and recovery data for these groups. Additionally, the probability of the proposed proportion of young in the harvest was calculated using wing data in a binomial likelihood. As illustrated in (6), is also related to differential vulnerability and age ratio. The final component is the likelihood of the observed population given the predicted value of the model. In each year, the likelihood of the observed population in each area was calculated based on a multivariate log–normal density centered at the array of predicted values. The sum of logarithms for these likelihoods were passed to the Metropolis–Hastings sampler in PyMC after every proposal of a new parameter; this joint log–likelihood was used to either accept or reject the proposed value. Because all parameters are assumed to have equal priors, these cancel out from the ratio of posterior densities used to calculate . The proposal distributions q( (t) i * ) and q( * (t) i ) ]similarly drop out, since the probability of jumping from (t)i to is equal to the reverse jump, under the random walk strategy. Model scenarios A suite of 10 distinct model scenarios was specified. The first was a null model which separately estimated parameters of the age–ratio reproduction function, then used this function to predict area–specific

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black duck populations for 11 years (1991–2001). Four additional scenarios each used the integrated approach described above to simultaneously estimate reproduction model parameters and predict subsequent population size. Each of the integrated models estimated some combination of vital rate bias parameters described in (10): Model 10 assumed reproduction bias, but no survival bias; Model 01 assumed survival bias, but no reproduction bias; Model 11 estimated both bias terms; while Model 00 estimated neither. These five models were replicated under two alternative biological models for black duck reproduction, one incorporating a Mallard competitive effect, the other excluding this effect, for a total of 10 scenarios. The Metropolis–Hastings sampler in PyMC produced predictions and estimates for the model set. A total of 100,000 sampling iterations for each model were executed, with the first 50,000 conservatively discarded as "burn–in" samples, assuming that convergence had been achieved by that stage. Manual inspection of sample traces suggested convergence and adequate mixing of each chain. We compared reproduction model parameter estimates among model scenarios, as well as bias factor estimates, where relevant, using 95% Bayesian credible intervals derived from the posterior distribution of the final 50,000 samples. We also calculated the log–bias of each population i = 1,2,3 in each of t = 1,...,10 years: (11) Model selection was performed using Akaike’s Information Criterion (Burnham & Anderson, 2002; Akaike, 1973, AIC), calculated for each model at each iteration. Burnham & Anderson (2002) illustrate the equivalence of AIC model weights and Bayesian posterior model weights, provided that model priors are equivalent (as we have specified). Use of AIC greatly simplifies model selection in a Bayesian framework relative to other approaches, such as reversible jump MCMC (King & Brooks, 2002). The lack of random effects and the relatively small set of models in this study eliminated the need for procedures that are far more complex to implement. The calculation of AIC at each MCMC iteration yielded a distribution of values, rather than the typical scalar value, which explicitly characterizes parametric uncertainty and its interaction with model selection uncertainty. Results Figure 7 illustrates the systematic over–prediction resulting from separately estimating vital rates, then using those rates in a predictive model. This effect is most severe in the Western and Central populations; moreover, over–prediction is higher in earlier years relative to later. Predictions derived from an integrated framework (fig. 8) are only subtly less positively biased overall. The ad-

dition of a reproduction bias term (fig. 9) or both bias terms (fig. 11) produces a more dramatic reduction in prediction bias, particularly earlier in the time series. Here, the relative over–prediction of the West and Central populations is balanced by under–prediction in the East to achieve relative unbiasedness overall. The addition of the survival bias parameter did not appreciably improve prediction (fig. 10). Calculated AIC–based model weights reinforce the influence of reproduction bias on prediction, as these models account for over 90% of total weight (table 1). Much biological uncertainty also remains in the form of the reproduction function, with neither model dominating the other consistently, with respect to AIC weight. Estimates of reproduction and survival rate bias factors are summarized in table 2. A positive reproduction bias was estimated when survival bias was assumed absent, under Model 10, while a negative survival bias is discovered in the absence of reproduction bias using Model 01 (though 95% credible intervals include zero). Specifying dual bias results in a positive reproduction bias under the no–competition reproduction model (Model 11), virtually no bias when competition is assumed (Model 11c), and a positive survival bias estimate in either case. Complementary to these estimates are those of the reproduction model parameters (table 3). The age ratio reproduction model parameter estimates are strikingly similar among statistical bias models, and between integrated and non–integrated models. All have approximately equivalent intercepts and a pattern of increasing negative density dependence west to east. Stronger differences are evident between biological models, where mallard competition effects are balanced by generally larger intercept values relative to those of the non–competitive models. Discussion The pattern of over–prediction that pervades black duck population models may well be independent of the quality of the model parameter estimators, or the data used by them. A potential explanation for these systematic and unidentified biases in observed population size is a flawed breeding survey (or wintering survey, depending on which model is employed (Conroy et al., 2002)). Some have suggested that the current survey design is inadequate for reliably estimating the breeding population (Bordage, 2000); a non–standard survey over the past decade may in fact be responsible for the particularly acute over–prediction in the early part of the time series (D. Bordage, pers. comm.). This type of bias cannot be accounted for by our integrated models. Assuming, however, that an important component of the existing bias is due to the models or data for estimating vital rates, an integrated framework such as that presented here may prove beneficial.

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log (pred) – log (obs)

1.0

0.5

0.0

–0.5

–0.5 2

log (pred) – log (obs)

Time 1.0

1.0

0.5

0.0

–0.5

–0.5 2

6 Time

Figs. 7, 8. Bias in population predictions from non–integrated model (7) and from integrated model with no bias factors (Model 00) (8), relative to actual population estimates for production with mallard competition (A) and without (B) over 11 years (1991–2001). Plots indicate differences log(predicted) – log(observed) for Western (dashed line), Central (dot–dashed) and Eastern (solid) populations of black ducks. Figs. 7, 8. Sesgos en las predicciones de población a partir del modelo no integrado (7) y a partir del modelo integrado sin factores de sesgo (modelo 00), con relación a las estimaciones de población actuales para la producción con competencia de ánades reales (A) y sin competencia (B) a lo largo de 11 años (1991–2001). Las representaciones gráficas indican diferencias logarítmicas (previstas) – logarítmicas (observadas) para las poblaciones de ánades sombríos del estrato geográfico occidental (línea discontinua), central (línea discontinua punteada) y oriental (línea continua).

Model parameter estimates that were informed by the consequent population prediction resulted in less biased population estimates relative to their non–integrated counterparts, particularly when reproduction bias parameters were specified. Remaining bias showed spatial and temporal patterns of heterogeneity. In particular, estimates were more positively biased in the first half of the time series; where negative biases occurred, they tended to be in the second half. Again, this trend may be due to inconsistencies in survey methodologies over this time period. Spatially, more over–prediction occurred in the Western and Central populations, balanced by relative unbiasedness or even under– prediction in the East. This may also be generally related to survey problems in these areas, because the West and Central regions are characterized by large, unsurveyed areas in Ontario and Quebec, in

contrast to the relatively smaller, well–surveyed Atlantic provinces in the East. The quality of predictions among integrated models were not obviously different according to which vital rate bias parameters were estimated. Each of the integrated models not only represents a different explanation for the source of over–prediction, but also for the factors influencing production in general. Having quantified structural and biological uncertainty, future development could link weighted predictions across models to a dynamic decision optimization procedure, thereby providing a complete decision analysis system to inform management. Though not currently incorporated into the modeling framework presented here, methods of dynamic optimization exist which complement the integrated, stochastic simulation approach outlined thus far (e.g. reinforcement learning).

