Animal Biodiversity and Conservation issue 27.1 (2) (2004) pp 297-572 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

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Dispersal and migration C. Schwarz & F. Bairlein

Schwarz, C. & Bairlein, F., 2004. Dispersal and migration. Animal Biodiversity and Conservation, 27.1: 297–298. Ringing of birds unveiled many aspects of avian migration and dispersal movements. However, there is even much more to be explored by the use of ringing and other marks. Dispersal is crucial in understanding the initial phase of migration in migrating birds as it is to understand patterns and processes of distribution and gene flow. So far, the analysis of migration was largely based on analysing spatial and temporal patters of recoveries of ringed birds. However, there are considerable biases and pitfalls in using recoveries due to spatial and temporal variation in reporting probabilities. Novel methods are required for future studies separating the confounding effects of spatial and temporal heterogeneity of recovery data and heterogeneity of the landscape as well. These novel approaches should aim a more intensive and novel use of the existing recovery data by taking advantage of, for instance, dynamic and multistate modeling, should elaborate schemes for future studies, and should also include other marks that allow a more rapid data collection, like telemetry, geolocation and global positioning systems, and chemical and molecular markers. The latter appear to be very useful in the delineating origin of birds and connectivity between breeding and non–breeding grounds. Many studies of migration are purely descriptive. However, King and Brooks (King & Brooks, 2004) examine if movement patterns of dolphins change after the introduction of a gillnet ban. Bayesian methods are an interesting approach to this problem as they provide a meaningful measure of the probability that such a change occurred rather than simple yes/no response that is often the result of classical statistical methods. However, the key difficulty of a general implementation of Bayesian methods is the complexity of the modelling —there is no general userfriendly package that is easily accessible to most scientists. Drake and Alisauskas (Drake & Alisauskas, 2004) examine the philopatric movement of geese using a classic multi–state design. Previous studies of philopaty often rely upon simple return rates —however, good mark–recapture studies do not need to assume equal detection probabilities in space and time. This is likely the most important contribution of multi–state modelling to the study of movement. As with many of these studies, the most pressing problem in the analysis is the explosion in the number of parameters and the need to choose parsimonious modelss to get good precision. Drake and Alisauska demonstrate that model choice still remains an art with a great deal of biological insight being very helpful in the task. There is still plenty of scope for novel methods to study migration. Traditionally, there has been a clear cut distinction between birds being labelled as "migrant" or "resident" on the basis of field observations and qualitative interpretations of patterns of ring–recoveries. However, there are intermediate species where only part of the population migrates (partial migrants) or where different components of the population migrate to different extents (differential migrants). Siriwardena, Wernham and Baillie (Siriwardena et al., 2004) develop a novel method that produces a quantitative index of migratory tendency. The method uses distributions of ringing–to–recovery distances to classify individual species’ patterns of movement relative to those of other species. The areas between species’ cumulative distance distributions are used with multi–

C. J. Schwarz, Dept. of Statistics and Actuarial Science, Simon Fraser Univ., 8888 University Drive, Burnaby BC, V5A 1S6 Canada. E–mail: cschwarz@stat.sfu.ca Franz Bairlein, Inst of Avian Research, Vogelwarte Helgoland, An der Vogelwarte 21, Wilhelmsaven 26386, Germany. E–mail: franz.bairlein@ifv.terramare.de ISSN: 1578–665X

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dimensional scaling to produce a similarity map among species. This map can be used to investigate the factors that affect the migratory strategies that species adopt, such as body size, territoriality and distribution, and in studies of their consequences for demographic parameters such as annual survival and the timing of breeding. The key assumption of the method is the similar recovery effort of species over space and time. It would be interesting to overlay maps of effort to try and remove any induced artefacts in the data. Differences in timing or the route of migration has often been studies separately. Lokki and Saurola (Lokki & Saurola, 2004) develop an omnibus procedure to test if the migration timing and/or route differ among two populations of birds (e.g. males vs females). It uses a randomization test to calibrate the test statistic. However, it makes the key assumptions about equal recovery effort in time and space so that the method may be most applicable to comparison among species with similar migration timing and movement to keep differential sighting/recovery rates from affecting the result. Of course, it is in these cases where it is most difficulty to separate the groups which will require substantial samples to have good performance. Thorup and Rahbek (Thorup & Rahbek, 2004) provide a framework for accounting for unequal spatial recovery probability investigating the geometric influence of ocean and sea on observed migratory patterns. Taking the data set of Pied Flycatchers (Ficedula hypoleuca) ringed as nestlings in Scandinavia and recovered en route on their initial migration and using a model based on the clock–and–compass innate navigation hypothesis they are showing that geometric constraints explain quite a bit of the variation in ring–recoveries. The model also shows that ring recovery patterns do reflect the migratory patterns, and that they are suitable for an analysis of the concentration of the migratory route which is important for the general use of ringing data in studies of migration. This is important for the general use of ringing data in studies of migration and dispersal. The new approach has also implications for understanding the migratory orientation program. The compiled papers highlight some novel ideas of how to analyse band recoveries to investigate migration routes and migration behaviour as well as dispersal patterns among birds and dolphins. Multistate modeling appears as a valuable tool as it enables to include various covariates and to analyse patterns of movement that change in time, are influenced by weather, or are different between age classes or sex. References Drake, K. L. & Alisauskas, R. T., 2004. Breeding dispersal by Ross’s geese in the Queen Maud Gulf metapopulation. Animal Biodiversity and Conservation, 27.1: 331–341. King, R. & Brooks, S. P., 2004. Bayesian analysis of the Hector's Dolphin data. Animal Biodiversity and Conservation, 27.1: 343–354. Lokki, H. & Saurola, P., 2004. Comparing timing and routes of migration based on ring encounters and randomization methods. Animal Biodiversity and Conservation, 27.1: 357–368. Siriwardena, G. M., Wernham, C. V. & Baillie, S. R., 2004. Quantifying variation in migratory strategies using ring–recoveries. Animal Biodiversity and Conservation, 27.1: 299–317. Sokolov, L., Chernetsov, N., Kosarev, K., Leoke, D., Markovets, M., Tsvey, Q. & Shapoval, A., 2004. Spatial distribution of breeding Pied Flycatchers Ficedula hypoleuca in respect to their natal sites. Animal Biodiversity and Conservation, 27.1: 355–356 (Extended abstract). Thorup, K. & Rahbek, C., 2004. How do geometric constraints influence migration patterns? Animal Biodiversity and Conservation, 27.1: 319–329. Traylor, J. J. , Alisauskas, R. T. & Kehoe, F. P., 2004. Multistate modeling of brood amalgamation in White– winged Scoters Melanitta fusca deglandi. Animal Biodiversity and Conservation, 27.1: 369–370 (Extended abstract).

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

Siriwardena, G. M., Wernham, C. V. & Baillie, S. R., 2004. Quantifying variation in migratory strategies using ring–recoveries. Animal Biodiversity and Conservation, 27.1: 299–317. Abstract Quantifying variation in migratory strategies using ring–recoveries.— Bird populations have traditionally been labelled as "migrant" or "resident" on the basis of field observations and qualitative interpretations of patterns of ring–recoveries. However, even such a non–systematic approach has identified many intermediate species where only part of the population migrates (partial migrants) or where different components of the population migrate to different extents (differential migrants). A method that would allow a quantitative definition of migratory tendency to be derived for many species would facilitate investigations into the ecological causes and life–history consequences of migratory behaviour. Species or populations could then be placed objectively into the continuum between true residency and an obligate, long–distance migratory habit. We present a novel method for the analysis of ring–recovery data sets that produces just such a quantitative index of migratory tendency for British birds, developed as part of the BTO’s Migration Atlas project (Wernham et al., 2002). The method uses distributions of ringing–to–recovery distances to classify individual species’ patterns of movement relative to those of other species. The areas between species’ cumulative distance distributions are treated as inter–species dissimilarities and a one–dimensional map is then constructed using multi–dimensional scaling. We have used the method in example analyses to show how it can be used to investigate the factors that affect the migratory strategies that species adopt, such as body size, territoriality and distribution, and in studies of their consequences for demographic parameters such as annual survival and the timing of breeding. We have also conducted initial analyses to show how temporal changes in the indices could reveal otherwise unmeasured population consequences of environmental change and thus have an important application in conservation science. Finally, we discuss how our approach to producing indices of migratory tendency could be enhanced to reduce the bias that follows from spatial or temporal variation in reporting rates and how they could be made more broadly valuable by incorporating other data sets and recovery data from other countries. Key words: Migration, Partial migration, Birds, Strategies, Ecology, Demography. Resumen Cuantificación de la variación en las estrategias migratorias mediante la recuperación de anillas.— Tradicionalmente, las poblaciones de aves se han definido como "migratorias" o "residentes" en función de las observaciones de campo y las interpretaciones cualitativas de las pautas de recuperación de anillas. Sin embargo, incluso un enfoque no sistemático de estas características ha sido capaz de identificar numerosas especies intermedias, en las que sólo una parte de la población emigra (especies migratorias parciales), o en las que distintos componentes de la población emigran en mayor o menor grado (especies migratorias diferenciales). Un método que permitiera derivar una definición cuantitativa de la tendencia migratoria de numerosas especies facilitaría las investigaciones de las causas ecológicas y de las consecuencias vitales del comportamiento migratorio. De este modo, las especies o poblaciones podrían situarse objetivamente en el continuo entre verdadera residencia y un hábito migratorio forzoso que obliga a recorrer largas distancias. En este estudio presentamos un método innovador desarrollado como parte del proyecto Migration Atlas del British Trust for Ornithology (BTO) (Wernham et al., 2002), que permite analizar conjuntos de datos obtenidos mediante la recuperación de anillas y elaborar un índice cuantitativo de la tendencia migratoria ISSN: 1578–665X

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de las aves británicas. Para ello se emplean distribuciones de distancias entre el lugar de anillamiento y el de recuperación, pudiendo así clasificar las pautas de movimiento de especies individuales con respecto a las de otras especies. Las áreas entre las distribuciones de distancias acumulativas de las especies se tratan como diferencias interespecíficas, para posteriormente elaborar un mapa unidimensional utilizando una escala multidimensional. Hemos utilizado este método para analizar varios ejemplos que ilustran cómo puede emplearse en la investigación de los diferentes factores que afectan a las estrategias migratorias adoptadas por las especies, tales como el tamaño corporal, la territorialidad y la distribución; y en los estudios que evalúan sus repercusiones en los parámetros demográficos, como la supervivencia anual y el momento de la reproducción. Asimismo, hemos realizado varios análisis iniciales para demostrar de qué modo los cambios temporales en los índices podrían revelar consecuencias poblacionales originadas por el cambio medioambiental, que de otro modo no podrían medirse, lo que nos permite contar con una importante aplicación en la biología de la conservación. Por ultimo, debatimos de qué forma podría perfeccionarse nuestro enfoque para la construcción de índices de tendencias migratorias, de manera que pudiera reducirse el sesgo provocado por la variación espacial o temporal en la tasa de recapturas, y cómo la incorporación de otros conjuntos de datos y de datos de recuperación de otros países podría mejorar significativamente su validez. Palabras clave: Migración, Migración parcial, Aves, Estrategias, Ecología, Demografía. G. M. Siriwardena, C. V. Wernham & S. R. Baillie, British Trust for Ornithology, Thetford, Norfolk IP24 2PU, U.K.

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Introduction

Methods

Migratory strategies are frequently described using a classification of species as migrants or residents. With respect to the birds of Britain & Ireland, such a scheme can separate species that clearly always migrate long distances (such as Swallow Hirundo rustica and Swift Apus apus) from others that are rarely found to venture more than a few kilometres from their breeding areas (such as Blue Tit Parus caeruleus and Dunnock Prunella modularis). However, between species that do not migrate at all and species that can be considered to be obligate migrants there are many "partial migrants". A "partially migrant" species can be defined conservatively as one in which different individuals in a single breeding population have different migratory strategies. Partial migrants might then include species in which almost all individuals migrate long distances, while a minority remain close to their breeding areas, and species in which only a minority leave the breeding grounds. Although such mixed strategies have long been recognized, the classification of migratory behaviour has usually been an ad hoc process, making qualitative use of records of greatly reduced winter numbers, of observations in overseas wintering areas and of any ring–recoveries that have occurred. Because the range of migratory strategies is, in reality, a continuum, such a simple classification cannot describe it in full. Recoveries, recaptures and resightings of ringed birds provide an invaluable tool in research into migration, allowing the locations of individuals at (at least) two points in time to be determined. As part of the research underpinning the British Trust for Ornithology’s Migration Atlas project (Wernham et al., 2002), we investigated more rigorous methods for interpreting ring–recovery data than had been used previously. One aim was to develop a quantitative method for the definition of migratory tendency, i.e. a method by which we could identify where each species lies in the continuum between true residents and true migrants. As well as allowing us to classify strategies objectively, a quantitative system of this kind would allow us to conduct statistical tests to explore the ecological and life– history causes and consequences of variations in migratory tendency. In this paper, we introduce our new approach to describing a species’ migratory behaviour quantitatively, examine some first results of applying the technique to the birds of Britain and Ireland and explore how it might be used further in order to shed light on the evolutionary, ecological and life–history causes and consequences of migration. We also show how the method can be adapted to investigate changes in migratory strategy over time and variation between the strategies of different populations. Our aim here is to provide an overview of the approach and its potential value, describing what it tells us about patterns of migration across species and asking what evidence it can contribute to comparative studies in evolution and ecology.

Quantifying migratory strategy We used the differences between the patterns of distances moved by individuals of different species between their breeding and wintering areas to reveal each species’ migratory tendency relative to that of each other species. Data from the literature on the biology of all species that breed in Britain and Ireland were used to identify seasons (at a resolution of a half–month) during which the majority of birds could safely be assumed to be on their breeding grounds and in the wintering areas that form the end–point of their migration in the non–breeding season: the remaining periods represented conservative definitions of the spring and autumn passage "seasons" (see Wernham & Siriwardena, 2002 for details). Recoveries of dead, ringed birds that had been ringed in each species’ breeding season and recovered during its winter season were then extracted from the BTO’s data archives. Live recaptures and resightings were omitted to avoid the problems caused by the potentially large spatial and temporal variations in sampling effort and the absence of movements under 10 km from the database (see Wernham & Siriwardena, 2002). Problems with bias due to spatial and temporal variation in sampling effort are also likely to affect dead recovery data, but will not be as extreme because they depend on reporting by the general public rather than by specialist ringers or observers, whose activities are likely to be highly concentrated in reserves and other "hotspots", rather than diffuse across habitats frequented by humans. Only birds ringed as adults were included so that movements that could have incorporated an element of natal dispersal were omitted. Recoveries that were not considered to incorporate accurate timing information or that were otherwise potentially compromised by irregular ringing or finding circumstances were also omitted (details of the criteria for inclusion are given in Wernham & Siriwardena, 2002). Reasonable recovery sample sizes were required to produce reliable frequency distributions, so analyses were restricted to the 91 species for which 20 or more suitable recoveries were available. The pattern of migratory movements by each species was interpreted through frequency distributions with respect to distance. Obligate migrants would be expected always to move large distances, true residents always to move only short distances and partial migrants to be recovered at a range of distances. Therefore, a left–skewed frequency distribution, showing a preponderance of short–distance recoveries, would suggest a more resident species (fig. 1C) while a right–skewed distribution, showing that long–distance movements dominate, would suggest an obligate migrant (fig. 1A). A partial migrant would have an intermediate frequency distribution of numbers of recoveries with distance (fig. 1B). The variation in migratory strategy could hence be quantified using the exact shape of the frequency distribution of distances moved.

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We constructed movement distance frequency distributions in two ways that differed in the definition of what constitutes a "long distance". The "absolute" method considered that a given distance is equally meaningful for each species, i.e. that a species in which the whole population moved 10,000 km was more migratory than one in which all birds moved 2,000 km. To produce absolute distributions, the interval between zero and the approximate maximum distance travelled by a passerine (10,000 km: a Swallow movement) was divided into 100 equal segments, such that recoveries of each species were assigned to quantitatively similar categories; any recovery at a distance of more than 10,000 km was assigned to the maximum distance category. The "standardized" method defined long distances on a species–specific basis, as the maximum over which the species had been recorded to move, i.e. it considered species where more individuals migrate over distances approaching the species’ maximum to be more migratory than species that rarely move distances close to the maximum. To produce standardized distributions, the interval between zero and the maximum distance moved by each species was divided into 20 equal segments, such that recoveries were assigned to categories that varied by species, in terms of absolute values, but were a fixed percentage of the maximum distance (5%) in width. Each of the two approaches described above has its strengths and weaknesses, and each emphasizes a different aspect of a species’ migratory behaviour. The "standardized" method allows the distance that constitutes "a long way" to vary between species. This means that two species that differ in the absolute distances that they move but that have similar proportions of their populations stopping to over–winter, say halfway to their most distant wintering areas, will be regarded as equally partially migrant. This should clearly be a desirable property in an index of migratory behaviour (differences in absolute distances moved can be tested independently). A problem with the standardized approach is that it relies on the existence of unusually distant recoveries of truly resident species to generate a left–skewed frequency distribution. These are rare, by definition, in resident species and large recovery samples must often exist before they are found. In their absence, maximum movement distances could be small, indicating no real migration at all but generating a frequency distribution suggesting a partially migrant strategy. The "absolute" method aids the detection of truly resident species by allowing them to have extremely left– skewed distributions but could hinder the detection of species with truly partially migrant strategies when the absolute distances moved are small. However, the use of a large number of divisions of the interval between zero and the maximum distance should maximize the sensitivity of any comparison of "absolute" distributions. Neither the standardized nor the absolute type of frequency distribution provides a clearly superior measure of migratory behaviour, so we analysed the two in parallel and interpreted the results in the light of their properties.

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Once frequency distributions of each type had been produced for each of the 91 species considered, they were converted into cumulative proportions of the sample (fig. 1D, 1E) as in the first step in a Kolmogorov–Smirnov test of the homogeneity of two frequency distributions (Sokal & Rohlf, 1995). This removed any effect of sample size on differences between distributions. The difference between each pair of species was then quantified by calculating the area between the cumulative frequency curves of the two species (fig. 1F). A matrix of "dissimilarity" coefficients was thus generated between each species and each of the others. We used multi–dimensional scaling (MDS; Everitt, 1978; Kruskal & Wish, 1978; Manly, 1986) to express the differences between species in terms of a single index for each of the "standardized" and "absolute" methods. MDS finds the orientation of a set of points, in a specified number of dimensions, that distorts the original between–point distances as little as possible. The distortion of the original dissimilarities in producing a fit with the required dimensionality is measured using a quantity called "stress": values of around 0.05 or less are generally considered to indicate a good fit of the derived locations to the original dissimilarities (Kruskal & Wish, 1978). The dissimilarity matrices derived from the standardized and absolute frequency distributions were analysed using the MDS procedure of SAS (SAS Institute, 1996). Each analysis allowed only a one– dimensional fit (i.e. the expression of the differences between distributions in terms of values on a single linear scale). Because the distributions concerned were generally simple and of a standard shape (sigmoid and asymptotic), differences between them could be readily interpreted as differences in the species’ tendencies to migrate. The "stress" measures for one–dimensional MDS solutions indicated an acceptable fit for both the standardized and absolute approaches to index calculation (0.061 and 0.025, respectively). Testing the causes and consequences of migratory behaviour For the purposes of the Migration Atlas project, the indices were used to place each species in a simple, objective classification according to migratory tendency (Wernham & Siriwardena, 2002). In addition, we conducted a range of exploratory, comparative, multi–species analyses designed to investigate some of the physical, social and ecological factors that might influence or be affected by the choice of a migratory strategy. These analyses were intended to illustrate the potential of the method: there are a number of caveats and unresolved analytical issues that would have to be addressed before definitive results could be produced. We analysed the variation in each of the absolute and standardized indices of migratory tendency between species by testing it against vari-

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A. Swallow

20 15 10 5

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Fig. 1. Producing indices of migratory tendency using the "absolute" method. The graphs show frequency distributions of recoveries with respect to distance category (1–100; ringing to recovery distance): A. Swallow, a migrant; B. Goldfinch, a partial migrant; C. Blue Tit, a sedentary species; D. Swallow, data as a cumulative distribution; E. Goldfinch, data as a cumulative distribution; F. Swallow and Goldfinch, cumulative distributions superimposed to show how the area between the curves provides a quantitative measure of their dissimilarity. Fig. 1. Elaboración de índices de tendencia migratoria mediante el empleo del método "absoluto". El gráfico indica las distribuciones de frecuencia de recuperaciones con respecto a la categoría de distancias (1–100; distancia entre el anillamiento y la recuperación): A. Golondrina común, migradora; B. Jilguero, migrador parcial; C. Herrerillo común, sedentario; D. Golondrina común, datos expresados como una distribución acumulativa; E. Jilguero, datos expresados como una distribución acumulativa; F. Golondrina común y jilguero, distribuciones acumulativas superpuestas a fin de demostrar cómo el área comprendida entre las curvas proporciona una medida cuantitativa de su disimilitud.

ous key variables using non–parametric tests (table 1). The tests were conducted using general linear models in which the ranks of the species with respect to each index of migratory tendency were either regressed against the ranks with respect to a continuous predictor or compared between the alternative classifications of a categori-

cal predictor. The regression tests were identical computationally to Spearman rank correlations (but were, philosophically, regressions) and the latter comparisons formed Kruskal–Wallis non–parametric analyses of variance (Sokal & Rohlf, 1995). The analyses were conducted using the GLM procedure of SAS (SAS Institute, Inc., 1990). We

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used non–parametric methods because we did not know how the indices were distributed and because parametric analyses would emphasize the great variation we found among the more migratory species and the differences between these species and more sedentary ones, rather than the variation among all 91 species considered (see Results). The analyses were then repeated using only subsets of the data: first, using only passerines and birds of prey and, second, using only those species whose recovery sample sizes included 50 or more breeding season–to–winter movements (considering all species, and then passerines and birds of prey alone). In each case, we ran a new comparison of recovery distance distributions and calculated new standardized and absolute indices. Birds of prey and passerines together represent a reasonably homogeneous group of terrestrial species whose migratory strategies are likely to be driven by similar factors. Some relationships might only be detectable using a more homogenous set of species like this. We conducted the tests based only upon larger recovery samples because the shapes of recovery distance frequency distributions are likely to reflect real populations more closely as sample sizes increase. These analyses considered the data on individual species to be independent measures of the relationships in question, an assumption that is unlikely to be strictly true. It was beyond the scope of our exploratory analyses to conduct formal phylogenetic analyses using independent nodes in evolutionary trees (Harvey & Pagel, 1991). Instead, we included in our analyses a specific control for the potentially confounding effects of phylogeny that controlled for interspecific relatedness at (approximately) the superfamily level (Sibley et al., 1988). "Superfamily", a categorical variable, was added into the general linear models relating the migratory tendency indices (transformed into ranks) to each continuous variable (the latter also being transformed into ranks where appropriate). We present results both including and excluding this control, because the disappearance of a significant difference after the introduction of a control for phylogeny would not necessarily indicate that it had been false: it would merely show that the relationship were confounded with phylogenetic differences. Conversely, any significant effects that are detected only after controlling for phylogeny should not be considered to be less important biologically than effects that are detectable in the presence of phylogenetic variation. Such a pattern would occur where a relationship with migratory tendency is significant within phylogenetic groups but is obscured by the variation between the groups when all species are pooled. Phylogenetic controls were not applied to the analyses using categorical variables because many of the categories in each case were entirely confounded with superfamily. The details of the memberships of the superfamily classifications that we used are given in Appendix 3a of Wernham et al. (2002).

Siriwardena et al.

Changes in migratory tendency over time The quantitative indices of migratory tendency described above could readily be applied to any set of sub–divisions of a population, sample sizes permitting, provided that ring–recovery data can reliably be assigned to the sub–divisions. We carried out a first exploration of variation in migratory strategy over time for each species by dividing the larger data sets used in the analyses described above (40 or more recoveries) into two equal parts (around the median recovery year). A total of 73 species had sufficiently large recovery sample sizes for these distributions to be produced. The "standardized" frequency distributions for the periods both before and after the median year were generated using the species–specific maximum distance across the whole data set (early and late combined). We then tested the significance of the differences between the early and late recovery distance distributions using Kolmogorov–Smirnov tests (Sokal & Rohlf, 1995) for each species. This rather coarse temporal analysis had the benefit of maximizing the number of species that could be tested. Ring–recovery sample sizes represent the only constraint on how time periods might be chosen to test more specific or complex temporal hypotheses. The results presented here provide only a guide to what is possible and to where interesting changes might have occurred. Differences between species’ breeding and wintering populations We compared the recovery frequency distributions for the breeding and wintering populations of species present in Britain & Ireland all year to ask whether we could formally identify partial migrants (species whose breeding populations were more migratory than their wintering ones) and species whose British and Irish populations are augmented by winter immigrants. This comparison used standardized and absolute frequency distributions for birds ringed in winter in Britain & Ireland and recovered in the breeding season, in conjunction with the breeding–to–winter distributions used in our other analyses. As previously, tests using absolute distributions will have revealed differences in the absolute distances moved, while those using standardized distributions were sensitive to differences in the proportion of the population that moved. Once again, these analyses were conducted using two–sample Kolmogorov–Smirnov tests (Sokal & Rohlf, 1995). A total of 47 species were considered, all those for which at least 20 suitable recoveries were available from each of the breeding and wintering populations. Results Variations in migratory tendency The range of variation in migratory tendency revealed by the "standardized" and "absolute" forms of

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100

5.5

Hirundo rustica

Catharacta skua

4.5

Absolute index

Puffinus puffinus

Apus apus

Absolute index

Sterna paradisaea

3.5 Sterna hirundo Anas crecca

2.5

Sterna hirundo

Sterna dougalii

Sterna hirundo

1.5

Sylcia atricapilla

Rissa tridactyla

Anthus pratensis

Fulmarus gracilis

Charadrius hiaticula

Gavia stellata Anas strepera Tadorna tadorna

Falco columbarius Larus ridibundus

Larus marinus

Ardea cinerea Turdus merula

Motacilla cinerea

Phalacrocorax aristotelis

Branta canadiensis

Falco peregrinus Mergus merganser

Carduelis chloris

Carduelis carduelis

0.5

Catharacta skua

Larus fuscus

Anthus pratensis

Morus bassanus Alcedo athis

–0.5 –1

Carduelis cannabina Gavia stellata

1 2 3 Standardized index

Streptopelia decaocto

Sylvia atricapilla

Corvus corax

Parus caeruleus

0 0

Sitta europaea Pica pica Athene noctua

Parus major Passer domesticus

Alcedo atthis

20 30 40 50 60 70 80 Standardized index

90 100

Fig. 2. Plot of indices of migratory tendency derived by the absolute method against those derived by the standardized method. Selected species’ data points are identified by their Latin names.

Fig. 3. Plot of indices of migratory tendency derived by the absolute and standardized methods, transformed into ranks. Selected species’ data points are identified by their Latin names.

Fig. 2. Representación gráfica de los índices de tendencia migratoria, derivados mediante el método absoluto, frente a los derivados mediante el método estandarizado. Los puntos correspondientes a los datos de las especies seleccionadas se identifican por sus nombres en latín.

Fig. 3. Representación gráfica de los índices de tendencia migratoria, derivados mediante los métodos absolutos y estandarizados, transformados en rangos. Los puntos correspondientes a los datos de las especies seleccionadas se identifican por sus nombres en latín.

our index is illustrated in figure 2. Each index shows a gradient from more migratory (highly positive values) to more sedentary (highly negative values), so that the concentration towards the bottom left of the graph (especially along the "absolute" axis) shows a large number of species with comparatively sedentary strategies. Obligate long–distance migrants can be seen to form a group discrete from more resident species, especially with respect to the absolute index (more positive index values, towards the top right of figure 2). Most of the variation in strategy therefore separates the few long–distance migrants with sufficiently large sample sizes for analysis from the majority of relatively non–migrant species. Within the broad group of "migrants", there is also greater variation in index values than there is among the more resident species. The uneven spread of species in figure 2 and, in particular, the proximity of genuine partial migrants such as Goldfinch Carduelis carduelis and Linnet C. cannabina to the "sedentary" species cluster suggested that examining the relative indices for each species as they are presented was unlikely to be very informative. The index values were, therefore, transformed into ranks, generating a clearer picture

of the relative positions of each species (fig. 3). These ranks then formed the basis for statistical tests examining the causes and consequences of migratory strategies. Figure 3 shows that the position of most species in the rank order of most to least migrant tends not to be dissimilar in terms of the two indices. The bias in the sample of species towards relatively sedentary species means that only a small proportion of the variation in strategy illustrated in figure 3 is made up of obligate migrants, so partial migrants, such as Meadow Pipit Anthus pratensis, and shorter–distance obligate migrants, such as Blackcap Sylvia atricapilla, can be found towards the "highly migrant" end of the range. Correlates of migratory tendency Body size (length, wingspan, wing length and weight) was a significant predictor of migratory tendency only for passerines and birds of prey (table 2). The significant results with no control for phylogeny suggested that larger species were less migratory, but the opposite pattern was found in the four results that were significant after the control was added (table 2).

306

Body shape had a clearer relationship with migratory tendency: more migratory species tended to have larger wing:body ratios and lower wing–loadings, reflecting morphological adaptations to promote efficient long–distance flight (table 2). There were also consistent, negative relationships between population size and migratory tendency (table 2), suggesting that rarer species tend to be more migratory. However, the effect was limited to the results for the standardized index, indicating that it was not related to the distances moved and that it depended on the input of the relatively sedentary species. A concordant, even stronger pattern was found with respect to distribution (number of occupied breeding atlas squares): species with restricted distributions tended to be more migratory (table 2). This variable had significant effects on the absolute indices as well as the standardized ones, especially when all species were included (table 2). The extent of territoriality was highly significantly related to migratory tendency, with strongly territorial species tending to be less migratory than colonial and solitary or weakly territorial species (table 3). This pattern was stronger in the analyses using absolute recovery distance distributions, suggesting that shorter migratory distances, in absolute terms, are particularly associated with strong territoriality. Nesting habit was also significantly related to migratory tendency, with open–nesting species tending to be more migratory than hole–nesting species by both standardized and absolute approaches, but the pattern was much less clear when passerines and birds of prey were examined alone (table 4). Of the tests exploring the possible effects of migratory strategy on demography (table 5), one very clear result indicated that more migratory species tend to begin to breed later: six of the eight indices of migratory tendency tested gave rise to such a result and five of these gave rise to similar, significant results after the incorporation of controls for phylogeny. This may be unsurprising because migrants are likely to leave their breeding grounds in autumn because conditions are becoming unsuitable and are therefore unlikely to return until the annual improvement in spring is well advanced. However, a correspondingly strong effect with respect to the lengths of breeding seasons was not found, reflecting a tendency for many migrants to finish breeding later as well. When all species were considered, all the indices of migratory tendency indicated that more migratory species had higher survival rates (table 5). However, this pattern was entirely confounded with phylogeny, so may be more related to factors such as body size than to migratory strategy per se. Among the passerines and birds of prey, only one test was significant, but it showed that more migratory species had lower survival rates, and this result was robust to the control for phylogeny (table 5). Changes in migratory tendency We found significant or near–significant temporal shifts in recovery distance distribution by one or both of the standardized and absolute methods of

Siriwardena et al.

calculating frequency distributions for 22 of the 73 species tested (table 6). Of the 51 species for which there was no significant change with time, 23 were species identified as sedentary in tests of the differences between ringing and recovery locations (Wernham et al., 2002). Note, however, that although Mute Swan and Buzzard were classified as non–migratory in the overall analyses, we found significant changes in migratory tendency over time for these species, suggesting that the lack of a clear pattern overall may have masked potentially important temporal variation for some species. More generally, for 14 of the species whose migratory tendency changed over time, the difference found was significant by both methods, the other eight cases involving small shifts in distribution or differences in the short–distance movements undertaken by very sedentary species (table 6). It was commoner for species to have shifted towards shorter than towards longer migratory distances (15 and 5 species, respectively), but there was no clear taxonomic pattern with respect to the direction of the changes (see table 6). Two other species have undergone more complex changes in distribution (table 6), in which the early and frequency distributions were significantly different, but not in ways that can be interpreted as simple changes in the migratory strategy of an average individual. Differences in migratory tendency between breeding and wintering populations Breeding and wintering populations differed in migratory tendency in terms of one or both forms of frequency distribution for 31 of the 47 species tested (table 7). Of these 31 differences, ten were significant only when standardized distributions were used while only one (Mistle Thrush Turdus viscivorus) was significant only when absolute distributions were used. For 11 species, the breeding population was the more migratory, indicating that the species concerned are partial migrants, but for 20 species it was the wintering one, reflecting that immigrants join resident populations in winter (table 7). Non– significant results could indicate entirely sedentary populations or, hypothetically, British & Irish populations that migrate annually between breeding and wintering grounds within the islands, with no immigration from overseas in either season. Discussion Variation in migratory strategy within and between species Our new, quantitative method has identified a wide range of variation in migratory strategy among the species we were able to test and more would doubtless be added if larger recovery samples were available for a larger number of species. We have produced the first objective and quantitative definition of the strategies intermediate between seden-

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Animal Biodiversity and Conservation 27.1 (2004)

Table 1. Variables used in tests of the influences on and of indices of migratory tendency. Tabla 1. Variables empleadas para determinar las influencias de los índices en y de tendencia migratoria.

Factor of interest

Variable(s) used

Categories/derivation

Sources

Potential influences on migratory tendency Body size

Length, wingspan, wing length, weight

Averages for adult birds

Snow & Perrins, 1998

Body shape

Wing–loading and wing:body ratio

Weight/wing length and wingspan/body length, respectively

Snow & Perrins, 1998

Social organization

Coarse classification of breeding strategy

Colonial, intermediate, territorial

Snow & Perrins, 1998

Nesting strategy

Type of nest built/used

Open, hole

Snow & Perrins, 1998

Population density

UK population size, no. of individuals

Distribution

Ubiquity of species within Britain & Ireland

Stone et al., 1997 Number of occupied 10 x 10 km squares in Britain & Ireland, 1988–1991

Gibbons et al., 1993

Potential effects of migratory tendency Survival

Average adult annual survival rate

Timing of breeding

Laying date, length of breeding season

tary residency and an obligate, long–distance migratory habit. Within this spread of strategies (figs. 2, 3), there are interesting patterns with respect to ecology and phylogeny. Many of the most migratory species were waterbirds, waders and seabirds, which reflects both their true strategies and a degree of reporting bias: many are conspicuous or are quarry species at various points along their migration routes, making recovery more likely. The highly migratory position of many seabirds reflects the dispersal of almost all species away from their breeding colonies in winter. We did identify some passerines, such as Swallow and Swift, as being highly migratory, but many migrant passerines (e.g. many warblers) were absent from the data set because ring–recoveries are increasingly scarce further from the British Isles, especially south of the Sahara. Notwithstanding this bias, the pattern in figures 2 and 3 also reflects the frequent occurrence of residency in Britain & Ireland: species such as Greenfinch Carduelis chloris and Chaffinch

Balmer & Peach, 1997; Siriwardena et al., 1998 95th percentile for start Campbell of egg–laying, and 95th percentile for end of breeding season minus 95th percentile for start for length of season (both British & Irish breeders only)

Fringilla coelebs are far more migratory where a continental climate of harsher winters and hotter summers prevails. Conditions in Britain & Ireland are also mild enough to allow some species that always migrate away from breeding grounds elsewhere to become partial migrants (Lundberg, 1988). There is a notable group of species off the main diagonal in figure 3, towards the standardized index axis, that includes Yellowhammer Emberiza citrinella, Buzzard Buteo buteo and Little Owl Athene noctua. These are species for which the standardized index suggests that migratory tendency is somewhat stronger than does the absolute index. In practice, these species mayt demonstrate why the absolute index is needed in addition to the standardized one: their maximum recovery distances are somewhat short, leading to "flat", stretched–out standardized recovery distributions that suggest more partially migratory species–specific strategies than probably exist in reality, at least in terms of the recovery sample.

308

Siriwardena et al.

Table 2. Relationships between indices of migratory tendency and continuous ecological/life–history variables reflecting potential influences on them. Results are shown for indices derived from all species and from raptors and passerines only and from species with 20 or more recoveries and those with 50 or more recoveries within each of these sets. Results (slope parameters and Pvalues) are presented for both the standardized (Stan) and absolute (Abs) methods. P(phylog) shows the P–value for the same relationship when a control for phylogenetic relatedness was incorporated. Slope parameters from univariate tests are shown unless the test was significant only after the control for phylogeny was applied, when the parameter from the latter is given; where both tests produced significant slopes, slope signs were always the same. * Slopes for which the test with or without a control for phylogeny was significant or near–significant at the 5% level; N. Number of species. Tabla 2. Relaciones entre índices de tendencia migratoria y variables continuas ecológicas/ vitales que reflejan las influencias potenciales a las que pueden estar sometidas. Los resultados se indican para índices derivados de todas las especies y únicamente de aves rapaces y paseriformes, así como de especies con 20 o más recuperaciones y de aquellas con 50 o más recuperaciones en cada uno de estos conjuntos. Los resultados (parámetros de pendiente y valores P) se indican tanto para los métodos estandarizados (Stan) como absolutos (Abs). P (phylog) indica el valor P para la misma relación tras haberse incorporado un control para la relación filogenética. Se indican los parámetros de pendiente de los tests univariantes, salvo que el test sólo haya sido significativo tras haber aplicado el control para la filogenia en los casos en que se ha proporcionado el parámetro de ésta; cuando ambos tests dieron pendientes significativas, los signos de pendiente siempre fueron iguales. * Pendientes en los que el test con o sin control para la filogenia fue significativo o casi significativo en el nivel del 5%; N. Número de especies.