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log (pred) – log (obs)

A 1.0

1.0

0.5

0.0

–0.5

log (pred) – log (obs)

6 Time

1.0

0.5

0.0

–0.5

–0.5 2

log (pred) – log (obs)

6 Time

1.0

0.5

0.0

–0.5

–0.5 2

6 Time

Figs. 9–11. Bias in population predictions from integrated model with reproduction bias (Model 10) (9), from integrated model with survival bias (Model 01) (10), and from integrated model with both reproduction and survival bias (Model 11) (11), relative to actual population estimates for production with mallard competition (A) and without (B) over 11 years (1991–2001). Plots indicate differences log (predicted) – log (observed) for Western (dashed line), Central (dot–dashed) and Eastern (solid) populations of black ducks. Figs. 9–11. Sesgos en las predicciones de población a partir del modelo integrado con sesgos de reproducción (Modelo 10) (9), a partir del modelo integrado con sesgos de supervivencia (Modelo 01) (10) y a partir del modelo integrado con sesgos de reproducción y sesgos de supervivencia (Modelo 11) (11), con relación a las estimaciones de población actuales para la producción con competencia de ánades reales (A) y sin competencia (B) a lo largo de 11 años (1991–2001). Las representaciones gráficas indican diferencias logarítmicas (previstas) – logarítmicas (observadas) para las poblaciones de ánades sombríos del estrato geográfico occidental (línea discontinua), central (línea discontínua punteada) y oriental (línea continua).

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Table 1. Mean AIC values, along with AIC values and associated model weights of 10 competing models for black duck population dynamics, based on the final 50,000 of 100,000 total MCMC iterations.

Table 2. Production and survival bias parameter estimates (log scale) for each of eight integrated model scenario combinations (95% Bayesian credible intervals in parentheses). Zero values indicate no estimate for given model scenario.

Tabla 1. Valores medios de AIC junto con valores de AIC y pesos de modelos asociados de 10 modelos alternativos para la dinámica poblacional del ánade sombrío, a partir de las 50.000 de un total de 100.000 iteraciones MCMC finales.

Tabla 2. Estimaciones de parámetros de producción y sesgos de supervivencia (escala logarítmica) para cada una de las ocho combinaciones de modelos integrados (ente paréntesis los intervalos bayesianos creíbles al 95%). Los valores cero no indican ninguna estimación para el modelo dado.

Model

AIC

Weight

10c

25827.94

0.00

0.519

Model

25828.59

0.65

0.375

00,00c

25831.61

3.67

0.082

0.701(0.641, 0.762)

0.696(0.638, 0.750)

–0.085(–0.478, 0.250)

Production Bias

Survival Bias

25835.88

7.94

0.010

10c

00c

25836.61

8.67

0.007

01c

25838.70

10.76

0.002

01c

–0.121(–0.473, 0.186)

0.679(0.573, 0.754)

0.051(–0.128, 0.251)

25838.77

10.83

0.002

11c

25839.86

11.92

0.001

11c

Null

25877.29

49.35

0.000

Nullc

25884.37

56.43

0.000

–0.100(–0.211, 0.027) 0.241(–0.298, 0.760)

Table 3. Production model parameter estimates for each model scenario (95% Bayesian credible intervals in parentheses). Spatially–explicit parameters listed on multiple lines for each model, where appropriate: West (top), Central (middle), East (bottom). Tabla 3. Estimaciones de parámetros del modelo de producción para cada modelo (entre paréntesis los intervalos bayesianos creíbles al 95%). Los parámetros espacialmente explícitos se detallan en líneas múltiples para cada modelo: estrato geográfico occidental (línea superior), estrato geográfico central (linea media), estrato geográfico oriental (línea inferior). Model 00

Null

Intercept

Black Duck effect

1.872 (1.402,2.316)

–0.501 (–0.724,–0.256)

1.473 (1.060,1.870)

–0.617 (–0.957,–0.275)

1.627 (1.376,1.871)

–0.877 (–0.993,–0.759)

1.843 (1.379,2.299)

–0.478 (–0.713,–0.238)

1.478 (1.050,1.888)

–0.608 (–0.952,–0.248)

1.640 (1.395,1.886)

–0.879 (–0.987,–0.766)

1.914 (1.367,2.462)

–0.494 (–0.794,–0.185)

1.284 (0.819,1.828)

–0.700 (–1.074,–0.360)

1.893 (1.679,2.095)

–0.928 (–1.022,–0.837)

1.843 (1.272,2.379)

–0.457 (–0.752,–0.142)

1.269 (0.761,1.775)

–0.671 (–1.032,–0.294)

1.805 (1.545,2.056)

–0.883 (–0.985,–0.763)

1.495 (0.955,2.019)

–0.255 (–0.531, 0.038)

1.655 (1.220,2.058)

–0.738 (–1.070,–0.386)

1.679 (1.421,1.938)

–0.902 (–1.013,–0.792)

Mallard effect 0

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Table 3. (Cont.) Model 00c

Intercept 2.727 (2.391,3.063)

Black Duck effect

Mallard effect

–0.794 (–0.887,–0.692)

–0.265 (–0.482, –0.048)

–0.787 (–0.884,–0.693)

–0.253 (–0.478, –0.033)

–0.837 (–0.935,–0.721)

–0.283 (–0.503, –0.036)

–0.800 (–0.898,–0.701)

–0.275 (–0.519, –0.025)

–0.805 (–0.904,–0.707)

–0.220 (–0.503,0.065)

1.705 (1.471,1.934) 1.493 (1.260,1.719) 01c

2.710 (2.351,3.072) 1.704 (1.470,1.921) 1.482 (1.260,1.704)

10c

2.820 (2.452,3.176) 1.541 (1.338,1.748) 1.693 (1.466,1.911)

11c

2.757 (2.358,3.141) 1.716 (1.477,1.947) 1.523 (1.292,1.759)

Nullc

2.729 (2.325,3.116) 1.736 (1.511,1.969) 1.507 (1.277,1.731)

Though incomplete, we have presented an integrated framework for modeling population dynamics. The feedback between predictions and parameter estimates achieved by sampling from a full joint posterior via Markov chain Monte Carlo results in vital rate estimates that are better predictors of population change. This holistic approach is a more efficient use of all available information, relative to standard modeling procedures that estimate parameters and project population states in serial. The availability of complementary procedures for dynamic decision analysis hold promise for the development of a truly integrated natural resource decision–making tool. References Akaike, H., 1973. Information theory as an extension of the maximum likelihood principle. In: Second International Symposium on Information Theory: 267–281 (B. N. Petrov & F. Csaki, Eds.). Akademiai Kiado, Budapest. Bordage, D., 2000. Black duck joint venture helicopter survey – Québec. Technical report, Canadian Wildlife Service, Québec Region, Environment Canada, Sainte–Foy, QC. Burnham, K. P. & Anderson, D. R.. 2002. Model Selection and Multi–Model Inference: A Practical, Information–theoretic Approach. Springer, New York.

Conroy, M. J., Miller, M. W. & Hines, J. E., 2002. Identification and synthetic modeling of factors affecting American black duck populations. Journal of Wildlife Management 6 (Wildlife Monograph No.150). Gamerman, D., 1997. Markov Chain Monte Carlo: statistical simulation for Bayesian inference. Chapman and Hall, London, first edition. Hastings, W. K., 1970. Monte carlo sampling methods using markov chains and their applications. Biometrika, 57: 97–109. King, R. & Brooks, S. P., 2002. Model Selection for Integrated Recovery/Recapture Data. Biometrics, 58: 841–851. Lubow, B. C., 1995. SDP: Generalized software for solving stochastic dynamic optimization problems. Wildlife Society Bulletin, 23: 738–742. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E., 1953. Equations of state calculations by fast computing machine. J. Chem. Phys. 21: 1087–1091. Robert, C. P. & Casella, G., 1999. Monte Carlo statistical methods. Springer–Verlag, New York. Wikle, C. K., 2003. Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology, 84: 1382–1394. Williams, B. K., Nichols, J. D., & Conroy, M. J., 2002. The Analysis and Management of Animal Populations. Academic Press, San Diego, CA. Zimpfer, N. L., 2004. Estimating movement and production rates in American black ducks. Master’s thesis, Univ. of Georgia, Athens.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

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A decision model for the efficient management of a conservation fund over time M. Drechsler & F. Wätzold

Drechsler, M. & Wätzold, F., 2004. A decision model for the efficient management of a conservation fund over time. Animal Biodiversity and Conservation, 27.1: 283–285. Extended abstract A decision model for the efficient management of a conservation fund over time.— An important task of conservation biology is to assist policy makers in the design of ecologically effective conservation strategies and instruments. Various decision rules and guidelines originate, e.g., from the Theory of Island Biography (MacArthur & Wilson, 1967) and metapopulation theory (Hanski, 1999). Designing effective strategies and instruments, however, is only part of the solution to problems of biodiversity conservation. In the real world, financial resources are scarce, and it is not only important that policies are ecologically effective but also that they are economically efficient, i.e. lead to maximum ecological benefit for a given resource input. Efficiency has been analysed, e.g. in the context of the spatial allocation of conservation funds (Wu & Bogess, 1999) and of the spatial design of compensation payments for biodiversity enhancing land–use measures (Wätzold & Drechsler, 2002). Decision analysis is a helpful tool for integrating knowledge from different disciplines and identifying optimal strategies and policies (e.g., Drechsler & Burgman, 2003). Methods of decision analysis, such as optimisation procedures, are often a core component of ecological–economic models that bring together ecological and economic knowledge via formal models (e.g., Ando et al., 1998, Drechsler & Wätzold, 2001, Johst et al., 2002). Such models do not only allow a static integration of economic and ecological aspects but also to describe the dynamics of ecological and economic systems in an integrated manner (Perrings, 2002). Examples of such dynamic modelling approaches are Richards et al. (1999), Costello & Polasky (2003) and Shogren et al. (2003). In the present paper we investigate a dynamic conservation management problem different from those of the above mentioned authors and tackle the problem of long–term conservation when future financial budgets are uncertain. The background for this problem is that many species can only survive if certain types of biodiversity–enhancing land–use measures are carried out on a regular basis, such as regularly mowing meadows to create habitat for butterflies (Settele & Henle, 2002). This means that funds have to be regularly available over time, because a temporal gap in the availability of funds may irrevocably drive a species to extinction. While over the last two decades or so a growing commitment of society and governments to conserve biodiversity could be observed, that in many cases also included the increasing provision of funds for this purpose, there are signs that this commitment is currently weakening. An example of such signs are opinion polls in some countries (e.g. Germany) showing that environmental and resource protection issues are given a lower priority by the general public than ten years ago. This implies that there is an increasing risk that conservation funds will be lower in the future than today either through a decrease in political support for such funds or through a decline in donations for private organisations that finance conservation funds.

Martin Drechsler, Dept. of Ecological Modelling, UFZ–Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany. E–mail: martind@oesa.ufz.de Frank Wätzold, Dept. of Economics, Sociology and Law, UFZ–Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany. ISSN: 1578–665X

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This risk forces governments and conservation organisations concerned by the long–term prevention of species loss to explore options which ensure that their policy aims will be achieved even if future funds are lower than today’s. Obviously, one important option is to save part of the current financial resources to counterbalance possible future budget cuts. In this context, the problem arises which proportion of the available budget should be spent now and which proportion later. In summary, there is the problem of efficiently allocating a conservation budget over time to maximise the survival probability of an endangered species, where the current budget is reasonably high and future conservation budgets are expected to decline in the medium term (although the size of these budgets is not known with any certainty). The aim of this paper is to address this problem on a conceptual level. To have a mechanism that is able to transfer current money to the future regardless of subsequent governments’ preferences and policies, we make the assumption that a conservation fund is being established that is independent of any future government’s decisions and administered by an independent agency with the time–consistent objective function to allocate financial resources over time such that the survival probability of an endangered species is maximised. The probability t of a population surviving T + 1 periods, each of length t can be written as the product of the probabilities of surviving each individual period (the complete description of the model including a more in–depth discussion of the results than presented here can be found in Drechsler & Wätzold, 2003):

(1)

where a is some species specific parameter and K(0) is the habitat capacity when no conservation measures are carried out (Lande, 1993; Grimm & Wissel, 2004). Conservation measures increase K(0) by t which costs an amount of money pt = b t with b constant. Parameter depends on the species and is inversely proportional to the coefficient of variation of the population growth rate (Lande 1993; Grimm & Wissel, 2004). Each year an amount of money gt = ht + t is granted to the conservation manager where ht is the deterministic component and t c [– , + ] is random and uniformly distributed to describe uncertainty in the future budgets. Money that is not spent can be moved into a fund Ft from which money can be drawn in later periods. The fund thus develops like (2) Borrowing is excluded, such that in each period only up to an amount Ft + gt can be spent t = 0,...,T

(3)

In each period the conservation manager has to decide how much money (pt) to spend for conservation in the present period and how much to allocate into the fund F and save for future periods. This inter– temporal optimisation problem is solved via stochastic dynamic programming (e.g., Clark, 1990). Due to the constraint (3) the solution is not straightforward. In each period, two possible solutions may formally occur: a corner solution where all available money is spent (pt = Ft + gt) and an interior solution where less then that is spent and some money is transferred to the next period. It turns out that the optimal payment in a certain period t depends on the number l of consecutive periods following the present period that have an interior solution:

(4)

One can see that the optimal payment increases with increasing fund Ft but decreases with increasing uncertainty in the grants. The latter has been shown by Leland (1968) in a 2–period model without constraint (3), denoted as “precautionary” saving and explained from the particular shape of the objective function. From eq. (4) one can also see that more money is saved when is large, i.e., when the aim is to conserve species with weakly fluctuating population growth. One can further show that it is optimal to allocate the payments as even over time as far as the constraint (3) allows. If, e.g., we have constantly decreasing grants it is optimal to save in the beginning and spend the saved money in the final periods. The problem now is that the number l depends on the future grants and if these are not known l is not known and can only be approximated by a probability distribution P(l). For the case where a negative trend is expected in the grants, such that gt = h0 – t + t we have determined P(l) and the expected optimal payment

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(5) It turned out that if the uncertainty in the grants is large or small compared to their deterministic trend one obtains a solution that is structurally similar to eq. (4), i.e. we have a situation of precautionary saving. In contrast, if the uncertainty was about of the order of magnitude of the trend we found cases where uncertainty increased the optimal payments. The reason is that the uncertainty has two contrary effects. The one is the standard "precautionary saving" effect caused by the shape of the benefit function. The other, opposing effect is that uncertainty may reduce the (expected) number l and thus increase the optimal payment. Sometimes the latter effect is stronger. However, we found strong evidence that the magnitude of such "precautionary spending" is negligibly small and for practical purposes we conclude that uncertainty generally reduces the optimal payment and more money should be saved. References Ando, A., Camm, J., Polasky, S. & Solow, A., 1998. Species Distributions, Land Values, and Efficient Conservation. Science, 279: 2126–2128. Clark, C. W., 1990. The optimal management of renewable resources. John Wiley & Sons, New York. Costello, C. & Polasky, A., 2003. Dynamic reserve site selection. Resource and Energy Economics, in press. Drechsler, M. & Burgman, M. A., 2003. Combining population viability analysis with decision analysis. Biodiversity and Conservation, in press. Drechsler, M. & Wätzold, F., 2001. The importance of economic costs in the development of guidelines for spatial conservation management. Biological Conservation, 97(1): 51–59 – 2003. Species conservation in the face of political uncertainty. UFZ discussion papers 4/2003, Leipzig. Grimm, V. & Wissel, C., 2004. The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations. Oikos, in press. Hanski, I.,1999. Metapopulation Ecology. Oxford University Press, Oxford. Johst, K., Drechsler, M. & Wätzold, F., 2002. An ecological–economic modelling procedure to design effective and efficient compensation payments for the protection of species. Ecological Economics, 41: 37–49. Lande, R., 1993. Risks of population extinction from demographic and environmental stochasticity and random catastrophes. American Naturalist, 142(6): 911–927. MacArthur, R. H. & Wilson, E. O., 1967. The Theory of Island Biogeography. Princeton Univ. Press, Princeton. Perrings, C., 2001. Modelling sustainable ecological–economic development. In: International Yearbook of Environmental and Resource Economics 2001/2: 179–201 (H. Folmer & T. Tietenberg, Eds), Elgar, Cheltenham. Richards, S. A., Possingham & H. G, Tizard, J., 1999. Optimal Fire Management for Maintaining Community Diversity. Ecological Applications, 9(3): 880–892. Settele, J. & Henle, K., 2002. Grazing and Cutting Regimes for Old Grassland in Temperate Zones. In: Biodiversity Conservation and Habitat Management (F. Gherardi, C. Corti & M. Gualtieri, Eds.) in Encyclopedia of Life Support Systems (EOLSS; chapter E1–67–03–02). Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford, U.K., [http://www.eolss.net]. Shogren, J. F., Parkhurst, G. M. & Settle, C., 2003. Integrating economics and ecology to protect nature on private lands: models, methods and mindsets. Environmental Science and Policy, 6: 233–242. Wätzold, F. & Drechsler, M., 2002. Spatial differentiation of compensation payments for biodiversity – enhancing land–use measures. UFZ discussion papers 4/2002, Leipzig. Wu, J. & Boggess, W. G., 1999. The Optimal Allocation of Conservation Funds. Journal of Environmental Economics and Management, 38: 302–321.