All species

Passerines & birds of prey

>20 Recoveries >50 Recoveries >20 Recoveries Variable N

>50 Recoveries

Stan

Abs

Stan

Abs

Stan

Abs

Stan

Abs

Body size Length

Slope

–

0.346

0.304

0.212

0.490

0.738

0.628

0.218

0.035

P(phylog) 0.460

0.159

0.914

0.131

0.138

0.508

0.037

0.399

–

0.246

0.094

0.114

0.158

0.508

0.891

0.218

0.061

P(phylog) 0.336

0.273

0.887

0.250

0.124

0.432

0.024

0.519

–

0.296

0.126

0.159

0.246

0.356

0.860

0.107

0.154

P(phylog) 0.429

0.399

0.930

0.368

0.063

0.230

0.006

0.859

–

0.251

0.200

0.125

0.253

0.822

0.243

0.503

0.010

P(phylog) 0.749

0.244

0.728

0.436

0.506

0.524

0.159

0.111

–

P Wingspan

Slope P

Wing length

Slope P

Weight

Slope P

0.139*

–

0.211* –0.352*

0.211* –0.316*

0.273* –0.243

–0.422*

Body shape Wing–loading Slope P

– –0.280*

–

–0.480*

0.314

0.313

0.165

0.299

0.362

0.060

0.958

0.003

P(phylog) 0.516

0.100

0.962

0.293

0.338

0.610

0.163

0.099

Wing:body

Slope

0.259*

0.352*

0.337* 0.357*

–

ratio

0.013

0.001

0.004

0.002

0.193

0.303

0.176

0.402

P(phylog) 0.245

0.030

0.140

0.061

0.259

0.210

0.108

0.937

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Animal Biodiversity and Conservation 27.1 (2004)

Table 2. (Cont.)

All species

Passerines & birds of prey

>20 Recoveries >50 Recoveries >20 Recoveries Variable

Stan

Abs

Stan

Abs

Stan

Abs

>50 Recoveries Stan

Abs

Population Population

Slope

size

–0.312* 0.003

0.111

–

–0.308* 0.009

0.099

– –0.432* 0.003

0.718

– –0.469* 0.004

0.885

–

P(phylog)

0.003

0.463

0.058

0.156

0.001

0.654

0.012

0.268

Distribution No. atlas

Slope

-0.495* –0.408* –0.551* –0.448* –0.323* 0.261* –0.313*

squares

<0.001

occupied

P(phylog) <0.001

0.128

<0.001 <0.001 0.001

0.047

–

0.029

0.080

0.063

0.150

0.031

0.352

0.145

0.739

Table 3. Tests of the variation in migratory tendency between species with territorial, weakly territorial or semi–colonial and colonial habits. The categories were compared using Kruskal–Wallis (K–W) tests; N. Numbers of species; MR. Mean rank. (Results are shown for both the standardized and absolute indices.) Tabla 3. Test de variación en la tendencia migratoria entre especies con hábitos territoriales, territoriales débiles o semicoloniales y coloniales. Las categorías se compararon utilizando tests de Kruskal–Wallis (K–W): N. Número de especies. MR. Rango medio. (Se indican los resultados para los índices estandarizados y absolutos.)

Standardized >20 Recoveries Social organization

Absolute

>50 Recoveries MR

>20 Recoveries

Colonial

52.5

43.1

Intermediate

54.5

41.1

Territorial

37.6

28.1

>50 Recoveries

58.0

45.8

52.6

40.9

34.2

25.7

All species

K–W H (2 d.f.) P

7.94 0.019

8.69 0.013

16.0 < 0.001

14.9 < 0.001

Passerines & birds of prey only Colonial

23.1

16.8

21.9

13.8

Intermediate

27.8

24.1

29.9

26.1

Territorial

22.5

17.2

22.2

17.3

K–W H (2 d.f.) P

0.985 0.611

2.50 0.287

2.19 0.335

5.02 0.081

310

Siriwardena et al.

Table 4. Tests of the variation in migratory tendency between species with open– and hole–nesting strategies. (For abbreviations, see table 3.) Tabla 4. Tests de variación en la tendencia migratoria entre especies con estrategias de nidificación en nido abierto o en agujero. (Para las abreviaturas, ver la tabla 3.)

Standardized >20 Recoveries Nesting strategy

Absolute

>50 Recoveries

Open

49.0

38.6

Hole

33.1

22.1

>20 Recoveries N

>50 Recoveries

49.9

38.6

29.1

All species

K–W H (1 d.f.) P

4.95 0.026

5.91 0.015

8.60 0.003

21.9 6.07 0.014

Passerines & birds of prey only Open

24.3

19.9

26.1

19.7

Hole

22.2

15.3

19.0

15.8

K–W H (1 d.f.) P

0.262 0.609

Our quantitative indices could prove most valuable in exploring intraspecific and interspecific variations in the patterns of movement of species with strategies intermediate between genuine residents like House Sparrow Passer montanus and Bullfinch Pyrrhula pyrrhula and the long–distance migrants in figures 2 and 3. These species include Goldfinch, Linnet, Puffin Fratercula arctica, Tufted Duck Aythya fuligula, Oystercatcher Haematopus ostralegus and Lesser Black–backed Gull Larus fuscus: it is from studying them that new evidence as to how bird migration in general is controlled is most likely to come and quantifying their strategies has been a first step. The consensus of research to–date is that migratory tendency is under strong genetic control (Berthold, 1996, 2001), although the detail of how the distances individuals move are determined remains unclear. Intraspecific variation in strategy may derive from effects of genetic diversity, dominance or factors related to life–histories, such as age– or sex–specific variation. Our indices do not currently allow these factors to be separated, but physical and social influences, for example, are often confounded and the distinction between "differential migration" with respect to age or sex and socially–mediated "partial migration" may be more semantic than biologically meaningful, because variation in migratory strategy is likely to be correlated with some genetic or demographic variation even if social dominance is the real determining factor (Siriwardena & Wernham, 2002).

1.49 0.223

3.03 0.082

1.06 0.303

Temporal changes in migration patterns Changes in migratory tendency over time are of intrinsic interest in biology and can have important implications for conservation, for example, as either causes or consequences of variations in abundance. For example, climate change could cause a decrease in the proportion of a partial migrant’s population that migrates, making it less vulnerable to hunting pressure overseas, or migration to distant wintering grounds could be density–dependent such that breeding population increases in Britain have little effect on winter abundance. In general, any improvements to our understanding of past influences on changes in abundance would aid the development of conservation policies for the future. There were no clear associations between changes in migratory behaviour and population trends (from Baillie et al., 2001; Gibbons et al., 1993), but our analyses revealed an interesting tendency for migratory movements to have become shorter, as would be predicted of an effect of global warming. Migratory populations or individuals might have begun to migrate shorter distances or the proportion of a population that migrates might have fallen. (Closer scrutiny of the recovery distance distributions would clarify this for each species.) However, examination of the species involved suggests that an effect of global warming is unlikely to be a general explanation: they include several seabirds and waders, but not the partially migrant terrestrial species like Meadow Pipit, Linnet and Goldfinch that might be expected to be respond to a warmer

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Table 5. Relationships between indices of migratory tendency and continuous ecological/life–history variables reflecting their potential effects on demography. (For abbreviations, see table 2.) Tabla 5. Relaciones entre los índices de tendencia migratoria y variables continuas ecológicas/vitales que reflejan sus posibles efectos en la demografía. (Para las abreviaturas, ver la tabla 2.)

All species

Passerines & birds of prey

>20 Recoveries >50 Recoveries Variable

>20 Recoveries > 50 Recoveries

Stan

Abs

Stan

Abs

Stan

Abs

Stan

Abs

Timing of breeding Sample size First

Slope

0.465

0.540

0.500

0.540

–

0.283

–

0.318

egg date

<0.001

0.282

0.054

0.305

0.056

P(phylog)

0.002

<0.001

0.002

<0.001

0.634

0.088

0.741

0.252

Slope

–0.173

–

of breeding P

0.105

0.303

0.260

0.541

0.986

0.535

0.613

0.793

season

0.079

0.120

0.642

0.595

0.673

0.755

0.096

0.561

Length

P(phylog)

Survival Sample size Annual

Slope

0.258

0.343

0.338

0.356

–

–0.480

survival

0.025

0.002

0.008

0.005

0.984

0.603

0.923

0.006

rate

P(phylog)

0.786

0.744

0.698

0.558

0.742

0.869

0.542

0.078

climate with a simple reduction in migratory distance (table 6). It is also notable that no temporal change was detectable for 51 of the 73 species tested (only half of which were identified as sedentary by Wernham et al., 2002; table 6), suggesting that large and uniform effects of climate change on migratory behaviour have not occurred, although more changes might have been found with larger recovery sample sizes. However, changing migratory distance is just one of several potential responses of bird populations to climate change, and one which, like the others, could be constrained because the behaviours concerned are controlled by endogenous rhythms and photoperiodic cues that are unrelated to climate (Coppack & Both, 2002). It is interesting that a relatively large proportion (9/22) of the species for which significant changes were found were seabirds and we found several complex patterns of change (table 6), which may reflect a tendency for only some parts of the population to become more migratory (for example, young birds or birds from further north). These patterns need further investigation, probably on a species–by–species basis. In fact, because climate change, at least, is likely to affect different species differently, future analyses might best be species– specific. The year around which distributions were divided could clearly have been critical in determin-

ing what temporal changes were revealed and this would be the first parameter to vary in future analyses. Adequately controlled tests, focusing on the species and time periods most appropriate for testing specific hypotheses, are now needed to provide a definitive answer about the possible effects of global climate change on migration patterns. Such research could build on our method and on the results presented here. Comparisons of breeding and wintering populations Our analyses provided a formal identification of a range of species as having either partially migrant breeding populations or winter populations that are augmented by immigrants (table 7). Passerines were significantly more likely to fall into the former category (G–test, Sokal & Rohlf (1995): G = 11.50, 1 d.f., P < 0.001), reflecting the use of Britain & Ireland as a wintering ground by many non– passerines that also breed in the islands. Large numbers of Fennoscandian breeding Blackbirds Turdus merula and Chaffinches also winter in the British Isles, joining the breeding populations (Wernham et al., 2002), but these species are unusual. All these patterns are already well–known so, apart from the new, quantitative information that our analyses provide, the more interesting

312

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Table 6. Results of tests of temporal changes in migratory tendency. Species are shown in taxonomic order. Sample sizes (S) refer to total numbers of breeding–to–winter movements that were divided in two around the median recovery years (Y) (19…) shown to produce "early" and "late" data sets. K–S test results (K–S) show significance levels from two–sample Kolmogorov–Smirnov tests as follows: * P < 0.1, ** P < 0.05, *** P < 0.01. Direction (D) indicates the direction of the effects: a minus (–) indicates that movements have become shorter with time and a plus (+) that they have become longer. Differences significant in one analysis only: #. Significant only in the standardized analysis because differences are small in terms of absolute distance; ##. A slight shift towards shorter migratory distances which was detected only by the greater distance resolution in the absolute analysis. Complex effects: ‡. Most recoveries have been in the middle of the range of distances (by both methods), wherein they became less distant after the median year, but other recoveries have been spread throughout the range of distances in both periods and many more minor shifts in the distribution have also occurred; †. Movements before the median year had a bimodal distribution; subsequently, the shorter distance movements became rarer and the longer distance movements more spread out, i.e. commonly both longer and shorter than the previous longer mode. Tabla 6. Resultados de tests de cambios temporales en la tendencia migratoria. Las especies se indican por orden taxonómico. Los tamaños de las muestras (S) se refieren al número total de movimientos entre la reproducción y el invierno, que se dividieron en dos en torno a los años medios de recuperación (19...), indicados para elaborar conjuntos de datos anteriores y posteriores. K–S test results indica los niveles significativos obtenidos en tests de Kolmogorov–Smirnov para dos muestras según lo indicado a continuación: * P < 0,1, ** P < 0,05, *** P < 0,01. Direction indica la dirección de los efectos: un signo menos (–) indica que los movimientos se han reducido con el tiempo, mientras que un signo más (+) revela que se han prolongado. Diferencias significativas en un único análisis: #. Significativa sólo en el anáisis estandarizado porque las diferencias son pequeñas en términos de distancia absoluta; ##. Un ligero cambio hacias distancias migratorias más pequeñas detectadas solo por la mayor resolución de distancia en el análisis absoluto. Efectos complejos: ‡. La mayoría de recuperaciones se encuentran en el centro de la gama de distancias (en ambos métodos), donde llegaron a ser menos distantes despues de la mitad del año, pero otras recuperaciones se han dispersado a través de la gama de distancias en ambos períodos y además se han producido más cambios de menor importancia en la distribución; †. Los movimientos antes de la mitad del año tenían una distribución bimodal; posteriormente, los movimientos de corta distancia se convirtieron en más raros y los de larga distancia más dispersos, es decir comunmente tanto más largo y más corto que la moda previa más larga.

K–S Species

Fulmar

111

Manx Shearwater 61

Gannet

*** ***

387

Cormorant

2,006

Shag

1,745

Grey Heron

462

Mute Swan

1,997

Greylag Goose

114

88.5

Canada Goose

1,902

74.5

2,492

Shelduck Mallard

Stan

K–S

Abs D

Roseate Tern

179

†

Common Tern

172

***

–

Arctic Tern

Guillemot

1,830

–

Razorbill

710

Puffin

143

–

Stock Dove

70.5

***

–

Species

Wood Pigeon

559

1,207

Tawny Owl

253

Kingfisher

Barn Owl

Pochard

Tufted Duck

261

***

Eider

524

***

Hen Harrier

***

Swallow

68.5

–

Meadow Pipit

–#

Pied Wagtail

308

Wren

109

Stan

Abs D *** –## +#

313

Animal Biodiversity and Conservation 27.1 (2004)

Table 6. (Cont,)

K–S Species

Stan

K–S Abs

Species

Stan

Abs D

Sparrowhawk

522

Dunnock

321

Buzzard

110

***

–#

Robin

590

Kestrel

863

***

–#

Blackbird

1,525

Merlin

122

Song Thrush

963

***

–

Peregrine

Mistle Thrush

Moorhen

65.5

Blue Tit

621

Coot

73.5

Great Tit

308

Oystercatcher

383

Ringed Plover

67.5

Lapwing

701

Snipe

Woodcock

277

Curlew

181

Redshank

125

Great Skua

240

Black–headed Gull 1,134

Common Gull

Lesser B–b Gull

654

Herring Gull

977

***

*** ***

** ***

** **

– +

–

Jay

129

Magpie

Jackdaw

113

Rook

108

Crow

156

73.5

Raven

Starling

1,433

65 66

–

House Sparrow

358

Chaffinch

154

Greenfinch

382

Goldfinch

118

Linnet

197

Great B–b Gull

220

Bullfinch

338

Kittiwake

240

Reed Bunting

1,043

Sandwich Tern

***

–#

‡

results may involve the species for which such patterns have not been described before. However, it is important to note that the results for these species, which include Mute Swan Cygnus olor, Coot Fulica atra, Blue Tit Parus caeruleus and Rook Corvus frugilegus, may only reflect effects of dispersal or of geographical or habitat differences between the breeding and wintering samples. For example, any dispersal in autumn will have been confounded with movements to wintering grounds and the breeding and winter samples may differ, for example, if the predominant catching methods or types of site for ringing differ markedly between seasons. Causes and consequences of differences in migratory tendency Our analyses of some of the potential ecological and life–history correlates of interspecific variation in migratory tendency revealed some interesting patterns. Comparing the results with and without con-

trols for phylogenetic relatedness provides a particularly useful perspective on the taxonomic level at which relationships occur and on the possibility that they are artefacts of other differences between taxonomic groups. With respect to body size, some results indicated that larger species are less migratory, as predicted, but the relationship was detectable only among passerines and birds of prey, related only to the absolute distances that they moved rather than to the proportions of their populations that migrate and was confounded with phylogeny. This result suggests that body size is related to the dichotomy between migrant and resident strategies as either is adopted at the superfamily level. After controlling for phylogeny, i.e. within superfamilies, larger proportions of the populations of larger species tend to migrate. This could indicate that larger species are more likely than their close relatives to move away from their breeding areas, although not necessarily moving very far, perhaps because their size allows them greater mobility.

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There were no such apparent contradictions in the results with respect to body shape, but the effect of wing–loading was detectable only with the absolute index for passerines and birds of prey and the effect of wing:body ratio was detectable only when all species were included (table 2). Further, only the relationships between the absolute indices and wing:body ratio were still significant after controls for phylogeny had been added (table 2). The effect of wing–loading on passerines and birds of prey was therefore confounded with phylogenetic variation and was either unimportant or, perhaps more probably, obscured by other factors when all species were included. The wing:body ratio results showed that individuals of relatively longer–winged species tend more often both to go further and to adopt a migrant strategy, but that only the former pattern was robust to phylogenetic variation. This suggests that morphology is related to the absolute distances moved by individual species, but that it is also related to whether species migrate or not at a higher taxonomic level. The relationships between population size and distribution and migratory tendency (table 2) probably reflect the fact that species near the northern edges of their breeding ranges are more likely to have to move south and west in winter and that the same species to tend to be rarer than species that are able to remain in the British Isles all year round. This test does not examine the effect of population "density", however, because different species’ populations will be distributed differently in the landscape according to social organization and variations in habitat. Higher quality individuals of species that are both territorial and migratory tend to arrive on their breeding grounds earlier (e.g. Francis & Cooke, 1986; Møller, 1994; Lozano et al., 1996), probably reflecting a system in which individuals compete for access to high quality territories via arrival times (Kokko, 1999). A logical corollary of this is that residency is promoted by territoriality and that there is a concomitant selection pressure for remaining closer to breeding areas in winter or returning to them earlier in spring. Among all species, our results supported this hypothesis (table 3), but the pattern was much weaker when passerines and birds of prey were considered alone, perhaps because most of these species are territorial. However, it probably also reflects the extent to which social organization is confounded with phylogeny. Whether species build open–cup or hole nests could be an important determinant of migratory behaviour, because nest–holes may be limiting for the latter, potentially providing a selective pressure for residency, shorter migration distances or an early return to the breeding grounds (Von Haartman, 1968; Alerstam & Högstedt, 1981). Our results support the idea that open–cup nesters are less constrained in annual movements by their breeding strategy (table 3). The pattern was weaker when passerines and birds of prey were examined alone

Siriwardena et al.

(table 3), suggesting that the pattern across all species could largely reflect more frequent hole– nesting among the passerines and birds of prey included, which tend to be relatively sedentary (figs. 2, 3). Nevertheless, the prevalence of hole–nesting among the passerines and raptors could also be a key factor driving their relatively low average migratory tendency. Our results suggest that demography is strongly influenced by migratory strategy, in terms of both the timing of breeding and survival: migrants bred later and tended to have higher mortality rates (after controlling for phylogeny; table 5). No effect of laying date was found in the standardized indices for passerines and birds of prey, suggesting that, at least among these species, later laying is associated more with migratory distance than with the selection of migratory strategy. This makes intuitive sense given that a major constraint on the date at which breeding can begin is likely to be travelling time. The effect of survival was also detected with the absolute index, suggesting that species that migrate further tend to have lower survival rates (presumably reflecting the hazards of migration). We also conducted tests of relationships with reproductive effort (egg volume and clutch size) but these were inconclusive, probably because this demographic variable has other components in addition to the ones that we were able to test (Siriwardena & Wernham, 2002). Many possible influences on, and consequences of, migratory tendency remain to be tested. These tests have not been conducted here either because they would require analyses too complex for an exploratory analysis or because the data needed were not available to us. It would also be useful to re–run our analyses using more formal phylogenetic controls and to investigate possible interactions between the relationships we have found using multivariate analyses. The potential of the approach: caveats and future work Analyses of ring–recovery data are generally dependent on the spatial and/or temporal distributions of ringing and recovery effort, depending on the question being asked, and the present study is no exception. Our results refer only to the populations of birds that the relevant ringing and recovery activity has sampled and the extent to which this is representative of the relevant British and Irish population will vary between species, regions and time periods. Specifically, ringing activity will vary spatially with human population density and habitat (proportionally more being conducted in gardens than in mature woodland, for example), and it will also vary over time (both between seasons and in the long term) as catching methods change and particular projects begin and end. The occurrence of recoveries will also vary spatially and temporally because of variation in reporting probabilities. Many species will have a reporting probability close to zero at sea, but it will also vary along migration routes

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Table 7. Results of statistical tests for differences between the migratory tendencies of populations using Britain & Ireland in the breeding season and in winter. Sample sizes (numbers of recoveries) are shown for birds ringed in the breeding season and recovered dead in winter ("B–W") and for birds ringed in winter and recovered dead in the breeding season ("W–B"). "K–S test results" show significance levels from two–sample Kolmogorov–Smirnov tests as follows: * P < 0.1, ** P < 0.05, *** P < 0.01. The final column (M) indicates which of the British & Irish breeding (B) or wintering (W) populations appeared more migratory. Tabla 7. Resultados de tests estadísticos para las diferencias entre las tendencias migratorias de poblaciones de Gran Bretaña e Irlanda en la estación de reproducción y en invierno. Se indican los tamaños de las muestras (número de recuperaciones) para las aves que fueron anilladas en la estación de reproducción y que se recuperaron muertas en invierno ("B–W"), así como para las aves anilladas en invierno y recuperadas muertas en la estación de reproducción ("W–B"). "K–S test results" indica los niveles significativos obtenidos en tests de Kolmogorov–Smirnov para dos muestras según lo indicado a continuación: * P < 0,1, ** P < 0,05, *** P < 0,01. La última columna (M) indica cuáles de las poblaciones reproductoras británicas e irlandesas (B) o invernantes (W) mostraron una mayor tendencia migratoria.

Sample sizes

K–S test results

Species

B–W W–B Stan Abs

Mute Swan

1,997

Canada Goose

1,902

Mallard Pochard

Teal

722

Sample sizes Species

B–W

Wren

109

K–S test results M

W–B Stan Abs

***

Dunnock

321

545

211

*** ***

Robin

590

448

2,492

844

*** ***

Blackbird

105

*** ***

Song Thrush

1,525 3,528 ** 963

875

Tufted Duck

261

248

*** ***

Mistle Thrush

Eider

524

103

***

Blue Tit

621

Sparrowhawk

522

Great Tit

308

529

Moorhen

Nuthatch

***

160

Jay

129

Oystercatcher

383

635

*** ***

Magpie

Dunlin

*** ***

Jackdaw

113

***

108

181

*** ***

Rook

Redshank

125

*** ***

Starling

1,134 1,635 *** ***

House Sparrow

358

673

Tree Sparrow

***

664

1,433 6,214 ***

B B

Curlew

Common Gull

1,349 **

Coot

Black–headed Gull

***

B ***

*** ***

Lesser Black–backed Gull 654

141

*** ***

Chaffinch

154

Herring Gull

977

438

***

Greenfinch

382

Greater Black–backed Gull 220

*** ***

Goldfinch

118

***

Wood Pigeon

105

***

Linnet

197

***

Bullfinch

338

270

***

Yellowhammer

111

Reed Bunting

Collared Dove

559 25

1,207

Tawny Owl

253

Pied Wagtail

308

278

Barn Owl

*** ***

W ***

3,679 ***

W B

over land, both due to the population density of potential human reporters and cultural factors such as interest in wildlife and knowledge of conservation

activity in general and ringing in particular. Variations in hunting activity in time and space are also a key influence. As well as determining which species

316

had sufficient data to be investigated at all, these variations in ringing and reporting will have determined the (sub–) populations that our analyses considered and will also have influenced the shapes of the recovery distance distributions that we used to produce indices of migratory tendency. Some of the biases they have caused may be unimportant because all samples in a given comparison were affected similarly (for example, a temporal comparison where the same spatial fraction of the population was sampled in both periods compared). Others, however, may have had a critical influence on our results (for example, if hunting pressure has changed over time such that a major source of recoveries from a given region that contribute to a temporal comparison cease). Such problems should be taken as a general caveat to our results and will have affected some analyses (such as temporal and breeding–winter comparisons) more than others, but they are not insurmountable. We have investigated the magnitude of, and variation in, biases such as those described above elsewhere (Wernham et al., 2002) and the results of such studies could be used to develop controls for variations in reporting rate and ringing activity, which could then be used to correct, or to modify, recovery distance distributions. For example, one approach would be to formulate reporting probability profiles with respect to distance along key migration routes and to apply these to the observed recovery distributions. Such profiles could be developed from spatial analyses of ring–recoveries or recaptures, perhaps building on the random–walk approach of Manly & Chatterjee (1993) or the approach for estimating dispersal distances from mark–recapture data developed by Thomson et al. (2003). Other forms of location or distance data could also be added to distributions such as we have used if they could be translated into distance– frequency profiles for individual species or produced as modifiers of the existing frequency distributions. A different approach, in the long term, would be to target ringing activity would to make the spread of sampling effort more even. Our indices of migratory tendency represent the first attempt to–date to quantify avian migratory strategies. The method has several caveats and further refinements are needed, but nonetheless we believe that it represents a significant advance and that it has already produced useful and interesting results, starting with an objective classification of the migratory behaviour of all British and Irish birds for which sufficient data are available (Wernham et al., 2002). Future applications of the methods, or adaptations of it, could include further comparative studies of the evolutionary influences on migratory behaviour and tests of the effects of temporal changes in climate, habitat or population density on the migratory tendency of bird populations. In addition, more powerful tests of hypotheses are likely to be possible by carrying out comparative analyses on data sets from different ringing schemes or by developing the technique to produce pan–European indices and analyses.

Siriwardena et al.

Acknowledgements We thank Mike Toms, John Marchant, Jacquie Clark and all the other contributors to the Migration Atlas project, as well as all those who have contributed to the ringing and recovery of British and Irish birds over the years. This work was funded by the BTO, generous donations from its members, the Heritage Lottery Fund and the Central Science Laboratory. The BTO Ringing Scheme is funded by the BTO, the Joint Nature Conservation Committee (on behalf of English Nature, Scottish Natural Heritage and the Countryside Council for Wales, and also on behalf of the Environment and Heritage Service in Northern Ireland) and Duchas The Heritage Service (in the Republic of Ireland). References Alerstam, T. & Högstedt, G., 1981. Evolution of hole– nesting in birds. Ornis Scand., 12: 188–193. Baillie, S. R., Crick, H. Q. P., Balmer, D. E., Bashford, R. I., Beaven, L. P., Freeman, S. N., Marchant, J. H., Noble, D. G., Raven, M. J., Siriwardena, G. M., Thewlis, R. & Wernham, C. V., 2001. Breeding birds in the wider countryside: their conservation status 2000. Research Report 252. British Trust for Ornithology, Thetford. www.bto.org/birdtrends Balmer, D. E. & Peach, W. J., 1996. Review of Natural Avian Mortality Rates. BTO Research Report 175. BTO, Thetford, Norfolk. Berthold, P., 1996. Control of Bird Migration. Chapman & Hall, London. – 2001. Bird Migration: a general survey. Second edition. Oxford University Press, Oxford. Campbell, B. & Ferguson–Lees, J. R., 1972. A Field Guide to Birds’ Nests. Constable, London. Coppack, T. & Both, C., 2002. Predicting life–cycle adaptation of migratory birds to climate change. Ardea, 90: 369–378. Everitt, B. S., 1978. Graphical Techniques for Multivariate Data. Heinemann, London. Francis, C. M. & Cooke, F., 1986. Differential timing of spring migration in wood–warblers (Parulinae). Auk, 103: 548–886. Gibbons, D. W., Reid, J. B. & Chapman, R. A., Eds., 1993. The New Atlas of Breeding Birds in Britain and Ireland: 1988–1991. T & A D Poyser, London. Harvey, P. H. & Pagel, M. D., 1991. The Comparative Method in Evolutionary Biology. Oxford Univ. Press, Oxford. Kokko, H., 1999. Competition for early arrival in migratory birds. J. Anim.. Ecol., 68: 940–950. Kruskal, J. B. & Wish, M., 1978. Multidimensional Scaling. Sage University Paper Series on Quantitative Applications in the Social Sciences, 07– 011. Sage Publications, Beverly Hills, USA. Lozano, G. A, Perrault, S. & Lemon, R. E., 1996. Age, arrival date and reproductive success of

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male American Redstarts, Setophaga ruticilla. J. Avian Biol., 27: 164–170. Manly, B. F. J., 1986. Multivariate statistical methods: a primer. Chapman & Hall, London. Manly, B. F. J. & Chatterjee, C., 1993. A model for mark–recapture data allowing for animal movement. In: Marked Individuals in the Study of Bird Population: 309–322 (J.–D. Lebreton, & P. M. North, Eds.). Birkhauser Verlag, Basel, Switzerland. Møller, A. P., 1994. Phenotype–dependent arrival time and its consequences in a migratory bird. Behav. Ecol. Sociobiol., 35: 115–122. SAS Institute, Inc., 1990. SAS/STAT User’s Guide, Version 6. Fourth edition. Volume 2, GLM– VARCOMP. SAS Institute, Cary, North Carolina. Sibley, C. G., Ahlquist, J. E. & Monroe, B. L. Jr., 1988. A classification of the living birds of the world based on DNA–DNA hybridization studies. Auk, 105: 409–423. Siriwardena, G. M., Baillie, S. R. & Wilson, J. D., 1998. Variation in the survival rates of some British farmland passerines with respect to their population trends on farmland. Bird Study, 45: 276–292. Siriwardena, G. M. & Wernham, C. V., 2002. Synthesis of the migration patterns of British and Irish birds. In: The Migration Atlas: movements of the birds of Britain and Ireland: 70–102. (C. V. Wernham, M. P. Toms, J. H. Marchant, J. A. Clark, G. M. Siriwardena & S. R. Baillie, Eds). T

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& A D Poyser, London. Sokal, R. R. & Rohlf, F. J., 1995. Biometry, 3rd Edition. Freeman, London. Snow, D. W. & Perrins, C. M., Eds., 1998. The Birds of the Western Palearctic. Concise edition. Oxford Univ. Press, Oxford. Stone, B. H., Sears, J., Cranswick, P. A., Gregory, R. D., Gibbons, D. W., Rehfisch, M. M., Aebischer, N. J. & Reid, J. B., 1997. Population estimates of birds in Britain and in the United Kingdom. British Birds, 90: 1–22. Thomson, D. L., van Noordwijk, A. & Hagemeijer, E. J. M., 2003. Estimating avian dispersal distances from data on ringed birds. J. Appl. Stat., 30: 1003–1008. Von Haartman, L., 1968 The evolution of resident versus migratory habit in birds: some considerations. Ornis Fenn., 45: 1–7. Wernham, C. V. & Siriwardena, G. M., 2002. Analysis and interpretation of ring–recovery data. In: The Migration Atlas: movements of the birds of Britain and Ireland: 44–69 (C. V. Wernham, M. P. Toms, J. H. Marchant, J. A. Clark, G. M. Siriwardena & S. R. Baillie, Eds.). T & A D Poyser, London. Wernham, C. V., Toms, M. P., Marchant, J. H., Clark, J. A., Siriwardena, G. M. & Baillie, S. R., Eds., 2002. The Migration Atlas: movements of the birds of Britain and Ireland. T & A D Poyser, London.

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"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

Animal Biodiversity and Conservation 27.1 (2004)

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How do geometric constraints influence migration patterns? K. Thorup & C. Rahbek

Thorup, K. & Rahbek, C., 2004. How do geometric constraints influence migration patterns? Animal Biodiversity and Conservation, 27.1: 319–329. Abstract How do geometric constraints influence migration patterns?— Null models exclusively invoking geometric constraints have recently been demonstrated to provide new insight as to what explains geographic patterns of species richness and range size distribution. Analyses of migration patterns have traditionally been conducted in the absence of appropriate simulations and analytical models. Here we present a null model exclusively invoking geometric constraints and a more advanced analytical model incorporating spread along a migration direction that allow investigation of the influence of physiographical and physiological boundaries for terrestrial taxa, with ocean and sea as geometric constraints, in relation to observed patterns of migration. Our models take into account the low recovery probability of terrestrial taxa over sea. The null model was not found to explain any of the directional variation in the ring–recoveries, but when comparing the distribution of data modeled using a simple clock–and–compass model with distributions of ring–recoveries, geometric constraints were found to explain up to 22% of the variation in ring–recoveries. However, the assumed directional concentrations per step used in the model were much higher than expected, and the qualitative fit of the model was rather poor even when non–terrestrial sites of recoveries were excluded. Key words: Bird migration model, Geometric constraints, Ring–recovery probability, Pied Flycatcher, Ficedula hypoleuca. Resumen ¿Cómo influyen las limitaciones geométricas en las pautas de migración?— Recientemente se ha demostrado que los modelos nulos que recurren exclusivamente a las limitaciones geométricas proporcionan nuevas aportaciones para explicar las pautas geográficas que definen la riqueza de las especies y la distribución por tamaños según el rango. Tradicionalmente, los análisis de pautas de migración se han realizado sin emplear simulaciones ni modelos analíticos apropiados. En este estudio presentamos un modelo nulo que se basa exclusivamente en limitaciones geométricas, así como un modelo analítico más avanzado que incorpora la dispersión y una dirección de migración, lo que permite investigar la influencia de los límites fisiográficos y fisiológicos en los taxones terrestres, tomando el océano y el mar como limitaciones geométricas, con relación a las pautas de migración observadas. Los modelos que hemos empleado tienen en cuenta la baja probabilidad de recuperación de los taxones terrestres en el mar. El modelo nulo no pudo explicar ninguna de las variaciones direccionales en las recuperaciones de anillas; sin embargo, al comparar la distribución de los datos modelados utilizando un modelo simple de reloj y brújula con distribuciones de recuperaciones de anillas, se constató que las limitaciones geométricas podían explicar hasta el 22% de la variación en las recuperaciones de anillas. Pese a ello, las concentraciones direccionales por pasos que se presupusieron en el modelo fueron muy superiores a lo previsto, y el ajuste cualitativo del mismo resultó bastante deficiente cuando se excluyeron los emplazamientos de recuperaciones no terrestres. Palabras clave: Modelo de migración de aves, Limitaciones geométricas, Probabilidad de recuperación de anillas, Papamoscas cerrojillo, Ficedula hypoleuca. Kasper Thorup & Carsten Rahbek, Vertebrate Dept., Zoological Museum, Univ. of Copenhagen, Universitetsparken 15, DK–2100, Denmark. Corresponding author: K. Thorup. E–mail: kthorup@zmuc.ku.dk ISSN: 1578–665X

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Introduction Migration is an ecological and evolutionary important phenomenon (e.g. Alerstam, 1990). Especially in some of the most mobile terrestrial animals, the birds, it is common to explore spatially separated regions by moving long distances. It is estimated that approximately half of the world’s roughly 9,000 currently recognized species of birds, corresponding to individuals in a magnitude of 50,000,000,000 perform some kind of migratory movement (Berthold, 2001), and Moreau (1972) estimated that in total 5,000,000,000 Palearctic landbirds leave their breeding grounds for Africa. Similar phenomena occur between North America and Central and South America and in Asia. Such behavior poses special demands for behavioral and physiological adaptations. Research on orientation in birds helps us to understand the constraints on evolution of bird migration, but bird migration is itself constricted by geographic boundaries. Nevertheless, most theoretical and modeling studies on bird migration, including analyses of distribution of ring–recoveries, typically assumes that the areas where birds are migrating are homogeneous or use randomly modeled landscapes (but see Erni et al., 2002). This is evidently not the case for continental and cross–continental migrants that experience dramatic changes in landscape most noticeable when facing open sea at continental coastlines. Recently, much attention has been paid to macroecological null models to examine the expected effect of geometric constraints on patterns of many different "traits" (Colwell & Lees, 2000; Colwell et al., 2004). It has been shown that non– even distribution (i.e., a peak) of species richness along e.g. latitudinal, longitudinal and peninsular gradients and across continents (Colwell et al., 2004) can arise through simple geometric constraints on species range boundaries, in the absence of any environmental or historical mechanisms. Continental shape has also been shown to be a potential constraint on spatial distribution of ranges sizes of breeding and migratory birds (Jetz & Rahbek, 2002; Bensch, 1999; respectively). Open sea does impose a migratory barrier of terrestrial migrants and there is a low recovery probability of terrestrial taxa over sea (e.g. Wernham et al., 2002). Hence, even the largely deterministic processes of adaptation and/or evolution of dispersal traits governing migratory choices of individuals may in principle produce geographical patterns of recoveries that have a non–deterministic (stochastic) explanation. Unless models invoking deterministic mechanisms of orientation (e.g. vector navigation) can be shown to fully predict the patterns of recoveries, geometric constraints on recoveries could emerge as a contributory explanation. In this study we provide a framework for investigating the geometric influence of ocean and sea on observed migratory patterns (i.e. taking the very low recovery probabilities over water into account). The approach is illustrated using an

empirical data set on Pied Flycatchers Ficedula hypoleuca, ringed as nestlings in Scandinavia and recovered en route on their initial southwestern migration before turning south on the Iberian Peninsula to wintering grounds in West Africa. Here we investigate the predictive power of simulated migration with and without incorporating the effect of geometric constraints. As a first step (the null model) we simulate migration with birds choosing random directions. As a second step (the analytical model) we simulate migration according to the simple clock–and–compass or vector navigation hypothesis (i.e. birds migrate for an endogenously controlled period of time in an endogenously controlled direction). Methods Modeling procedure The simulated data sets were constructed by using a computer model of a vector navigation system (a clock–and–compass strategy) using vector summation (Rabøl, 1978; Mouritsen, 1998; Sandberg & Holmquist, 1999; Thorup et al., 2000, 2003; Thorup & Rabøl, 2001), where each migratory step is considered a vector with a fixed length and a direction picked randomly (and independently) from a circular distribution. The circular distribution is characterized by its directional concentration measure r (where r = 0 for a uniform circular distribution and r = 1 for a unidirectional distribution without scatter; cf. Batschelet, 1981). Thus in the vector summation model the orientation is considered to vary around the inherited mean direction between each flight step according to a circular probability distribution whose concentration reflects the combined influence of internal factors (the birds capability of flying in the inherited direction) and external factors, as e.g. wind drift. The model has 3 parameters: (1) rstep. Variation in directional choice within individuals (between steps) with one picked for each migratory step, (2) rbetween. Variation in directional choice between individuals, which is picked once for each modeled individual, and (3) step length, which is held constant for the whole migratory journey and for all individuals. Random directions were drawn from von Mises distributions (the distributions normally used in circular statistics), which were simulated following Fisher (1993). The mean of the directional distribution in the model is assumed to equal the mean migration direction. Variation between individuals was included by adding the direction picked from the between–individuals distribution to the resulting sample mean vector after the number of migratory steps had been summed (Thorup et al., 2000). The simulated data sets including non–terrestrial points were constructed by simulating tracks for 5,000 birds flying each number of steps from 1 to 70 steps with a step length of 125 km, resulting in a point sky of 700,000 endpoints. These dis-

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Fig. 1. Endpoints of simulated tracks of Pied Flycatchers Ficedula hypoleuca originating in the center of mass of ringing site locations using a directional concentration per step of 0.7: Circles. Number between–individuals variation included (N = 401); Plusses. A between–individuals variation of 0.99 included (N = 401); The thick line. Average direction used (see fig. 3). Circles show distances of 100, 150, 250, 350, 450, 550, 1,000, 2,000 and 3,000 km from the center of mass; circles from 100–550 indicate the distance intervals used. Fig. 1. Puntos de equivalencia de trayectorias simuladas de papamoscas cerrojillos Ficedula hypoleuca originadas en el centro másico de los emplazamientos de anillamiento utilizando una concentración direccional progresiva de 0,7: Círculos. Se incluye el número de la varicaión entre individuos (N = 401); Signos más: se incluye una variación entre individuos de 0,99 (N = 401); La línea gruesa indica la dirección media utilizada (ver la fig. 3). Los círculos indican distancias de 100, 150, 250, 350, 450, 550, 1.000, 2.000 y 3.000 km desde el centro másico. Los círculos desde 100 hasta 550 indican los intervalos de distancia empleados.

tance coordinates were then grouped into distance intervals and used for calculation of directional concentrations. The step length considered alone is not crucial for the behavior of the model, rather it is the combination rstep/step length (see Thorup et al., 2003). To cope with the uncertainty in these two parameters we investigated the effects of a large range of rstep–values while keeping step length constant. The step length used (125 km) is the same as the one used by Mouritsen (1998), and it is assumed to correspond to one nights migration distance. Our procedure is somewhat different from the procedure used by Mouritsen (1998), who calculates a directional concentration for a specific number of steps. However, that procedure disregards the resulting spread of the endpoints in

the migration direction (see Mouritsen & Mouritsen, 2000; Thorup & Rabøl, 2001), an effect, which the procedure used here takes into account. To model the influence of geometric constraints as imposed by open sea, non–terrestrial points were removed using ArcView GIS 3.2 software (see fig. 1). The underlying assumption for this procedure is that birds, if not over land after completing a migratory step, embark on a new migratory step choosing direction anew from the circular distribution. From the simulated data sets directional concentrations for comparison with the empirical data set were derived using a bootstrap method. For each distance interval, 200 estimates of the expected directional concentration were calculated by repeatedly drawing the number of random angles, corre-

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Empirical data

Proportion of recoveries

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The null model was run with birds choosing random directions. First it was run as a random walk with total variation within individuals (rstep= 1) with no variation between individuals included. The model was also run with no variation in directional concentration per step (rstep= 1) but with total variation between individuals (rbetween = 0).