Editor executiu / Editor ejecutivo / Executive Editor Joan Carles Senar

Secretaria de Redacció / Secretaría de Redacción / Editorial Office

Secretària de Redacció / Secretaria de Redacción / Managing Editor Montserrat Ferrer

Museu de Zoologia Passeig Picasso s/n 08003 Barcelona, Spain Tel. +34–93–3196912 Fax +34–93–3104999 E–mail mzbpubli@intercom.es

Consell Assessor / Consejo asesor / Advisory Board Oleguer Escolà Eulàlia Garcia Anna Omedes Josep Piqué Francesc Uribe

Animal Biodiversity and Conservation 27.1 (2004)

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Costs of detection bias in index–based population monitoring C. T. Moore & W. L. Kendall

Moore, C. T. & Kendall, W. L., 2004. Costs of detection bias in index–based population monitoring. Animal Biodiversity and Conservation, 27.1: 287–296. Abstract Costs of detection bias in index–based population monitoring.— Managers of wildlife populations commonly rely on indirect, count–based measures of the population in making decisions regarding conservation, harvest, or control. The main appeal in the use of such counts is their low material expense compared to methods that directly measure the population. However, their correct use rests on the rarely–tested but often–assumed premise that they proportionately reflect population size, i.e., that they constitute a population index. This study investigates forest management for the endangered Red–cockaded Woodpecker (Picoides borealis) and the Wood Thrush (Hylocichla mustelina) at the Piedmont National Wildlife Refuge in central Georgia, U.S.A. Optimal decision policies for a joint species objective were derived for two alternative models of Wood Thrush population dynamics. Policies were simulated under scenarios of unbiasedness, consistent negative bias, and habitat–dependent negative bias in observed Wood Thrush densities. Differences in simulation outcomes between biased and unbiased detection scenarios indicated the expected loss in resource objectives (here, forest habitat and birds) through decision–making based on biased population counts. Given the models and objective function used in our analysis, expected losses were as great as 11%, a degree of loss perhaps not trivial for applications such as endangered species management. Our analysis demonstrates that costs of uncertainty about the relationship between the population and its observation can be measured in units of the resource, costs which may offset apparent savings achieved by collecting uncorrected population counts. Key words: Wildlife surveys, Detection bias, Opportunity costs, Optimization, Uncertainty, Decision making. Resumen Costes de los sesgos de detección en el monitoreo de poblaciones basado en índices.— Los gestores de poblaciones de fauna silvestre a menudo toman decisiones relativas a la conservación, recolección o control a partir de medidas indirectas de la población basadas en recuentos. El principal atractivo que presenta este tipo de recuentos son los bajos costes de material, en comparación con otros métodos que miden la población de forma directa. Sin embargo, el correcto uso de los mismos depende de una premisa que suele darse por sentada, aunque rara vez se comprueba, y que consiste en suponer que reflejan proporcionalmente el tamaño de la población; es decir, que constituyen un índice poblacional. El presente estudio investiga la gestión forestal de dos especies en peligro de extinción: el pájaro carpintero de cresta roja (Picoides borealis) y el zorzal mustelino (Hylocichla mustelina) en la Reserva Nacional de Animales Salvajes de Piedmont, en Georgia central, Estados Unidos. Se simularon varias políticas de conservación bajo escenarios referentes a las densidades del zorzal mustelino insesgados, con un consistente sesgo negativo y con un sesgo negativo dependiente del hábitat. Las diferencias obtenidas con respecto a los resultados de simulación entre los escenarios de detección sesgados y los no sesgados indicaron la pérdida prevista en los objetivos en materia de recursos (en este caso, el hábitat y las aves del bosque) a través de una toma de decisiones basada en los recuentos poblacionales sesgados. Teniendo en cuenta los modelos y la función de los objetivos que hemos empleado en nuestro análisis, las pérdidas previstas ascendieron al 11%, lo que supone un porcentaje bastante significativo en aplicaciones tales como la gestión de especies en peligro de extinción. Nuestro análisis demuestra que los costes de incertidumbre ISSN: 1578–665X

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acerca de la relación entre la población y su observación pueden medirse en unidades del recurso dado; es posible que estos costes compensen los ahorros aparentemente conseguidos mediante la recopilación de recuentos poblacionales no corregidos. Palabras clave: Estudios de fauna silvestre, Sesgos de detección, Costes de oportunidad, Optimización, Incertidumbre, Toma de decisiones. Clinton T. Moore, USGS Patuxent Wildlife Research Center, Warnell School of Forest Resources, University of Georgia, Athens, GA 30602–2152, U.S.A.– William L. Kendall, USGS Patuxent Wildlife Research Center, 11510 American Holly Drive, Laurel, MD 20708–4017, U.S.A. Corresponding author: C. T. Moore. E–mail: clinton_moore@usgs.gov

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Introduction Managers of animal populations and wildlife habitats often rely on indirect measures of population abundance to support decision making in conservation, harvest, or control. For example, conservation decisions made by an agency may be based on numbers of singing birds detected during a roadside survey rather than on a direct estimate of population abundance. The use of such count–based measures is common because they are popularly perceived to be substantially less expensive to collect than data that allow the direct estimation of population abundance, while almost as informative. But this perception is frequently unfounded. Ancillary data that permit direct estimation of abundance (or density) often can be collected at marginal additional expense to that of the original survey (Anderson, 2003): counts by paired observers (Nichols et al., 2000) and distances to subjects (Buckland et al., 2001) are two of many types of such data sources (Williams et al., 2002). More importantly, the fact that bias in the count (as a measure of population status) is almost never quantified renders its information content questionable (Anderson, 2001, 2003). The use of an unadjusted count as a measure of relative population abundance is valid only if the measure is strictly proportional to population size. In other words, the expected detection rate of the population (i.e., average of count / population) must remain constant over all conditions for the count to constitute a valid index (Anderson, 2001; Williams et al., 2002). However, this assumption is rarely tested in practice, and where it has been tested, detection rate is often found to vary (Williams et al., 2002). Factors associated with variation in detection rate include habitat features, environmental conditions, sampling and observer characteristics, and population abundance itself (Verner, 1985; Anderson, 2001). In conservation decision making, one possible consequence of using counts unadjusted for detection rate is that decisions that appear best (i.e., optimal for some objective outcome) on the basis of such counts may not be the same as those that would have been chosen had true abundance been known or estimated. Therefore, an opportunity cost, measurable in units of the resource, may be associated with the use of unadjusted count data. The opportunity cost could be commensurate with, or even greater than, the cost of obtaining the ancillary measurements to permit direct abundance estimation. For some problems in wildlife conservation, for example, endangered species recovery, opportunity costs may not be inconsequential. Thus, the total cost of a monitoring program is equal to the cost of collecting the unadjusted counts plus the expected opportunity cost of either failing to estimate detection probability or establishing its constancy. If opportunity costs can be shown to be small over a range of plausible departures from the constant detection probability assumption, then the collection of unadjusted counts may be justified as

more efficient than alternative approaches. Our study analyzes this problem in the context of forest management on the Piedmont National Wildlife Refuge (PNWR) in central Georgia (U.S.A.). Here, the joint objective of management is provision of habitat for an endangered species, the Red– cockaded Woodpecker (Picoides borealis), and the maintenance of a population of a shrub–nesting neotropical migratory bird, the Wood Thrush (Hylocichla mustelina). However, response of the Wood Thrush population to silvicultural actions is largely unknown, therefore the degree to which management for the woodpecker conflicts with management for the Wood Thrush is uncertain. We built a dynamic optimization model in which we specifically addressed this form of structural uncertainty. We also addressed uncertainty in the constancy of Wood Thrush detection rate among habitats. Simulating the decision making process under alternative forms of the optimization model yields a statistic, the expected value of information, that represents the opportunity cost of management under population measurement uncertainty. Although the models we will describe lack certain details that would make them useful in a management application, they are sufficiently useful for the purpose of illustrating the idea that real management costs may be incurred whenever there is considerable uncertainty about what an unadjusted population count is measuring. Methods Study area and description of management The PNWR is a 14,136–ha unit of the U.S. National Wildlife Refuge System. The site supports a second–growth mixed pine (Pinus taeda, P. echinata) and hardwood (Quercus spp., Carya spp.) forest that regenerated naturally on severely eroded farmland abandoned in the 1930s (Gabrielson, 1943; Czuhai & Cushwa, 1968). Forest management is directed towards the maintenance of all native flora and fauna, sustenance of important ecosystems, and provision of public recreation, including wildlife viewing and sport harvest of some wildlife species (U.S. Fish and Wildlife Service, Piedmont National Wildlife Refuge, URL: http://piedmont.fws.gov). PNWR is also a designated recovery site for the Red–cockaded Woodpecker (U.S. Fish and Wildlife Service, 2000). The woodpecker’s preferred foraging and breeding habitat consists of pure, open stands of mature (80 or more years) pine with a fire– maintained herbaceous understory (Loeb et al., 1992). But these forest habitats have become highly fragmented or have disappeared altogether, particularly since the early 20th century, as intensification of management on industrial forest lands emphasized shorter timber rotations and as exclusion of fire from all forest lands permitted increased hardwood succession (Ligon et al., 1986). Because the PNWR is identified as a woodpecker recovery site, forest

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management is oriented towards increasing the abundance of this species. To this end, forest managers conduct aggressive regimes of thinning and regeneration cutting, prescribed burning, and mechanical vegetation removal to promote the creation of pure, open stands of mature pine with a herbaceous understory and reduced hardwood midstory (U.S. Fish and Wildlife Service, Piedmont National Wildlife Refuge, Habitat Management Plan, 1982; unpublished report). Whereas these conditions are most favorable for production and survival of woodpeckers, their suitability for other forest wildlife species is mostly unknown. In particular, reductions in the hardwood midstory and the overstory canopy would be expected to be detrimental to the persistence of the Wood Thrush, a neotropical migrant species commonly associated with dense understory and midstory conditions of closed–canopy forest interiors (Weaver, 1949; Hamel et al., 1982; Roth et al., 1996). This species of management concern (Hunter et al., 1992) is thought to be declining over its range (Peterjohn et al., 1995), with fragmentation of interior forest conditions across the eastern U.S. implicated (Whitcomb et al., 1981; Temple & Cary, 1988; Hansen & Urban, 1992). Therefore, a concern for refuge managers is that silvicultural actions targeted for the woodpecker do not cause excessive harm to populations of nontarget species such as the Wood Thrush. Comparison of Wood Thrush population parameters on control and silviculturally–treated forest compartments, both pre– and post–treatment, found no detectable effect of treatments on population growth of the Wood Thrush at PNWR (Powell, 1998; Powell et al., 2000). In fact, point estimates of many parameters and of the population growth rate were greater following treatments than before (Powell, 1998). However, parameter estimates had considerable sampling variability, therefore definitive conclusion of a treatment effect remains somewhat equivocal (Powell et al., 2000). Decision model We linked a model of annual Wood Thrush population dynamics to a forest management model, where both models were deterministic. The forest model expressed the quantities of and transitions among three PNWR forest seral types: pine regeneration (F1, age 0–20), mature mixed pine–hardwood (F2, age 20–90), and open pine forest suitable for woodpecker utilization (F3, age 20–90). Although pine habitat younger than approximately age 40 is seldom used by foraging woodpeckers (Epting et al., 1995) and age 16 is considered the transition point between the regeneration class and the poletimber (mature) class, our use of these assumptions greatly simplified the model and did not diminish its instructional value. At any time t, the proportional amounts of forest in each seral type was indicated in the vector

ft = [ XF1(t), XF2(t), XF3(t)]' In the absence of forest management, annual rates of natural transition from the mature classes into the regeneration class were 12 = 13 = 1/70, merely the inverse of the length of the mature age class (fig. 1A). Similarly, the graduation rate from the 20–year regeneration class to the F2 class was = 1/20. When unmanaged, F3 forest can be21 come unsuitable for woodpecker use in as few as 4 years (Piedmont National Wildlife Refuge, 1982, unpublished report), therefore, we assumed that F3 forest is lost at the rate of 23 = 1/4 per year in the absence of management. However, management can cause regeneration to exceed the natural rates (through regeneration cutting), can effect the transfer of forest from the F2 class into the F3 class (thinning and burning), and can reduce attrition from F3 into F2 (thinning and burning) (fig. 1B). Regeneration cuts from types F2 and F3 in year t are denoted d12t and d13t, respectively, creation of new F3 habitat is denoted d32t, and re–treatment of F3 habitat is denoted d23t. The thinning–burning decisions are expressed in the transition matrix:

where columns and rows represent pre– and post– transition states, respectively, and the regeneration decisions are expressed as follows:

Given the decisions dijt at time t, the state of the forest ft is transformed to a new state ft+1 by first applying the thinning–burning treatments, then the regeneration treatments: ft+1 = Ut Vt ft We used alternative forms of a simple exponential growth model to express our uncertainty regarding dynamics of the Wood Thrush population occurring in habitats considered favorable (designated F) and unfavorable ( ). Under one alternative, we assumed that the population of Wood Thrushes in F3 habitat increased (i.e., F = F3), whereas that in F2 habitat decreased ( = F2). We used growth rates for the favorable ( F = 1.012) and unfavorable ( = 0.949) habitat quality types consistent with point estimates provided by Powell et al. (2000) for treated and untreated areas, respectively. However, because of high sampling variability, their findings also were consistent with the converse proposition that habitat F2 was favorable and habitat F3 was not. Therefore, the alternative population model used the same parameter values for F and as

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A 1–

1–

(1 –

F2 Mature forest non–RCW

F1 Immature forest (< 20yr)

) (1 –

)

F3 Mature forest RCW

(1 –

)

B 1 –

[1 – max(

, d12)] (1 – d32)

F2 Mature forest non–RCW

F1 Immature forest (< 20yr)

, d13)] [1 – (

– d23)]

F3 Mature forest RCW

max(

, d12) (1 – d32) + max(

max(

, d12) (

, d13) d32

[1 – max(

– d23) + max(

, d13)] d32 , d12)] (

, d13) [1 – (

– d23)

– d23)]

Fig. 1. Model of dynamics among regeneration (F1), mature untreated (F2), and mature forest treated for Red–cockaded Woodpecker use (F3) at PNWR. Portions of the mature forest naturally regenerate every year, and mature classes are augmented by recruitment from the regeneration class. In the absence of management (A), the treated stage F3 reverts back to the F2 stage and ultimately disappears. However, management (B) can reduce this attrition either through treatment of F2 forest or re–treatment of F3 forest. Furthermore, regeneration rate can be increased by cutting from the mature classes. Fig. 1. Modelo de dinámica entre la regeneración (F1), bosque maduro no tratado (F2), y bosque maduro tratado para su uso en el pájaro carpintero de cresta roja (F3) en PNWR. Algunas áreas del bosque maduro se regeneran anualmente de forma natural, y las clases maduras se aumentan mediante el reclutamiento procedente de la clase de regeneración. En ausencia de gestión (A), la fase tratada F3 revierte a la fase F2 y finalmente desaparece. Sin embargo con gestión (B) se puede reducir este desgaste, bien a través del tratamiento del bosque F2 o del retratamiento del bosque F3. Además, la tasa de regeneración se puede incrementar reduciendo las clases maduras.