The empirical data set consisted of ring–recoveries of Pied Flycatchers ringed as nestlings in Finland and recovered the following autumn or winter before 1 march (fig. 3; N = 415). Typically, modeling studies on migratory orientation considers ring–recoveries as vectors, assuming that it is meaningful to use only the distance and direction between ringing and recovery sites regardless of the ringing site. This corresponds to parallel displacement of recovery vectors to a common, but imaginary, ringing site resulting in imaginary recoveries from non–terrestrial sites. This procedure obscures the effect of birds being ringed at different sites facing different distributions of main barriers. To avoid this bias, we assumed that all birds had been ringed at the same site. This center of mass of the latitudinal and longitudinal coordinates of the ringing sites of nestlings in Finland was used as starting location for all calculations (and as starting point for the simulated tracks, see fig. 1). The minor drawback of this approach is that it results in a too low estimate of the directional concentration on short distances (less than 500 km), but a minor (and more concentrated, see discussion) error on the estimate on long distances (contrary to parallel displacing recoveries). To reduce these biases, recoveries from a comparatively small geographical area was used (Finland). Directional concentrations r of points were calculated for each distance interval 100–149, 150–249, 250–349, 350–449,..., 3,850–3,949 and 3,950–4,049 km using loxodromic (constant compass courses) distances for both the empirical and the simulated data set. For the distance intervals 450–549, 550– 649, 950–1,049, 1,050–1,149,..., 1,650–1,749, 2,450–2,549, 2,650–2,749, 2,750–2,849, 3,550– 3,649, 3,650–3,750 and more than 4,049 km there were less than three recoveries, and the points were therefore omitted from the analysis. Birds recovered less than 1,000 km from the ringing site were generally from around the Baltic Sea; 1,800–2,500 km from France and Italy; 2,700– 3,500 km from the Iberian Peninsula, and more than 3,500 from North Africa. We excluded the 2,600 km interval including the very aberrant recoveries in Greece (fig. 3) due to uncertainties regarding exact recovery locations. Birds recovered more than 3,500 km from the ringing site relating to endpoints from North Africa, where we expect the birds to have changed their migration direction, were also excluded.

The analytical model

Statistical analysis

The analytical model was run with a variation within individuals (rstep) of 0.60, 0.70, 0.80, 0.90 and 0.95 with no variation between individuals included (random walk with drift). The model was also run with no variation in directional concentration per step but with a variation between individuals (rbetween) of 0.98 and 0.99.

For comparison between simulated and empirical data, we calculated Pearson’s product moment correlation coefficients between mean angular deviation (s) times distance (cf. Thorup et al., 2000) for the two data sets. Using Spearman’s correlation coefficients instead yielded very similar correlation coefficients. Models were evaluated using the amount of explained variation (r2)

0.15 0.1 0.05 0 0

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Fig. 2. Relative distribution of the number of recoveries of Pied Flycatchers Ficedula hypoleuca in each distance interval (+) compared with simulated data with non– terrestrial points removed (line; rstep = 0.80; rbetween = 1). Fig. 2. Distribución relativa del número de recuperaciones de papamoscas cerrojillos Ficedula hypoleuca en cada intervalo de distancias (+), en comparación con datos simulados en los que los puntos no terrestres habían sido eliminados (línea; rstep = 0,80; rbetween = 1).

sponding to the number of ring–recoveries in the same distance category. From these directional concentrations both median and confidence intervals were derived. Distributions resulting from this modeling approach, incorporating or excluding the effect of geometric constraints, was then compared with the ring–recovery distributions based on the assumption that all birds had been ringed at the same location defined as the center of mass of actual ringing sites (see fig. 1). The null model

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Recovery density 2–10 11–20 21–30 31–49

Fig. 3. Ring–recoveries of Pied Flycatchers Ficedula hypoleuca ringed as nestlings in Finland and recovered within the same autumn/winter (N = 415): Plusses. Single recoveries; Crosses. Ringing sites; Large dot. Center of mass of ringing sites; Thick line. Rhumbline (constant compass) course from center of mass of ringing sites to large concentrations of recoveries in Iberia (fitted by eye). Circles show distances from the center of mass of ringing sites of 100, 150, 250, 350, 450, 550, 1,000, 2,000 and 3,000 km; circles from 100–550 indicate the distance intervals used. Fig. 3. Recuperaciones de anillos de papamoscas cerrojillos Ficedula hypoleuca que fueron anillados en Finlandia siendo crías y que se recuperaron el mismo otoño/invierno (N = 415): Signos más. Recuperaciones únicas; Cruces. Emplazamientos de anillamiento; Punto grande. Centro másico de emplazamientos de anillamiento; Línea gruesa. Rumbo loxodrómico (brújula constante) desde el centro másico de los emplazamientos de anillamiento hasta amplias concentraciones de recuperaciones en la península Ibérica (ajustados visualmente). Los círculos indican distancias desde el centro másico de los emplazamientos de anillamiento de 100, 150, 250, 350, 450, 550, 1.000, 2.000 y 3.000 km. Los círculos desde 100 hasta 550 indican los intervalos de distancia empleados.

There was a rather low correspondence between the number of recoveries in each distance category compared to modeled numbers. Figure 2 shows this for a model run with a moderate directional concentration per step, no variation between individuals in migration direction (rstep = 0.80; rbetween = 1, respectively) and with non–terrestrial points removed. This strongly suggests, that migrants use some longer steps or alternatively, shorter stopover times, in the first part of the migration. The model procedure then corresponds to birds using a constant interval of changing directions throughout the migratory journey.

Results The ring–recoveries of Finnish Pied Flycatcher nestlings show a distinct southwest migration through Europe (fig. 3). The null models with random orientation do not explain significant amounts of the observed directional variation in the ring–recoveries (table 1, fig. 4). In general only little directional variation is explained by the analytical model even with non–terrestrial points removed (table 1, fig. 5). When non–terrestrial points are included (fig. 5A), none of the directional concentrations per step explained any amount of directional variation. The

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Table 1. Pearson’s product moment correlation coefficient (r) and explained variation (r2) between mean angular deviation times distance for empirical and simulated data, respectively. Data for various model runs with different values of rstep and rbetween (see methods) when non–terrestrial points are included or removed. The aberrant 2,600 km distance interval with two outlying Greek recoveries has been excluded (see text). If r–values are less than 0 only signs are given. Only P–values less than 0.10 are given. Tabla 1. Coeficiente de correlación momento–producto (r) de Pearson y variación (r2) explicada entre el promedio de distancia de tiempos de la desviación angular para datos empíricos y simulados, respectivamente. Los datos correspondientes a varios modelos presentan valores diferentes de rstep y rbetween (ver los métodos), según se hayan incluido o eliminado los puntos no terrestres. Se ha excluido el intervalo de distancias atípicas de 2.600 km con dos recuperaciones periféricas Greek (consultar el texto). Si los valores r son inferiores a 0, sólo se indican los signos. Únicamente se facilitan los valores P inferiores a 0,10.

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addition of even a small amount of variation between individuals (fig. 5C) results in a too large scatter on long distances (more than 2,500 km) corresponding to the birds having reached as far as the Iberian Peninsula. With non–terrestrial points removed (fig. 5B), the qualitative fit tends to be somewhat better, especially if the very aberrant point at 2,600 km (resulting from two recoveries in Greece) is removed, with model output lines mirroring the shape of the ringing recovery points, but only the directional concentrations per step of 0.8, 0.90 and 0.95 had significant Spearman rank correlation coefficients and explained 0.15, 0.22 and 0.14, respectively of the directional variation (table 1). However, these models tend to fit well neither in France and Italy nor on the Iberian peninsula (fig. 6). Still the addition of small amounts of variation between individuals results in a too large scatter on long distances (more than 2,500 km) even with non–terrestrial points removed (fig. 5D). Due to the use of the center of mass as reference point, the scatter of the recoveries closer than 500 km is expected to be too large, but removing these from analysis did not change the pattern. Though simulated data sets with non–terrestrial points removed generally explained small amounts of the observed directional variation the correlation coeffi-

0.01

cients were significantly larger than without removing the non–terrestrial points (mean r = –0.093 and 0.169, with terrestrial points included and removed, respectively; P = 0.007; paired t–test) Discussion Despite decades of research, we still lack knowledge about how free–flying birds orientate on migration (Alerstam, 1996). Since Perdeck’s (1957) paradigmatic displacement experiments and the formulation of the "clock–and–compass" hypothesis we have learned much about behavioral responses in caged migrants from controlled experiments using mostly Emlen funnels (e.g. Berthold, 1996, 2001). Additional theories and hypotheses on how birds perform migratory navigation —such as "goal–area navigation" (Rabøl, 1978)— have been formulated, but only recently have research been directed at testing these at larger scales (Wehner, 2001). Because controlled, manipulative experiments are usually impossible or impractical at large scales in nature for more than a few species, factorial designs intended to evaluate the role of competing explanations for large–scale dispersal, migration and orientation, are limited to what nature happens

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to have produced. In that respect, the vast amount of ring–recoveries (more than 1 million from birds ringed in Britain and Ireland alone, Wernham et al., 2002) provide unique opportunities to build predictive models to evaluate hypotheses. Extensive effort has been done recently to evaluate orientation hypotheses using ring–recoveries (Mouritsen, 1998; Mouritsen & Mouritsen, 2000; Thorup et al., 2000, 2003), but none of these has taken into account the potential effect of geometric constraints. Summing up With the analytical model simulating a simple migratory orientation strategy, geometric constraints were found to explain up to 22% for the best fitted model, which did not include any variation between individuals and had a rather low variation per step (rstep = 0.90). Assuming no variation between individuals is, however, unrealistic as variation between individuals forms the basis of evolution of new migratory traits (Helbig, 1994, 1996; Helbig et al., 1994). Furthermore, overall removing non–terrestrial points lead to more explained variation (table 1). This indicates that a significant amount of the observed concentration of migratory paths is explained by geometric constraints and shows the importance of including geometric constraints in future studies on large–scale migration patterns. Previous modeling studies on land bird migratory orientation have generally not taken this into account and it is usually not included in studies on optimal migration. However, we also found a presumably important general pattern of qualitative lack of fit of the presented models, with the empirical data being more concentrated than simulated either in the middle part of the migration or in the last part. Possible reasons for this lack of fit could be biased empirical data, birds using more of the landscape geometry than included in this study or that the underlying migration model (here the clock–and–compass strategy) used is not reasonable or it is insufficient.

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Fig. 4. Comparison of spread of empirical data set (ring–recoveries of Pied Flycatchers Ficedula hypoleuca; filled squares) with values from various null models (lines). Terrestrial points have been removed in the two lower lines. Lines being above squares indicate a larger spread of modeled data than observed in the empirical data set at that particular distance. Fig. 4. Comparación de la dispersión del conjunto de datos empíricos (recuperaciones de anillos de papamoscas cerrojillos Ficedula hypoleuca; cuadros negros) con valores procedentes de varios modelos nulos (líneas). Los puntos terrestres se han eliminado en las dos líneas inferiores. Las líneas situadas encima de los cuadros indican una mayor dispersión de datos modelados en comparación con lo observado en el conjunto de datos empíricos a esa distancia concreta.

Are observed patterns real Due to differences in reporting rates between different regions it is often questioned to what degree ring– recoveries do reflect the true migratory patterns (summarized in e.g. Perdeck, 1977; Nichols, 1996). However, quite pronounced species–specific differences in ring recoveries, even for species with rather similar migratory routes do exist. Furthermore, such differences can be detected even at a fine scale (within regions). An example of this is shown on figure 7, which shows the distributions of ring–recoveries of Scandinavian Pied Flycatchers and Scandinavian and Finnish Redstarts for comparison with those of the Finnish Pied Flycatchers (fig. 3). A significant difference in the recovery patterns is evident from these maps, with Scandinavian Pied Flycatchers showing more recoveries to the north–west compared to the Finnish, and recoveries of Scandinavian and Finnish

Redstarts being concentrated south–east of the Finnish Pied Flycatchers. This indicates, that ring recovery patterns do reflect the migratory patterns, and that they are suitable for an analysis of the concentration of the migratory route. This is important for the general use of ringing data in studies of migration. Ringing data are currently being used for a lot of different kinds of analyses, as it is, for most species, the only possible method for gathering data. Furthermore, bird ringing has been performed for more than 100 years, which means that an impressive amount of data has already been collected (Bairlein, 2001). This makes it possible to use existing data for analysis of migratory patterns, which has become important in the case of possible effects of climate change.

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Fig. 5. Comparison of deviation of empirical data set (ring–recoveries of Pied Flycatchers Ficedula hypoleuca) with values from various model runs. Lines show modeled values for indicated rstep/rbetween values and filled squares show ring–recoveries. Lines being above squares indicate a larger spread of modeled data than observed in the empirical data set at that particular distance: A. No variation between individuals; non–terrestrial points included; B. No variation between individuals; non– terrestrial points removed; C. Variation between individuals included; non–terrestrial points included; D. Variation between individuals included; non–terrestrial points removed. Fig. 5. Comparación de la desviación del conjunto de datos empíricos (recuperaciones de anillas de papamoscas cerrojillos Ficedula hypoleuca) con valores procedentes de varias aplicaciones de modelos. Las líneas indican valores modelados para los valores rstep/rbetween indicados, mientras que los cuadros negros indican recuperaciones de anillos. Las líneas situadas encima de los cuadros indican una mayor dispersión de datos modelados en comparación con lo observado en el conjunto de datos empíricos a esa distancia concreta: A. Ninguna variación entre individuos; puntos no terrestres incluidos; B. Ninguna variación entre individuos; puntos no terrestres eliminados; C. Variación entre individuos incluida; puntos no terrestres incluidos; D. Variación entre individuos incluida; puntos no terrestres. eliminados

Implications for our understanding of the migratory orientation program The rather low scatter of ring recoveries described in this study suggests that the rather wide scatter found in the study by Mouritsen (1998) could be

caused partly by not taking geography into account (i.e. by using the parallel displacement of recoveries). Most studies of free–flying migrants do find a much higher directional concentration of tracks of migrants (Bäckman & Alerstam, 2003). However, this cannot explain the amount of variation on

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Fig. 6. Confidence intervals for the best fitting model with little variation within individuals and no variation between individuals (rstep = 0.90 and rbetween = 1). For explanation see figure 5. Fig. 6. Intervalos de confianza para el modelo que presenta un mejor ajuste, con escasa variación en el conjunto de los individuos y ninguna variación entre individuos (rstep = 0,90 y rbetween = 1). Para detalles al respecto, ver la figura 5.

Recovery density 2–10 11–20 21–30 31–47

pull. (SF) Recovery density 1 2–3 4–5 6–7 8–9 10–11

Fig. 7. Ring–recoveries of: A. Pied Flycatchers Ficedula hypoleuca ringed as nestlings in Scandinavia (N = 486); B. Redstarts Phoenicurus phoenicurus ringed as first–years (N = 93) or nestlings (N = 82) in Scandinavia or Finland, and recovered within the same autumn/winter: Crosses. Ringing sites; Plusses. Single recoveries; Open circles. Ringed as nestling in Finland (Redstart only; N = 32). Large dot and thick line as for figure 2. Fig. 7. Recuperaciones de anillos de: A. Papamoscas cerrojillos Ficedula hypoleuca que fueron anillados en Escandinavia siendo crías (N = 486); B. Colirrojos reales Phoenicurus phoenicurus anillados durante el primer año de vida (N = 93) o como crías (N = 82) en Escandinavia o Finlandia, y recuperados el mismo otoño/invierno. Cruces. Emplazamientos de anillamiento; Signos más. Recuperaciones únicas; Círculos blancos. Anillados siendo crías en Finlandia (sólo colirrojos reales; N = 32). El punto grande y la línea gruesa significan lo mismo que en la figura 2.

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short–distance recoveries, which is nevertheless found in the ring–recoveries (Mouritsen, 1998). It is generally believed that young migrants use a vector navigation program for finding their species–specific wintering area, and the present study is based on such a program. The rather high concentration of migratory steps needed to provide a reasonable fit suggests that following coastlines could play a significant role in guiding migrants. However, this is contradicted by e.g. Redstarts Phoenicurus phoenicurus being concentrated away from coastal areas within the Iberian Peninsula (fig. 7B). Thus the lack of qualitative fit for the model may indicate that a simple form of vector navigation program is only part of the program used by free–flying migrants. Moreover, the assumptions underlying the best fitted model are not realistic (i.e. no variation between individuals in migratory direction: rbetween = 0), but the degree as to how much this assumption is violated is not yet known (Thorup & Rabøl, 2000; Thorup et al., 2003). The combined effect of geography and simple factors, such as compensation for previous drift, to correct their course along the migration route, can probably account for the failure of the simple vector navigation program to satisfactorily encompass the observed patterns. Importance of taking geometry into account The present study shows a significant role of geometric constraints, thereby stressing the importance of including such constraint in analyses. It is possible to extend the modeling procedure used here to include most factors important in migration (e.g. flight range) and to use mortality for evaluating the model. Such a model has already been employed (Erni et al,. 2002), though the primary focus of that model is physiological and not the migratory orientation program. However, at present estimates of mortality are heavily dependent on estimated flight ranges, which in turn rely on equations whose parameters are difficult to estimate (Pennycuick et al., 1996; Rayner & Maybury, 2003). Furthermore, the actual mortality associated with migration is very difficult to assess (Nichols, 1996), though recent estimates suggest that it is high (Sillett & Holmes, 2002). Nevertheless such approaches are likely to improve our understanding of the migratory orientation program and can guide further research. Acknowledgements We are especially grateful to to the Finnish ringing scheme (Pertti Saurola and Jukka Happala) for help and permission to use their recoveries for this project. Also we wish to thank Jesper Johannes Madsen for his help with obtaining the recovery data, and the Swedish ringing schemes (Bo Sällström) for permission to use their recoveries for this project.

References Alerstam, T., 1990. Bird Migration. Cambridge University Press. – 1996. The geographical scale factor in orientation of migrating birds. Journal of Experimental Biology, 199: 9–19. Bairlein, F., 2001. Results of bird ringing in the study of migration routes and behaviour. Ardea, 89 (special issue): 7–19. Bensch, S., 1999. Is the range size of migratory birds constrained by their migratory program? Journal of Biogeography, 6: 1225–1237. Berthold, P., 2001. Bird Migration. A General Survey (2nd ed.). Oxford University Press, Oxford. – 1996. Control of Bird Migration. Chapman & Hall, London. Bäckman, J. & Alerstam, T., 2003. Orientation scatter of free–flying nocturnal passerine migrants: components and causes, Animal Behaviour, 65: 987–996. Colwell, R. K. & Lees, D. C., 2000. The mid-domain effect: Geometric constraints on the geography of species richness. Trends in Ecology and Evolution, 15(2): 70-76. Colwell, R. K., Rahbek, C. & Gotelli, N. J., 2004. The mid-domain effect and species richness patterns: What have we learned so far? American Naturalist, 163: E1-E23. Erni, B., Liechti, F. & Bruderer, B., 2002. Stopover Strategies in Passerine Bird Migration: A Simulation Study. Journal of Theoretical Biology, 219: 479–493. Fisher, N. I., 1993. Statistical analysis of circular data. Cambridge University Press, Cambridge. Helbig, A. J., 1994. Genetic basis and evolutionary change of migratory directions in a European passerine migrant Sylvia atricapilla. Ostrich, 65: 151–159. – 1996. Genetic basis, mode of inheritance and evolutionary changes of migratory directions in palearctic warblers (Aves: Sylviidae). Journal of Experimental Biololgy, 199: 49–55. Helbig, A. J., Berthold, P., Mohr, G. & Querner, U., 1994. Inheritance of a novel migratory direction in central European Blackcaps (Aves: Sylvia atricapilla). Naturwissenschaften, 81: 184–186. Jetz, W. & Rahbek, C., 2002. Geographic Range Size and Determinants of Avian Species Richness. Science, 297: 1548–1551. Moreau, R. E., 1972. The Palaearctic–African Bird Migration Systems. Academic Press, London. Mouritsen, H., 1998. Modelling migration: the clock– and–compass model can explain the distribution of ringing recoveries. Animal Behaviour, 56(4): 899–907. Mouritsen, H. & Mouritsen, O., 2000. A mathematical expectation model for bird navigation based on the clock–and–compass strategy. Journal of theoretical Biology, 207: 283–291. Nichols, J. D., 1996. Sources of variation in migratory movements of animal populations: Statistical inference and a selective review of

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empirical results for birds. In: Population dynamics in ecological space and time: 147–197 (D. E. Rhodes, R. K. Chesser & M. H. Smith, Eds.). The University of Chicago Press. Pennycuick, D. J., Klaassen, M., Kvist, A. & Lindström, Å., 1996. Wingbeat frequency and the Wingbeat frequency and the body drag anomaly: wind–tunnel observations on a thrush nightingale (Luscinia luscinia) and a teal (Anas crecca). Journal of Experimental Biology, 199: 2757–2765. Perdeck, A. C., 1958. Two types of orientation in migrating starlings, Sturnus vulgaris L., and chaffinches Fringilla coelebs L., as revealed by displacement experiments. Ardea, 46: 1–37. – 1977. The analysis of ringing data: pitfalls and prospects. Vogelwarte, 29 Sonderheft: 33–44. Rabøl, J., 1978. One–direction orientation versus goal area navigation in migratory birds. Oikos 30, 216–223. Rayner, J. M. & Maybury, W. J., 2003. The drag paradox: Measurements of flight performance and body drag in flying birds. In: Avian Migration: 543–562 (P. Berthold, E. Gwinner & E. Sonnenschein, Eds). Springer Verlag. Sandberg, R. & Holmquist, B., 1998. Orientation and long–distance migration routes: An at-

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tempt to evaluate compass cue limitations and required precision. Journal of Avian Biology, 29(4): 626–636. Sillett, T. S. & Holmes, R. T., 2002. Variation in survivorship of a migratory songbird throughout its annual cycle. Journal of Animal Ecology, 71: 296–308. Thorup, K. & Rabøl, J., 2001. The orientation system and migration pattern of long–distance migrants: conflict between model predictions and observed patterns. Journal of Avian Biology, 32: 111–119. Thorup, K., Rabøl, J. & Madsen, J. J., 2000. Can clock–and–compass explain the distribution of ringing recoveries of pied flycatcher? Animal Behaviour, 60: F3–F8. Thorup, K., Alerstam, T., Hake, M. & Kjellén, N., 2003. Can vector summation describe the orientation system of juvenile ospreys and honey buzzards? – An analysis of ring recoveries and satellite tracking. Oikos, 103: 350–362. Wehner, R., 2001. Bird navigation – computing orthodromes. Science, 291: 264–265. Wernham, C. V., Toms, M. P., Marchant, J. H., Clark, J. A., Siriwardene, G. M. & Baillie, S. R., Eds., 2002. The Migration Atlas: movements of the birds of Britain and Ireland. T & A D Poyser, London.

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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Breeding dispersal by Ross’s geese in the Queen Maud Gulf metapopulation K. L. Drake & R. T. Alisauskas

Drake, K. L. & Alisauskas, R. T., 2004. Breeding dispersal by Ross’s geese in the Queen Maud Gulf metapopulation. Animal Biodiversity and Conservation, 27.1: 331–341. Abstract Breeding dispersal by Ross’s geese in the Queen Maud Gulf metapopulation.— We estimated rates of breeding philopatry and complementary dispersal within the Queen Maud Gulf metapopulation of Ross’s Geese (Chen rossii) using multistate modeling of neckband observations at five breeding colonies, 1999– 2003. Probability of philopatry was female–biased, but varied among colonies. Probabilies of annual movement among breeding colonies ranged 0.02 to 0.14 for females and 0.12 to 0.38 for males and was substantially higher than expected. These estimates (1) underscore the potential for dispersal to alter breeding distribution, (2) demonstrates that the influence of immigration on colony–specific rates of population growth is nontrivial, and (3) provides behavioral evidence for extensive gene flow among subpopulations. Sex differences in apparent survival estimated from multistate models likely resulted from a combination of higher rates of neckband loss by males compared to females, and higher rates of permanent emigration by males from our study area. Key words: Dispersal, Multistate, Philopatry, Ross's Goose, Chen rossi. Resumen Dispersión de los reproductores del ansar de Ross en la metapoblación del golfo de la Reina Maud.— Estimamos las tasas de filopatría de reproducción y la dispersión complementaria del ansar de Ross (Chen rossii) en la metapoblación del golfo de la Reina Maud utilizando la modelación multiestado a partir de las observaciones de animales marcados en el cuello en cinco colonias de reproducción, 1999–2003. La probabilidad de filopatría presentaba un sesgo a favor de las hembras, pero variaba de una colonia a otra. Las probabilidades de movimiento anual entre las colonias de reproducción oscilaban entre el 0,02 y el 0,14 para las hembras, y entre el 0,12 y el 0,38 para los machos, siendo considerablemente superiores a lo previsto. Estas estimaciones 1) subrayan las posibilidades de que la dispersión modifique la distribución de reproducción, 2) demuestran que la influencia de la inmigración en las tasas de crecimiento poblacional de cada colonia no es irrelevante y 3) proporcionan evidencia conductual acerca de un amplio flujo genético entre subpoblaciones. Las diferencias por sexo en la supervivencia aparente estimadas a partir de modelos multiestado probablemente fueron debidas a una combinación de tasas más elevadas de pérdida de marcaje en el cuello por parte de los machos en comparación con las hembras, y a tasas más elevadas de emigración permanente por parte de los machos de nuestra área de estudio. Palabras clave: Dispersión, Multiestado, Filopatría, Ansar de Ross, Chen rossi. Kiel L. Drake, Dept. of Biology, Univ. of Saskatchewan, 115 Perimeter Road, Saskatoon, Saskatchewan S7N 50X4, Canada.– Ray T. Alisauskas, Canadian Wildlife Service, 115 Perimeter Road, Saskatoon, Saskatchewan, S7N 0X4, Canada.

ISSN: 1578–665X

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Introduction Species distributions often encompass broad geographic ranges that include great spatial variability in landscape characteristics. Corresponding variability in ecological conditions leads to uneven distributions of density throughout a species’ range, because animals congregate in areas where habitats are suitable. Such subpopulations are often geographically separated from each other by areas of less suitable habitats (Weins, 1997). Nevertheless, almost all species have evolved mechanisms that allow dispersal across unsuitable or less optimal habitats. Consequently, disjunct conspecific populations are potentially interconnected through migration networks or dispersal to new breeding areas. Such potential for movement among subpopulations is key to the concept of metapopulations (Gilpin & Hanski, 1991; Hanski & Gilpin, 1997), where persistence is a function of not only survival and recruitment of individuals but also of immigration and emigration between component subpopulations (Pulliam, 1988). In North America, breeding and wintering distributions of continental populations of the closely– related Ross’s Goose (Chen rossii) and Lesser Snow Goose (Chen caerulescens, hereafter Snow Goose, collectively referred to as ‘light geese’) are such that they fall within the conceptual domain of a metapopulation. Both species breed at spatially discrete colonies in arctic and subarctic habitats and winter in allopatric subpopulations across a broad range in southern North America (Ryder & Alisauskas, 1995; Mowbray et al., 2000). Despite spatial segregation of breeding subpopulations, there is tremendous potential for exchange of light geese because of mixing during migration when long– term pair bonds begin to form during late winter and continue through spring migration (Ryder & Alisauskas, 1995; Mowbray et al., 2000). Much attention in North America has focused on the exponential population increase of light geese and their potential to damage breeding habitats (Batt, 1997; Moser, 2001). Regardless of causes resulting in unchecked population growth, light goose populations are markedly larger and occur over much broader winter ranges than they did 50 years ago. Winter range expansion highlights the ability of Snow Geese and Ross’s Geese to adapt to changing landscape conditions (Alisauskas et al., 1988; Alisauskas, 1998). Despite these species’ apparently adaptive nature during the nonbreeding period, female Snow Geese were thought to be generally philopatric to breeding colonies (Cooke et al., 1995), even when the consequences of philopatry appeared to be maladaptive such as when population densities exceed carrying capacity (Cooch et al., 1989; Cooch et al., 1991; but see Cooch et al., 2001). No information about vagility of Ross’s Geese was available. We estimated rates of movement among breeding colonies by Ross’s Geese to gain insight about the potential for breeding dispersal to influence species distribution and gene flow in light geese.

We focused the current analysis on Ross’s Geese because of uninterrupted marking within the Queen Maud Gulf Bird Sanctuary (QMGBS) since 1989 that resulted in a substantial marked population at the outset of this study. Efforts to neckband Snow Geese in the QMGBS, and their subsequent resightings, have recently increased (Drake & Alisauskas, unpubl. data), but there remains insufficient data to include them in the current analysis. Nonetheless, Ross’s Geese and Snow Geese associate throughout their annual cycles (Alisauskas, 2002), and the extent of such associations during breeding likely has increased recently with the growth in number of Snow Geese in the QMGBS where > 95% of the continental population of Ross’s Geese breeds (Kerbes, 1994). Materials and methods Study area Data were collected annually at five breeding colonies within the QMGBS (fig. 1) during 1999–2003. Colony 3 (hereafter, Karrak Lake 67 o 14’ N, 100o 15’ W) contains the Karrak Lake Research Station where investigations of Ross’s Goose breeding ecology have occurred continuously since 1991. Surveys for neckbanded geese began in 1994 at Karrak Lake and were extended to other colonies (nine, 10, 46 and 81) within QMGBS starting 1999. We selected these colonies because they represent some of the largest known colonies within QMGBS. We suggest that these colonies collectively account for ~90% of the known continental breeding population of Ross’s Geese (Ryder & Alisauskas, 1995; Alisauskas et al., 1998). Marking efforts and surveys for neckbanded geese Following methods used by Alisauskas & Lindberg (2002), we captured and neckbanded adult and gosling Ross’s Geese within brood–rearing habitats (fig. 2) during August 1991–2003. Marking efforts during 1991–1998 focused on areas north of Karrak Lake along the Karrak and Simpson River drainages, because we reasoned that most of the geese using these areas were from Karrak Lake. During 1999–2003, we continued to mark most geese north of Karrak Lake, but also began to neckband geese near colony 10, and the mouth of the McNaughton River (fig. 2). Because we could not assign with certainty Ross’s Geese captured on brood–rearing areas in August to colonies in which they nested the previous June, we included only those birds that were resighted at breeding colonies in June 1999–2003. Only 501 goslings were neckbanded during 1999–2002 so we excluded these from consideration; we further judged that their inclusion would have doubled the number of parameters to be estimated while increasing the sample size only by ~15%. Hence, our analysis included adult birds

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Animal Biodiversity and Conservation 27.1 (2004)

km 10 0

100 km

68 86 99 93

98 58 57 59

101

100

85 84 91

11 62 12 14 13 15 29 30 102

95 96

3 50

71 24 37

40 25 26

21 22 4

9 17

43 63

16 31 42 48 32 41

34 35 19

Fig. 1. Light goose colonies within the Queen Maud Gulf Bird Sanctuary, Nunavut, Canada. Numbers correspond to locations of goose colonies and black areas depict spatial extent of larger colonies. Surveys for neckbanded geese were conducted at colonies 3 (Karrak Lake), 9 (Simpson River), 10 (East McNaughton), 46 (West McNaughton), and 81 (Reference Lake) 1999–2003. Fig. 1. Colonias de gansos menos pobladas de la Reserva Ornitológica del golfo de la Reina Maud, Nunavut, Canadá. Los números corresponden a emplazamientos de colonias de ánsares, mientras que las áreas en negro representan la extensión espacial de colonias más amplias. Los estudios correspondientes a los ánsares marcados en el cuello se llevaron a cabo en las colonias 3 (lago Karrak), 9 (río Simpson), 10 (McNaughton Este), 46 (McNaughton Oeste) y 81 (lago de referencia) 1999–2003.

(n = 3,233) sighted at least once at one of the sampled colonies during 1999–2003, regardless of year of marking. We restricted observations for neckbanded geese to their 22–day incubation period (Ryder, 1972) because of our interest in estimating dispersal between breeding attempts. Observations strictly during incubation reduced potential for bias caused by sampling non–breeding adults, as territorial breeders displace most non–breeders from colonies by the onset of incubation (Ryder & Alisauskas, 1995). Extent of breeding distribution at each colony was mapped each year in June from a helicopter. Data were digitized and imported into SPANS GIS study area with Albers equal area projection. Layers showing colony extent were overlaid with a layer showing land and water (30 m resolution from LandSat imagery) to calculate the area of terrestrial habitat at

each colony occupied by nesting geese. In 2002, for example, terrestrial habitat occupied by nesting geese was 164.9 km2 at colony 3, 10.3 km2 at colony 9, 151.2 km2 at colony 10, 39.8 km2 at colony 46, and 1.9 km2 at colony 81. Due to this vast area in which neckbands could only be search for on foot, we were unable to survey all colonies entirely. Instead, we selected areas within each colony thought to have the highest nesting densities, generally in the center, to maximize efficiency at detecting neckbands. We maintained consistency among years by searching for neckbands in defined study areas in each colony. We assumed that the ratio of neckbanded to unmarked birds remained consistent regardless of variation in nesting density. Any broken neckbands found on the ground by observers in the course of travel through colonies during neckband surveys were noted.