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before, but switched assignment of F2 and F3 as the unfavorable and favorable habitats. Because Wood Thrushes are not known to commonly use areas of pine regeneration (Roth et al., 1996), we assumed that seral type F1 provided no suitable habitat for Wood Thrushes and therefore ignored this type in the population models. The general expression of the population model was: Nq(t+1) =

Nq(t) =

(t) Xq(t+1), q = F,

;

the latter form expressing population size N at time t+1 in terms of bird density in habitat quality type q at time t and projected proportion X of habitat quality type q at time t+1. Thus, the state of the modeled system is described at any time t by the amount of forest in each seral stage and by the density of Wood Thrushes in types F2 and F3. Given a selection of one of the population models, the steps of the model were as follows: (1) obtain the current forest state (proportional amount of each seral type) and current Wood Thrush densities within types; (2) obtain the current set of forest management decisions given those states; (3) project the state of the forest to the next decision period, given decision set; (4) calculate bird abundances and densities at next decision period, given population model and future forest state; and (5) move to next decision period, repeat 1–5. Optimization For both population models, we used dynamic programming to search for a stationary, state–specific optimal decision policy for a suitable conservation– based objective function (Dreyfus & Law, 1977; Williams et al., 2002). We converted expected Wood Thrush abundance at the next time period to a value bounded between 0 and 1:

where the constants in the function were based on a minimum acceptable density of 0.05 pairs/ha and a maximum satisfying density of 2.0 pairs/ha over the entire refuge. The objective function was Jt = XF3(t + 1)0.5 Wt 0.5 i.e., an equally–weighted geometric average of future proportion of F3 habitat (for woodpecker population viability) and future Wood Thrush abundance value. If decisions drive either component of this objective function to zero, then Jt = 0. Because both XF3(t + 1) and Wt are bound between 0 and 1, Jt is also bounded between 0 and 1. Dynamic programming seeks those decisions that maximize the sum of this return value over an infinite time frame of decision making ( ). The decision policy found by dynamic programming will be state–dependent,

that is, a set of optimal decisions is associated with each possible system state. A useful decision policy should also be stationary, that is, it should be invariant with regard to which time period a decision is to be made. We used program ASDP (Lubow, 1995, 1997) to perform the optimization. Because of the unit–sum constraint, compositional state variables such as amounts of forest in three habitat types are not easy to represent in a rectangular matrix of state variable combinations. For example, if XF1 and XF2 are both allowed to range between 0.0 and 1.0, then the combination (XF1, XF2) = (0.6, 0.2) is a valid system state (XF3 = 0.2, by subtraction), whereas (XF1, XF2) = (0.6, 0.5) is not. We defined a two–part transformation that expressed the three dependent compositional states in terms of two independent integer states. First, integer indices I = 1,2,...,11 are converted to logits L through a power function: L = p [ (I – a) / b ]q where a, b, p, and q are constants chosen to appropriately center, scale, and shape the relationship. Next, the logits are converted to forest type proportions in the usual way: Xi = exp (Li) / [1 +

3exp (Lj)]

Furthermore, because we expected the decision policy to be most sensitive to the Wood Thrush population states when densities were low rather than high, we used square–root transformations of observed densities as state variables. Program ASDP processes only discrete values of state and decision variables. We discretized both forest composition state variables into 9 levels each and both Wood Thrush state variables into 11 levels each (observed density ranging between 0– 2.3 pairs/ha, approximately), yielding 9,801 state combinations. Decisions d32, d12, and d13 were each discretized into steps of 0.2 over the range 0.0–1.0, and decision d23 was discretized into steps of 0.05 over the range 0.0–0.25. The highly nonlinear structure of the state variables caused overestimates of the objective values under ASDP’s linear interpolation and extrapolation features. Though the errors were slight, the compounding of errors over the course of the program’s iterations caused difficulty in convergence to stationary policies. Therefore, we imposed a small (0.999) discounting rate on the objective value and terminated the program after 100 iterations, a point at which the number of optimal decisions changing between iterations reached a minimum (16–19 of 9,801 states) and where objective values had not appreciably exceeded their mathematical bounds. Computation of expected opportunity costs Our interest was in assessing the outcome of decision making based on observed states of the system

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(i.e., observed Wood Thrush densities obtained from unadjusted counts) that may or may not correspond to unobservable true states. We considered four plausible cases of how the relationship between observed densities and true densities of Wood Thrush may vary among habitats: (1) birds are perfectly detectable in F2 and F3 habitats (detection rate = 1.0), (2) birds are equally detectable in F2 and F3 habitats, but not perfectly detectable (detection rate = 0.5), (3) birds are more detectable in F3 habitat (detection rate = 0.7) than in F2 habitat (0.3), and (4) birds are more detectable in F2 habitat (detection rate = 0.7) than in F3 habitat (0.3). Our focus, then, was the assessment of the quality of decision making for cases 2–4 as if case 1 was the operative mode of detection. For each population model and its associated optimal decision policy, we simulated decision making under each detection model over a 100–year time period. We chose five starting forest states for each simulation: (1) high F1 (initial forest composition ft' = [0.60, 0.35, 0.05]), (2) balanced age class / low F3 (ft' = [0.22, 0.73, 0.05]), (3) balanced age class / high F3 (ft' = [0.22, 0.38, 0.40]), (4) low F1 / low F3 (ft' = [0.05, 0.90, 0.05]), and (5) low F1 / high F3 (ft' = [0.05, 0.55, 0.40]). For starting Wood Thrush density states, we chose 0.5 pairs/ha for both F2 and F3 habitats. Because the decision models were all deterministic, multiple simulation runs were not required. Cumulative 100–year values of J were obtained for each detection scenario, annualized (divided by 100), and compared to the annualized cumulative J value for the model of perfect detection. Thus, three comparisons were available for each population model and starting state. If j A – j B represents the difference in annualized J values for two scenarios A and B, then the maximum difference attributable to either the XF3(t+1) or the W components of the objective function (when the other component is held fixed) is 1 – (jB / jA)2. Expressed as a percentage, this value can be interpreted as the maximum percent loss, or partial opportunity cost, incurred by one of the resources under suboptimal management. For example, a partial opportunity cost of 10% implies that either woodpecker habitat amount or Wood Thrush abundance is reduced by as much as 10% if decision making is made under an inappropriate detectability assumption. Also, we averaged the jA – jB differences over the alternative detection scenarios to represent expected resource cost under complete uncertainty with respect to the detection models. This statistic closely resembles the value of information (Lindley, 1985), an estimate of expected loss when one lacks the information to distinguish among alternative expected outcomes of decision making (here, alternative detection scenarios). For example, an expected partial opportunity cost of 10% implies that the sacrifice in either woodpecker habitat or Wood Thrush abundance is expected to average 10% over several plausible but uncertain detection relationships.