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Analysis We used multistate modeling (Arnason, 1973; Hestbeck et al., 1991; Brownie et al., 1993; Schwarz et al., 1993) in Program MARK to analyze resight data of neckbanded Ross’s Geese for estimation of dispersal and complementary philopatry probabilies. Multistate models allow estimation of probabilies for apparent survival, , detection, , and movement among states, . We considered variation by colony, sex, and year for each of these parameters, subscripted as { c, pc, c}, { x, px, x}, or { t, pt, t}, respectively. Thus, our fully parameterized global model { c·x·t, pc·x·t, c·x·t} had 240 potentially estimable parameters. The iterative routine used during the maximum likelihood function failed to converge numerically for this model, so we re–examined input data and found that movement was not detected in 99 of the 160 possible colony–, sex–, and time–specific movements. We obtained convergence after fixing these parameters to zero, but we were warned by Program MARK that convergence was suspect. Numerical estimation for a few models from the candidate set resulted in inconsistent deviances relative to the number of parameters being estimated (White, G. W., pers comm). Inconsistency in changes of deviance confirmed that the global model failed to converge properly even after fixing parameters. Consequently, we reduced the number of strata from five to three, thus reducing the number of parameters being estimated while maintaining biologically relevant models, as follows. We constrained movement to occur among three strata only: Karrak Lake, Colony 10, and other colonies combined (nine, 46, and 81; hereafter, other colonies). Strata were redefined based on colony sizes (fig. 1) and sampling effort. Karrak Lake and colony 10 represent the two largest colonies within the QMGBS (~433,000 and ~386,000 breeding Ross’s Geese in 1998, respectively), and colonies nine, 46, and 81 are substantially smaller (ranging between ~30,000 and ~95,000 in 1998; Alisauskas et al., 1998). Sampling effort varied somewhat among years at different colonies, from the interplay between spring phenology and availability of aircraft with which to visit study colonies. Sampling effort was highest at Karrak Lake, where unlike other colonies, it could be accessed entirely by foot or by boat from our permanent research facility. All other colonies were accessed by helicopter, and so neckband observations there ranged from one to four days. Relative effort among colonies was consistent such that sampling effort at Karrak Lake > colony 10 > other colonies, for all years of the study. Such reduction of structure of the global model resulted in numerical convergence. Our reduced global model had 96 potential parameters, including all sources of variation and all possible interactions. Our modeling approach was to test fit of the global model to the data and then, based upon biological knowledge of the study

Drake & Alisauskas

organism and differences in sampling effort, compare a set of candidate models with reduced numbers of parameters to assess parsimony and fit of models to the data using AICc (Burnham & Anderson, 1998). Program MARK does not provide goodness–of–fit test specifically for multistate data sets, so we parameterized the data as a Cormack–Jolly–Seber (CJS) data set and tested for goodness–of–fit of { x*t, px*t} (Lebreton & Pradel, 2002) using 1000 iterations of the parametric bootstrap available for such global models in Program MARK. Deviance of the global model was less than 85% of the simulated deviance indicating that the data were not overdispersed so a variance inflation factor ( ) was not used (Burnham & Anderson, 1998). We considered 15 models in our candidate set. Movement probability was our primary parameter of interest, so our approach to hypothesis testing and parameter estimation was to sequentially reduce sources of variation in probabilities of resighting and then survival, while retaining full– structured variation in movement probabilities. First, we reduced sources of variation in . We retained effects of colony and time in all parameterizations of because sampling effort varied among colonies, and because we suspected temporary emigration as the size of breeding populations at colonies varies annually (Alisauskas & Rockwell, 2001). We considered 4 additional parameterizations of including (1) a multiplicative interaction between colony and additive effects of sex and time ( c·[x+t]), (2) a completely additive model ( c+x+t), and (3) additive ( c+t) and (4) multiplicative ( c·t) models without sex effects. We used the parameterization of from the best of these models in all subsequent modeling of survival and movement probabilities. Including the structure within the global model, we considered six parameterizations of . Breeding colonies represent subpopulations where the has potential for colony–specific differences in implications for colony–specific growth rates as well as potential fitness costs to individuals. There is considerable clinal variation in winter and migration affinities of Ross’s geese marked in the QMGBS over a narrow range of ~200 km of longitude (Alisauskas et al., 2005); thus it is likely that different segments of the QMGBS metapopulation are subject to geographically variable harvest pressure (Moser & Duncan, 2001), so we tested for colony specific rates of survival ( c·x·t) vs. ( x·t). Most evidence suggests that true survival, , does not vary between sexes in most species of geese (Melinchuk & Ryder, 1980; Alisauskas & Lindberg, 2002; but see Francis & Cooke, 1992). However, sex differences in fidelity, F, to breeding colonies may still result in sex–specific , because = S * F. So, we considered { c·x·t} vs. { c·t}. After testing for colony and sex effects, we considered models with additive effects of sex and time ( x+t), a linear time trend ( x+T), and a model that included only the effect of sex ( x).

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Animal Biodiversity and Conservation 27.1 (2004)

McMaughton River

Sim o ps ive nR r

Karrak River

km 10 0

50 km

Fig. 2. Locations of Ross’s Goose banding efforts within brood rearing areas in the Queen Maud Gulf Bird Sanctuary, Nunavut, Canada, 1999–2003. Banding drive locations are shown as dots, while stars delineate locations of breeding colonies that were surveyed for neckbanded geese. Fig. 2. Emplazamientos de las campañas de marcaje en el cuello del ansar de Ross en áreas de crianza de la Reserva Ornitológica del golfo de la Reina Maud, Nunavut, Canadá, 1999–2003. Los emplazamientos de las campañas de marcaje en el cuello se indican como puntos, mientras que las estrellas definen los emplazamientos de las colonias de reproducción que se investigaron con respecto a los gansos marcados en el cuello.

We proceeded to estimate starting with models optimally structured for and . We retained colony structure in in all models because of our interest in stratum–specific estimates. These included fully multiplicative effects of colony, sex and year { c.x.t}, additive effects of sex and time specific to each colony { c.[x+t]}, complete additivity { c+x+t}, a multiplicative model excluding the effect of sex { c.t}, and an additive model with colony and sex effects { c+x}. All manipulations of model structure were done using the design matrix in Program MARK, and all models were fit using the logit link function (White & Burnham, 1999). Results Model { x+t, c+t, c+x} was clearly best supported by our data (wAICc = 0.993, table 1); thus, all estimates were based on this model. This model showed that apparent survival varied over time in parallel between sexes, but that survival was equal among

colonies. Predictably, recapture probabilities varied among colonies, but differences were consistent for all years of study. Movement probability was constant but varied among colonies in parallel between sexes. Estimates of apparent survival ranged between 0.631 ± 0.038 (SE) and 0.682 ± 0.033 for females, and between 0.489 ± 0.034 and 0.546 ± 0.044 for males (table 2). Recapture probabilities varied in an additive fashion among colonies and years, but were as low as 0.069 ± 0.025 for colonies nine, 46 and 81, and as high as 0.612 ± 0.037 at Karrak Lake (fig. 3). Colony– and sex–specific dispersal probabilities ranged from 0.023 ± 0.024 to 0.344 ± 0.085 for females and from 0.122 ± 0.063 to 0.376 ± 0.074 for males (fig. 4a). We found 44 broken neckbands that had fallen off of male Ross’s Geese but only 12 from females. Compared to 7,904 males and 7,718 females that had been marked with neckbands, this represents a strong male bias in apparent rates of neckband loss (likelihood ratio 2 = 8.76, df = 1, P < 0.005)

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Table 1. Model structure, AICc, AICc, model weight (wAICc), number of parameters (K), and model deviance, for multistate modeling of apparent survival ( ), recapture (p), and dispersal ( ) probabilities of neckbanded Ross’s Geese within the Queen Maud Gulf metapopulation, 1999–2003. Dots indicate multiplicative interactions between colony (c), sex (x), time (t) and time–trend (T). Plus signs indicate an additive model. Tabla 1. Estructura de los modelos, AICc, AICc, peso de los modelos (wAICc ), número de parámetros (K), y desviación de los modelos, para la modelación multiestado de probabilidades de supervivencia ( ), recaptura (p) y dispersión ( ) de ánsares de Ross marcados en el cuello en la metapoblación del golfo de la Reina Maud, 1999–2003. Los puntos indican interacciones multiplicativas entre la colonia (c), el sexo (x) el tiempo (t) y la tendencia temporal (T). Los signos más indican un modelo aditivo.

Model {

x+t

px+t

c+x

{

c·x

}

AICc

wAICc

Model deviance

7045.07

0.00

0.99

471.50

x+t

px+t

7055.23

10.16

0.01

495.81

{

x+t

px+t

} c·(x+t)

7059.02

13.95

0.00

450.85

{

x+t

px+t

c·x·t

}

7069.92

24.85

0.00

437.14

{

x+t

px+t

c+x+t

}

7071.52

26.45

0.00

508.06

{

x·t

pc+t

}

7072.85

27.78

0.00

433.88

} c·x·t

7073.95

28.88

0.00

410.15

}

7075.55

30.48

0.00

444.82

7077.50

32.43

0.00

444.72

{

c·x·t

{

x+T

{

c·x·t

pc+t

}

AICc

c·x·t

pc+t

c·x·t

}

c·x·t

pc+x+t

c·x·t

}

7078.21

33.14

0.00

406.10

{

c·x·t

pc·(x+t)

} c·x·t

7078.46

33.39

0.00

398.02

{

c·x·t

pc·t

}

7081.36

36.29

0.00

405.08

{

c·x·t

p c·x·t

{ {

c·x·t

7081.86

36.79

0.00

405.58

c·t

px+t

} c·x·t

7094.46

49.39

0.00

443.10

x+t

px+t

c·t

}

7127.93

82.86

0.00

532.02

c·x·t

}

Discussion Movement probability Until development of methods for unbiased estimation of philopatry (and it’s complement, dispersal) that account for detection probability, inferences were often based on return rates (Geramita & Cooke, 1982; Anderson et al., 1992: table 11–3). Although our conclusion about female–biased breeding philopatry in Ross’s Geese qualitatively is consistent with general patterns for waterfowl, our results demonstrate that return rates offer only tentative inference about philopatry, similar to other investigations that have used mark–recapture methods (Lindberg et al., 1998; Doherty et al., 2002; Blums et al., 2003). Estimates of breeding philopatry for both male and female Ross’s Geese were substantially higher than sex–specific return rates reported for many goose species (Anderson et al., 1992: table 11–3).

Our results were consistent with the general prediction for female–biased breeding philopatry based upon the mating–system hypothesis (Greenwood, 1980; Rohwer & Anderson, 1988). Nevertheless, female philopatry was highly variable among colonies, and was less than absolute in all cases. This underscores the importance of dispersal in colony–specific population dynamics of Ross’s Geese in the QMGBS, and its potential to influence breeding distribution and gene flow. We applied estimates of female dispersal probabilities to size of breeding subpopulations and found that they represent a large numbers of birds that switch colonies annually. We used breeding population estimates of light geese at Karrak Lake (~866,000) from Alisauskas et al. (1998) and assumed that about 50% are Ross’s Geese, half of which are females (216,500). Assume that 216,500 females nest at Karrak Lake during year i, survive at a rate of 0.83 (Alisauskas et al., 2005), and breed during i + 1 at a hypothetical rate of 0.75. Thus, at

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i + 1 there are ~179,700 (216,500 x 0.83) surviving individuals of which 134,800 (179,700 x 0.75) will breed. Of those breeders, ~14,600 (134,800 x 0.108) will disperse from Karrak Lake and breed at another colony. Accordingly, assuming that average nesting density is equal among colonies, then based on colony area (km2, Alisauskas et al., 1998), colony 10 had ~772,000 geese (~193,000 female Ross’s Geese) and colonies 9, 46 and 81 combined represented ~339,000 geese (~84,800 female Ross’s Geese). Assuming the same rates of survival and breeding probability, we applied stratum–specific estimates of dispersal from the other colonies to Karrak Lake and found that ~17,400 (193,000 x 0.83 x 0.75 x 0.145) females emigrate from colony 10, and ~18,200 (84,800 x 0.83 x 0.75 x 0.344) females emigrate from the other combined colonies to Karrak Lake. Such calculations suggest a net increase of ~21,000 (35,600–14,600) females to Karrak Lake within a given year due to breeding dispersal alone (fig. 4b). Based upon limited information about movements of geese to and from La Pérouse Bay (LPB), and overwhelming female–bias in re–encounters at the colony, Cooke et al., (1995) argued that gene flow was male–mediated among breeding subpopulations of Snow Geese, while acknowledging that females showed some dispersal (Geramita & Cooke, 1982). More recently, Cooch et al. (2001) used a retrospective analysis to analyze life table response of the LPB colony and showed that emigration of adults had increased over time. Their results suggested that philopatry to brood rearing areas may be more flexible than fidelity to nesting areas. Our results suggest that breeding philopatry is also a flexible trait in closely–related Ross’s Geese. Given similarities in life histories of these congeners and their sympatry throughout the annual cycle, we suspect that female dispersal in Snow Geese is more common that previously thought. Although our analysis was focused on Ross’s Goose movement between breeding colonies within the QMGBS metapopulation, 21% (680/3233) of the Ross’s Geese used in our analysis were immigrants to the QMGBS that were banded along the West Coast of Hudson Bay (WHB), and 8.5% (58/ 680) of these were females. Movement of WHB geese represent breeding dispersal of distances ranging 500–800 km depending upon colony of settling, but because observations have not yet been done there, we could not estimate reverse movements. Our estimates of annual dispersal among breeding colonies by Ross’s Geese provide strong behavioral evidence for extensive gene flow among breeding subpopulations. These findings for Ross’s Geese are consistent with genetic studies of Snow Geese which suggested little or no phylogeographic structure in frequency of mtDNA haplotypes detected in Snow Geese from different breeding areas across North America (Avise et al., 1992; Quinn, 1992). Additionally, based on recoveries from each of the Pacific, Central and Mississippi Flywyas (Alisauskas et al., in review), there is great overlap

Table 2. Apparent survival estimates from multistate modeling of neckbanded adult Ross’s Geese breeding within the Queen Maud Gulf metapopulation, 1999–2003. Survival was best modelled by including sex and time effects, but was equal among sampled colonies. Probabilities are given ± SE. Tabla 2. Estimaciones de supervivencia aparente a partir de la modelación multiestado de la reproducción de ansar de Ross adultos marcados en el cuello en la metapoblación del golfo de la Reina Maud, 1999–2003. La mejor modelación de supervivencia se obtuvo incluyendo los efectos del sexo y del tiempo, pero fue similar entre las colonias muestreadas. Las probabilidades se indican ± SE.

Survival probability Year

Female

Male

1999

0.682 ± 0.038

0.546 ± 0.044

2000

0.657 ± 0.035

0.516 ± 0.040

2001

0.653 ± 0.033

0.489 ± 0.035

2002

not estimated

in winter range used by Ross’s Geese marked in different brood–rearing areas used by the QMGBS metapopulation. Overall, Ross’s Geese from QMGBS now have one of the most extensive winter ranges of any arctic–nesting goose species from a single arctic region. As well, high rates of movement by both sexes of Ross’s Geese among colonies in QMGBS, hint at considerable movement by Ross’s Geese to QMGBS from WHB. Shared winter areas of Ross’s Geese with different breeding locations suggest that subpopulations of light geese are extensively interconnected by broadly overlapping migration networks which likely enhances likelihood of breeding dispersal. Such movement patterns are consistent with the "considerable population connectedness" inferred by Avise et al. (1992). Studies of other colonial geese have shown that dispersal increases with increasing population density (Lindberg et al., 1998) and, that emigration can be an adaptive response to habitat degradation (Cooch et al., 1993; Cooch et al., 2001). We were unable to estimate population density at breeding colonies within the QMGBS, other than for Karrak Lake, and so were precluded from directly assessing breeding dispersal as a function of breeding density. Nevertheless, we found an asymmetry favouring movement toward Karrak Lake despite it being the most expansive colony of the ones studied. Slattery & Alisauskas (2002) detected density dependent effects on growth and survival of gos-

Drake & Alisauskas

Detection probability (p)

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0.70 0.60 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.5 0.0 1998

Karrak Lake Colony 10 Other colonies

1999

2000

2001

2002

2003

Year Fig. 3. Colony–specific detection probabilities from multistate modeling of neckbanded adult Ross’s Geese resighted at breeding colonies within the Queen Maud Gulf metapopulation, 1999–2003. Bars represent standard error of the estimate. Fig. 3. Probabilidades de detección para cada colonia a partir de la modelación multiestado de ánsares de Ross adultos marcados en el cuello que fueron reavistados en colonias de reproducción en la metapoblación del golfo de la Reina Maud, 1999–2003. Las barras representan el error estándar de la estimación.

lings marked on brood–rearing areas north of Karrak Lake, so other factors may override a connection to dispersal probabiltiy. For example, there is a strong cline in chronology of snowmelt with that in the west of QMGBS consistently far in advance (e.g., ~5% snow cover in 2003) compared to that 300 km to east (> 75% snow cover in 2003, Alisauskas, pers. obs.). Early nesting by arctic–breeding geese has strong fitness benefits (Cooke et al., 1984) because of the short time available for goslings to attain flight before freeze–up (Raveling, 1978). Hence, geese at Karrak Lake consistently may enjoy more favourable snow–free nesting conditions compared to most other colonies to the east. Consistent with this idea is that female emigration from colony 10, the most eastward colony examined, exceeds female immigration to colony 10 for both Karrak and other colonies. The pattern of asymmetry in movement to Karrak Lake may also relate to east–west differences in likelihood of mate loss and subsequent repairing by widowed Ross’s Geese. Alisauskas et al. (2005) found that Ross’s Geese banded in the vicinity of colony 10 in eastern QMGBS were most likely to be recovered in the Central and Mississippi Flyway’s; whereas, geese banded north of Karrak Lake were more likely to be recovered in the Pacific Flyway. Due to changes in management of light geese (Moser & Duncan, 2001), harvest in the Central and Mississippi Flyway has increased while it has re-

mained relatively stable in the Pacific Flyway. Breeding geese mix non–homogenously among different wintering and migration locations such that breeding populations from eastern QMGBS show greater affinities to Central and Mississippi Flyway winter areas. These geese are subjected to higher hunting mortality and mate loss than populations toward the west of QMGBS, which have greater affinities to the Pacific Flyway winter areas. Thus, our suggestion for higher probability of mate loss by geese from Colony 10 may lead to repairing with maturing geese from Karrak Lake, and so partially account for asymmetry in movement from Colony 10 to Karrak Lake than vice versa. Apparent survival rates Sex differences in apparent survival provide insights to potential sources of bias when we interpret our results, given what is known about true survival estimates. Our estimates of apparent survival for neckbanded Ross’s Goose females corresponded closely with estimates for true survival rates of neckbanded females from band recovery models (Alisauskas et al., 2005). Accordingly, we suggest that at least part of differences in apparent survival of males and females from multistate modeling resulted from violation of model assumptions rather from differences in true survival between sexes. Multistate models are constrained to = 1, which

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Animal Biodiversity and Conservation 27.1 (2004)

F: 0.75 ! 0.07 M: 0.63 ! 0.08

A F: 0.89 ! 0.03 M: 0.66 ! 0.08

Karrak Lake

F: 0.04 ! 0.01 M: 0.21 ! 0.04

F: 0.15 ! 0.03 M: 0.25 ! 0.04

F: 0.34 ! 0.09 M: 0.38 ! 0.07

Colony 10

F: 0.02 ! 0.02 M: 0.21 ! 0.07

F: 0.07 ! 0.03 M: 0.12 ! 0.05 F: 0.11 ! 0.05 M: 0.12 ! 0.06

Colonies 9,46,81

F: 0.63 ! 0.09 M: 0.41 ! 0.12

immigration < emigration

B F: 5,000 M: 28,800

immigration > emigration

Karrak Lake

F: 17,400 M: 30,400

Colony 10

F: 7,200 M: 11,300

F: 18,200 M: 19,900 F: 9,600 M: 16,400

F: 12,700 M: 14,700

Colonies 9,46,81 immigration l emigration

Fig 4. Breeding philopatry and dispersal of female (F) and male (M) Ross’s Geese from multistate modeling of neckband resightings at breeding colonies within the Queen Maud Gulf metapopulation, 1999–2003: A. Dispersal probabilities ± SE; B. Calculated numbers combining movement probabilities with estimates of population size for each colony. Fig. 4. Filatropía reproductora y dispersión de ansar de Ross hembras (F) y machos (M) a partir de la modelación multiestado de reavistajes de marcas en el cuello en colonias de reproducción en la metapoblación del golfo de la Reina Maud, 1999–2003: A. Probabilidades de dispersión ± SE; B. Números calculados que combinan probabilidades de movimiento con estimaciones del tamaño poblacional para cada colonia.

allows separate estimation of otherwise confounded probabilities of survival and movement. Under this necessary restriction, multistate models will produce survival estimates that are biased low if move-

ment of individuals to an unobserved state occurs (i.e., permanent emigration from the sampled areas and/or marker loss). We suggest that the sex differences in estimates of apparent survival resulted

340

from a combination of (1) higher rates of permanent emigration by males from surveyed areas than by females and (2) higher rates of neckband loss by males. Higher rates of permanent emigration by males from our study area is consistent with our finding of greater vagility within our study area by male than by females. Additionally, rates of neckband loss are generally higher for males in numerous other goose species (Alisauskas & Lindberg, 2002 and references therein). Low recapture probability of Ross’s Geese during annual banding efforts precluded direct estimation of neckband loss as done by Alisauskas & Lindberg (2002). Nevertheless, our discovery of nearly 4 times as many broken neckbands from males than from females, despite similar numbers marked, is in agreement with the general pattern of higher neckband loss by males. Most of these neckbands were lost probably during aggressive behaviour by males on breeding territories. The close correspondence between estimates of survival for neckbanded females from multistate modeling to those for neckbanded females from band recovery models suggests that the probability of permanent emigration from QMGBS by females was close to zero during the course of this study. Acknowledgements This study was funded through the Arctic Goose Joint Venture by the Canadian Wildlife Service, the Institute for Wetlands and Waterfowl Research, Delta Waterfowl Foundation, California Department of Fish and Game, and the Polar Continental Shelf Project. We thank the following people for field assistance during neckband surveys and banding efforts: I. Adams, C. Brayton, J. Charlwood, D. Clark, J. Conkin, K. Ermel, D. Johns, D. Kellett, D. Knudson, J. Leafloor, S. Lawson, M. Lindberg, R. Lorenz, A. Lusignan, K. Mehl, F. Moore, R. Olson, M. Roberts, S. Slattery, K. Spragens, D. Spencer, D. Stern, C. Swoboda, J. Traylor, N. Wiebe, and C. Woods. We thank D. Kellett and F. Cooke for helpful comments on previous versions of the manuscript. KLD received personal support through a University of Saskatchewan Graduate Scholarship, IKON Fellowship from the Institute for Wetlands and Waterfowl Research, and a research stipend from Delta Waterfowl Foundation. References Alisauskas, R. T., 1998. Winter range expansion and relationships between landscape and morphometrics of midcontinent Lesser Snow Geese. Auk, 115: 851–862. – 2002. Arctic climate, spring nutrition, and recruitment in midcontinent Lesser Snow Geese. Journal of Wildlife Management, 66: 181–193. Alisauskas, R. T., Ankney, C. D. & Klaas, E. E., 1988. Winter diets and nutrition of midcontinent

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Lesser Snow Geese. Journal of Wildlife Management, 51: 403–414. Alisauskas, R. T., Drake, K. L., Slattery, S. M. & Kellett, D. K., 2005. Neckbands, harvest and survival of Ross’s Geese from Canada’s central arctic. Journal of Wildlife Management, 69 (in press). Alisauskas, R. T. & Lindberg, M. L., 2002. Effects of neckbands on survival and fidelity of White– fronted and Canada Geese captured as non– breeding adults. Journal of Applied Statistics, 29: 521–537. Alisauskas, R. T. & Rockwell, R. F., 2001. Population dynamics of Ross’s Geese. In: The status of Ross’s Geese: 55–67 ( T. J. Moser, Ed.). Arctic Goose Joint Venture Special Publication, U.S. Fish and Wildlife Service, Washington, D.C., and Canadian Wildlife Service, Ottawa, Ontario. Alisauskas, R. T., Slattery, S. M., Kellett, D. K., Stern, D. & Warner, K. D., 1998. Spatial and temporal dynamics of Ross’s and Lesser Snow Goose colonies in Queen Maud Gulf Bird Sanctuary, 1966–1998, September 1998. Canadian Wildlife Service, Saskatoon. 21p. Unpublished report. Available at the Prairie and Northern Research Centre, Canadian Wildlife Service, 115 Perimeter Rd., Saskatoon, Saskatchewan, Canada, S7N 0X4. Anderson, M. G., Ryhmer, J. M. & Rohwer, F. C., 1992. Philopatry, dispersal, and the genetic structure of waterfowl populations. In: Ecology and management of breeding waterfowl: 365–395 (B. J. D. Batt, A. D. Afton, M. G. Anderson, C. D. Ankney, D. H. Johnson, J. A. Kadlec & G. L. Krapu, Eds.). Univ. of Minnesota Press, Minneapolis. Arnason, A. N., 1973. The estimation of population size, migration rates and survival in a stratified population. Researches on Population Ecology, 15: 1–8. Avise, J. C., Alisauskas, R. T., Nelson, W. S. & Ankney, C. D., 1992. Matriarchal population genetic structure in an avain species with natal philopatry. Evolution, 46: 1084–1096. Batt, B. J. D., Ed., 1997. Arctic ecosystems in peril: report to the Arctic Goose Habitat Working Group. Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington D. C. & Canadian Wildlife Service, Ottawa, Ontario. Blums, P., Nichols, J. D., Lindberg, M. S., Hines, J. E. & Mednis, A., 2003. Factors affecting breeding dispersal of European ducks on Engure Marsh, Latvia. Journal of Animal Ecology, 73: 292–307. 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., 1998. Model selection and inference: A practical information theoretic approach. Springer–Verlag, New York, Inc.

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Cooch, E. G., Jefferies, R. L., Rockwell, R. F. & Cooke, F., 1993. Environmental changes and the cost of philopatry: an example in the Lesser Snow Goose. Oecologia, 93: 128–138. Cooch, E. G., Lank, D. B., Rockwell, R. F. & Cooke, F., 1989. Long–term decline in fecundity in a snow goose population: evidence for density dependence? Journal of Animal Ecology, 58: 711– 726. – 1991. Long–term decline in body size in a snow goose population: evidence of environmental degradation? Journal of Animal Ecology, 60: 483–496. Cooch, E., Rockwell, R. F. & Brault, S., 2001. Retrospective analysis of demographic responses to environmental change: a Lesser Snow Goose example. Ecological Monographs, 71: 377–400. Cooke, F., Findlay, C. S. & Rockwell, R. F., 1984. Recruitment and the timing of reproduction in Lesser Snow Geese (Anser caerulescens caerulescens). Auk, 101: 451–458. Cooke, F., Rockwell, R. F. & Lank, D. B., 1995. The Snow Geese of La Pérouse Bay: Natural selection in the wild. Oxford Univ. Press, Oxford. Doherty, P. F. Jr., Nichols, J. D., Tautin, J., Voelzer, J. F., Smith, G. W., Benning, D. S., Bentley, V. R., Bidwell, J. K., Bollinger, K. S., Bradza, A. R., Buelna, E. K., Goldsberry, J. R., King, R. J., Roetker, F. H., Solberg, J. W., Thorpe, P. P. & Wortham, J. S., 2002. Source of variation in breeding–ground site fidelity of Mallards (Anas platyrhnchos). Behavioral Ecology, 13: 543–550. Francis, C. M. & Cooke, F., 1992. Sexual differences in survival and recovery rates of Lesser Snow Geese. Journal of Wildlife Management, 56: 287–296. Gilpin, M. & Hanski, I., Eds., 1991. Metapopulation dynamics: Empirical and theoretical investigations. Academic Press, London. Geramita, J. M. & Cooke, F., 1982. Evidence that fidelity to natal breeding colony is not absolute in female Snow Geese. Canadian Journal of Zoology, 60: 2051–2056. Greenwood, P. J., 1980. Mating systems, philopatry, and dispersal in birds and mammals. Animal Behaviour, 28: 1140–1162. Hanski, I. & Gilpin, M. E., Eds., 1997. Metapopulation biology: ecology, genetics, and evolution. Academic Press, San Diego, California. 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. Kerbes, R. H., 1994. Coloies and numbers of Ross’s Geese and Lesser Snow Geese in the Queen Maud Gulf Migratory Bird Sanctuary. Canadian Wildlife Service Occasional Paper, 81: 1–47. Lebreton, J. D. & Pradel, R., 2002. Multistate capture recapture models: modeling incomplete individual histories. Journal of Applied Statistics, 29: 353–369.

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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. Melinchuk, R. & Ryder, J. P., 1980. The distribution, fall migration routes and survival of Ross’s Geese. Wildfowl, 31: 161–171. Moser, T. J., 2001. The status of Ross’s Geese. Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington D.C., and Canadian Wildlife Service, Ottawa, Ontario. Moser, T. J. & Duncan, D. C., 2001. Harvest of Ross’s Geese. In: The Status of Ross’s Geese: 43–54 (Moser, T. J., Ed.). Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington DC, and Canadian Wildlife Service, Ottawa, Ontario. Mowbray, T. B., Cooke, F. & Ganter, B., 2000. Snow Goose (Chen caerulescens). In: The birds of North America, 514: 1–40 (A. Poole & F. Gill, Eds.) Philadelphia: The Academy of Natural sciences; Washington DC: The American Ornithologists’ Union. Pulliam, H. R., 1988. Sources, sinks, and population regulation. The American Naturalist, 135: 652–661. Quinn, T. W., 1992. The genetic legacy of Mother Goose: phylogeographic patterns of Lesser Snow Goose Chen caerulescens caerulescens maternal lineages. Molecular Ecology, 1: 105–117. Raveling, D. G., 1978. The timing of laying by northern geese. Auk, 95: 294–303. Rowher, F. C. & Anderson, M. G., 1988. Female– biased philopatry, monogamy, and the timing of pair formation in waterfowl. Current Ornithology, 5: 187–221. Ryder, J. P., 1972. Biology of nesting Ross’s Geese. Ardea, 60: 185–215. Ryder, J. P. & Alisauskas, R. T. 1995. Ross’ Goose (Chen rossii). In: The birds of North America, 162: 1–29 (A. Poole & F. Gill, Eds.) Philadelphia: The Academy of Natural sciences; Washington DC: The American Ornithologists’ Union. Schwarz. C. J., Schweigert, J. F. & Arnason, A. N., 1993. Estimating migration rates using tag–recovery data. Biometrics, 49: 177–193. Slattery, S. M. & Alisauskas, R. T., 2002. Use of the Barker model in an experiment examining covariate effects on first–year survival in Ross’s Geese (Chen rossii): a case study. Journal of Applied Statistics, 29: 497–508. Weins, J. A., 1997. Metapopulation dynamics and landscape ecology. In: metapopulation biology: ecology, genetics, and evolution: 43–62 (I. Hanski & M. E. Gilpin, Eds.). Academic Press, San Diego, California. White, G. W. & 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

Animal Biodiversity and Conservation 27.1 (2004)

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Bayesian analysis of the Hector’s Dolphin data R. King & S. P. Brooks

King, R. & Brooks, S. P., 2004. Bayesian analysis of the Hector's Dolphin data. Animal Biodiversity and Conservation, 27.1: 343–354. Abstract Bayesian analysis of the Hector's Dolphin data.— In recent years there have been increasing concerns for many wildlife populations, due to decreasing population trends. This has led to the introduction of management schemes to increase the survival rates and hence the population size of many species of animals. We concentrate on a particular dolphin population situated off the coast of New Zealand, and investigate whether the introduction of a fishing gill net ban was effective in decreasing dolphin mortality. We undertake a Bayesian analysis of the data, in which we quantitatively compare the different competing biological hypotheses, determining the effect of the sanctuary upon the dolphin population. Key words: Capture–recapture, Posterior model probabilities, Management schemes, Markov chain Monte Carlo. Resumen Análisis bayesiano de datos del delfín de Héctor.— En los últimos años, ha aumentado la preocupación por muchas poblaciones de fauna, como consecuencia del descenso observado en sus tendencias poblacionales. Ello ha llevado a la aplicación de programas de gestión orientados a aumentar las tasas de supervivencia y, por consiguiente, el tamaño de la población de numerosas especies de animales. Nos concentramos en una población de delfines concreta, situada en la costa de Nueva Zelanda, investigamos si la aplicación de una ley que prohíbe la utilización de redes de enmalle ha resultado eficaz a la hora de reducir la mortalidad de los delfines. Llevamos a cabo un análisis bayesiano de los datos, en el que comparamos cuantitativamente distintas hipótesis biológicas alternativas, y determinamos el efecto de la reserva en la población de delfines. Palabras clave: Captura–recaptura, Probabilidades del modelo posterior, Programas de gestión, Cadena Monte Carlo de Markov. R. King, CREEM, Univ. of St. Andrews, Buchanan Gardens, St. Andrews, Fife KY16 9LZ, U.K.– S. P. Brooks, Statistical Lab., Univ. of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, U.K.

ISSN: 1578–665X

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Introduction We consider a detailed study relating to a population of Hector’s dolphin (Cephalorhynchus hectori) located around the coast of New Zealand. These dolphins have been listed as an endangered species, and so there is particular concern and interest in identifying factors that influence their survival and the corresponding effectiveness of any management schemes to protect them. Our data concerns a colony of dolphins from the Southern Island of New Zealand around Bank’s Peninsula off the coast of Akaroa. In this area, commercial gill nets are commonly used for fishing, and it is believed that these may be a contributor to dolphin mortality (Dawsson & Slooten, 1993; Slooten & Dawson, 1994). In an attempt to reduce this threat to the dolphin population, a sanctuary was placed around the peninsula in 1988. The sanctuary imposed a ban on the use of gill nets within 4 nautical miles of the shore between November to February, coinciding both with the peak inshore commercial gill netting season and the period when dolphins move closest to shore (Dawson, 1991; Cameron et al., 1999). We wish to investigate whether this sanctuary was effective in terms of decreasing dolphin mortality. We begin in "Material and modelling" section by introducing the data that we have concerning the dolphin population and by describing the management scheme. We then describe the methodology that we use in order to answer the question of whether or not the sanctuary was effective. In "Results" section we present the results, before concluding with a Discussion. Material and modelling Data Our data comprise multi–site capture–recapture records, collected annually between 1985 and 1993. The study site was divided into three locations: locations 1 and 3 lie either side of the peninsula to the South and West, respectively, and are separated by region 2, which is the harbour area around Akaroa. The data collection process involves a boat going out to observe the dolphins within the inshore waters of the different areas over a number of days each year. Within each trip, individual dolphins sighted are uniquely identified via markings on their dorsal fin and/or body (see Hammon et al., 1990 for example). Our data comprises the capture histories of each of 102 individuals, detailing the years and locations that each dolphin is observed. A total of 668 days are spent observing the dolphins throughout the study period, with most data collected within 4 month periods during the summer season. The length of time spent by the observers at each of the sites over the different years is far from uniform. For example in 1986, 92 days are spent observing in area 2, whereas in 1989, no effort is expended in area 3. The number

of days that are spent each year observing dolphins in each of the areas is given in table 1 With the large variation in the amount of effort expended over the different years, and between the different sites, there is also a large variation in the number of dolphins that are observed over time and location. In Section 2.2, we show how we are able to explicitly account for catch–effort information in modelling the recapture rates for the dolphins. Modelling and notation The data are assumed to be well described by the Arnason–Schwarz model (see for example Schwarz et al., 1993; Brownie et al., 1993; Dupuis, 1995; Dupuis et al., 2002; King & Brooks, 2003b for further details). Essentially, we assume that dolphins may move freely between the different areas (independently of one another), but do not emigrate outside the study area. In addition, an individual dolphin’s movement between the different areas is assumed to have a Markovian structure so that the migration of an animal depends only upon its current location and not upon its previous migration history. As usual, we make the further assumption that the observed individuals are representative of the whole population. Under the Arnason–Schwarz model, we express the likelihood for our data as a function of the survival, recapture and migration rates (see for example, King & Brooks, 2003a). For this dataset the parameters in the model are the survival, recapture and migration rates, which we assume may depend upon time, location, neither, or both. We define: t (r) – Prob (a dolphin in area r c R at time t survives until time t + 1); pt+1 (r) – Prob (a dolphin in area r c R at time t + 1 is resighted at this time); t (r, s) – Prob (a dolphin in area r c R at time t is in area s c R at time t + 1, given that it survives until time t + 1) where R = {1,2,3} denotes the regions that the study area is divided into, and t = 1985,...,1992. For notational convenience, we set ={

(r), r c R, t = 1985,...,1992}

and similarly for p and . We represent different competing models, in terms of the dependence of the parameters upon the time and/or location, by placing different restrictions upon the parameters. For example, the biological hypothesis that the survival rate remains constant throughout the study, and is common to all areas would be represented by the restriction, t

(r) = , for all r c R and t = 1985,...,1992

Clearly, this model implies that the sanctuary had no effect upon the survival rate of the dolphins. Conversely, if we believed that the survival rate did change at the time that the sanctuary was introduced, then we may wish to consider the model

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Table 1. The amount of effort (in days) spent sighting dolphins in each of the study areas and years. Tabla 1. Cuantificación del esfuerzo (en días) dedicado a avistar delfines en cada unas de las áreas de estudio y años.

Year Site

1985

1986

1987

1988

1989

1990

1991

1992

1993

with a change–point, so that (r) = (r) = t t

(r), for t = 1985,...,1987 (r), for t = 1988,...,1992 b a

for example. In this study, we are particularly interested in whether or not the sanctuary that was introduced in 1988 had a significant impact upon the survival rates of the dolphins. Thus, we impose the constraint that there may be at most one change–point, which may or may not occur at the time that the sanctuary was introduced. Our class of models therefore comprises models for which there is either no change–point over time; or where there are the restrictions, t

(r) = a(r), for t = 1985,..., T–1 (r) = b(r), for t = T,...,1992 t

for T = 1986,...,1992. Finally, we need to define the possible location dependencies. We allow all possible restrictions upon the locations e.g., a common survival rate for all locations, distinct survival in each area, or a common survival rate for two different areas which is distinct from the third. We identify these models by defining the sets of areas with common survival rates. For example, the model with a common survival rate over areas is denoted by {1,2,3}, while the model {1}, {2,3} denotes that there is a common survival rate in areas 2 and 3, which is distinct to that in area 1, and so on. Since we consider change–point models here, we allow the location restrictions to be placed independently on the survival rates both before and after the change–point, if there is one. Thus, the dependence of the parameters upon location is conditional on the year. For the recapture rates, we have additional information relating to the amount of effort that was expended in each location and year of the study. We wish to incorporate this into the analysis, making full use of all available information. Thus, we specify the recapture rates as a function of the corresponding effort taken in that year and location. In particular, we assume that sightings in year

t = 1986,...,1992 and location r c R occur as a Poisson process with general underlying recapture intensity rate t (r). Thus, we set, pt (r) = 1 – exp [– t (r) xt (r)] where xt (r) denotes the catch–effort in year t and location r. This can be reparameterised in the form pt (r) = 1 – [1 –

(r)]xt(r)

where t (r) is directly interpretable as the underlying recapture rate per unit time (i.e. day). Allowing a complete spatio–temporal dependence for the recapture intensity rates (i.e. having distinct t (r) parameters for each t = 1985,...,1992 and r c R) essentially reduces the model to that with arbitrary recapture rates pt (r). However, we consider special cases for the t (r) parameter, representing different possible models directly analogous to those considered above for the survival rates, (i.e. a maximum of a single change– point and all possible location dependencies). For example, the model, t

(r) =

(r), for t = 1986,...,1993

represents the system where the recapture intensity rates depend only upon the location of the dolphin, suggesting that the dolphins may be inherently more observable in some areas than others; or that the observers themselves have more (or less) information relating to the areas where dolphins are most likely to be seen in each of the designated areas. Note that for all models we implicitly assume that the underlying recapture rate for any given year is homogeneous over days within that year. We impose the same year constraints upon the migration rates, as for the survival rates for similar reasons as those discussed above. See King & Brooks (2002) for further discussion of model structures. Discriminating between these competing models tells us about the underlying dynamics of the system in terms of the possible effect of the intro-

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duction of the sanctuary upon the survival and behaviour of the dolphins. We adopt a Bayesian approach here. Bayesian analysis Here we undertake a Bayesian analysis of the data, in which inference is based upon the posterior distribution of the parameters obtained by combining the likelihood of the data given the parameters, with the corresponding prior distribution placed upon the parameters (independently of the data), via Bayes’ theorem. The prior distribution represents our beliefs concerning the parameters before observing any data. The posterior distribution is then an update of these prior beliefs having observed the data. The posterior distribution is, in general, very complex and so to obtain any inference based upon the posterior distribution we use Markov chain Monte Carlo (MCMC) in order to obtain estimates of the parameters of interest (see for example Brooks, 1998). Within our analysis, the nature and number of parameters depends upon the model and, as well as parameter estimation, we have the additional issue of model uncertainty. This is of particular interest in our case, since the different models represent different competing hypotheses relating to the effectiveness of the sanctuary —our primary question of interest. Within the Bayesian paradigm, we are able to incorporate model uncertainty by considering the model itself to be an unknown parameter which we wish to estimate. We are then able to form the joint posterior distribution over both parameter and model space. Since the posterior distribution is defined over different dimensions (i.e. for the different models), we use Reversible jump MCMC (RJMCMC) in order to explore the distribution. See Green (1995), Richardson & Green (1997), and also King & Brooks (2003b) —in the context of multi–site capture–recapture data— for example. Using these methods, we are able to construct a single Markov chain which can explore the posterior distribution, and estimate summary statistics, such as: (1) posterior model probabilities for time and/or location dependence of the parameters; (2) the posterior probability of a change in survival rate when the sanctuary is introduced; and (3) model averaged statistics of interest, such as posterior model–averaged survival rates, taking into account model uncertainty (in addition to parameter uncertainty). However, before we can undertake our analysis we need to place suitable priors on the models and their associated parameters. Prior distributions We do not have any prior information relating to the survival or migration rates, and so we use vague priors. In particular, for each model, we place a U[0, 1] prior on each of the survival rates. For the migration rates we use a Dir (2,2,2,) prior, corre-

sponding to an uninformative Jeffrey’s prior (Carlin & Thompson, 1998, section 2.2.3; Jeffreys, 1961, p 181). However, we consider the prior on the recapture intensities in more detail. Placing a prior on the recapture intensity rates implicitly imposes a prior on the recapture rates, which may be more directly interpretable. Thus, we wish to place a prior on the recapture intensity which is consistent with our beliefs concerning the recapture rates. Since the recapture intensity is positive, an obvious prior that we may wish to use is a Gamma prior. We are then able to calculate the corresponding prior on the recapture rates, using the usual transformation of variables, as follows. Suppose that we specify a Γ(a,b) prior on t (r). For ease of notation, we assume that there is a common underlying for each year and location, and note that the same prior will be used for the recapture intensity rates across all possible models (i.e. area and time dependence structures). Then, it can be easily shown that the corresponding prior on the recapture rates are of the form,

Placing a vague prior on does not impose a similarly vague prior on p t (r); for example, Γ(0.001, 0.001) a prior is often considered to be vague. However, in this case using such a prior on β produces a prior on pt (r) (assuming xt (r) … 0), which essentially places point masses at zero and one. Conversely, it is obvious from equation (1) that if we set = 1 in the prior for , then,

Suppose that we set,

then, the prior for the recapture rate will be flat (i.e. U[0,1]), when the mean amount of time is spent observing dolphins in the given year and location, i.e. when xt (r) = 0. If a larger amount of time is spent (i.e. xt (r) > 0), then the prior for the recapture rate is skewed to the right, and there is more prior mass on larger recapture rates. Otherwise, if a lower amount of time is spent (i.e. xt (r) < 0), then the prior for pt (r) is skewed to the left, and there is more prior mass on lower recapture rates. We can also explicitly calculate the prior mean and variance for the recapture rates, since,

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Animal Biodiversity and Conservation 27.1 (2004)

Clearly, there are many possible values of b that we may wish to consider. For example, we may wish to set b = median xt (r), or b = maxt,r xt (r). Prior information may be able used to discern the most appropriate form of b to take. Since we do not have any prior information, we specify that ~ Γ(1, 0) a priori, but note that the posterior distribution is fairly insensitive to sensible choices of prior. Finally, we need to place a prior on the models themselves. For each of the parameters, we place an equal prior probability on each possible age dependence structure. For the recapture intensities and the survival rates we place a flat prior across each possible combination of strata independently within each set of ages. Note that for the recapture intensities and survival rates, the prior is not flat over each individual model. Placing a flat prior over the whole of the model space, so that each individual model was equally likely a priori, would result in a greater amount of prior mass on models with a single change–point compared to those with a constant rate over time. For example, when there are no change–points there are a total of five possible location dependence structures as illustrated by the column headings of table 3. However, when there is a single change–point there are five location–dependent models both for before and after that change–point and so 52 models in total. Thus, putting a flat prior across all models would make a change–point model five times more likely than having no change–point, since there are five times more change–point models than there are models without a change–point.

We can then calculate the posterior distribution of the parameters and use an MCMC procedure to obtain estimates of the posterior statistics that we are interested in. We assume that there is model uncertainty relating to the survival and migration parameters, and use the reversible jump algorithm to move between the different possible models, as above. Details of the MCMC procedure are given in the appendix. Results We ran the simulations for a total of 1 million iterations, with the first 100,000 discarded as burn–in, and consider each set of parameters in turn. Convergence is rapid for these simulations and this is confirmed using standard diagnostic techniques, see Brooks & Roberts (1998). Survival rates For the survival rates, we are particularly interested in whether there is any evidence that they were increased by the introduction of the sanctuary. Thus, we are particularly interested in whether there is a change–point in the survival rates, and if so, when this change occurred in relation to the introduction of the sanctuary. Within the Bayesian framework, we are able to quantitatively compare the different possible models, in terms of their posterior (marginal) model probabilities. These are presented in table 2.

Table 2. The posterior marginal probabilities of no change–point and a change–point in each year for the survival probabilities, recapture intensities and migration rates. Recall that the survival and migration rates are defined for years 1985–1992, and the recapture intensities for years 1986–1993. Tabla 2. Probabilidades marginales posteriores de ausencia de punto de cambio y presencia de punto de cambio en cada año para las probabilidades de supervivencia, intensidades de recaptura y tasas de migración. Recuérdese que las tasas de supervivencia y migración se han definido para los años 1985–1992, y las intensidades de recaptura para los años 1986–1993.

Posterior probability Change–point

Survival rates

Recapture intensities

Migration rates

None

0.314

0.175

0.180

1986

0.088

–

0.036

1987

0.036

0.241

0.155

1988

0.160

0.035

0.549

1989

0.136

0.421

0.041

1990

0.112

0.087

0.024

1991

0.069

0.010

1992

0.084

0.012

0.005

1993

–

0.012

–

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1.0 0.8 Survival rate

The marginal model with most posterior support has a constant survival rate over all time, suggesting that the introduction of the sanctuary had no impact upon the survival rates of the dolphins. However, the second most probable model a posteriori places a change–point at the time that the sanctuary is introduced. The model–averaged estimates of the survival rates (together with 95% highest posterior density intervals – HPDI’s) are given in figure 1, and suggest a slow decreasing trend post–1987, in each area. Thus, there appears to be little evidence that the sanctuary was effective in decreasing the mortality rate of the dolphins. The similarity between survival rates within the different areas across all years are indicative of the large posterior probabilities that the survival rates are constant across areas. These are presented in table 3). Clearly, the most posterior support has a common survival rate over the different areas: approximately 50% for all years. Conversely, there is very little posterior support for distinct survival rates within each area.

0.6 0.4 0.2 0.0 1985 1986 1987 1988 1989 1990 1991 1992 Year Fig. 1. The posterior model–averaged mean (*) and 95% HPDI (vertical lines) for the survival rates over time: — Area 1; - - - Area 2; ····· Area 3.

Recapture intensities There is some evidence that the recapture intensities change over time within the study, as shown in table 2. In particular the strongest support is for a change–point in 1989 (42%) or 1987 (24%). The evidence for a change–point is clearly seen in the posterior means for the recapture intensities over time in figure 2A, where there appears to be a large increase in the recapture intensity from 1989 in area 3. This not only coincides with the sanctuary being introduced the previous year but also to a relatively small amount of effort made in area 3 from 1989 onwards, compared with previous years (see table 1), resulting in a much greater number of dolphins observed in this area. Before 1989 only 3 dolphins are observed in area 3; from 1989, 24 dolphins are observed in area 3. There also appears to be a change in the posterior probabilities for the location dependence from 1989 onwards for the recapture rates, as shown in table 3. The corresponding mean recapture rates are given in figure 2B. We can see that the recapture rates in area 2 are generally higher than those in other areas. This is the harbour area of the peninsula, where most effort was actually made in observing the dolphins. The dip in the recapture rates for all areas within the middle of the study corresponds to generally decreased effort spent in observing the dolphins (see table 1). Migration rates The marginal posterior model probabilities for the change–point models for the migration rates are also given in table 2. Most posterior support (55%) is for the model where there is a change– point in the migration rates the year that the sanctuary is introduced. This suggests that the sanctuary may have influenced the behaviour of

Fig. 1. Media posterior del modelo promediado (*) y 95% de HPDI (intervalos máximos de densidad posterior) (líneas verticales) para las tasas de supervivencia a lo largo del tiempo: — Área 1; - - - Área 2; ···· Área 3.

the dolphins, in terms of their movement around the peninsula. The change in the movement of the dolphins is clearly seen in the model–averaged estimates of the migration rates, presented in figure 3. One possible reason for this may be an indirect link with possible changes in fish stocks in the area, due to the gill net fishing ban, which may also have an impact upon the abundance of different breeds of fish and marine animals. However, further research would be necessary in order to investigate this possibility. The dolphins appear to predominantly stay in the same areas, although there are some movements between the different regions. This demonstrated by the general dominance of the top line in each of figures 3A–C. Discussion We consider multi–site capture–recapture data of the Hector’s dolphins around the Bank’s Peninsula in New Zealand. This species is endangered, and we wish to assess the impact (if any) of the management scheme placing a fishing gill net ban around the shore–line of the peninsula during the summer months. We have applied a Bayesian analysis to the data collected. However, within our analysis, there is little evidence to suggest that

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Table 3. Posterior marginal probabilities or the arrangement of strata over time for the survival probabilities and recapture intensities. Tabla 3. Probabilidades marginales posteriores o disposición de los estratos a lo largo del tiempo para las probabilidades de supervivencia y las intensidades de recaptura.

Strata groupings Survival probabilities Year 1985 1986 1987 1988 1989 1990 1991 1992 Recapture intensities 1986 1987 1988 1989 1990 1991 1992 1993

{1}, {2}, {3} 0.046 0.035 0.035 0.041 0.051 0.059 0.063 0.070

{1,2}, {3} 0.138 0.138 0.140 0.152 0.166 0.170 0.172 0.178

{1,3}, {2} 0.144 0.132 0.133 0.134 0.135 0.136 0.137 0.142

{1}, {2,3} 0.135 0.133 0.132 0.138 0.145 0.154 0.159 0.164

{1,2,3} 0.537 0.562 0.561 0.534 0.503 0.482 0.468 0.446

0.294 0.322 0.323 0.322 0.324 0.324 0.325 0.324

0.084 0.055 0.054 0.252 0.260 0.262 0.264 0.263

0.308 0.243 0.232 0.047 0.014 0.012 0.010 0.011

0.251 0.356 0.370 0.368 0.396 0.398 0.397 0.396

0.062 0.025 0.022 0.011 0.001 0.005 0.003 0.005

B 1.0 0.8

0.20 Recapture rate

Recapture intensity

A 0.25

0.15 0.10

0.6 0.4 0.2

0.05

0.0

0.00 1985 1986 1987 1988 1989 1990 1991 1992 Year

1985 1986 1987 1988 1989 1990 1991 1992 Year

Fig. 2. The posterior model–averaged mean (*) and 95% HPDI (vertical lines) for the recapture intensities (A) and recapture rates (B) over time: — Area 1; - - - Area 2; ···· Area 3. Fig. 2. Media posterior del modelo promediado (*) y 95% de HPDI (líneas verticales) para las intensidades de recaptura (A) y las tasas de recaptura (B) a lo largo del tiempo: —. Área 1; - - Área 2; ···· Área 3.

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0.8

0.8 Migration rate

Migration rate

350

0.6 0.4

0.2

0.0

0.0 1985 1986 1987 1988 1989 1990 1991 1992 Year

1985 1986 1987 19881989 1990 1991 1992 Year

1.0

Migration rate

0.8 0.6 0.4 0.2 0.0 1985 1986 1987 1988 1989 1990 1991 1992 Year Fig. 3. The model-averaged (*) and 95% HPDI for the migration rates from area 1 (A), area 2 (B) and area 3 (C). Lines denote movement to : — Area 1; - - - Area 2; ···· Area 3. Fig. 3. Modelo promediado (*) y 95% de HPDI para las tasas de migración de (a) el área 1 (A), el área 2 (B) y el área 3 (C). Las líneas indican movimiento hacia: — Área 1; - - -. Área 2; ···· Área 3.

this sanctuary achieved its aim. On the contrary, if there has been a trend in the survival rate this appears to have been negative overall. This, perhaps surprising, result may be a consequence of other factors that may not have been accounted for. For example, it is unclear as to the effect that such a sanctuary would have upon the whole of the local ecosystem. The increased fish stocks that would be present within the inshore waters once the ban was imposed may impact upon both predators higher in the food chain (possibly attracting them to the area), and also other marine animals competing for the same resources. In addition, Slooten & Dawson (1994) identify pollution as influencing the survival rates of the dolphins. Thus, the sanctuary alone appears to be ineffective in its attempt to improve the survival

rates of the dolphins, although it is possible that had the sanctuary not have been introduced, there may have been an even greater decline in the survival rate of the dolphins. Some or all of these factors could have been investigated by adopting an alternative design incorporating controls for each of these potential effects. Such designs (often referred to as BACI —before, after, control and impact— designs) have significant advantages for assessing management impacts and would have improved the ability of the study to determine the true effect of the sanctuary. The movement of the dolphins between the different areas does appear to change at the same time that the sanctuary is placed around the peninsula. Within our Bayesian analysis, this marginal model has a posterior probability of 55%. This may be a

Animal Biodiversity and Conservation 27.1 (2004)

result of the effect that the sanctuary had upon the fish stocks in the study area, which may be more evenly spread around the peninsula with the imposed fishing bans. Alternatively, the dolphins may move more freely to areas which were previously heavily populated with gill nets. Clearly, any such hypothesis would need further investigation. Finally, we note that there is some evidence that the recapture intensities changed from 1989. It is clear from figure 2A that this appears to be largely due to the change in the recapture intensity in area 3. This may be as a result of the observers learning where the dolphins are more abundant in area 3 from previous boat surveys in the study; or possibly as a result of the sanctuary being introduced and affecting the dolphins behaviour (possibly moving further inshore or to slightly different waters), and making them easier to observe. Clearly, the sanctuary placed around the peninsula does not appear to be enough to improve the survival rates of the dolphins. It would appear unlikely that it would increase the mortality rate of the dolphins, so this analysis suggests that there are other (more predominant) factors that are affecting the dolphins survival rates which need to be addressed, in order to conserve this dolphin population. Acknowledgements This analysis would not have been possible without the data, which were colleted by Stephen Dawson, Elisabeth Slooten and Stefan Bräger. Their fieldwork was funded by Reckitt and Colman (NZ) Ltd, NZ Whale and Dolphin Trust, Department of Conservation and University of Otago. References Brooks, S. P., 1998. Markov Chain Monte Carlo Method and its Application. The Statistician, 47: 69–100. Brooks, S. P. & Roberts, G. O., 1998. Diagnosing Convergence of Markov Chain Monte Carlo Algorithms. Statistics and Computing, 8: 319–335. 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. Cameron, C., Barker, R., Fletcher, D., Slooten, E. & Dawson, S., 1999. Modelling Survival of Hec-

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tor’s Dolphins around Banks Peninsula, New Zealand. Journal of Agricultural, Biological and Environmental Statistics, 4: 126–135. Carlin, B. P. & Louis, T. A., 1998. Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall/CRC Dawson, S., 1991. Incidental catch of Hector’s dolphin’s in inshore gillnets. Marine Mammal Science, 7: 283–295. Dawson, S. & Slooten, E., 1993. Conservation of Hector’s Dolphins: The Case and Process which led to Establishment of the Bank’s Peninsula Marine Mammal Sanctuary. Aquatic Conservation: Marine and Freshwater Ecosystems, 3: 207–221. Dupuis, J. A., 1995. Bayesian Estimation of Movement and Survival Probabilities from Capture– recapture Data. Biometrika, 82: 761–772. Dupuis, J. A., Badia, J., Maublanc, M. & Bon, R., 2002. Survival and Spatial Fidelity of Mouflons (Ovis gmelini): a Bayesian Analysis of an Age– dependent Capture–Cecapture Model. Journal of Agricultural, Biological and Environmental Statistics, 7: 277–298. Green, P. J., 1995. Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination. Biometrika, 82: 711–732. Hammon, P. S., Mizroch, S. A. & Donovan, G. P., 1990. Individual Recognition of Cetaceans: Use of Photo–Identification and other techniques to estimate Population Size. Technical report, International Whaling Commission. Jeffreys, H., 1961. Theory of Probability, third edition. Oxford, Univ. Press. King, R. & Brooks, S. P., 2002. Bayesian Model Discrimination for Multiple Strata Capture–Recapture Data. Biometrika, 89: 785–806. – 2003a. A Note on Closed Form Likelihoods for Arnason–Schwarz models. Biometrika, 90: 435–444. – 2003b. Survival and Spatial Fidelity of Mouflon: the Effect of Location, Age and Sex. Journal of Agricultural, Biological and Environmental Statistics, 8: 486–513. Richardson, S. & Green, P. J., 1997. On Bayesian Analysis of Mixtures with an Unknown Number of Components. Journal of the Royal Statistical Society, Series B, 59: 731–792. Schwarz, C. G., Schweigert, J. F. & Arnason, A. N. , 1993. Estimating Migration Rates using Tag– Recovery Data. Biometrics, 49: 177–193. Slooten, E. & Dawson, S., 1994. Hector’s Dolphin, in Delphinidae and Phocoenidae (Vol V. of Handbook of Marine Mammals). Academic Press, N.Y.

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Appendix. In order to explore and summarise the posterior distribution we use a reversible jump Markov chain Monte Carlo algorithm. Essentially this involves constructing a Markov chain with stationary distribution equal to the posterior distribution. Then, following an initial burn–in period, so that the stationary distribution is reached, realisations of the chain can be regarded as a sample from the posterior distribution and used to estimate summary statistics of interest. The reversible jump MCMC algorithm consists of two different move types: one for updating the parameters; and the other for updating the model itself. Apéndice. Con la finalidad de explorar y resumir la distribución posterior, hemos utilizado un algoritmo de Monte Carlo de cadena de Markov de salto reversible. En esencia, este algoritmo implica construir una cadena de Markov con una distribución estacionaria equivalente a la distribución posterior. Posteriormente, y después de un periodo inicial que permita llegar a una distribución fija, las partes finales de la cadena pueden ser utilizadas como una muestra de la distribución posterior, siendo entonces utilizada para estimar los estadísticos resumen en los que estamos interesados. El algoritmo reversible de la cadena de Monte Carlo es de dos tipos: uno para actualizar los parámetros y el otro para actualizar al propio modelo.

Within–model MCMC updates We update each of the parameters in the model using the standard Metropolis–Hastings algorithm, within each iteration of the Markov chain. In particular, we use a random walk Metropolis–Hastings algorithm. Suppose that we are proposing to update the parameter t (r). Then we propose parameter 't (r), such that 't (r) = 't (r) + where, ~ U[– , ], with δ chosen via pilot tuning. We accept the new proposed value with the standard acceptance probability (see for example Brooks, 1998). In practice = 0.1 appears to work well for both the survival rates and recapture intensities. This is simply generalised when we have sets of times and/or strata grouped together. We use an alternative Metropolis–Hastings update for the migration rates, since we need to retain the sum to unity constraints. Suppose that for a given time t, we wish to update the parameter t (r, s). Then we randomly choose u c R \ s, and set, 't (r, s) = 't (r, u) =

t t

(r, s) + (r, u) –

where ~ U[–v, v] In practice we set v = 0.05. This move is accepted with the standard probability. Within each iteration of the Markov chain we cycle through each age group and propose to update all r, s c R using the above procedure. Between–model (RJ) MCMC updates To update the dependence structure of the recapture parameters, we need to use the reversible jump algorithm to move between the different models, since the models are of different dimensions. The reversible jump algorithm can be seen to be an extension of the Metropolis–Hastings algorithm, allowing for movements between different states. Within each step of the Markov chain, we propose to update: (i) the number of change–points on the survival rates, recapture intensities and migration rates; (ii) the location of a change–point (if any) for each of the parameters; and (iii) the area dependence on the survival rates and recapture intensities for each age group. We consider the different types of reversible jump updates in turn. Adding/removing change–point We initially consider the survival rates. If there is a constant survival rate, we propose to add a change–point. Otherwise if there is already a change–point, we propose to remove it, since we only consider models with a maximum of a single change–point. Initially, suppose that we propose to add a change–point, so that the current model, denoted by m has a common survival rate over time, and for simplicity we assume that the survival rate is also common over all areas, i.e.,

Animal Biodiversity and Conservation 27.1 (2004)

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Appendix. (Cont.)

(r) =

for all r c R and t = 1985,...,1992

Then we propose to move to the new model, m', with parameters, 't (r) = + for all r c R and t = 1985,...,T–1 't (r) = – for all r c R and t = T,...,1992 where T is randomly chosen in the interval {1986,...,1992}, and ~ N(0, 2), for 2 chosen via pilot tuning. Then, if any 't (r) v [0,1] the move is automatically rejected, else it is accepted with probability,

where π(⋅|⋅) is the posterior distribution over parameter and model space evaluated at the given parameter values; |J| a Jacobian term, which is equal to 2 in this case; P(m'|m) = 1/7 is the probability that given that we are in model m, we propose to move to model m' with a change–point at time T (for which there are seven possible choices, each chosen with equal probability), and q( ) is the corresponding Normal proposal density. Clearly this approach can be generalised for any given location dependence, with the restriction that the new survival rates before and after the change–point have this same location dependence, with the corresponding changes to the Jacobian term. Note that in general |J| = 2k, where k is simply the number of distinct survival rates over the locations. Alternatively, in the reverse move, to retain the reversibility conditions, we only propose to remove the change–point if the location dependence is the same over all times. Then the proposed survival rate is simply taken to be the mean of the survival rates either side of the change–point for each area. The corresponding acceptance probability is simply the reciprocal of equation (2). The analogous update is used for the recapture intensity, with the restriction that the proposed recapture rates are positive. We set 2 = 0.1 and 0.01 for the survival rates and recapture intensities respectively. However, we need to consider a different updating procedure for the migration rates, since we need to retain the restriction that the migration rates sum to unity. Again, suppose that we propose to add in a change–point to the current model m, with migration rates (r, s), r, s c R (recall that R = {1,2,3}). Then, for r c R and s = 1,2, we propose the parameters in the new model to be, 't (r, s) = t (r, s) + (r, s) for all r c R and t = 1985,...,T–1 't (r, s) = t (r, s) for all r c R and t = T,...,1992 with probability 1/2; else we set, 't (r, s) = t (r, s) for all r c R and t = 1985,...,T–1 't (r, s) = t (r, s) + (r, s) for all r c R and t = T,...,1992 Here, T chosen uniformly in [1986,...,1992], and, (r, s) ~ N (0, 2), where 2 is chosen via pilot tuning. In this case we set 2 = 0.1. Essentially, we are simulating a new set of migration rates for either before or after the change–point, which are similar to their current values, while the others remain the same. We also set,

to ensure that the migration rates sum to unity. If any 't (r, s) v [0,1], then we automatically reject the move, else we accept the move with the acceptance probability,

Here |J| = 1, P(m'|m) = 1/2 x 1/7; P(m'|m) = 1/2 and q( ) denotes the Normal proposal density for the set of parameters = { (r, s): r c R, s = 1,2}.

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Appendix. (Cont.)

Location of change–point Initially, consider the survival rates. Then, within this updating procedure, we propose to update the location of the change–point if there is one present. Suppose that we are in the model with the change–point at time T, so that, t

(r) = 't (r) =

(r) for all r c R and t = 1985,...,T–1 (r) for all r c R and t = T,...,1992.

Then we propose to move to model m' by updating the change–point to time T' = T ! 1. If T' v [1985,...,1992], we reject the proposal; else we set, 't (r) = 't (r) =

(r) for all r c R and t = 1985,...,T'–1 (r) for all r c R and t = T',...,1992 b

We then accept the proposed move with the standard Metropolis–Hastings acceptance probability, i.e.,

We use the analogous updating procedure for the recapture intensities and migration rates. Updating area dependence Initially consider the survival rates and assume that there is no change–point; else, if there is a change–point, then we update the location dependence before and after the change–point independently of each other. We assume that there is a common survival rate over time and location, i.e. the area dependence is denoted by {1,2,3}. Then, we propose to update this area dependence, by splitting the group into two: there are three possibilities {1}, {2,3}; {1,3}, {2}; and {1,2}, {3}. We choose each one with equal probability, without loss of generality suppose that we propose to move to model {1}, {2,3}. Then, for the new model m', we propose the parameters, 't (r) = t (r) + 't (r) = t (r) –

for r = 1 and t = 1985,...,1992 for r = 2, 3 and t = 1985,...,1992

where, ~ N (0, τ), for τ chosen via pilot tuning. We reject the proposed move if any 't (r) v [0,1], else, we accept the move with probability,

where the Jacobian |J| = 2; P(m'|m) denotes the probability of moving to model m' from model m and q( ) the Normal proposal density for the simulated parameter . Here, P(m'|m) = 1/3, since we can only increase the dimension of the model, and, P(m|m') = 1/2, if in model m' = {1}, {2,3} we propose to decrease, or increase, the dimension of the model to {1,2,3}, or {1}, {2}, {3}, with equal probability. Once more, to retain the reversibility condition, the reverse move is deterministically defined by this move. In particular, we set the new survival rate to be the mean of the current values, and accept the move with probability equal to the reciprocal of the above. The analogous move holds when proposing to move to the model with distinct survival rates over all areas. We apply the analogous move to the recapture intensities, restricting the parameter values to be simply > 0. Pilot tuning suggests setting τ = 0.1 for both the survival rates and recapture intensities.

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Spatial distribution of breeding Pied Flycatchers Ficedula hypoleuca in respect to their natal sites L. Sokolov, N. Chernetsov, V. Kosarev, D. Leoke, M. Markovets, A. Tsvey & A. Shapoval

Sokolov, L., Chernetsov, N., Kosarev, V., Leoke, D., Markovets, M., Tsvey, A. & Shapoval, A., 2004. Spatial distribution of breeding Pied Flycatchers Ficedula hypoleuca in respect to their natal sites. Animal Biodiversity and Conservation, 27.1: 355–356. Extended Abstract Spatial distribution of breeding Pied Flycatchers Ficedula hypoleuca in respect to their natal sites.— Study of philopatry and dispersal of pied flycatchers Ficedula hypoleuca was launched on the Courish Spit (SE Baltic) in 1981. Since then, ca. 9,000 nestlings were ringed at different sites in the Russian part of the Courish Spit. A total of 557 individuals ringed as pulli were recaptured in subsequent seasons in the study area. Both males and females are more often recaptured in the plots where they were ringed than in other plots. These results were interpreted in the framework of the hypothesis forwarded by Löhrl (1959) and supported by Berndt & Winkel (1979). These authors suggested that cavity nesters (pied flycatchers and collared flycatchers F. albicollis) imprint their future local breeding area during the period of postfledging exploration. Birds that survive until the next spring, return to these imprinted areas to breed. A similar study done by Sokolov et al. (1984) on the Courish Spit in an open nesting species, the chaffinch Fringilla coelebs, confirmed this finding. We assumed that juvenile pied flycatchers disperse for varying distances during their postfledging movements and imprint a local area, some 1–5 kilometres in diameter. This area is the goal of their migration next spring. It is suggested that in spring, yearlings are non–randomly distributed in respect to the area they have imprinted as juveniles. Recently, Vysotsky (2000, 2001) re–analysed the same data on philopatry of pied flycatchers on the Courish Spit and forwarded an alternative hypothesis. He suggests that juveniles, both males and females, do not imprint any local area during the postfledging period, but are distributed randomly across the area of several dozens of kilometres in spring. Vysotsky was able to show that distribution of distances of natal dispersal did not differ from the random pattern the study plot which was an 8.5 km long line of nest boxes along the Courish Spit. The aim of this study was to test these two alternative hypotheses. To do so, we set up nine new study plots in 2000. Over 800 nest–boxes were made available for the birds (in addition to the old 400) in the 44 km long area. We recaptured pied flycatchers returning for breeding during four years, 2000–2003. The distribution of natal dispersal distances was compared with the null model which assumes that pied flycatchers settle randomly in the study area. We took all nest boxes from which pied flycatchers successfully fledged in a particular year and all next boxes where we were able to capture either a male or a female in the subsequent year, and calculated the distances between each pair of such nest boxes. Simulations were run separately for each sex. Theoretical distributions already include control efficiency. If some nest boxes were not checked in some year, or if we failed to capture one or both members of a breeding pair, we did not include this nest box in

Leonid Sokolov, Nikita Chernetsov, Vladislav Kosarev, Dmitry Leoke, Mikhail Markovets, Arseny Tsvey & Anatoly Shapoval, Biological Station Rybachy, Rybachy 238535, Kaliningrad Region, Russia. Correesponding author: L. Sokolov. E–mail: lsok@bioryb.koenig.ru ISSN: 1578–665X

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the model. Some birds could settle outside the study plot. Therefore, the theoretical distribution may underestimate the actual range of natal dispersal, but is unlikely to overestimate it. The number of females ringed as nestlings and recaptured as one–year–old birds was 43. The distribution of their natal distances (mean 6,8 km, SE = 0,81; median 5,4 km) was not significantly different from the pattern predicted by the null model (Wilcoxon matched pairs test: z = 1,25; p = 0,21). Conversely, males settled significantly closer to their natal nest box (n = 83; mean 4,3 km, SE = 0,57; median 2,5 km) than predicted by the model (Wilcoxon matched pairs test: z = 2,45; p = 0,014). For example, 24% of males settle within one km from their natal site, as compared with 7% predicted by the model. Males are found with a greater than chance probability within the 7 km zone around their natal site. The hypothesis by Vysotsky (2000) can thus be rejected for pied flycatcher males. Pied flycatcher females are known to settle at larger distances from their natal nest box. The very fact that were controlled 83 males and only 43 females suggests, assumed that sex ratio at fledging is close to being equal and that true survival rates during the first year of life do not differ greatly between the sexes, that many females emigrated from of our study plot. This does not mean that juvenile females do not imprint a home area during the postfledging period, as suggested by Vysotsky (2000). We think that the reason for this is not the inadequate navigational ability of the females but the fact that they were attracted by a prospecting male at some distance from their migratory destination and settle there. Such intercepting was suggested by Fedorov (1996) for Acrocephalus warblers, and it may exist in other migratory passerines. This is supported by the data on natal site fidelity from Spain which show that in Spanish pied flycatcher populations, recruitment rate did not differ between female and male juveniles (Potti & Montalvo, 1991). Females from these southern populations have a limited chance to be attracted by prospecting males in even more southern areas. References Berndt, R. & Winkel, W., 1979. Verfrachtungs–Experimente zur Frage der Geburtsortspragung beim Trauerschnäpper (Ficedula hypoleuca). J. Ornithol., 120: 41–53. Fedorov, V. A., 1996. To the question of formation of territorial connections in some Acrocephalus warbler species. Russ. J. Ornithol., express issue 1: 8–12 (in Russian). Löhrl, H., 1959. Zur Frage des Zeitpunktes einer Prägung auf die Heimatregion beim Halsbandschnäpper, Ficedula albicollis. J. Ornithol., 100: 132–140. Potti, J. & Montalvo, S., 1991. Return rate, age at first breeding and natal dispersal of pied flycatchers Ficedula hypoleuca in central Spain. Ardea, 79: 419–428. Sokolov, L. V., Bolshakov, K. V., Vinogradova, N. V., Dolnik, T. V., Lyuleeva, D. S., Payevsky, V. A., Shumakov, M. E., Yablonkevich, M. L., 1984. Testing of the ability for imprinting and finding the site of future nesting in young chaffinches. Zool. zh., 73: 1671–1681 (in Russian). Vysotsky, V. G., 2000. Structure of the local population in birds: example of the pied flycatchers Ficedula hypoleuca. Ph. D. thesis, Zoological Institute, St. Petersburg (in Russian). – 2001. Do the data on post–fledging dispersal of pied flycatchers Ficedula hypoleuca support the concept of imprinting a local area? Avian Ecol. Behav., 6: 83–84.

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Comparing timing and routes of migration based on ring encounters and randomization methods H. Lokki & P. Saurola

Lokki, H. & Saurola, P., 2004. Comparing timing and routes of migration based on ring encounters and randomization methods. Animal Biodiversity and Conservation, 27.1: 357–368. Abstract Comparing timing and routes of migration based on ring encounters and randomization methods.— A method for comparing two migration routes is introduced. In the method are needed approximations of the average daily positions which are computed based on averages of dates and of positions in a window sliding through the encounters ordered in ascending order by date. The method contains two tests. The first, global, test statistic compares the entire migration routes and is the average of the distances between the daily positions of the two routes to be compared. The second test is used to identify sections during migration where the routes may deviate and is based on consecutive averages of the distances of short time periods between the daily positions of the two routes. A randomization test is used to assess the statistical significance of the test statistics in both components. The methods are applied to artificial and real data. Examples of the use of the method are computed with data sets of ring recoveries of Common terns (Sterna hirundo) and Ospreys (Pandion haliaetus) ringed in Finland in 1930–2002. Key words: Migration route, Migration timing, Ring encounters, Randomization. Resumen Estudio comparativo de la adecuación temporal y de las rutas migratorias mediante el empleo de hallazgos de anillas y métodos de randomización.— El trabajo presenta un método para comparar rutas migratorias. El método utiliza aproximaciones para el cálculo de las localizaciones medias diarias basadas en las fechas y localizaciones promedio, en una ventana que se va desplazando a lo largo de las distintas observaciones ordenadas en orden ascendente según fecha. El método incluye dos pruebas estadísticas. La primera compara de forma global todas las rutas migratorias, y lo que se compara es la media de las distancias entre las distintas posiciones diarias de las dos rutas. La segunda prueba permite identificar tramos durante la migración en los que las rutas se pueden desviar, y se basa en promedios consecutivos de las distancias de breves períodos de tiempo entre las posiciones diarias de las dos rutas. Se utiliza una prueba de randomización para evaluar la relevancia estadística de las pruebas en ambos componentes. Los métodos se aplican a datos artificiales y reales. Los ejemplos del empleo del método se calculan con conjuntos de datos de recuperaciones de anillas de charranes comunes (Sterna hirundo) y de águilas pescadoras (Pandion haliaetus) anillados en Finlandia entre 1930 y 2002. Palabras clave: Ruta migratoria, Adecuación temporal de las migraciones, Hallazgos de anillas, Randomización. Heikki Lokki, Dept. of Computer Science, P. O. Box 68, FIN–00014 Univ. of Helsinki, Finland.–Pertti Saurola, Finnish Museum of Natural History, P. O. Box 17, FIN–00014 Univ. of Helsinki, Finland. Corresponding author: H. Lokki. E–mail: Heikki.Lokki@cs.Helsinki.Fi

ISSN: 1578–665X

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Introduction One of the early goals of bird ringing was to study migration. In this paper we introduce a method for comparing timing and routes of migration based on encounters along birds’ flyways from two groups of birds. The two groups A and B to be compared could be different sexes, different age classes of a species, different causes of recoveries such as "found dead" or "killed", or encounters from different decades. The difference in timing of migration has been studied by examining differences in peak passages of birds (e.g. Saurola, 1981; Jenni & Kéry, 2003). Fourier analysis has been used in describing migration patterns and plotting the position– date relation (Perdeck, 1977). Perdeck & Clason (1982, 1983) analyzed flyways and sex-based differences in migration and winter quarters of Anatidae ringed in the Netherlands on the basis of average positions of recoveries in disjoint time intervals. Munro & Kimball (1982) used the Mardia–Watson–Wheeler test (Batschelet, 1981) in testing for similarity in ring recovery patterns. It has been shown (Lokki & Saurola, 1987) that the Mardia–Watson–Wheeler test does not perform well in two–sample location problems and significance can also be caused by unequal variances in the ring recovery patterns to be compared. In addition Munro & Kimball (1982) compared recovery dates of harvested mallards by testing the mean dates of recoveries of different groups and they found out significant differences between groups in the average arrival time to the areas where the mallards were hunted. The distributions of Common tern (Sterna hirundo) and Arctic tern (Sterna paradisaea) recoveries of birds ringed in Finland have been plotted according to recovery month and latitude in order to illustrate the temporal patterns of migration (Saurola, 1978). In Swedish Bird Ringing Atlas (Fransson & Petterson, 2001) the differences between mean positions of recoveries of various time periods are tested by a randomization test (Lokki & Saurola, 1987). An extensive overview of many aspects of the study of migration routes has been written by Bairlain (2001). In the literature cited, the geographical and temporal differences of migration are studied separately. The geographical difference is often studied using algorithms for two–sample location problems and the temporal component is excluded by restricting the time of the encounters to one month. The timing of arrival of two groups of birds to a study area (e.g. ringing station or hunting area) has been used as an indication of a possible temporal difference in migration. In our method both geographical and temporal differences are considered simultaneously. We show how to identify the reason (geographical or temporal) of difference. In addition we show how to find time periods during migration in which there are differences in mean geographical locations.