Results In the population model that considered F3 as favorable habitat for Wood Thrushes, optimal annualized cumulative returns for the model of perfect detection ranged 0.412–0.466 over the five starting states (table 1). Corresponding values for the models of equal detection and higher detection in F3 were comparable, even greater in some cases (table 1). This outcome suggests that our decision policy was suboptimal, likely a consequence of difficulties in the optimization routine described earlier. However, the returns under the alternative detection models are in close agreement, suggesting that, at least for these two models of imperfect detection, management based on uncorrected counts or densities was as profitable as management under perfect detection. However, the model of lower detection in F3 generated lower annualized returns than did the model of perfect detection (table 1; detection model M3). Differences in returns appear small (0.007–0.014), but they translate into partial opportunity costs of 3–6% for either of the resource components (table 1). Averaging the differences over the three models of imperfect detection suggests expected partial opportunity costs of 1–3% under model uncertainty (table 1). For the model proposing F2 as favorable habitat, annualized returns for the model of perfect detectability ranged 0.236–0.268 (table 1). Simulation of each of the models of imperfect detection provided lower annualized returns in all cases (table 1). The smallest differences in returns occurred for the model of lower detection in F2 (0.0–0.003), intermediate differences were observed for the model of equal detectability between habitats (0.002– 0.008), and the greatest differences occurred for the model of lower detection in F3 (0.003–0.015) (table 1). The largest value of partial opportunity cost was 11% (table 1). Averaging over all the models of imperfect detection yielded 2–6% in expected partial opportunity cost (table 1). Discussion Whether unadjusted counts constitute reliable indicators of wildlife population abundance has been an issue of recent intensive debate (Thompson et al., 1998; Hutto & Young, 2002; Engeman, 2003; Anderson, 2003). Perhaps one reason that the arguments persist is that the extra costs associated with collecting the ancillary data to estimate detection rate are tangible and easy to perceive, whereas consequences of decision making based on faulty detectability assumptions are not. This "invisible cost" of misled management may be inappropriately taken by some as evidence that such costs are negligible and perhaps contributes to a complacency toward the problem of unmeasured detection biases. Our analysis of a very simple population model under quite reasonable alternative patterns

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Table 1. Annualized cumulative objective values for a combined model of forest management and Wood Thrush response, under each of four alternative forms of Wood Thrush detection rate, two alternative Wood Thrush population models, and for five initial forest states. One population model projected positive population growth in forest type F3 and negative growth in type F2, and the other model projected the converse. Detection models were: M0. Perfect detectability; M1. Detection rate of 0.5 in F2 and F3 habitats; M2. Detection rate of 0.3 in F2 habitat and 0.7 in F3 habitat; M3. Detection rate of 0.7 in F2 and 0.3 in F3; Av. Averaged. Partial cost is the maximum difference in objective function value for either of the contributing components of the objective function (amount of F3 habitat, Wood Thrush abundance value), holding the other component fixed. Tabla 1. Valores anualizados de los objetivos acumulativos para un modelo combinado de gestión forestal y respuesta de los zorzales mustelinos, según cada una de las cuatro formas alternativas de su tasa de detección según dos modelos alternativos de población y según cinco estados forestales iniciales. Un modelo poblacional proyectó un crecimiento de población positivo en el bosque del tipo F3 y un crecimiento negativo en el del tipo F2, mientras que el otro modelo proyectó lo opuesto. Los modelos de detección fueron: M0. Detectabilidad perfecta; M1. Tasa de detección de 0,5 en los hábitats F2 y F3; M2. Tasa de detección de 0,3 en el hábitat F2 y de 0,7 en el hábitat F3; M3. Tasa de detección de 0,7 en F2 y de 0,3 en F3 (M3); Av. Promedio. El coste parcial es la diferencia máxima en el valor de la función de objetivos para cada uno de los componentes que contribuyen a dicha función (total de hábitat F3, valor de abundancia de los zorzales mustelinos), manteniendo fijo el otro componente.

Population model Initial forest state F3

> 1,

Cumulative value (annualized) M0

Partial cost M1

High F1

0.412 0.407

0.409

0.404

0.024

0.014

0.042 0.027

Balanced age, low F3

0.442 0.440

0.443

0.429

0.007 –0.007

0.058 0.020

Balanced age, high F3

0.441 0.440

0.443

0.427

0.007 –0.007

0.063 0.021

Low F1, low F3

0.459 0.462

0.459

0.452

–0.012 –0.001

0.031 0.006

Low F1, high F3

0.466 0.467

0.466

0.459

–0.005 –0.002

0.028 0.007

High F1

0.236 0.234

0.234

0.233

0.016

0.021

0.028 0.021

Balanced age, low F3

0.256 0.254

0.256

0.249

0.018

0.002

0.053 0.025

Balanced age, high F3

0.255 0.254

0.255

0.248

0.014

0.000

0.057 0.024

Low F1, low F3

0.268 0.259

0.265

0.253

0.061

0.019

0.108 0.063

Low F1, high F3

0.268 0.264

0.265

0.257

0.025

0.019

0.075 0.040

< 1,

of detection bias demonstrates that uncertainty with respect to detectability results in some degree of opportunity cost that is measurable in units of the resource. The opportunity costs we observed were small (all partial costs < 11%), but two points are worth noting. First, opportunity costs will certainly vary according to choice of population model, detection model, and objective function. For example, partial costs become as large as 24% if we reduce detection rate from 0.3 to 0.1 in one of the detection models and increase its counterpart rate from 0.7 to 0.9. Second, a given cost may have different implications in different management settings. For example, an 11% cost in habitat or population value may be inconsequential in many management settings, but not in all, particularly in endangered species management. If F2 habi-

tat is favorable for Wood Thrushes and yields greater detectability than does F3 habitat, our model suggests that up to 11% of Red–cockaded Woodpecker habitat (type F3) could be needlessly sacrificed if management decisions are inappropriately based on the assumption that Wood Thrushes are equally detectable in both habitats. In this case, the erroneous implication of observing equal densities of Wood Thrushes in F2 and F3 habitats is that the total population density is too far below the satisfying density and should be increased by removing some of the unfavorable (F3) habitat and creating more of the favorable (F2) habitat. Our analysis dealt only with detection bias in the form of undercounting, whereas many monitoring programs may collect information also prone to overcount bias. For example, in many programs,

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an individual or its sign (tracks, fecal pellets, etc.) may be counted more than once, and these counts are often considered to represent a consistent over– representation of a smaller number of individual animals (Diefenbach et al., 1994). We know of no reason to expect that such biases would result in opportunity costs of similar magnitude to our findings; indeed, it is the opinion of one reviewer that undercounting, constant or otherwise, may be the least problematic form of count bias. Confirmation of such speculation must wait until these detection scenarios are explicitly modeled. When a monitoring program of uncorrected counts is considered for the support of management decision making, we recommend the construction and exploration of decision models such as these to fully estimate the true costs of conducting that form of monitoring. Our suspicion is that opportunity costs of biased monitoring often will not be negligible, even under mild departures from the assumed proportionality relationship. We also suggest that costs of estimating detection rate will often be less than the total of the collection cost of uncorrected counts plus the opportunity cost of decision making based on those counts. Even where the cost of uncorrected counts is anticipated to be low relative to the cost of correcting them, the relationship between the count and true density should at least occasionally be monitored. Previous authors have suggested approaches to determining if or how much detection probability varies across time or space (see Skalski & Robson, 1992; MacKenzie & Kendall, 2002). However, this paper is a first attempt at directly evaluating the implication of this variation in terms of management objectives More generally, this study was a first step in exploring the proper consideration of monitoring effort and design in making decisions. We have focused on the effect of bias in indices of system state on an optimal policy and its consequences for the system. We have not considered the relative precision of adjusted and unadjusted counts, and its impact on system response in terms of objectives. Wildlife systems are managed in the face of four sources of uncertainty (Nichols et al., 1995): environmental variation, partial controllability (a given decision has variable impact on the system), structural uncertainty about the key factors that drive the system, and partial observability (sampling variability and estimation bias). Reduction of partial observability should improve decisions directly by giving the manager a clearer picture of system status. This reduction also provides the indirect benefit of helping to reduce structural uncertainty. The cost/benefit of different approaches to monitoring (both in terms of sampling effort which controls precision and the types of data collected to reduce bias) could be incorporated directly into the decision model by making these alternative monitoring decisions part of the general decision space, and incorporating the cost of monitoring into the objective function.