We present the route comparison method and study its performance with artificial and real data. The real data used are recoveries of Common terns (Sterna hirundo) and Ospreys (Pandion haliaetus) ringed in Finland in 1930–2002. The biological interpretations of the findings are not presented in this paper. Material and methods Comparison of two migration routes for geographical or temporal differences A population of migrants consists of a set of triplets P(t,x(t),y(t)) containing all the locations (x(t),y(t)) of all individual birds of the group. Time t can be restricted to a period of calendar years or whole years can be considered. Here we consider time t as a discrete variable so that each individual is recorded once a day. The population is divided into two groups, A and B. For each group, the average daily position can with a simibe simply computed as lar average for group B. Then the distance between the average positions of the two routes at each time is defined as

and finally, the average of the differences is defined as

where m is the number of days in the migration route. Two populations P(t,x(t),y(t)), summarized in daily positions of migration routes, can differ in two ways: geographically and/or temporally. When two routes differ only geographically the tracks of the routes are different but the timing of migration of the two groups do not differ. When two routes differ only temporally, route points of the groups are at the same location on different dates. The temporal difference or difference in timing of the migration can originate in many ways: the start of the migration of birds of the two groups can occur in different periods of time and/ or the speed of the migration can be different even in different periods during migration. In the scope of this paper comparisons of two populations are reasonable only if the starting and ending areas of the migration are the same for both populations. The encounters along migration routes are often made by the general public and are not sampled systematically. The following assumptions are necessary: (1) the dates and locations of encounters are recorded accurately; (2) encounters are independent of each other; (3) individuals of both populations are encountered with the same pattern. We expect that the assumption

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(3) is valid when the encounters of both groups originate from at least partially overlapping areas during migration. In a similar fashion, sample quantities can also be computed —these are denoted using the letter S. Real groups of encounters may contain several records for a given day and days without records may occur. We first summarize each sample to an approximate average route with exactly one position for each day. We denote the average daily positions as . Next we compute

and finally, the average difference in positions as

We can formulate the null hypothesis: DP = 0. The sampling distribution of the test statistic DS is found by randomizing the inclusion of the encounters to groups. The method has three parts. In part 1, the individual encounters are converted to average daily positions. The basis for the algorithm is that we slide a time–window through the set of encounters sorted in ascending order by date. In each time–window we compute the averages of dates and of geographical positions. In part 2, the randomization test is performed. In part 3, the data is examined more closely to identify where differences in the migration route or timing occurred. Part 1 1. Sort the encounters in ascending order by date. 2. Starting with the first encounter, create a time–window with at least wp encounters and at least wd days. Include all encounters of the last day in the window. In the current window, compute and store the average of the dates (= day1) and the average of the positions. 3. Repeat until the end of the encounters: move the time window forward by excluding all encounters of the earliest day and including encounters of the following days as in step 2. In the current window, compute and store the average of the dates and the average of the positions. 4. Compute exactly one route point for each date in the period of $1,…,$m ($m is the average date of the last time–window in step 3). If there are more than one ( ) positions for a date $i, compute the average of those positions and replace the block of records with this one average. If there is no route point for a date $1,…,$m, use linear interpolation to compute one. The averages in steps 2, 3 and 4 are computed as arithmetic means of dates and centers of gravity of positions (see Perdeck, 1977). Also medians could be used. Geographical distances must be

computed on the earth’s surface. The reason for time–windows of varying sizes is to ensure a minimum number of encounters in each window so that the averages are computed on the base of samples of reasonable sizes. Part 2 Here we assume that the encounters of groups A and B have been converted to average daily positions of each migration route. 1. Compute DS as the mean of the differences of the approximate daily average positions along the two migration routes. 2. Randomize the inclusion of encounters into the two groups A and B by pooling both groups and dividing randomly the pooled group into two subgroups of the sizes of the original groups nA and nB. Compute the mean DS of step 1. Repeat step 2 many times (N) to create the empirical distribution (D*) of the mean DS. 3. Calculate p = /N, where is the number of values in D* which are greater than the observed DS. The distances in the average daily position are computed using the Haversine formula (Sinnot, 1984). The number of randomizations N in step 2 depends on the significance level required. A reasonable relation is N m 10/ . Part 3 The statistic DS tests if average of the differences of the daily positions of the entire routes is zero. Unfortunately, if the p–value of the test is small, it does not identify where a difference may have occurred. A formal test can be constructed by examining the averages of the daily distances piece–wise in periods of k (e.g. k = 10) days. For each block, a p–value is obtained. A correction for multiple testing is done using the Single–Step Resample Method presented in Westfall & Young (1993). Westfall & Young (1993) suggest that the number of resamples (N*) should be m 10,000 and we use N* = 10,000. Small multiplicity–adjusted p–values identify blocks where differences may have occurred. If the test statistic D S indicates that the null hypothesis should be rejected, the reason can be a geographical or temporal difference along the migration route. In cases where there are differences in some part of the migration routes and elsewhere the migration routes are very similar, the test statistic D S can fail to detect the difference. These differences will be found by examining the differences of migration routes piece–wise. When there are geographical differences in two migration routes, the lines connecting the route points on the same days are perpendicular to the migration direction. When the difference in migration routes is temporal, the connecting lines are parallel to the migration direction.

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Data sets 60

We will use three data sets to evaluate and illustrate the tests. Data set 1 is artificial and data sets 2 and 3 are real recoveries.

The artificial data set represents fast long–distance migrants. We simulated the migration route of this fictitious species by lat(t) = (latN – latS) / 2 * 1 / sin(sin(...sin( /2)...)) * sin(sin(...sin(2 * /365 * (t + 3/4 * 365))...)) + (latN + latS ) / 2 (1) where t is date, latN and latS are the latitudes in the north (breeding area) and in the south (wintering area) respectively. Each addition of the repetitive sin function in (1) simulates a faster migrant and we have used 24 sin functions. The longitude of the records is modeled otherwise correspondingly, but we added a term: (latN – lat(t)) / (latN – latS) * (lonN – lon(t)) (2) to the longitude in order to create a curved migration route (see figure 1). In addition, to each latitude and longitude value a random number from the normal distribution N(0,1) was added. The population set consisted of 1,000 birds on each of 185 days. This population represents a fast migrant and the birds follow a curved and relatively narrow flyway with only a small position-date variation. A sample of the simulated records is plotted in figure 1. Example 1a For this data set we drew two samples of sizes 100, and shifted one of the samples 3° to the east in order to create geographical difference between the two samples. Example 1b For this data set we drew two samples of sizes 100, and subtracted 5 days from all observations of one of the samples in order to create temporal difference between the two routes. Example 2 Recoveries of young (1 yr) and adult (+2 yr) Common terns (Sterna hirundo) ringed in Finland in 1930–2002 and encountered during 1 VII–28 II. Common terns follow closely the coast lines of Western Europe and Africa and the migration route includes curves and embodies large position–date variation. The 495 records of young and 230 records of adult Common terns are shown in figure 2. If (1) the date of the recovery is less accurate than 3 days, or (2) the place is

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Fig. 1. The shape of the migration route shown by randomly drawn 300 locations of the artificial data set 1. The geographical variation is due to a standard deviation of 1 degree in longitude and the temporal variation due to a standard deviation of 1 degree in latitude. Fig. 1. La forma de la ruta migratoria se indica mediante 300 emplazamientos dibujados al azar del conjunto 1 de datos artificiales. La variación geográfica obedece a la desviación estándar de un grado en longitud, mientras que la variación temporal obedece a la desviación estándar de un grado en latitud.

inaccurate, or (3) the condition of the bird is unknown, or (4) the bird had been dead for a long time, the recovery was excluded. Example 3 Recoveries of young (1 yr) and adult (+3 yr) Ospreys (Pandion haliaetus) ringed in Finland in 1932–2002 and encountered during 1 VIII–31 I. Ospreys migrate over a broad area and winter mainly in Western Africa but also in South Africa and the Mediterranean area. The 521 records of young and 201 records of adult Ospreys are shown in figure 3. The recoveries were selected according to the same criteria as in example 2. In addition we compared recoveries of young Ospreys reported "killed" and "found dead" along migration routes. Results In this chapter we present results of examinations of the method for comparing timing and routes of migration. We also show how to distinguish geographical difference from temporal dif-

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1b. In the left part the points of the same dates of the two migration routes are connected at intervals of 10 days. When the reason for difference is the temporal deviation the connecting lines are parallel to the migration direction as figure 5 (left) demonstrates. In the right part are shown the original and multiplicity–adjusted p–values (denoted ) of piece–wise analysis of data set 1b. As can be seen the probabilities that the null hypothesis hold are low in the period of 28 VIII–17 X and high elsewhere. The same time period is marked in the left part of figure 5.

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Fig. 4. When two migration routes deviate geographically the lines connecting route points of the same dates are perpendicular to the migration direction. The migration direction is a Fourier curve fitted to the combined data of both sets of encounters. Fig. 4. Cuando dos rutas migratorias se desvían geográficamente, las líneas que conectan los puntos de la ruta de las mismas fechas son perpendiculares a la dirección de la migración. La dirección de la migración es una curva de Fourier ajustada a los datos combinados de ambos conjuntos de hallazgos.

ference in migration. Further we will show how to examine the differences in migration piece–wise in addition to the average difference over the entire migration period. Distinguishing between geographical and temporal differences in migration routes and finding periods where differences occur Examples 1a and 1b were created only to show how geographical difference can be distinguished from difference of timing in migration. In analysis of example 1b we also show how periods of indication of difference in migration can be found. Figure 4 illustrates the analysis of example 1a. The points of the same dates of the two migration routes are connected at intervals of 5 days. When the reason for difference is the geographical deviation, the connecting lines are perpendicular to the migration direction as figure 4 demonstrates. Figure 5 illustrates the analysis of example

We compared the migration routes of young and adult Common terns (example 2 and figure 2). The observed value of the test statistic D S is located to the right of the empirical distribution of the test statistic (1,000 randomizations), giving a very small p–value (p < 0.001). The difference between the routes of young and adult Common terns is temporal because the lines joining the approximate positions at the same dates are parallel to the migration direction as can be seen in figure 6 (left). In figure 6 (right) are shown the results of the piece– wise test. In the period of 8 IX–26 XI the – values show that the null hypothesis should be rejected at significance level = 0.05. The same period is shown in the left part of figure 6. At the beginning of the migration of young and adult Common terns no indication of difference in timing was found, but soon after they have left Finland the young birds lag behind the adult ones until the routes converge time–wise in wintering areas in Southern Africa. Comparison of migration routes of young and adult Ospreys We compared the migration routes of young and adult Ospreys (example 3 and fig. 3). Only two values of the test statistic in 1,000 randomizations was above the observed value DS. Thus we reject the null hypothesis of similar routes of young and adult Ospreys at high significance level (p = 0.002). The observed difference in migration is illustrated in figure 7. In figure 7 (right) two sections can be localized where the null hypothesis should be rejected at significance level = 0.05: (1) at the beginning of migration in 29 VIII– 7 IX and later in 18 X–6 XI. The sections are shown also in the left part of figure 7 where it can be seen that young Ospreys leave Finland earlier than adult ones, in most of Central Europe there is no difference in migration patterns of the two groups, but later adult Ospreys cross the Mediterranean and Sahara before the young birds. In the wintering area South of Sahara after 6 XI the recoveries do not show any difference.

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Fig. 5. When two migration routes deviate temporally the lines connecting route points of the same dates are parallel to the migration direction (left figure). The migration direction is a Fourier curve fitted to the combined data of both sets of encounters. In the section of 28 VIII–17 X, where the values < 0.05 the null hypothesis should be rejected at the significance level of = 0.05. The section is marked also in the left part of the figure (lines of + signs). In the right figure dotted segments of lines are the original and solid segments multiplicity–adjusted p–values . Fig. 5. Cuando dos rutas migratorias se desvían temporalmente, las líneas que conectan los puntos de las rutas de las mismas fechas son paralelas a la dirección de la migración (figura de la izquierda). La dirección de la migración es una curva de Fourier ajustada a los datos combinados de ambos conjuntos de hallazgos. En la sección de 28 VIII–17 X, donde los valores < 0,05, la hipótesis nula debería rechazarse en el nivel de significación de = 0,05. La sección también está marcada en la parte izquierda de la figura (líneas de signos +). En la figura de la derecha, los segmentos punteados son los valores originales, mientras que los segmentos sólidos son los valores p ajustados para multiplicidad.

Comparison of recoveries of young Ospreys reported killed or found dead In figure 8 are shown the results of testing whether reported killing along the Osprey’s migration route varies. The data set contains 241 recoveries that were reported as "killed" and 274 recoveries of Ospreys reported as "found dead". The observed test statistic D S is in the middle of the empirical distribution of the test statistic (p = 0.54) and the null hypothesis cannot be rejected. However, in figure 8 (right) can be seen that in the period of 2 IX–17 IX the null hypothesis of the first time period should be rejected at significance level = 0.05. In figure 8 (left) can be seen that the lines joining the average positions of the same days are perpendicular to the migration direction indicating that the killing of Ospreys is emphasized in the South–Eastern part of the beginning of migration. We repeated the test to the same data where recoveries South of Sahara (or latitude 25) were excluded. This test indicated that the null

hypothesis must be rejected at a statistically highly significant level because the observed difference DS was greater than all values in 1,000 randomizations. Discussion Geographical differences in sets of encounters have most often been studied in the literature as a twosample location problem and it has been shown (Lokki & Saurola, 1987) that tests based on randomization perform well in this problem. Temporal differences have been studied separately for example by comparing arrival dates to a study location. With the method introduced in this paper one can compare both geographical and temporal differences of two populations of encounters at the same time and later distinguish between the two causes of difference. In addition, the progress of migration can be studied in more detail in order to locate periods of differences along the migration route. In a way the method can be considered to be

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Fig. 6. Young and adult Common terns ringed in Finland start migration at the same time and follow the same geographical route. Later young birds lag behind adult ones. In the wintering area no geographical differences were found. The period 8 IX–26 XI, when the multiplicity–adjusted values < 0.05 indicating that the null hypothesis should be rejected, is marked in both left and right figures. In the right figure dotted segments of lines are the original and solid segments multiplicity–adjusted p– values . Fig. 6. Los charranes comunes jóvenes y adultos anillados en Finlandia inician la migración a la vez y siguen la misma ruta geográfica. Posteriormente, las aves jóvenes van a la zaga de las adultas. En el área invernal no se encontraron diferencias geográficas. El período de 8 IX–26 XI, cuando los valores p ajustados para multiplicidad < 0,05 indican que la hipótesis nula debería rechazarse, está marcado tanto en la figura de la izquierda como en la de la derecha. En la figura de la derecha, los segmentos p punteados son los valores originales, mientras que los segmentos sólidos son los valores ajustados para multiplicidad.

an extension to the two–sample location test based on randomization. Here the third dimension, in addition to the latitude and longitude, is time, which brings complexity to the analysis. The migration route comparison method works in two phases, both of which are based on approximations of the average daily positions. In the first phase the analyst computes the mean DS of the distances of the daily positions of the two routes. The test statistic DS is a mean. Therefore it is possible that a difference in one part of the migration route will not result in rejection of the null hypothesis, if in another part of the routes the daily positions are exceptionally close to each other. The comparison of "killed" and "found dead" young Ospreys is an example of this phenomenon. In the second phase of the method these kinds of exceptional situations should be noticed. In the second phase the analyst follows the migration routes piece–wise in time periods of k days and uses the mean of the distances of k days as test statistics. Correction for multiple testing was performed. By connecting the locations on the

same dates of the two routes the analyst can infer the cause of the observed difference: if the connecting lines are perpendicular to the migration direction the reason for rejecting the null hypothesis is geographical and in case of temporal difference the connecting lines are parallel to the migration direction. In the piece–wise examination we used periods of k = 10 days. If k ^ 10 the piece–wise test would lose power, because more tests would be performed and the correction for multiple testing would dilute the original p–values more. If k p 10 the test may fail to detect indications of short–term differences but the power of the test rises. In our examples k = 10 was satisfactory because in all periods where the original p–values were < 0.05 at least part of the p–values corrected for multiple testing were also < 0.05. What would be reasonable sizes of samples or sets of encounters? If one is analyzing migration routes that are several thousands of kilometers long, it is obvious that with only a few dozens of encounters the migration route will not be covered

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Fig. 7. Young Ospreys ringed in Finland seem to start migration before adult ones according to these data. It can be seen from the values < 0.05 in the period of 29 VIII–7 IX (right figure) and connecting lines of route points of the same period in the corresponding period in the left figure. In the mid–part of the migration (Central Europe) there is no indication of difference in the migration pattern of young and adult Ospreys which is illustrated by the –values in the period 8 IX–17 X (right figure). Later, in period 18 X–6 XI, the adult birds cross the Mediterranean and Sahara before young birds. This is seen from value < 0.05 in that period (right figure) and connecting lines of route points that are parallel to the migration direction (left figure). South of Sahara no evidence of different populations was found. In the right figure dotted segments of lines are the original and solid segments multiplicity–adjusted p– values . Fig. 7. Según estos datos, las águilas pescadoras jóvenes anilladas en Finlandia parecen iniciar la migración antes que las adultas. Ello puede apreciarse por los valores < 0.05 en el período de 29 VIII– 7 IX (figura de la derecha) y las líneas de conexión de los puntos de la ruta del mismo período en el período correspondiente de la figura de la izquierda. En la parte central de la migración (Europa Central) no existe ninguna indicación de la diferencia en la pauta migratoria de las águilas pescadoras jóvenes y adultas, según lo ilustrado por los valores del período 8 IX–17 X (figura de la derecha). Posteriormente, en el período 18 X–6 XI, las aves adultas cruzan el mar Mediterráneo y el Sahara antes que las aves jóvenes. Ello puede apreciarse por el valor < 0,05 de ese período (figura de la derecha) y las líneas de conexión de los puntos de la ruta que son paralelas a la dirección de la migración (figura de la izquierda). Al sur del Sahara no se encontró ninguna evidencia de poblaciones distintas. En la figura de la derecha, los segmentos punteados son los valores originales, mientras que los segmentos sólidos son los valores p ajustados para multiplicidad.

well. Our tests of the route comparison method showed that the results are satisfactory even with sample sizes of 100 for even 10,000 km long routes. What would be an optimal size of the sliding window? Our algorithm requires that a predefined minimum number of encounters must be included in each window. This adaptive nature of the window size means that the time span that the windows cover varies with the time spacing of the data. We have tested the method with window sizes of at least 10, 20 or 30 records in each window. When we compared the routes of young and adult Common terns the sliding window of size m 30 recoveries corresponds to a time window of 43 days on

average. Accordingly, in comparing routes of young and adult Ospreys the window size of m 30 records corresponds to a time window of 36 days on average. It is possible that time windows covering such long periods of time are not flexible enough to follow the migration in sensible details. For example, in the Common tern case the comparison test would indicate temporal difference in migration in a longer period than in using a smaller window size. With a window size of m 10 recoveries in both cases of comparing routes of young and adult Common terns and Ospreys the corresponding time window is on average 15 days and the actual number of recoveries in the windows is on average 14–15. With these numbers the sliding window can follow

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Fig. 8. Young Ospreys’ recoveries are reported "killed" (n = 241) and "found dead" (n = 274) in different geographical areas in period of 2 IX–17 IX. In that period the values < 0.05 (right figure) and connecting lines of route points of the same dates are perpendicular to the migration direction (left figure). In the right figure dotted segments of lines are the original and solid segments multiplicity– adjusted p–values . Fig. 8. Las recuperaciones de águilas pescadoras jóvenes se indican como "muertas" (n = 241) y como "encontradas muertas" (n = 274) en distintas áreas geográficas en el período de 2 IX–17 IX. En dicho período, los valores < 0,05 (figura de la derecha) y las líneas de conexión de los puntos de la ruta correspondientes a las mismas fechas son perpendiculares a la dirección de la migración (figura de la izquierda). En la figura de la derecha, los segmentos punteados son los valores originales, mientras que los segmentos sólidos son los valores p ajustados para multiplicidad.

the migration route in reasonable detail and at the same time the number of records in each window is reasonably high for computing the averages. So we conclude that a window size of the order of magnitude 10 is preferable over a window size of 30 because a smaller time window is more flexible in following the migration and at the same time the number of encounters in each window is reasonably high for computing means. The Fourier analysis has been suggested for summarizing encounters in average position–date relation (Perdeck, 1977). We have fitted a partial sum of Fourier series with three harmonics to the data sets of this paper. These Fourier curves can roughly summarize geographical routes. The timing of migration is not followed with reasonable precision because during migration the birds may pass some sections of the migration fast and then have lengthy stopovers. The sliding window technique can follow this pattern more closely than the Fourier curves that we have fitted. Adding more harmonics to the Fourier series may give a better fit. When biologically interpreting the findings that the route comparison method produces, the analyst must keep in mind the possible effects on

results that differences in recovery rates in different areas or periods of time may have. When the recoveries are generated by the general public and the routes are geographically close to each other, the recovery rates of both data sets are often roughly the same and the findings obtained by the method introduced in this paper are likely to reflect possible differences in migration of the two populations in question. When comparisons are made on the basis of ring recoveries generated by hunters the different hunting regulations in different areas must be taken into account. When encounters of live birds are used, observations that are not independent must be excluded. Such records could be e.g. observations in a short period of time of the same individual in the same location. Acknowledgements We thank professor Carl Schwarz and an anonymous referee for helpful comments, Marina Kurtén MA for checking the language and Heidi Björklund MSc for producing the maps of figures 2 and 3.

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References Bairlain, F., 2001. Results of bird ringing in the study of migration routes. Ardea, 89(1) (special issue): 7–19. Batschelet, E., 1981. Circular Statistics in Biology. Academic Press. Fransson, T. & Petterson, J., 2001. Swedish Bird Ringing Atlas (In Swedish with English summaries) Vol. 1. Stockholm. Jenni, L. & Kéry, M., 2003. Timing of autumn bird migration under climate change: advances in longdistance migrants, delays in short–distance migrants. Proc. R. Soc. Lond. B, 270: 1467–1471. Lokki, H. & Saurola, P., 1987. Bootstrap methods for two–sample location and scatter problems. Acta Ornithologica, 23(1): 133–147. Munro, R. E. & Kimball, C. F., 1982. Population Ecology of the Mallard VII. Distribution and Derivation of the Harvest. U.S. Wildl. Serv. Resour. Publ., 147: 1–127. Perdeck, A. C., 1977. The analysis of ringing data: pitfalls and prospects. Die Vogelwarte, 29 Sonderheft: 33–44.

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Perdeck, A. C. & Clason, C., 1982. Flyways of Anatidae ringed in the Netherlands —an analysis based on ring recoveries. In: Proc. 2nd technical meeting on Western Palearctic migratory bird management, Paris 11–13 Dec. 1979: 65–88 (D. A. Scott & M. Smart, Eds.). Slimbridge, I.W.R.B. Perdeck, A. C. & Clason, C., 1983. Sexual differences on migration and winter quarters of ducks ringed in the Netherlands. Wildfowl., 34: 137– 143. Saurola, P., 1978. Rengaslöytöjä tiirojen ja kihujen muuttoreittien varsilta. Lintumies, 13: 44–50. (In Finnish with English summary: Finnish recoveries of Sterna and Stercorarius.) Saurola, P., 1981. Varpushaukan muutto suomalaisen rengastusaineiston kuvaamana. Lintumies, 16: 10–18 (In Finnish with English summary: Migration of the sparrowhawk Accipiter nisus as revealed by Finnish ringing and recovery data.) Sinnot, R. W., 1984. Virtues of the haversine. Sky and telescope, 68(2): 159–170. Westfall, P. H. & Young, S. S., 1993. Resamplingbased multiple testing: examples and methods for p–value adjustments. John Wiley & Sons, Inc.

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Multistate modeling of brood amalgamation in White–winged Scoters Melanitta fusca deglandi J. J. Traylor, R. T. Alisauskas & F. P. Kehoe

Traylor, J. J. , Alisauskas, R. T. & Kehoe, F. P., 2004. Multistate modeling of brood amalgamation in White–winged Scoters Melanitta fusca deglandi. Animal Biodiversity and Conservation, 27.1: 369–370. Extended abstract Multistate modeling of brood amalgamation in White–winged Scoters Melanitta fusca deglandi.— Female waterfowl may lose or abandon offspring shortly after hatch often resulting in the phenomena of post–hatch brood amalgamation (PHBA; Eadie et al., 1988). Potential fitness implications of this behavior has generated considerable debate (Eadie et al., 1988; Pöysä, 1995; Savard et al., 1998) about physiological or ecological costs and benefits to ducklings in amalgamated broods. Several researchers have proposed that PHBA is a result of, but is not limited to, accidental mixing (i.e., accidental mixing hypothesis), initial brood size at hatch (i.e., brood size and success hypotheses), or maternal female condition at hatch (i.e., energetic stress hypothesis) (Eadie et al., 1988; Bustnes & Erikstad, 1991; Pöysä, 1995). We studied PHBA in July and August, 2000–2001, in a population of White–winged Scoters on Redberry Lake, Saskatchewan, (52° 00' N, 107° 10' W), a 4,500 ha federal bird sanctuary and World Biosphere Reserve. Ducklings (n = 265 in 2000 and n = 399 in 2001) were captured in nests at hatch, given a uniquely– colored nape marker for individual identification, and re–observed during daily observation sessions. We were interested primarily in movement probabilities during the first two weeks after hatch, when most travel by ducklings occurs, and after which duckling survival was constant (Traylor, 2003). We used multistate modeling (Brownie et al., 1993) in Program MARK (White & Burnham, 1999) to test hypotheses concerning PHBA. We estimated probabilities of (1) staying with putative mothers and natal siblings, , or (2) movement to a foster brood followed by adoption by a foster mother and conspecific non–siblings, . We tested hypotheses about relationships between and hatch date, brood size at hatch, female condition at hatch and size, duckling condition and size at hatch, and weather within one week of hatching. An average of 37.7% and 9.6% of ducklings moved to foster broods in 2000 and 2001, respectively. PHBA was highest the first four days of duckling age in 2000 and the first ten days in 2001. Duckling movement to foster broods in 2000 was a function of hatch date ( = –1.24, 95% CL: –2.33, –0.15), female condition ( = –0.83, 95% CL: –1.52, –0.14), and female size ( = –1.26, 95% CL: –2.21, –0.32). In 2001, duckling movement probability was related to hatch date ( = 0.33, 95% CL: –0.07, 0.72), initial brood size ( = –0.69, 95% CL: –1.12, –0.25), female condition at hatch ( = 0.35, 95% CL: –0.08, 0.78), female size ( = –0.32, 95% CL: –0.74, 0.10), duckling condition ( = –0.54, 95% CL: –0.99, –0.10), and weather ( = 1.14, 95% CL: 0.63, 1.64); 95% CL ( , , ) included zero suggesting weak effects on , but their inclusion resulted in considerably better models than if these effects were ignored.

Joshua J. Traylor, Ray T. Alisauskas, Dept. of Biology, Univ. of Saskatchewan, 112 Science Place, Saskatoon, Saskatchewan, S7N 5E2, Canada.– R. T. Alisauskas, Prairie and Northern Wildlife Research Center, Canadian Wildlife Service, 115 Perimeter Road, Saskatoon, Saskatchewan, S7N 0X4, Canada.– F. Patrick Kehoe, Ducks Unlimited Canada, #200 10720–178 St., Edmonton, Alberta, T5S 1J3, Canada. Corresponding author: J. J. Traylor, 115 Perimeter Road, Saskatoon, Saskatchewan, S7N 0X4, Canada. E– mail: joshua.traylor@ec.gc.ca ISSN: 1578–665X

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Our estimates of movement probabilities to foster broods are unbiased by any failure to account for detection probability and are the first published for ducklings. Unlike some studies that suggest PHBA occurs later in brood rearing (Pöysä, 1995), we found that most PHBA occurred soon after hatch (< 4 days) in scoters. We found that PHBA conformed to predictions from multiple hypotheses and was a function of weather, hatching synchrony, nesting densities, brood rearing areas and their likely effects on predation efficiency of gulls. Predation is likely the most important factor in evolution of PHBA (Bustnes & Erikstad, 1991; Pöysä et al., 1997). Redberry Lake’s nesting gulls feed intensively on newly hatched ducklings. In addition, biotic and abiotic effects interact with disturbances (i.e., predation attempts, large wave action, conspecific interactions) that may have resulted in accidental mixing because ducklings have not fully imprinted on parents and are highly gregarious after hatch (Savard et al., 1998). The energetic stress hypothesis reasons that PHBA may result from state–dependent decisions of maternal and foster hens contingent upon nutrient reserves of females at hatch. We found that likelihood of offspring movement was inversely related to female condition. Such females may make decisions about abandonment of ducklings before ducklings are completely hatched to spare energy associated with brood rearing (Eadie et al., 1988). The brood size hypothesis predicts that small broods are most likely to be abandoned (Pöysä, 1995). We found that adopted ducklings were from smaller broods. Parental investment theory predicts less parental care for offspring experiencing high mortality as it is indicative of future survival prospects and low reproductive value (Carlisle, 1985; Pöysä, 1995). Our combined findings of high duckling mortality soon after hatch from intense gull predation at hatch (Traylor, 2003), significant amount of adoption after hatch, and positive effects of brood size on duckling survival support the brood success hypothesis (Pöysä, 1995; Pöysä et al., 1997). Use of multistate modeling enabled proper estimation of relationships between likelihood of duckling adoption and numerous ecological variables. PHBA in this population appears to be consistent with several hypotheses, although it is probably a complex function of numerous factors that vary year to year. References 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. Bustnes, J. O. & Erisktad, K. E., 1991. Parental care in the common eider (Somateria mollissima): factors affecting abandonment and adoption of young. Can. J. Zool., 69: 1538–1545. Carlisle, T. R., 1982. Brood success in variable environments: implications for parental allocation. Animal Behaviour, 30: 824–836. Eadie, J., Kehoe, F. P. & Nudds, T. D., 1988. Pre–hatch and post–hatch brood amalgamation in North American Anatidae: a review of hypotheses. Can. J. Zool., 66: 1709–1721. Pöysä, H., 1995. Factors affecting abandonment and adoption of young in common eiders and other waterfowl: a comment. Can. J. Zool., 73: 1575–1577. Pöysä, H., Virtanen, J. & Milonoff, M., 1997. Common goldeneyes adjust maternal effort in relation to prior brood success and not current brood size. Behav. Ecol. & Sociobiol., 40: 101–106. Savard, J–P. L., Reed, A. & Lesage, L., 1998. Brood amalgamation in surf scoters Melanitta perspicillata and other Mergini. Wildfowl, 49: 129–138. Traylor, J. J., 2003. Nesting and duckling ecology of white–winged scoters (Melanitta fusca deglandi) at Redberry Lake, Saskatchewan. M. Sc. Thesis, Univ. of Saskatchewan. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study Supplement, 46: 120–139.

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Analysis using large–scale ringing data S. R. Baillie & P. F. Doherty

Baillie, S. R. & Doherty, P. F., 2004. Analysis using large–scale ringing data. Animal Biodiversity and Conservation, 27.1: 371–373. Birds are highly mobile organisms and there is increasing evidence that studies at large spatial scales are needed if we are to properly understand their population dynamics. While classical metapopulation models have rarely proved useful for birds, more general metapopulation ideas involving collections of populations interacting within spatially structured landscapes are highly relevant (Harrison, 1994). There is increasing interest in understanding patterns of synchrony, or lack of synchrony, between populations and the environmental and dispersal mechanisms that bring about these patterns (Paradis et al., 2000). To investigate these processes we need to measure abundance, demographic rates and dispersal at large spatial scales, in addition to gathering data on relevant environmental variables. There is an increasing realisation that conservation needs to address rapid declines of common and widespread species (they will not remain so if such trends continue) as well as the management of small populations that are at risk of extinction. While the knowledge needed to support the management of small populations can often be obtained from intensive studies in a few restricted areas, conservation of widespread species often requires information on population trends and processes measured at regional, national and continental scales (Baillie, 2001). While management prescriptions for widespread populations may initially be developed from a small number of local studies or experiments, there is an increasing need to understand how such results will scale up when applied across wider areas. There is also a vital role for monitoring at large spatial scales both in identifying such population declines and in assessing population recovery. Gathering data on avian abundance and demography at large spatial scales usually relies on the efforts of large numbers of skilled volunteers. Volunteer studies based on ringing (for example Constant Effort Sites [CES]; Peach et al., 1998; DeSante et al., 2001) are generally co–ordinated by ringing centres such as those that make up the membership of EURING. In some countries volunteer census work (often called Breeding Bird Surveys) is undertaken by the same organizations while in others different bodies may co–ordinate this aspect of the work. This session was concerned with the analysis of such extensive data sets and the approaches that are being developed to address the key theoretical and applied issues outlined above. The papers reflect the development of more spatially explicit approaches to analyses of data gathered at large spatial scales. They show that while the statistical tools that have been developed in recent years can be used to derive useful biological conclusions from such data, there is additional need for further developments. Future work should also consider how to best implement such analytical developments within future study designs. In his plenary paper Andy Royle (Royle, 2004) addresses this theme directly by describing a general framework for modelling spatially replicated abundance data. The approach is based on the idea that a set of spatially referenced local populations constitutes a metapopulation, within which local abundance is determined as a random process. This provides an elegant and general approach in which the metapopulation model as

Stephen R. Baillie, British Trust for Ornithology, The Nunnery, Thetford, Norfolk IP24 2PU, U.K. E– mail: stephen.baillie@bto.org Paul F. Doherty, Dept. of Fishery and Wildlife Biology, Colorado State Univ., CO 80523–1474, U.S.A. E– mail: doherty@cnr.colostate.edu ISSN: 1578–665X

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described above is combined with a data–generating model specific to the type of data being analysed to define a simple hierarchical model that can be analysed using conventional methods. It should be noted, however, that further software development will be needed if the approach is to be made readily available to biologists. The approach is well suited to dealing with sparse data and avoids the need for data aggregation prior to analysis. Spatial synchrony has received most attention in studies of species whose populations show cyclic fluctuations, particularly certain game birds and small mammals. However, synchrony is in fact a much more widespread process, with bird populations across wide areas showing similar trends and fluctuations as a result of common climatic and environmental factors (Paradis et al., 2000). Dispersal may also play an important role in such synchrony but its role is less well understood. Nigel Yoccoz and Rolf Ims (Yoccoz & Ims, 2004) show how synchrony can be investigated using data at three spatial scales taken from their field studies of the population dynamics of small mammals in North Norway. Small mammal abundance was estimated from trapping data using closed population models and also from total numbers of individuals captured. They use simulated data to show that synchrony, measured by the correlation coefficients between time series, was biased low by up to 30% when sampling variation was ignored. Appropriate analysis of such data will require simultaneous modelling of process and sampling variation, for example through the use of state–space models (Buckland et al., 2004). This view links back nicely to the approaches proposed by Andy Royle (Royle, 2004). Cycles in the abundance of small mammals have major affects on the demography of their predators, as is shown in the paper by Pertti Saurola and Charles Francis (Saurola & Francis, 2004). They report on the design and results of large–scale, long–term studies of owl populations by a network of amateur bird ringers in Finland. They show that breeding success varies with the stage of the microtine cycle. They also show how their data can be used to estimate dispersal over large spatial scales and illustrate the importance of correcting for uneven spatial variation in sampling effort. Further results from this study are reported in a companion paper within the population dynamics session (Francis & Saurola, 2004). Multi–species analyses of population dynamics are developed further in the paper by Romain Julliard (Julliard, 2004). He combines counts from the French Breeding Bird Survey with survival and recruitment estimates from the French CES scheme to assess the relative contributions of survival and recruitment to overall population changes. He develops a novel approach to modelling survival rates from such multi–site data by using within–year recaptures to provide a covariate of between–year recapture rates. This provided parsimonious models of variation in recapture probabilities between sites and years. The approach provides promising results for the four species investigated and can potentially be extended to similar data from other CES/MAPS schemes. The final paper by Blandine Doligez, David Thomson and Arie van Noordwijk (Doligez et al., 2004) illustrates how large-scale studies of population dynamics can be important for evaluating the effects of conservation measures. Their study is concerned with the reintroduction of White Stork populations to the Netherlands where a re–introduction programme started in 1969 had resulted in a breeding population of 396 pairs by 2000. They demonstrate the need to consider a wide range of models in order to account for potential age, time, cohort and "trap–happiness" effects. As the data are based on resightings such trap–happiness must reflect some form of heterogeneity in resighting probabilities. Perhaps surprisingly, the provision of supplementary food did not influence survival, but it may have had an indirect effect via the alteration of migratory behaviour. Spatially explicit modelling of data gathered at many sites inevitably results in starting models with very large numbers of parameters. The problem is often complicated further by having relatively sparse data at each site, even where the total amount of data gathered is very large. Both Julliard (2004) and Doligez et al. (2004) give explicit examples of problems caused by needing to handle very large numbers of parameters and show how they overcame them for their particular data sets. Such problems involve both the choice of appropriate starting models for sparse data and the speed with which convergence is achieved. Further analytical and software developments are needed in order to make it easier to analyse such data sets. References Baillie, S. R., 2001. The contribution of ringing to the conservation and management of bird populations:a review. Ardea, 89: 167–184. Buckland, S. T., Newman, K. B., Thomas, L. & Koesters, N. B., 2004. State–space models for the dynamics of wild animal populations. Ecological Modelling, 171: 157–175. Desante, D. F., Nott, M. P. & O’Grady, D. R., 2001. Identifying the proximate demographic cause(s) of population change by modelling spatial variation in productivity, survivorship and population trends. Ardea, 89: 185–208. Doligez, B., Thomson, D. L. & Van Noordwijk, A. J., 2004. Using large–scale data analysis to assess life

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history and behavioural traits: the case of the reintroduced White stork Ciconia ciconia population in the Netherlands. Animal Biodiversity and Conservation, 27.1: 387–402. Francis, C. M. & Saurola, P., 2004. Estimating components of variance in demographic parameters of Tawny Owls, Strix aluco. Animal Biodiversity and Conservation, 27.1: 489–502. Harrison, S., 1994. Metapopulations and conservation. In: Large–scale Ecology and Conservation: 111–129 (P. J. Edwards, N. R. Webb & R. M. May, Eds.). Blackwell Science, Oxford. Julliard, R., 2004. Estimating the contribution of survival and recruitment to large scale population dynamics. Animal Biodiversity and Conservation, 27.1: 417–426. Paradis, E., Baillie, S. R., Sutherland, W. J. & Gregory, R. D., 2000. Spatial synchrony in populations of birds: effects of habitat, population trend and spatial scale. Ecology, 81: 2112–2125. Peach, W. J., Baillie, S. R. & Balmer, D. E., 1998. Long–term changes in the abundance of passerines in Britain and Ireland as measured by constant effort mist–netting. Bird Study, 45: 257–275. Royle, J. A., 2004. Generalized estimators of avian abundance from Count Survey data. Animal Biodiversity and Conservation, 27.1: 375–386. Saurola, P. & Francis, C., 2004. Estimating population dynamics and dispersal distances of owls from nationally coordinated ringing data in Finland. Animal Biodiversity and Conservation, 27.1: 403–415. Yoccoz, N. G. & Ims, R. A., 2004. Spatial population dynamics of small mammals: some methodological and practical issues. Animal Biodiversity and Conservation, 27.1: 428–435.

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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Generalized estimators of avian abundance from count survey data J. A. Royle

Royle, J. A., 2004. Generalized estimators of avian abundance from count survey data. Animal Biodiversity and Conservation, 27.1: 375–386. Abstract Generalized estimators of avian abundance from count survey data.— I consider modeling avian abundance from spatially referenced bird count data collected according to common protocols such as capture– recapture, multiple observer, removal sampling and simple point counts. Small sample sizes and large numbers of parameters have motivated many analyses that disregard the spatial indexing of the data, and thus do not provide an adequate treatment of spatial structure. I describe a general framework for modeling spatially replicated data that regards local abundance as a random process, motivated by the view that the set of spatially referenced local populations (at the sample locations) constitute a metapopulation. Under this view, attention can be focused on developing a model for the variation in local abundance independent of the sampling protocol being considered. The metapopulation model structure, when combined with the data generating model, define a simple hierarchical model that can be analyzed using conventional methods. The proposed modeling framework is completely general in the sense that broad classes of metapopulation models may be considered, site level covariates on detection and abundance may be considered, and estimates of abundance and related quantities may be obtained for sample locations, groups of locations, unsampled locations. Two brief examples are given, the first involving simple point counts, and the second based on temporary removal counts. Extension of these models to open systems is briefly discussed. Key words: Abundance estimation, Avian point counts, Detection probability, Hierarchical models, Metapopulation models, Population size. Resumen Estimadores generalizados de abundancia en aves a partir de datos de estudios de recuento.— En el presente estudio se analiza la modelación de la abundancia en aves mediante datos de recuento de aves, referenciados espacialmente y obtenidos a partir de protocolos comunes, como los de captura–recaptura, muestreo por observadores múltiples, muestreo por eliminación y recuentos de puntos simples. Las muestras de pequeño tamaño, así como el amplio número de parámetros, han propiciado numerosos análisis que no tienen en cuenta la indexación espacial de los datos y, por consiguiente, no proporcionan un tratamiento adecuado de la estructura espacial. En este trabajo se describe un marco general para la modelación de datos replicados en el espacio, que considera la abundancia local como un proceso aleatorio, todo ello basado en el punto de vista de que el conjunto de poblaciones locales referenciadas espacialmente (en los lugares de toma de muestras) constituye una metapoblación. De este modo, la atención puede centrarse en el desarrollo de un modelo para la variación en la abundancia local que sea independiente del protocolo de muestreo que se esté utilizando. La estructura del modelo metapoblacional, en combinación con el modelo de generación de datos, define un modelo jerárquico simple que puede analizarse mediante el empleo de métodos convencionales. El marco de modelación propuesto es de carácter general, en el sentido de que permite considerar amplias clases de modelos metapoblacionales, covariantes del nivel del emplazamiento sobre datos de detección, y la abundancia, pudiendo obtenerse estimaciones de abundancia y cantidades relacionadas para emplazamientos de muestreo, grupos de emplazamientos y emplazamientos no muestreados. A tal efecto, se incluyen dos breves ejemplos; el ISSN: 1578–665X

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primero trata de los recuentos de puntos simples, mientras que el segundo se basa en los recuentos por extracción temporal. También se apunta la posibilidad de ampliar estos modelos a sistemas abiertos. Palabras clave: Estimación de la abundancia, Recuentos de puntos aviares, Probabilidad de detección, Modelos jerárquicos, Modelos metapoblacionales, Tamaño poblacional. J. Andrew Royle, USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel MD 20708, U.S.A.

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Introduction The detectability of individuals is a fundamental consideration in many studies of animal populations. The need to properly account for detectability has given rise to an extensive array of sampling protocols and statistical procedures for estimating demographic parameters in the presence of imperfect detection (Williams et al., 2002). Conventional capture–recapture methods in which individual animals are marked, released, and recaptured (or resighted) constitute the most useful class of methods in terms of the information content provided by the data, and the complexity of detection process models that may be considered. In studies of avian populations, implementation of capture– recapture methods is often difficult in field situations. Because of this, there has been considerable recent interest in methods based on avian point counting that are capable of controlling for imperfect detection while remaining efficient to implement in field situations. These methods include those based multiple observer sampling (Cook & Jacobson, 1979; Nichols et al., 2000), temporary removal (Farnsworth et al., 2002), distance sampling (Rosenstock et al., 2002) and even simple point counts (Royle, 2004a). These and similar methods are also widely used in the study of other organisms including marine mammals, ungulates, and amphibians. My motivation derives from studies of bird populations, and so subsequent discussion and examples focus on bird sampling problems. Many small–scale studies of animal populations and large–scale monitoring efforts rely on sampling designs in which one (or more) of these common sampling protocols is replicated spatially. This is partly out of necessity —many species exist at low densities, and effective sampling areas are small— but often there is direct interest in characterizing spatial variation in abundance. These replicated surveys yield spatially indexed count data yi for sample location (or site) i = 1,2,...,R. In most sampling protocols for which it is possible to estimate abundance in the presence of imperfect detection, yi = {yik; k = 1,2,...,K} is a vector of counts. As an example, if a removal protocol is used then yi = (yi1, yi2, yi3) are the number of animals first observed (and "removed" from further counting) in consecutive time intervals of say 3 minutes. The precise nature of the count statistic under other common sampling protocols is described in "Notation and preliminaries" section. There are two objectives considered in many studies of animal populations that are demographically closed. First is estimation of "abundance", population size, or density. In the context of spatially replicated surveys, this is often defined as total or average abundance of the sampled area. The second objective is estimation of the effects of (site–specific) covariates on abundance or density. Typical covariates of interest are those that describe habitat or landscape structure. Interestingly, there have been few general suggestions for ac-

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commodating site–specific covariates in these common sampling protocols. Royle et al. (2004a) describe an approach for incorporating abundance covariate effects in distance sampling models that is related to the models described here. One important difficulty present in most spatially replicated bird counting surveys (of breeding birds) is that typical abundance at individual sampled locations is very small. Consequently, site–specific sample sizes (number of observed birds) are small. The general small sample situation is problematic when it comes to estimation because the likelihood contains many (abundance) parameters each of which is ill–informed by the available sparse data. Estimation of spatially explicit abundance is usually infeasible. A common solution is to aggregate data across sites and apply conventional estimation methods to the aggregate counts. In doing so, site– specific information is lost so that, for example, estimation and modeling of site–specific covariate effects on abundance and detection is infeasible. In addition, spatial scale becomes a concern when deciding how data should be aggregated. While it may be reasonable to combine multiple samples within a small forest or woodlot, additional considerations should be relevant at larger scales. Finally, the use of aggregated counts cannot generally be justified based on the likelihood for the observed data, i.e., the site–specific counts. That is, the aggregated counts are not sufficient statistics for the objective (total) abundance "parameter" under a sampling scheme involving spatial replication. Additional assumptions are required to formally justify aggregation of count statistics among sites; this is elaborated on in "The likelihood under spatial replication" section. These deficiencies motivate the need for a more general approach to dealing with spatially replicated count survey data. In this paper, I describe a general framework for modeling and estimation of abundance from spatially replicated animal count data. The key idea is introduction of a metapopulation model that characterizes the (spatial) variation in abundance of the spatially referenced populations being sampled. The metapopulation view provides a concise framework for combining the data collected at multiple sample locations, regardless of the sampling protocol used to collect data. Specification of a metapopulation model is a great advantage because it allows the biologist to focus on explicit formulation of the abundance model at the level of the sample unit, independent of the detection process. The main benefit of adopting the metapopulation view is that a broad class of more complex models are possible including models which describe variation in site–specific abundance explicitly (e.g., with covariates), and models which allow for latent spatial variation (overdispersion, spatial correlation) that is not modeled explicitly by covariates. These metapopulation models form the basis for the development of generalized estimators of abundance based on any of the previously mentioned protocols. These are generalized in the sense that they can accommodate variation in site–specific abundance,

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factors that influence detectability, and additional considerations described in "The metapopulation view" section. Under this metapopulation formulation, local abundance is regarded as a "random effect", and the general model structure is commonly referred to in statistics as a hierarchical model. Hierarchical models are commonly analyzed by either integrated likelihood or Bayesian methods. This is described in "Estimation and inference" section. Modeling abundance effects, estimating density, estimating local population size, and even predicting abundance at unsampled locations are straightforward problems under this hierarchical modeling framework.

expressed as the total count i yi. plus the sum of the unobserved individuals at each location y.0 = i yi0. The likelihood under spatial replication Hereafter, I assume that the observations yi are independent when conditioned on local population size Ni and parameters of the detection process. Under this assumption, the sampling distribution of the data y = {yi; i = 1,2,...,R} under most common sampling protocols is the product multinomial

(1)

Notation and preliminaries Let Ni be the number of birds available to be counted at location i; i = 1,2,...,R. Sampling yields the vector of counts yi for each sample location. The precise nature of the data vector yi depends on the sampling protocol used. For several of the more common sampling protocols, the data structure is as follows: (1) For independent double or multiple observer protocols, k indexes an "observer detection history". For example, with two independent observers, K = 3 and yi1, and is the number of birds seen by observer 1 (but not observer 2), yi2 is the number seen by observer 2 (but not 1), and yi3 is the number seen by both observers. In general, with T observers, there are K = 2T – 1 observable observer histories. (2) For a removal protocol, k indexes the time interval of (first) detection. i.e., yi1 is the number of birds first seen in interval 1, yi2 in interval 2, and yi3 in interval 3, etc. (3) For distance sampling, the count statistics are indexed by distance, so that yik is the number of birds seen in distance class k at site i. (4) For conventional "capture–recapture" experiments, the data structure is analogous to that obtained under multiple observer sampling except that the capture history is organized in time. For example, in a two period study, K = 3, and let yi1 be the number of individuals with capture history "10" (seen in the first interval, but not the second), yi2 be the number of individuals with capture history "01", and yi3 be the number of individuals with capture history "11". Various other protocols may also be considered, including that based on simple point counts (see "Point counts" section). A final bit of notation will be useful. In some applications that involve spatial replication, putative interest lies in estimation of the total abundance at the sampled sites: Ntotal = i Ni. The familiar "dot notation" will be used to indicate various sums. Let yi0 = Ni – yi. be the number of birds not detected at each site where (the total number detected at site i). Thus, Ntotal can be

for a sampling protocol yielding 3 observable frequencies (e.g., 3 period removal, 2 observers, etc). The cell probabilities, k, are functions of one or more detection probability parameters p (the precise function depends on the protocol being used) and (the probability that an individual is not captured). For example, under a removal sampling protocol with three removal periods, the cell probabilities have the following form when detection probability is assumed constant: 1 2 3 0

= = = =

p (1 – p)p (1 – p)2p (1 – p)3.

For other sampling protocols, these cell probabilities are different functions of various detection probability parameters but their precise form is not relevant in any of the following discussion. While the product multinomial likelihood (1) is not inherently intractable, in many practical situations there are important considerations that render it so. In particular, there are usually many unknown abundance parameters (the N i's), in addition to the parameters that describe the detection process. Also, local population sizes are frequently very small and, consequently, the sample sizes (number of captured individuals) for each location are small. In many surveys, there may in fact be many zero counts. One common solution to dealing with these problems is to aggregate the counts (i.e., pool data from multiple sample locations). This is discussed subsequently. Spatial aggregation of count statistics A common goal of many studies is estimation of the total abundance, Ntotal, across all sampled locations. Ignoring the fact that the data are indexed by sample location, one might focus on the likelihood of the aggregated count statistics:

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(2) To be more concise, using the notation introduce in "Notation and preliminaries" section, Eq. (2) is . (3) Interestingly, the use of aggregated counts cannot be justified under the (correct) likelihood for the disaggregated data given by (1) without some additional assumptions elaborated on shortly. While it is true that Eq. (3) is the correct likelihood for the total counts if the site–specific counts are unknown, it is not equivalent to the likelihood based on the disaggregated site–specific counts. That is, given the site–specific counts yik, the totals y.k = i yik are not the sufficient statistics for Ntotal when the Ni are viewed as fixed but unknown parameters. I believe that the idea of pooling the site–specific sufficient statistics is partially motivated by convenience. The main support for use of (3) over (1) seems to be that there are too many Ni parameters in the joint likelihood (1) and this motivates one to consider them as nuisance parameters. However, estimation based on aggregated data does not appear consistent with usual notions of the treatment of nuisance parameters. For example, integration of the nuisance parameters from the likelihood under a suitable prior distribution, or conditioning on sufficient statistics, both of which are fairly conventional treatments of nuisance parameters. It can be demonstrated that if Ni are assumed to have a Poisson distribution with mean , then one can justify aggregation (i.e., (3)) from likelihood (1). In this sense, estimation based on aggregated counts can be viewed as having implied a Poisson assumption on Ni with constant mean. Importantly, it precludes other possibilities: That Ni are over– dispersed relative to the Poisson, or that the mean is not constant. Thus, technical details aside, the important reason that one should not aggregate data is that it renders impossible the consideration of covariate effects on both abundance and detection probability, and consideration of more complex variance structure. The conditional likelihood under spatial aggregation As an alternative to using the likelihood (1), it is common to use so–called "conditional" estimators based on obtaining an estimate of 0 from the conditional likelihood (4) The likelihood given by (4) is motivated by noting that the sufficient statistic for Ni is yi., and so by conditioning on yi., Ni is removed from the problem. Estimators based on the conditional likelihood (4) and the "unconditional" likelihood (3) are asymptoti-

cally equivalent (Sanathanan, 1972), and both specifications are commonly used in practice. For the common parameterizations of k (under the sampling protocols described previously), it is clear that the aggregated counts are sufficient statistics for those model parameters contained in k, and hence use of aggregated counts can be justified under likelihood (4) if interest is focused on estimating detection probability parameters. Estimation of Ntotal is then based on the assertion that y.. = i yi is Binomial (Ntotal, 1– 0). While this may be true, it should be noted that it was not y.. that was conditioned on in order to obtain Eq. (4), but rather yi.. The neglected likelihood component is

Once again, there is no way to reformulate this in terms of Ntotal without additional model structure on Ni (e.g., if Ni has a Poisson distribution). The metapopulation view A more appealing and general solution to the problem of spatial replication can be achieved by regarding the collection of local populations as a metapopulation (Levins, 1969; Hanski & Gilpin, 1977). For the present purposes, a useful operational definition of metapopulation is simply "a population of (local) populations indexed by space". Interest in the study of metapopulation biology has exploded in recent years both in terms of theoretical development and applications of metapopulation concepts to many taxa. Patch occupancy, local extinction and local colonization are all metapopulation characteristics of some theoretical and practical interest. In the present context, that of demographically closed systems during the time of sampling, the local population trait in question is size, but in general (open systems) local population mortality and recruitment events are also of interest, the metapopulation summaries being local extinction and colonization probabilities, respectively. Note that mortality at the local population level is the aggregate of individual mortality and emigration processes, and recruitment at the local population level is the aggregate of individual recruitment and immigration processes. The relationship between local population processes and several important metapopulation parameters are given in table 1. Note that demographic closure during sampling is not inconsistent with metapopulation theory which requires that populations mix across time to some extent. In a demographically closed system, I view local population size as being a more general description of patch occupancy. The event that a patch is occupied is equivalent to the event that Ni > 0, and patch occupancy is Pr(N > 0) for a collection of homogeneous patches. In general, Pr(N > 0) is a function of density, and the variation in local population sizes as described shortly.

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Table 1. Summary of metapopulation concepts. Tabla 1. Resumen de conceptos metapoblacionales.

Time scale

Closure status

Local population attribute

Within year

Closed local pops.

Occurrence

Patch occupancy

Size

Density

Mortality

Extinction probability

Recruitment

Colonization probability

Across years

Open local pops.

Metapopulation parameter

Extinction and colonization events are also intimately linked to local abundance (among other things). Thus, models of variation in abundance are of more relevance than simply as a characterization of abundance per se.

distance sampling). More importantly, there is an obvious and simple extension to accommodate a non–uniform distribution of individuals. One can consider that varies spatially, for example:

Probabilistic characterization of metapopulations

where log ( I) = b0 + b1xi1 where xi1 is the value of some covariate at site i. Several further extensions are also obvious. One is to allow for excess Poisson variation by inclusion of a random effect, ei, as:

It is natural to express the notion of a metapopulation probabilistically, by imposing a probability distribution on abundance. This is expressed by Ni i g(Nl ) where "i" is read "is distributed as" and g(Nx ) is some discrete probability density. The local populations may be independent, or not, but considerable simplicity arises when they are independent. Practically, independence means that individuals cannot occur in more than one local population (i.e., the Ni's do not overlap). The generality that this probability characterization permits is that may be allowed to vary spatially in a number of ways, any discrete probability density may be considered for g(Nx ). The main practical benefit of this metapopulation view is that the metapopulation structure serves as a framework for combining a large number of spatially referenced count data surveys. In essence, this model is a prior distribution on abundance. More generally, I believe that the structure of the metapopulation is of fundamental interest. That is, the goal of many (if not most) studies of avian abundance can be formulated in terms of the metapopulation distribution or its summaries such as E[N] (density), covariate effects (on density), etc. The simplest example of a metapopulation model is that resulting from a uniform distribution of individuals across the landscape. Then, aggregating occurrence events into non–overlapping sample areas yields Ni i Poisson ( ). This seems a natural choice for describing variation in abundance because it arises under a homogeneous Poisson point process, the standard null distribution for the spatial arrangement of organisms. Moreover, it is also an assumption that underlies many common animal sampling methods (e.g.,

Ni i Poisson ( I)

log ( I) = b0 + ei where ei i Normal(0, 2). Alternatively, a more natural model of over–dispersion for Ni is the negative binomial distribution Ni i NegBin( , ) with variance + 2/ . In any spatial sampling problem, it is natural to consider the possibility that the spatial process is correlated. That is, that there exists latent structure beyond any covariates that are contained in the model. Royle et al. (2004b) consider a model in which the log–linear model for the mean contains a spatially indexed random effect that is (spatially) correlated. Such structure may be appealing in many animal abundance modeling problems where it is likely that habitat affinities are only known imprecisely, or there is limited ability to quantify the relevant habitat components. Open systems The focus of this paper is on modeling and estimation of abundance in demographically closed systems. The linkage between local abundance and patch occupancy in closed systems has been mentioned previously. However, similar relationships between other metapopulation attributes and abundance can also be made. For example, local colonization probability is Pr(Nt+1 > 0 l Nt = 0) and local extinction probability is Pr(Nt+1 = 0 l Nt > 0). In fact, one can characterize Pr(Nt+1 l Nt) in general, for each discrete state Nt, which represents an important generalization over the current treatments of the problem that characterize occurrence as being the binary event that N > 0, extinction as

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the event that Nt+1 = 0 l Nt > 0 and colonization as the event that Nt+1 > 0 l Nt = 0. Under this coarse characterization of metapopulation dynamics, there is no consideration of density dependent mechanisms, and variation in abundance leads to heterogeneity in detection probability (Royle & Nichols, 2003) which must then be modeled indirectly. These issues are beyond the scope of this paper. Estimation and inference The metapopulation description of local abundance as a random (spatial) process seems a natural way to describe spatially referenced populations and may be appealing to many ecologists. However, local abundance is never observed, instead being informed by survey data according to one of the many possible sampling protocols described in "Notation and preliminaries" section (among others). Thus, it is necessary to incorporate this metapopulation model into a framework that is amenable to estimation and inference from data. The metapopulation model is essentially a "random effects" distribution for local abundance, Ni. The classical approach to handling random effects (e.g., Laird & Ware, 1982) is to base inference on the marginal likelihood of the data, having removed the random effects from the likelihood by integration. In the multinomial sampling problems considered here, the integrated likelihood of yi is:

Integrated likelihood has been considered under similar models by Royle & Nichols (2003), Dorazio et al. (2004), Royle (2004b) and Royle et al. (2004). The Poisson distribution seems to be the de facto standard for the distribution g(Nl ) as it can be used to justify analysis based on the aggregated counts, and its motivation as a random distribution of individuals in space (a homogeneous point process) is appealing. Subsequently, I will focus on the Poisson case. In this case, the integrated likelihood is:

where k are functions of p (depending on the sampling protocol used). This does have a closed form that is more amenable to computation. In particular,

mates of or any covariate effects on abundance, and detection probability parameters. The fact that appears as a product with each k in Eq. (5) may lead one to question identifiability of model parameters. However, the k are not freely varying parameters, but instead are constrained by the sampling protocol to depend on a smaller set of detection probability parameters. One can easily write down consistent moment estimators for and detection probability parameters from Eq. (5) under the common sampling protocols. It is a simple matter to maximize Eq. (5) numerically using conventional methods found in many popular software packages. For example, the free software package R (Ihaka & Gentleman, 1996) was used in the analyses of "Applications" section (routines are available from the author upon request). A natural alternative to integrated likelihood for fitting random effects models is to adopt a Bayesian view and focus on characterizing the posterior distribution of the model unknowns conditional on the data using common Markov chain Monte Carlo (MCMC) methods. While this is straightforward in the present problem, I neglect those details here. While there are considerable philosophical differences between the two approaches, I believe that the main practical difference has to do with estimating the random effects (or summaries of them) and characterizing uncertainty in those estimates. This is discussed in the following section. In more complex models, such as when additional random effects are considered in a model for i, estimation by integrated likelihood becomes difficult and so adopting a Bayesian formulation of the problem might become necessary (see Royle et al., 2004b). Estimating abundance and related quantities The MLE of , , is an estimate of the prior mean abundance at a site. Or, in the case where i varies (e.g., covariates), one obtains as a function of abundance covariates. To estimate Ntotal note that, under the Poisson assumption on Ni, Ntotal i Poisson (R ) and so

where is the MLE from the integrated likelihood. Generally, interest may not be in the estimated prior means, but rather in estimating the realized abundance either for the collection of sample locations, or aggregated in some manner (over some spatial domain, or a collection of sample sites). For this, the classical method of estimating random effects is referred to as Best Unbiased Prediction (BUP). That is,

(5) This is just the product of (independent) Poisson random variables. Maximization of (5) yields esti-

where is used in place of . This is a simple calculation (see Royle, 2004a for an example).

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The Bayesian treatment of the problem is more general in the sense that variation in is directly considered. For example, the Bayes estimator of Ni is the posterior mean: . In effect, the dependence on has been removed by integration. Consequently, one could exto be more variable than in practical pect sample sizes. Estimates of patch occupancy, say , can also be obtained from these random effects models. For example, under the Poisson model for Ni, = 1– e – . Goodness–of–fit and model selection One convenient implication of the closed form likelihood given by Eq. (5) is that one can use conventional deviance statistics for Poisson data to assess goodness–of–fit (see Dorazio et al. [2005] and Royle et al. [2004a] for examples). Under negative binomial models, or when the likelihood is not multinomial, bootstrap procedures appear to be necessary (Dorazio et al., 2005; Royle, 2004b; Dodd & Dorazio, in press). Model selection based on integrated likelihood may be carried out using AIC (Burnham & Anderson, 1998) regardless of the form of the likelihood. Applications The modeling framework presented here can be easily applied to any of the common bird sampling protocols described previously. To illustrate, we consider application to data collected using conventional point count data, and also data collected according to temporal removal protocol. A comprehensive analysis of a large– scale capture–recapture data set is considered by Royle et al. (2004b) and an application to distance sampling data is given by Royle et al. (2004a). Dodd & Dorazio (in press) provide a comprehensive integrated–likelihood analysis of frog count data collected according a point counting protocol. Point counts Point counts are often considered to be of marginal value to statisticians with an interest in conventional modeling of marked animal data because there is a widespread misperception that information on abundance cannot be disentangled from detection probability. Royle (2004) showed that if point counts are spatially and temporally replicated within a demographically closed system, then the integrated likelihood methods described in "Estimation and inference" section can be used to effectively model both detection and abundance effects.

An important distinction between the point count protocol and the others considered previously is that temporal replication is necessary to estimate detection from simple point counts. This is because given simple binomial counts, yi, with index Ni and probability p, where Ni are independent random variables from g(N l ), p appears as a product with the location parameter of g in the integrated likelihood. For example, in the Poisson case with mean , the marginal distribution (the integrated likelihood) of yi is Poisson with mean p . Royle et al. (unpublished report) gave a heuristic explanation to demonstrate that additional information from spatial and temporal replication is available. In particular, a moment estimator for p is simply the correlation between counts made in successive sample periods. i.e., (6) for counts made at two sampling occasions. Then, is . More formally, the integrated likelihood under the replicated point count protocol is

(7) p can vary as a function of covariates, and even temporally, but we neglect that generality here. Note that Eq. (7) does not close, contrary to the multinomial likelihood case that yields Eq. (5). Data considered here are a subset of those analyzed by Royle (2004a) consisting of replicated point counts at 50 stops along a North American Breeding Bird Survey route. The point counts were replicated 11 times within approximately a one month period during the breeding season. Here, we consider only the first two counts for all 50 stops. Poisson and negative binomial models were considered for abundance. Under the Poisson model, the moment estimates of p and were also computed ( and based on Eq.(6)). For comparison, an abundance index being the mean (across sites) of the maximum count (over the two samples) was also computed. The 4 species considered are: Ovenbird (Seiurus aurocapillus), Hermit thrush (Catharus guttatus) Woodthrush (Hylocichla mustelina) and American robin (Turdus migratorius). Results of the model fitting are given in table 2, along with AIC scores (Burnham & Anderson, 1998). Generally, the overdispersed negative binomial appears favored (except for the Hermit thrush). Estimated mean abundance differs considerably from that reported by Royle (2004) based on analysis of all 11 replicates. This is consistent with lack of closure over the longer time period or higher rates of temporary emigration which is why I have restricted attention to the first two replicate observations here. The main purpose of this example is to demonstrate that it is feasible to estimate abundance

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Table 2. MLEs and AIC for Poisson and negative binomial hierarchical models fitted to the avian and are the Poisson moment estimates. point count data: Tabla 2. Estimaciones de los parámetros de máxima verosimilitud (MLE) y criterio de información de Akaike (AIC) para moldelos Poisson y modelos jerárquicos binomiales negativos ajustados a los datos de recuentos puntuales: y son las estimaciones del momento de Poisson.

Poisson

Negative binomial

Species

Index

Ovenbird

0.96

0.53

1.25

0.43

1.53

215.68

AIC 0.33

2.01

Hermit thrush

0.10

0.55

0.13

0.55

0.13

47.82

0.55

0.13

Woodthrush

0.52

0.60

0.63

0.58

0.65

150.59

0.53

0.72

0.78 147.44

Am. Robin

1.12

0.46

1.65

0.38

2.02

241.73

0.17

4.53

1.21 235.16

from simple point counts while controlling for (i.e., modeling) detection probability. In point count sampling, there is some advantage to reducing the time interval between counts to the extent possible in order to minimize temporary emigration which leads to some complication interpretas density (applicable to a known area). ing Thus, consecutive counts (e.g., consecutive three minute point counts) may be the best strategy for implementing the point count estimator. Removal counts

2 3 4

inf

49.82

motivation for considering the mixture models elaborated on in "Estimation and inference" section. Removal data from several sites are shown in table 3, highlighting the typical small sample data sets that arise from local scale bird counting. Models were fit using the Poisson metapopulation model assuming that Ni i Poisson ( i) where log ( i) = b0 + b1UFCi + b2BAi

Next we consider avian point count data collected in Frederick County, Maryland. The data were collected at 70 locations within a large forest tract, according to a conventional removal sampling protocol (Farnsworth et al., 2002) with four sample intervals of length three minutes. The main objective was to evaluate the effect of two habitat covariates: understory foliage cover (UFC) and the basal area of large trees (BA). See Royle et al. (2004) for further description and an alternative analysis of some of these data. We focus here on data for the Ovenbird (Seiurus aurocapillus). The data for each sample point are yi = (yi1, yi2,yi3,yi4) where yik is the number of males first seen in interval k. For this illustration, we assume that detection probability, p, is constant so that the multinomial cell probabilities are: 1

AIC 1.69 213.70

= = = =

p (1 – p)p (1 – p)2p (1 – p)3p

Several covariates were collected that are thought to influence p (e.g., time of day) and a more complete analysis of these data is in progress. Here, we consider only the habitat effects on abundance. That reasonable covariates on both detection and abundance can be identified is important

Results for several models are summarized in table 4, including AIC scores for evaluating the relative merits of each model. For example, under the constant model = 1.138 (SE = 0.093), or 1.138 male ovenbirds per point count sample. Point counts in this study were of radius 100 m, so one could interpret this as density if so inclined. More importantly, the habitat effects appear important so that density changes as a function of UFC and BA. There is a large positive effect of UFC and negative effect of BA. Because ovenbirds are ground nesters, and therefore would benefit from protection afforded by understory foliage, these results appear sensible. Also, the fact that the model containing both effects was not favored is not unexpected because UFC and BA are negatively correlated. These results are broadly consistent with those reported by Royle et al. (2004a) obtained using a distance sampling protocol (data were collected in a manner consistent with multiple protocols). Conclusions In this paper I have considered the problem of modeling spatially replicated avian count data that are collected according to many common sam-

Royle

384

Table 3: Ovenbird removal data (number first seen in four consecutive intervals). Tabla 3. Datos de extracción del tordo mejicano (primer número observado en cuatro intervalos consecutivos).

t=2

t=3

point 34

t=1 0

t=4 0

point 35

point 36

point 37

point 38

point 39

point 40

point 41

point 42

point 43

point 44

point 45

pling protocols. These include methods that yield a multinomial sampling distribution including conventional capture–recapture methods, multiple observer sampling, temporary removal and even simple point counts. One important statistical consideration is that data are frequently sparse (low counts and many zeros), owing to generally low densities of most breeding birds, and small sample areas. In addition, the likelihood under spatial replication may contain a large number of abundance parameters (Ni for each sample location) that render it intractable using conventional methods. Conventional methods of analyzing bird count data often focus on estimating total abundance

over the collection of sample locations. Under this limited treatment of the problem, variation at the level of the sample location is, in effect, averaged out. Covariates cannot be considered, and one must consider spatial scale in deciding how to aggregate data. Importantly, aggregation may only be justifiable under certain spatial homogeneity assumptions. For example, if local abundance (at the level of the sample locations) is assumed to be Poisson with constant mean, then aggregation can be justified. However, this may not be a reasonable assumption in many problems. Alternatively, the spatial attribution of the data is an important consideration in many studies, and can be exploited to develop more general models for describing abundance. For example, the goal of many studies is to estimate abundance covariate effects. And, factors that influence detectability may also vary among sample locations. Explicitly acknowledging spatial variation in local abundance facilitates investigation of these possibilities. The solution to the problem of modeling spatially replicated data proposed here is to view local abundance as a random process. Then, attention can be focused on developing a model for the variation in local abundance free of detection probability considerations. This is appealing in the context of familiar metapopulation ideas that seek to characterize the structure among spatially referenced (local) populations that constitute the metapopulation. Taken together, the data model (the multinomial likelihood) and metapopulation model define a simple hierarchical model for which formal and rigorous methods of analysis are possible. For example, one can estimate parameters and conduct inference based on the integrated likelihood (having removed the random effects by integration). Alternatively, Bayesian analysis based on the posterior distribution is relatively straightforward. The generality of the proposed modeling strategy is appealing. Mean abundance ( under the Poisson model) may be parameterized in terms of additional parameters that describe variation in the Poisson mean (and hence abundance), and

Table 4: Results of models fit to ovenbird counts obtained under a temporary removal protocol. Tabla 4. Resultados de los modelos ajustados a los recuentos del tordo mejicano obtenidos con arreglo a un protocolo de extracción temporal.

Model

Constant

0.572

0.130

+ UFC

0.572

0.113

+ BA

0.572

0.106

+ UFC + BA

0.572

0.102

UFC

AIC 303.86

1.859 1.042

303.32 –0.829

302.35

–0.643

303.72

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there is no need even to restrict attention to a Poisson random effects distribution. Such generality is easily dealt with formally within the context of the hierarchical model specification. Two brief examples were given to demonstrate how a classical analysis of such models might proceed. The first example made use of simple point counts (replicated temporally) and considered a simple constant detection model and both Poisson and negative binomial models for local abundance. In the second example, data collected according to a removal sampling protocol were considered. In that example, habitat covariates were considered as possible effects on local abundance.

the total of the survival and recruitment processes. It stands to reason that such models will yield improved estimates of (local) survival and recruitment parameters.

Extension to demographically open systems

Burnham, K. P. & Anderson, D. R., 1998. Model Selection and Inference: A Practical Information– Theoretic Approach. Springer–Verlag, New York. Cook, R. D. & Jacobson, J. O., 1979. A design for estimating visibility bias in aerial surveys. Biometrics, 35: 735–742. Dodd, C. K. & Dorazio, R. M., (2004). Using point– counts to simultaneously estimate abundance and detection probabilities in a salamander community. Herpetologica, 60: 68–78. Dorazio, R. M., Jelks, H. & Jordan, F., 2005. Improving removal–based estimates of abundance by sampling spatially distinct subpopulations. Biometrics (to appear). 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 prob abilities from point–count surveys. Auk, 119(2): 414–425. Ihaka, R. & Gentleman, R., 1996. R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5: 299–314. Laird, N. M. & Ware, J. H., 1982. Random–effects models for longitudinal data. Biometrics, 38: 963–974. Hanski, I. A. & Gilpin, M. E. (Ed.), 1997. Metapopulation biology: ecology, genetics, and evolution. Academic Press, San Diego, U.S.A. Levins, R., 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America,15: 237–240. MacKenzie, D. I., Nichols, J. D., Hines, J. E. Knutson, M. G. & Franklin, A. D., 2003. Estimating site occupancy, colonization and local extinction probabilities when a species is detected imperfectly. Ecology, 84: 2200–2207. 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(2): 393–408. 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(1): 46–53. Royle, J. A. & Nichols, J. D., 2003. Estimating

Considerable generality can be achieved by considering extensions of hierarchical abundance models to systems that are demographically open, such as might occur if sampling is conducted during the breeding season in multiple years. There are several interesting "open population" situations that may be considered: (1) Many monitoring programs that generate counts in multiple years may not yield information on individual animals across years. This is common of most "point counting" surveys. In this situation, a simple metapopulation model structure such as Ni,t i Poi( t Ni,t–1) may be useful for integrating data across years. Moreover, they facilitate a characterization of metapopulation dynamics that represents a generalization over methods considered by, for example, MacKenzie et al. (2003) that are based on detection/non–detection data; (2) A common lack of closure is due to the phenomenon of "temporary emigration". In this case, let Mi be the size of some super–population located at sample location i. Let Ni,t i Bin(Mi, ) be the number of individuals available for sampling during occasion t at site i. Finally, let yi,t be the multinomial data with index Ni,t collected according to one of the standard sampling protocols. Note that Ni,t may be removed by integration so that, marginally, yi,t; t = 1,2,... are multinomial random variables with index Mi and cell probabilities . Consequently, k the joint likelihood of the data is a product multinomial, similar to that described in "Point counts" section. However, here the temporal replication, combined with some protocol other than simple point counts, allows estimation of the additional parameter , which is 1 minus the temporary emigration probability; (3) The third type of open scenario is that in which there exists encounter information on individual animals across years such as that arising from sampling based on networks of mist net stations. In this case, Nt must be decomposed into a survival component and a recruitment component where the survival component is Bin(Nt–1, t) and the recruitment component is Poi(Nt–1 t). Note that individual encounter information is directly informative about whereas a spatial model for abundance is informative about

Acknowledgements The author would like to thank Deanna K. Dawson. USGS Patuxent Wildlife Research Center,, and Scott Bates, U.S. National Park Service, for the data collected in Frederick County, Maryland. References

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Abundance from Repeated Presence Absence Data or Point Counts. Ecology, 84: 777–790. – 2004a. N–Mixture Models for estimating population size from spatially replicated counts. Biometrics, 60: 108–115. – 2004b. Modeling abundance index data from anuran calling surveys. Conservation Biology (to appear). Royle, J. A., Connery, B. & Sharp, M. (2005). Estimating avian abundance from simple point counts. U. S. FWS (unpublished report). Royle, J. A., Dawson D. K. & Bates. S., 2004a.

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Estimating abundance effects in distance sampling models. Ecology (to appear). Royle, J. A., Kéry, M., Schmid, H. & Gautier, R., 2004b. Spatial modeling of avian abundance. Unpublished report. Sanathanan, L., 1972. Estimating the size of a multinomial population. Annals of Mathematical Statistics, 43: 142–152. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, San Diego, California, U.S.A.