Acknowledgements Forest models used in this work were motivated by the first author’s dissertation research, which received support by the USGS Georgia Cooperative Fish and Wildlife Research Unit (Research Work Order 34, Coop. Agreement #14–16–0009–1551), the U.S. Geological Survey Cooperative Research Units Center, the U.S. Fish and Wildlife Service, Region 4, the U.S. Environmental Protection Agency (STAR Fellowship U–915424–01–0), U.S. Geological Survey Patuxent Wildlife Research Center, and University of Georgia Warnell School of Forest Resources. R. Shell, C. Johnson, D. Metteaeur, and other PNWR staff offered invaluable advice, information, and help. We thank B. C. Lubow for his patient assistance with program ASDP. We also appreciate the efforts of E. Reed, D. R. Diefenbach, and M. J. Conroy in reviewing our work and in offering many helpful improvements to the paper. References Anderson, D. R., 2001. The need to get the basics right in wildlife field studies. Wildlife Society Bulletin, 29: 1294–1297. – 2003. Response to Engeman: index values rarely constitute reliable information. Wildlife Society Bulletin, 31: 288–291. Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L. & Thomas, L., 2001. Introduction to distance sampling. Oxford University Press, Oxford. Czuhai, E. & Cushwa, C. T., 1968. A resume of prescribed burnings on the Piedmont National Wildlife Refuge. USDA Forest Service Research Note SE–86. Diefenbach, D. R., Conroy, M. J., Warren, R. J., James, W. E., Baker, L. A. & Hon, T., 1994. A test of the scent–station survey technique for bobcats. Journal of Wildlife Management, 58: 10–17. Dreyfus, S. E. & Law, A. M., 1977. The art and theory of dynamic programming. Academic Press, New York. Engeman, R. M., 2003. More on the need to get the basics right: population indices. Wildlife Society Bulletin, 31: 286–287. Epting, R. J., DeLotelle, R. S. & Beaty, T., 1995. Red–cockaded woodpecker territory and habitat use in Georgia and Florida. In: Red–cockaded woodpecker: recovery, ecology and management: 270–276 (D. L. Kulhavy, R. G. Hooper & R. Costa, Eds.). Center for Applied Studies in Forestry, Stephen F. Austin State Univ., Nacogdoches, Texas. Gabrielson, I. N., 1943. Wildlife refuges. Macmillan, New York. Hamel, P. B., LeGrand Jr., H. E., Lennartz, M. R. & Gauthreaux Jr., S. A., 1982. Bird–habitat relationships on Southeastern forest lands. USDA Forest Service, Southeastern Forest Experiment Station, General Technical Report SE–22.

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Índex / Índice / Contents Animal Biodiversity and Conservation 27.1 (2004) ISSN 1578–665X

Actes del Congrès Internacional EURING 2003 / Actas del Congreso Internacional EURING 2003 / Proceedings of the EURING International Conference 2003

21–72 Evolutionary Biology Session

297–370 Dispersal and Migration Session

73–91 Random Effects Session

371–435 Analysis of Ringing Data Session

93–146 Multi–state Models Session

437–467 Abundance Estimation and Conservation Biology Session

147–173 Methodological Advances Session 175–228 Computing and Software Session 229–296 Decision Analysis Session

469–529 Population Dynamics Session 531–560 Poster Session 561–572 EURING Short Course

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Hansen, A. J. & Urban, D. L., 1992. Avian response to landscape pattern: the role of species’ life histories. Landscape Ecology, 7: 163–180. Hunter, W. C., Pashley, D. N. & Escano, R. E. F., 1992. Neotropical migratory landbird species and their habitats of special concern within the Southeast Region. In: Status and management of neotropical migratory birds: 159–171 (D. M. Finch & P. W. Stangel, Eds.). USDA Forest Service General Technical Report RM–229. Hutto, R. L. & Young, J. S., 2002. Regional landbird monitoring: perspectives from the Northern Rocky Mountains. Wildlife Society Bulletin, 30: 738–750. Ligon, J. D., Stacey, P. B., Conner, R. N., Bock, C. E. & Adkisson, C. S., 1986. Report of the American Ornithologists’ Union Committee for the Conservation of the Red–cockaded Woodpecker. Auk, 103: 848–855. Lindley, D. V., 1985. Making decisions, 2nd edition. Wiley, London. Loeb, S. C., Pepper, W. D. & Doyle, A. T., 1992. Habitat characteristics of active and abandoned red–cockaded woodpecker colonies. Southern Journal of Applied Forestry, 16: 120–125. Lubow, B. C., 1995. SDP: Generalized software for solving stochastic dynamic optimization problems. Wildlife Society Bulletin, 23: 738–742. – 1997. Adaptive stochastic dynamic programming (ASDP): Supplement to SDP user’s guide, Version 2.0. Colorado Cooperative Fish and Wildlife Research Unit, Colorado State Univ., Fort Collins, Colorado. MacKenzie, D. I. & Kendall, W. L., 2002. How should detection probability be incorporated into estimates of relative abundance? Ecology, 83: 2387–2393. Nichols, J. D., Hines, J. E., Sauer, J. R., Fallon, F. W., Fallon, J. E. & Heglund, P. J., 2000. A double–observer approach for estimating detection probability and abundance from point counts. Auk, 117: 393–408. Nichols, J. D., Johnson, F. A. & Williams, B. K., 1995. Managing North American waterfowl in the face of uncertainty. Annual Review of Ecology and Systematics, 26: 177–199. Peterjohn, B. G., Sauer, J. R. & Robbins, C. S., 1995. Population trends from the North American Breeding Bird Survey. In: Ecology and management of Neotropical migratory birds: 3–39 (T.

Moore & Kendall

E. Martin & D. M. Finch, Eds.). Oxford Univ. Press, New York. Powell, L. A., 1998. Experimental analysis and simulation modeling of forest management impacts on wood thrushes, Hylocichla mustelina. Ph. D. Thesis, Univ. of Georgia, Athens. Powell, L. A., Lang, J. D., Conroy, M. J. & Krementz, D. G., 2000. Effects of forest management on density, survival, and population growth of wood thrushes. Journal of Wildlife Management, 64: 11–23. Roth, R. R., Johnson, M. S. & Underwood, T. J., 1996. Wood Thrush (Hylocichla mustelina). In: Birds of North America, 246: 1–28 (A. Poole & F. Gill, Eds.). Academy of Natural Sciences, Philadelphia, Pennsylvania and American Ornithologists’ Union, Washington, D.C. Skalski, J. R. & Robson, D. S., 1992. Techniques for wildlife investigations. Academic Press, San Diego. Temple, S. A. & Cary, J. R., 1988. Modeling dynamics of habitat–interior bird populations in fragmented landscapes. Conservation Biology, 2: 340–347. Thompson, W. L., White, G. C. & Gowan, C., 1998. Monitoring vertebrate populations. Academic Press, San Diego. U. S. Fish and Wildlife Service, 2000. Technical/ agency draft revised recovery plan for the red– cockaded woodpecker (Picoides borealis). U.S. Department of Interior, Fish and Wildlife Service, Atlanta, Georgia. Verner, J., 1985. Assessment of counting techniques. In: Current ornithology, Volume 2: 247–302 (R. F. Johnston, Ed.). Plenum Press, New York. Weaver, F. G., 1949. Hylocichla mustelina: wood thrush. In: Life histories of North American thrushes, kinglets, and their allies: 101–123 (A. C. Bent, Ed.). U.S. National Museum Bulletin 196. [republished 1964 by Dover, New York] Whitcomb, R. F., Robbins, C. S., Lynch, J. F., Whitcomb, B. L., Klimkiewicz, M. K. & Bystrak, D., 1981. Effects of forest fragmentation on avifauna of the eastern deciduous forest. In: Forest island dynamics in man–dominated landscapes: 125–205 (R. L. Burgess & D. M. Sharpe, Eds.). Springer–Verlag, New York. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, San Diego.