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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 B. Doligez, D. L. Thomson & A. J. van Noordwijk

Doligez, B., Thomson, D. L. & van Noordwijk, A. J., 2004. 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: 387–402. Abstract 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.— The White stork Ciconia ciconia has been the object of several successful reintroduction programmes in the last decades. As a consequence, populations have been monitored over large spatial scales. Despite these intense efforts, very few reliable estimates of life history traits are available for this species. Such general knowledge however constitutes a prerequisite for investigating the consequences of conservation measures. Using the large–scale and long–term ringing and resighting data set of White storks in the Netherlands, we investigated the variation of survival and resighting rates with age, time and previous individual resighting history, and in a second step supplementary feeding, using capture–recapture models. Providing food did not seem to affect survival directly, but may have an indirect effect via the alteration of migratory behaviour. Large–scale population monitoring is important in obtaining precise and reliable estimates of life history traits and assessing the consequences of conservation measures on these traits, which will prove useful for managers to take adequate measures in future conservation strategies. Key words: Age and time effects on survival, Capture–resighting models, Migrating probability, Population dynamics, Supplementary feeding, Trap–dependence. Resumen Empleo de análisis de datos a gran escala para evaluar rasgos de historia vital y de comportamiento: el caso de la población de cigüeñas blancas Ciconia ciconia reintroducidas en los Países Bajos.— Durante las últimas décadas, la cigüeña blanca Ciconia ciconia ha sido objeto de diversos y satisfactorios programas de reintroducción, lo que ha permitido controlar poblaciones a grandes escalas espaciales. Pese a la intensidad de tales esfuerzos, se dispone de muy pocas estimaciones fiables acerca de los rasgos de la historia vital de esta especie. No obstante, estos conocimientos generales constituyen un requisito previo para investigar las consecuencias de las medidas de conservación. El empleo de datos de reavistaje y de anillamiento a largo plazo y a gran escala de las cigüeñas blancas de los Países Bajos nos ha permitido investigar la variación en las tasas de supervivencia y de reavistaje según la edad, el tiempo y la historia previa de reavistajes individuales. Asimismo, en una segunda fase, hemos analizado los efectos de la alimentación suplementaria a partir de modelos de captura–recaptura. Parece que la provisión de alimentos no incidió directamente en la supervivencia, pero es posible que tuviera un efecto indirecto como consecuencia de la alteración del comportamiento migratorio. El control de la población a gran escala es fundamental para obtener estimaciones precisas y fiables de rasgos de historia vital, así como para evaluar las consecuencias de las medidas de conservación de dichos rasgos, que resultarán de especial utilidad para los gestores a la hora de emprender iniciativas apropiadas con respecto a las estrategias de conservación futuras. Palabras clave: Efectos del tiempo y de la edad en la supervivencia, Modelos de captura–reavistaje, Probabilidad de migración, Dinámica poblacional, Alimentación suplementaria, Dependencia de las trampas. Blandine Doligez & A. J. van Noordwijk, NIOO–KNAW, Centre for Terrestrial Ecology (CTE), Boterhoeksestraat 48, Postbus 40, 6666 ZG Heteren, The Netherlands.– David L. Thomson, Max Planck Inst. for Demographic Research, Konrad–Zuse–Str. 1, D–18057 Rostock, Germany. Corresponding author: B. Doligez, Dept. of Biometry and Evolutionary Biology, CNRS UMR 5558, Univ. Claude Bernard–Lyon I, Bâtiment Gregor Mendel, 43 Blvrd du 11 novembre 1918, F– 69622 Villeurbanne Cedex, France. E–mail: doligez@biomserv.univ-lyon1.fr

ISSN: 1578–665X

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Introduction Habitat degradation and loss due to human activities has led to increased extinction risks especially for rare or highly specialised species and species using several habitat types at different times of the year, such as migratory species (e.g. Senra & Alés, 1992). Conservation biology aims at helping managers to (1) assess the status of populations and identify the risk factors these populations are subject to, (2) decide which conservation measures are optimal to restore or protect endangered populations, and (3) assess the effect of such measures in a feed–back process allowing efficient measures to be taken in further conservation phases (Lebreton & Clobert, 1991; Caswell, 2001). In the last decades, many conservation actions have been undertaken, but for practical reasons, conservation programmes are often limited in space and time. A conservation programme is often considered successful when the target population has been restored to at least the level before the decline within the duration of the programme, i.e. a short– term, numerical response. However, long–term sustainability of restored populations may not be met (Schaub et al., 2004). Conservation measures may affect population dynamics in many ways by acting on different life history or behavioural traits, and may durably change population composition (e.g. proportion of non–breeding individuals), affecting population processes such as sexual selection and intraspecific competition. A precise assessment of the success of conservation actions is an important step requiring the identification of traits affected by conservation measures (Sarrazin & Barbault, 1996; Caswell, 2001). Such assessments, however, are still scarce (Schaub et al., 2004). In particular, difficulties arise from the usually small size of the populations concerned, often preventing the computation of reliable estimates with limited confidence intervals. Large–scale, long– term population monitoring is critically needed to assess the impact of conservation actions, but is rarely compatible with financial and technical support limitations. The conservation of the White stork Ciconia ciconia in Europe is one example of a large spatial scale conservation action. White stork populations strongly declined all over Europe after 1945, and became extinct or nearly extinct in many Western European countries (Bairlein, 1991). Increased mortality, due to (i) starvation on the wintering grounds, and (ii) increase of risk of collision with power lines and electrocution along the migration route, has been identified as one of the causes of population decline (Barbraud et al., 1999). Between 1950 and 1970, conservation actions were implemented at a regional or national level in several countries (Bairlein, 1991; see e.g. Schaub et al., 2004). Consequently, White stork populations have been monitored over large spatial and temporal scales, with intense efforts of ringing nestlings and identifying breeders and wintering birds (see Senra

Doligez et al.

& Alés, 1992; Tortosa et al., 1995). In the Netherlands, an intensive reintroduction programme was initiated in 1969, consisting in (i) a captive breeding programme, with the release of juveniles produced by captive pairs (until 1995), and (ii) providing nest sites (poles) and food at the release stations for non–captive, independent individuals, both during the breeding season and winter. This programme resulted in restoring the White stork population in the Netherlands up to its level before 1945 (396 breeding pairs in 2000). An assessment of the consequences of this programme in terms of population dynamics and life history traits is now required to allow adequate further conservation measures to be taken. In particular, providing food could have altered life history traits directly and/or indirectly. Direct effects could include increased clutch size, brood size and fledgling number and body condition as well as increased juvenile and adult body condition before migration and survival (Brittingham & Temple, 1988; Hörnfeldt et al., 2000; Sasvari & Hegyi, 2001; Tortosa et al., 2002; Tortosa et al., 2003). Indirect effects could include individual behavioural changes (Bairlein, 1991), most importantly concerning foraging and migratory habits. Such changes might consist of alteration of migration route (Fiedler, 1998; Berthold et al., 2001) or wintering areas (Tortosa et al., 1995); or partial loss of the migratory habit, especially with food being provided all year round. If resident birds escape major sources of mortality linked to migration (Tortosa et al., 1995; Schaub & Pradel, 2004; Schaub et al., 2004), supplementary feeding could indirectly influence survival. Assessing the influence on survival rate of food provided at the stations, and investigating its mechanisms, thus appears of prime importance to assess the success of the reintroduction programme and predict future population dynamics under different conservation measures, and thus eventually identify further measures that are needed to keep the population size at a healthy level. In long–lived species such as the White stork, adult survival is a key life history trait for population dynamics, and thus constitutes a preferential target for conservation measures (Lebreton, 1978; Stearns, 1992; Schaub et al., 2004). We thus focused on survival rate, and took advantage of the high ringing and resighting efforts throughout the Netherlands to perform a survival analysis at a large spatial scale. We used capture–recapture methodology to identify the factors affecting survival probability (internal factors: age, previous individual history; and environmental factors: year, food availability), and obtain reliable estimates of this life history trait according to these factors. We discuss here the technical aspects of these analyses based on long–term, large–scale data sets, and provide elements of discussion for the influence of supplementary feeding on life history traits. The biological aspects will be developed and discussed in detail elsewhere. Precise estimates of life history traits are needed to build up integrated demographic

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population models allowing the assessment of the consequences of variation in life history traits on population dynamics (Caswell, 2001; Schaub & Pradel, 2004), and thus the investigation of short– and long–term population dynamics and the assessment of extinction risk under different conservation scenarios. Such integrated models constitute powerful tools to help in making optimal conservation decisions.

care in this species). However, survival rate has previously been found not to differ among sexes in various populations of the White stork (Kanyamibwa et al., 1993; Barbraud et al., 1999), although these studies were based on small sample sizes. The effects retained were then simplified by grouping a posteriori years, cohorts and age classes of similar survival and resighting estimates, and modelling age patterns with different relationships between age and survival (linear, quadratic, etc.).

Material and methods

Supplementary feeding

Species, data collection and selection for analyses

To assess the effect of providing food at the release stations, the distance from the nest to the nearest station was considered. During the nestling phase, foraging visits by parents occur mainly within 2 km from the nest (about 75% of the feeding visits by a pair), less frequently from 2 to 5 km from the nest (about 25%), and only exceptionally farther than 5 km from the nest (Dallinga & Schoenmakers, 1984; Carrascal et al., 1990; Alonso et al., 1991; see also Johst et al., 2001). Thus, the shorter the distance from the nest to the nearest station, the higher the potential influence of feeding on survival. Therefore, two different classes of distance relative to the location of the nest from the nearest release station were defined: class 1 (close): breeding adults whose nest is closer than 2 km from the nearest station (feeding in majority at the station); class 2 (distant): adults whose nests are farther than 2 km (rarely feeding at the station). These distance classes should reflect the percentage of feeding visits made by the parents at the stations.

The White stork is a long–lived migratory species breeding throughout Europe. Data on ringing and live resightings of White storks in the Netherlands have been gathered and monitored by Dutch ringers. Each year, volunteers followed active nests of White storks, and wherever possible, nestlings were given individually numbered metal rings that can be read with a telescope, and parents were identified when ringed. Resightings of all other adults during the breeding season were also collected, though it is difficult to establish with certainty their breeding status. Survival analyses were restricted to birds ringed as nestlings within the Netherlands between 1980 and 1999, using live resightings during the breeding season (April to July) for years 1981 to 2000, within the Netherlands only (less than 5% of resightings of Dutch–ringed storks during the breeding season are made outside the Netherlands). All individuals manipulated in any way were excluded from the analyses, because such manipulations may have long–term influences on survival (Sarrazin et al., 1994). We obtained 3,682 records of birds ringed as nestlings or fledglings between April and July, from 1980 to 1999. Among these birds, a total of 763 individuals have been resighted later as adults during the breeding season within the Netherlands, with a total of more than 5,700 resightings from 1981 to 2000. Effects considered on survival and resighting probability Age, time and cohort effects Age, time and cohort effects, which have previously been shown to influence survival rate (Kanyamibwa et al., 1990; Kanyamibwa et al., 1993; Barbraud et al., 1999), were included in the analyses. Here, a full age effect was considered, i.e. 21 age classes were defined in the starting models. Cohort effects can arise from long–term effects during adulthood of developmental conditions shared by individuals born in the same year (Lindström, 1999; Reid et al., 2003). A cohort effect could be tested only after removal of the age x time interaction if not significant. Conversely, no sex effect was considered because too few individuals could be sexed (no sexual dimorphism or sex differences in parental

Assessment of survival and resighting probabilities: capture–recapture analyses Resighting effort during the breeding season is mainly linked to breeding activities. Because (i) a given percentage of breeding birds are missed each year, and (ii) not all birds may engage in breeding activities each year (especially in the first years of their life), resighting probability is likely to be smaller than 1. In that case, the use of capture–recapture methodology is required to get unbiased estimates of survival probability (Lebreton et al., 1992; Clobert, 1995; Martin et al., 1995). Furthermore, the class of distance to the nearest release station of an individual is susceptible to change over the course of the individual’s lifetime. In this case, multi–state capture–recapture models, allowing the assessment of state–specific survival and resighting rates and transition probabilities between states (here, the distance classes), may be appropriate (Nichols et al., 1994; Nichols & Kendall, 1995). Goodness–of–fit (GOF) Goodness–of–fit tests were performed to ensure that the starting model (i.e. before selection) fits the data (Burnham et al., 1987; Lebreton et al., 1992). We used a modified version of Release tests (Burnham

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et al., 1987) to test for trap–dependence effects (Pradel, 1993; see below). The time effect model (CJS model) was tested cohort by cohort because a strong effect of age is expected in this long–lived species with delayed maturity (see Cézilly et al., 1996). The results of these cohort–by–cohort tests were then summed over all cohorts to compute the global tests for the whole data set. Accounting for trap–dependence Individuals resighted at time t may not have the same probability to be resighted at time t+1 as individuals not resighted at time t (Burnham et al., 1987; Cam & Monnat, 2000), an effect called short– term (or immediate) "trap–dependence" (Pradel, 1993). Data were prepared for analysis of trap– dependence as described in Pradel (1993), using U– Care software (Choquet et al., 2003). Capture– resighting histories were split after each capture to allow distinction between resighting rate immediately after a resighting and after no resighting. Trap– dependence was then modelled by an artificial age– dependence structure, with two age classes (one year after the previous resighting vs. later; Pradel, 1993; see also Pugesek et al., 1995). To account for real age–dependence simultaneously, individuals were re–injected in the data after splitting their capture–resighting history into groups corresponding to their actual age. Parameters were then equalised among these groups according to year and individual’s age and cohort (Appendix). Model selection and notation Model selection was based on the Akaike’s Information Criterion corrected for effective sample size (AICc; Lebreton et al., 1992; Burnham et al., 1995). The models selected were those whose AICc value differed by less than two units from the lowest AICc model. Deviances and AICc values of the different models were calculated using software MARK (White & Burnham, 1999). All effects could not be included at once in the starting model because the number of parameters required would exceed the upper limits of our available computer memory when using MARK (~1,000). Therefore, analyses were performed in two steps: (i) first, the effects of time, age, trap–dependence and cohort on survival and resighting rates were investigated and, when appropriate, simplified a posteriori on the basis of parameter estimates (see above); (ii) second, this simplified model, where the number of parameters had been reduced, was used as a starting model for assessing the influence of the variables linked to conservation measures (here supplementary feeding). Model notation has been extended from the notation defined in Lebreton et al. (1992) and in Nichols et al. (1994) for multi–state models. Sa×t (survival probability) is the probability that a bird of age a at time t–1 survives until time t (an additive effect of time and age on survival was

noted Sa+t). Pa×t (resighting probability) is the probability that a bird of age a is recaptured at time t, given that it is alive and present at time t. Subscripts c and m denote a cohort effect and an immediate trap–dependence effect on resighting rate respectively. In multi–state models, state– specific survival and resighting probabilities are noted Sa×t×s and Pa×t×s respectively. Ta×t×sr (transition probability) is the probability that a bird of age a in state s at time t–1 is in state r at time t, given that the individual has survived from time t–1 to time t. Results Simplification of time, age and cohort effects Goodness–of–fit tests To increase the sensitivity of the global Release test (see Horak & Lebreton, 1998), only tests for cohorts with sufficient data (i.e. at least 2 expected individuals per cell) were included. Goodness–of–fit of model Sa×t, Pa×t was strongly rejected ((² = 308.59, df = 64, p < 0.0001). This was due to a very high 2 CT test value ((² = 267.64, df = 40, p < 0.0001). Thus, resighting probability depended on previous resighting history of the individuals (trap–dependence; Burnham et al., 1987; Pradel, 1993). Here, birds resighted at time t were approximately twice as likely to be resighted again at time t+1 than birds not resighted at time t, i.e. a strong "trap–happiness" effect (Pradel, 1993; fig. 1). The other components of the GOF tests were not significant, except test 3 Sm, but this was due to a single cohort out of 14, i.e. close to the 1/20 expected by chance. When excluding this cohort, we obtained a non–significant 3 SR + 3 Sm + 2 Cm test ((² = 34.16, df = 23, p = 0.063; = 1.49). Thus trap–dependence was the major source of lack of fit. The model Sa×t, Pa×t×m fitted the data, and was used as the starting model for the model selection procedure. Model selection Resighting probability models were simplified first to keep greater power for survival modelling (Lebreton et al., 1992). Additive effects of age, time, trap–dependence and cohort on resighting probability were retained in the final model (table 1). Similarly, an additive effect of age and time on survival probability was retained, i.e. temporal variation of survival probability was parallel among age classes. No cohort effect could be detected on survival probability (table 1). Additional attempts to reduce resighting probability with the simplified survival model were unsuccessful (table 1). The final model retained at this stage was thus Sa+t, Pa+t+c+m, with a total of 97 parameters. No other model could compete with this model, as differences in AICc values were

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A Survival probability

1.0 0.8 0.6 0.4 0.2 0.0

B Resighting probability

1.0 0.8 0.6 0.4 0.2 0.0 1

7 8 Age

10 11 12 13 14

Fig. 1. Variation in survival and resighting probabilities with age: survival (A) and resighting (B) probability estimates (! SE) for year 1994 (medium survival and resighting rate–year, see fig. 2) are shown as examples (birds aged up to 14 years because only birds born after 1980 were included). Survival and resighting probabilities increase gradually with age up to 8 years, when they stabilise. This increase is modelled appropriately by a quadratic relationship for survival (A) and an inverse quadratic relationship for resighting (B), on a logit scale (see table 2). Because age and time effects were additive (see table 1), the age–specific patterns of variation in other years are parallel to those shown in the examples here. In B, trap–dependence on resighting rate is also illustrated: black squares, birds not resighted in the previous year; open squares, birds resighted in the previous year. (Survival and resighting estimates were obtained from model Sa+t, Pa+t+m.) Fig. 1. Variación según la edad en las probabilidades de supervivencia y de reavistaje: estimaciones de probabilidad (! EE) de supervivencia (A) y de reavistaje (B) para el año 1994 (supervivencia media y tasa de reavistaje–año, fig. 2) a modo de ejemplos (aves de más de 14 años de edad, dado que sólo se incluyeron aves nacidas después de 1980). Las probabilidades de supervivencia y de reavistaje aumentan gradualmente con la edad, hasta alcanzar los 8 años, que es cuando se estabilizan. Este aumento se modela adecuadamente mediante una relación cuadrática para la supervivencia (A) y una relación cuadrática inversa para el reavistaje (B), en una escala logit (ver tabla 2). Dado que los efectos de la edad y del tiempo fueron aditivos (tabla 1), las pautas de variación por edades en otros años son análogas a las indicadas en los ejemplos que se detallan aquí. En B, también se ilustra la dependencia de las trampas en la tasa de reavistaje: cuadros negros, aves no reavistadas el año anterior; cuadros blancos, aves reavistadas el año anterior. (Las estimaciones de supervivencia y de reavistaje se obtuvieron a partir del modelo Sa+t, Pa+t+m.)

always much larger than 2 (table 1). When only breeding birds were included in the analysis (i.e. excluding birds of unknown breeding status), the same final model was selected (results not detailed here).

A posteriori characterisation of the effects retained To simplify further the model selected, effects of age, time and cohort on survival and resighting probabilities were characterised explicitly (table 2). Both sur-

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Table 1. Steps of the simplification of the starting model Sa×t, Pa×t×m to model Sa+t, Pa+t+c+m. Model notation according to Lebreton et al. (1992): Np. Number of identifiable parameters; T. Time; A. Age; TD. Trap–dependence; *Indicates the model selected in each step. The deviance value given here is the relative deviance from the saturated model (deviance of the saturated model: 3,332.87). Tabla 1. Fases de la simplificación del modelo de inicio Sa×t, Pa×t×m al modelo Sa+t, Pa+t+c+m. Las anotaciones sobre el modelo se basan en Lebreton et al. (1992): Np. Número de parámetros identificables; T. Tiempo; A. Edad; TD. Dependencia de las trampas; * Indica el modelo seleccionado en cada fase. El valor de desviación indicado corresponde a la desviación relativa con respecto al modelo saturado (desviación del modelo saturado: 3.332,87).

Model

Deviance

AICc

Effect tested

First step, simplifying resighting probability (effects on resighting probability) Sa×t, Pa×t×m

592

5,384.35

10,036.11

Starting model

Sa×t, P(a×t)+m

401

5,634.91

9,829.52

Additive effect of TD

*Sa×t, Pa+t+m

249

5,921.36

9,774.67

Additive effects of A, T and TD

Sa×t, Pa×m

249

6,021.42

9,874.73

No T effect

Sa×t, Pt×m

248

7,003.65

10,854.78

No A effect

Sa×t, Pa×t

400

5,942.05

10,134.35

No TD

Sa×t, Pa+m

231

6061.95

9,876.07

No T effect, additive A and TD effects

Sa×t, Pt+m

230

7,072.13

10,884.08

No A effect, additive T and TD effects

Sa×t, Pa+t

248

6,251.79

10,102.91

No TD, additive effects of A and T

Sa×t, Pa

230

6,433.52

10,245.48

A effect only

Sa×t, Pt

229

7,177.83

10,987.61

T effect only

Sa×t, Pm

212

7,416.03

11,189.07

TD effect only

Sa×t, P·

211

7,567.42

11,338.30

Constant resighting probability

Second step, simplifying survival probability (effects on survival probability) Sa×t, Pa+t+m

249

5,921.36

9,774.67

Starting model

*Sa+t, Pa+t+m

6,087.32

9,578.34

Additive effects of A and T

Sa, Pa+t+m

6,157.79

9,611.93

No T effect

St, Pa+t+m

6,362.27

9,814.37

No A effect

S·, Pa+t+m

6,419.84

9,835.31

Constant survival probability

Third step, checking for effects on resighting probability (effects on resighting probability) *Sa+t, Pa+t+m

6,087.32

9,578.34

Starting model

Sa+t, Pa+t

6,428.87

9,917.84

No TD

Sa+t, Pa+m

6,239.43

9,693.58

No T effect

Sa+t, Pt+m

7,217.07

10,669.17

No A effect

Sa+t, Pa

6,622.48

10,074.59

No T effect nor TD

Sa+t, Pt

7,334.87

10,784.93

No A effect nor TD

Sa+t, Pm

7,577.01

10,992.48

No A nor T effects

Sa+t, P·

7,763.45

11,176.88

Constant resighting probability

Fourth step, testing for cohort effect (effects on survival S and resighting probability P) Sa+t+c, Pa+t+m+c

116

6,016.51

9,586.15

Additive cohort effect on S and P

Sa+t+c, Pa+t+m

6,068.91

9,599.11

Additive cohort effect on S

*Sa+t, Pa+t+m+c

6,037.40

9,567.60

Additive cohort effect on P

Sa+t, Pa+t+m

6,087.32

9,578.34

No cohort effect

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Table 2. A posteriori characterisation of age, time and cohort effects on survival and resighting probabilities in the previously selected model Sa+t, Pa+t+m+c: Np. Number of identifiable parameters. Simplification of the effects: A². Quadratic relationship with age on a logit scale (survival probability); 1/A². Inverse quadratic relationship with age on a logit scale (resighting probability); t3. Separation of survival probability in three time periods: 1981 to 1987, 1988 to 1995, and 1996 to 2000 (see fig. 2A); t '3. Separation of resighting probability in three categories of years: low resighting probability (1985 and 1987), high resighting probability (1992, 1995 and 1999), and medium resighting probability (other years) (see fig. 2B); c3. Separation of resighting probability in three types of cohort: low resighting rate–cohorts, high resighting rate–cohorts, and the 1982 cohort (with a very high resighting rate, see text). (Deviance and AICc values for the final simplified model S A²(8)+t3, P1/A²(8)+t’3+c3+m are given.) Tabla 2. Caracterización a posteriori de los efectos de la edad, el tiempo y las cohortes en las probabilidades de supervivencia y de reavistaje en el modelo previamente seleccionado Sa+t, Pa+t+m+c: Np. Número de parámetros identificables. Simplificación de los efectos: A²: Relación cuadrática con la edad en una escala logit (probabilidad de supervivencia); 1/A². Relación cuadrática inversa con la edad en una escala logit (probabilidad de reavistaje); t3. Separación de la probabilidad de supervivencia en tres períodos de tiempo: de 1981 a 1987; de 1988 a 1995; y de 1996 a 2000 (fig. 2A); t’3. Separación de la probabilidad de reavistaje en tres categorías de años: probabilidad de reavistaje baja (1985 y 1987), probabilidad de reavistaje alta (1992, 1995 y 1999), y probabilidad de reavistaje media (otros años) (fig. 2B); c3. Separación de la probabilidad de reavistaje en tres tipos de cohortes: cohortes con una tasa de reavistaje baja, cohortes con una tasa de reavistaje alta, y la cohorte de 1982 (con una tasa de reavistaje muy elevada; ver el texto). Se indican los valores de desviación y de AICc para el modelo final simplificado SA²(8)+t3, P1/A²(8)+t’3+c3+m. Effect simplified Starting selected model

Survival

Resighting

Deviance

AICc

Sa+t

Pa+t+c+m

6,037.40

9,567.60

Age effect on survival

SA²(8)+t

Pa+t+c+m

6,057.79

9,552.92

Time effect on survival

Sa+t3

Pa+t+c+m

6,059.98

9,557.17

Age effect on resighting

Sa+t

P1/A²(8)+t+c+m

6,050.49

9,543.57

Time effect on resighting

Sa+t

Pa+t’3+c+m

6,053.34

9,550.53

Cohort effect on resighting

Sa+t

Pa+t+c3+m

6,052.99

9,548.12

SA²(8)+t3

P1/A²(8)+t’3+c3+m

6,139.76

9,496.68

Final simplified model

vival and recapture probabilities increased with age up to a plateau (fig. 1). Age–dependence could a posteriori be modelled with a quadratic relationship on a logit scale for survival probability, and with an inverse quadratic relationship on a logit scale for resighting probability (table 2). This difference indicates that the increase of resighting probability with age is steeper than the increase of survival probability. Both survival and resighting probabilities reached a plateau at age 8 (stabilisation at age 6: AICc = 9,503.09; age 7: AICc = 9,498.23; age 8: AICc = 9,496.68; age 9: AICc = 9,496.26; age 10: AICc = 9,496.13). Time–dependence of survival probability could be modelled by considering three periods (1981 to 1987, 1988 to 1995, and 1996 to 2000; table 2, fig. 2A). The first period corresponded to high survival rates but low sample sizes, as reflected by large confidence intervals. Survival decreased during the second period, and again during the third one (fig. 2A).

The break in 1995–1996 corresponded to the end of the captive breeding phase of the reintroduction. A linear decline in survival probability over the 21 years of the period was tested and rejected, but other ways of modelling survival probability could also have been retained. In particular, time–dependence of survival probability could most likely be modelled parsimoniously using external meteorological variables, both during the breeding season and the winter season (Kanyamibwa et al., 1993; Barbraud et al., 1999). The separation in three time periods performed here is not claimed to be the most parsimonious nor best fitting the data. Resighting probability was rather constant over the 20 years, except in years 1985 and 1987 when it was particularly low, and in years 1992, 1995, and 1999 when it was particularly high (fig. 2B). Grouping years in these three categories (low–, medium– and high– resighting rates) appropriately modelled time–dependence of resighting rate (table 2). Similarly, the

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cohort effect on resighting probability could be modelled by considering three types of cohort: (i) high resighting probability–cohorts (1980, 1985, 1987, 1988, 1990, and 1993 to 1999), (ii) low resighting probability–cohorts (1981, 1983, 1984, 1986, 1989, 1991, 1992), and (iii) the 1982 cohort, with an exceptionally high resighting probability (table 2). The origin of these differences between cohorts in resighting probabilities is not clear. This a posteriori characterisation of age, time and cohort effects on survival and resighting probabilities thus decreased the final number of parameters to be estimated down to 12. The final model for this first step was noted S A²(8)+t3, P1/A²(8)+t’3+c3+m (table 2). Influence of distance to the nearest release station 75.0% of birds (2,760 out of 3,682) where ringed in nests located less than 2 km away from the nearest release station (distance class 1). This can be explained by a breeding activity quasi–exclusively located at the release stations during the first 8 years (until 1987), combined with high numbers of nest poles provided for the breeding birds close to the stations. Assessing variable status: dynamic vs. fixed over individuals’ lifetime We first assessed whether the distance class changed over time for a given individual in our data set. 40.6% of resighted birds (264 out of 651) changed class of distance either between hatching and first breeding attempt and/or between several breeding attempts. Thus, the influence of class of distance to the nearest release station on survival rate and resighting probability had to be investigated using a multi–state approach. Further simplifications of the starting model Further simplifications of the starting model were needed and made before investigating the influence of the distance class. Because variation in survival and resighting probabilities with age stabilised at age 8, capture–recapture histories were still split to account for trap–dependence, but after splitting, individuals were re–injected in a single group when aged 8 years or more. This left us with eight age groups instead of 21. Furthermore, the cohort effect was eliminated from this multi–state analysis, because the number of parameters required to parameter the PIMs in MARK would otherwise again exceed the limit. Because the cohort effect was only additive, we assumed that eliminating it should not strongly affect model selection. With no cohort effect, 402 parameters were needed. Survival and distance to the nearest release station The starting model was S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t×s, with two states (close and distant, i.e. two

classes of distance), thus two transitions possible from each state (i.e. one transition estimated per state, Tclose to distant and Tdistant to close, with Tclose to close = 1–Tclose to distant and Tdistant to distant = 1–Tdistant to close). Again, transition and resighting probabilities were simplified first to keep greater power for survival modelling. An additive effect of age, time and class of distance was retained on transition probabilities (table 3). Individuals were more likely to move and breed closer to the stations than the reverse. Storks were less likely to change class of distance with increasing age, and this variation was successfully modelled by an inverse relationship on a logit scale (table 3). Finally, transitions were particularly low (close to zero) in years 1984 and 1985, and particularly high in year 1997. Separating years in these three categories (low–, medium– and high–transition rates) appropriately modelled time–dependence of transition rate (table 3). With this simplified modelling of transition probability, an effect of class of distance was retained on resighting probability, in interaction with trap–dependence alone or trap– dependence and age (table 3; effects not detailed here). Finally, when simplifying survival probability, the models with (i) no effect of the class of distance and (ii) an additive effect of the class of distance to age and time effects on survival competed, the second being slightly less supported (AICc difference between both models: 1.81; table 3). This suggests that the class of distance only has at best a slight direct effect on survival, survival rates being lower for birds seen far from the stations (class 2, distant) than birds seen close (class 1, close) (results not detailed here). Discussion Our study was a first step in understanding the consequences of conservation measures on the White stork population biology and dynamics in the Netherlands through the identification of the factors responsible for the patterns of variation of survival and resighting probabilities. We focused on survival probability as the major life–history trait determining population dynamics in this long–lived species (Lebreton, 1978; Stearns, 1992; Schaub et al., 2004). We discuss below advantages and technical aspects of using large–scale data sets to test for the effect of many factors (age, time, previous history, etc) on survival and resighting probabilities. The biological implications of these effects, and the assessment of conservation measures (here, direct and indirect influences of providing food on survival), need further investigation, and will be discussed elsewhere. Estimates of survival rates with capture– resighting analyses and their variation according to different factors (age, density, meteorological variables) have previously been given for several European populations of White storks (Kanyamibwa et al., 1990; Kanyamibwa et al., 1993; Barbraud et al., 1999; Schaub et al., 2004). Except for the Swiss

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Survival probability

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 B

0.5 0.4 0.3 0.2

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

1989

1988

1987

1986

1985

0.0

1984

0.1 1983

Resighting probability

0.6

Year Fig. 2. Variation in survival and resighting probabilities with time. Third–year survival (A) and resighting (B) probability estimates are shown as examples (years 1983 to 2000 because only birds born after 1980 were included). For survival, three time periods were defined: 1981 to 1987 (high survival probabilities with high confidence intervals, black squares); 1988 to 1995 (medium survival probabilities, open circles); and 1996 to 2000 (low survival probabilities, grey triangles) (see text and table 2). For resighting, three categories of years were defined: low resighting rate years (1985 and 1987, open circles); high resighting rate years (1992, 1995 and 1999, grey triangles); and medium resighting rate years (other years, black squares) (see text and table 2). Temporal variation of survival and resighting probabilities for other age–classes again parallel those shown here because age and time effects were additive (see table 1). (Estimates were obtained from model Sa+t, Pa+t+m.) Fig. 2. Variación según el tiempo en las probabilidades de supervivencia y de reavistaje. La supervivencia durante el tercer año (A) y las estimaciones de probabilidad de reavistaje (B) se indican a modo de ejemplo (de 1983 a 2000, dado que sólo se incluyeron aves nacidas después de 1980). Para la supervivencia, se definieron tres períodos de tiempo: de 1981 a 1987 (probabilidades de supervivencia altas, con intervalos de confianza elevados, cuadros negros); de 1988 a 1995 (probabilidades de supervivencia medias, círculos blancos); y de 1996 a 2000 (probabilidades de supervivencia bajas, triángulos grises) (ver texto y tabla 2). Para el reavistaje, se definieron tres categorías de años: años con una tasa de reavistaje baja (1985 y 1987, círculos blancos); años con una tasa de reavistaje alta (1992, 1995 y 1999, triángulos grises); y años con una tasa de reavistaje media (otros años, cuadros negros) (ver texto y tabla 2). La variación temporal de las probabilidades de supervivencia y de reavistaje para otras clases de edad es análoga, una vez más, a las indicadas aquí, debido a que los efectos de la edad y del tiempo fueron aditivos (ver tabla 1). (Las estimaciones se obtuvieron a partir del modelo S a+t, P a+t+m.)

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Table 3. Steps of the simplification of the multi–state model S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t×s testing for an effect of the class of distance to the nearest release station (state s) on survival and resighting probabilities: Np. Number of identifiable parameters. * Indicates the model selected in each step. When simplifying resighting and survival probabilities before a posteriori characterising transition probability, the same models were selected for survival and resighting probabilities. Tabla 3. Fases de la simplificación del modelo multiestado S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t×s para comprobar un efecto de la clase de distancia con respecto a la estación de liberación más próxima (estado s) sobre las probabilidades de supervivencia y de reavistaje: Np. Número de parámetros identificables; * Indica los modelos seleccionados en cada fase. Cuando se simplificaron las probabilidades de reavistaje y de supervivencia antes de llevar a cabo una probabilidad de transición a posteriori mediante técnicas de caracterización, se seleccionaron los mismos modelos para las probabilidades de supervivencia y de reavistaje.

Model

Deviance

AICc

First step, simplifying transition probability S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t×s

284

5,318.16

9,390.9

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, T(a(8)+t)×s

5,468.83

9,091.8

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t+s

153

5,406.71

9,191.6

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×s+t

5,500.93

9,085.0

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+t×s

5,471.78

9,080.4

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×t

152

5,469.43

9,255.3

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)×s

5,561.09

9,108.5

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Tt×s

5,523.17

9,117.4

*S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+t+s

5,508.73

9,078.5

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+t

5,589.25

9,157.0

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+s

5,565.61

9,096.8

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Tt+s

5,565.43

9,121.0

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)

5,626.34

9,155.5

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Tt

5,617.33

9,170.8

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ts

5,614.50

9,131.6

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, T.

5,667.53

9,182.6

Second step, a posteriori characterisation of transition probability S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+t+s

5,508.73

9,078.5

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, Ta(8)+t’’3+s

5,522.37

9,057.6

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, T1/A(8)+t+s

5,509.81

9,067.4

*S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, T1/A(8)+t’’3+s

5,523.08

9,046.2

S(A²(8)+t3)×s, P(1/A²(8)+t’3+m)×s, T1/A(8)+t’’3+s

5,523.08

9,046.2

S(A²(8)+t3)×s, P1/A²(8)×s+t’3×s+m, T1/A(8)+t’’3+s

5,528.05

9,049.2

*S(A²(8)+t3)×s, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,524.34

9,043.4

S(A²(8)+t3)×s, P1/A²(8)+t’3×s+m×s, T1/A(8)+t’’3+s

5,523.46

9,044.6

S(A²(8)+t3)×s, P1/A²(8)×s+t’3+m, T1/A(8)+t’’3+s

5,529.35

9,046.4

S(A²(8)+t3)×s, P1/A²(8)+t’3×s+m, T1/A(8)+t’’3+s

5,528.11

9,047.2

*S(A²(8)+t3)×s, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,524.66

9,041.7

S(A²(8)+t3)×s, P1/A²(8)+t’3+m+s, T1/A(8)+t’’3+s

5,529.39

9,044.5

S(A²(8)+t3)×s, P1/A²(8)+t’3+m, T1/A(8)+t’’3+s

5,547.03

9,060.1

Third step, simplifying resighting probability

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Table 3. (Cont.)

Model

Deviance

AICc

Fourth step, simplifying survival probability S(A²(8)+t3)×s, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,524.66

9,041.7

SA²(8)×s+t3, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,525.76

9,038.8

SA²(8)+t3×s, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,525.00

9,038.1

*SA²(8)+t3+s, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,525.91

9,034.9

*SA²(8)+t3, P1/A²(8)+t’3+m×s, T1/A(8)+t’’3+s

5,526.12

9,033.1

S(A²(8)+t3)×s, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,524.34

9,043.4

SA²(8)×s+t3, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,525.40

9,040.5

SA²(8)+t3×s, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,524.78

9,039.8

SA²(8)+t3+s, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,525.64

9,036.7

*SA²(8)+t3, P1/A²(8)×s+t’3+m×s, T1/A(8)+t’’3+s

5,525.88

9,034.9

population, which also used large samples (more than 3,500 individual recovery histories over 28 years; Schaub et al., 2004), previous studies obtained much smaller adult survival estimates (0.65 to 0.78), despite higher resighting rates (0.85 to 0.95). The difference in survival between these studies and ours likely arises from the much smaller sample sizes on which the former were based, combined with differences in wintering conditions between years 1970’s (droughts in Africa) and 1990’s (increasing number of storks wintering in Spain), or different conservation measures in different populations. The large–scale and long–term monitoring of the White stork population in the Netherlands, in relation with the reintroduction programme, allowed the gathering of a very large and high quality resighting data set. Ringing and resighting efforts have been high over the whole study period. As a consequence, we were able to model fine patterns of variation in survival and resighting probabilities with age and time, detect small effects of these factors and obtain precise estimates of these parameters. In particular, our study is probably one of the first to include a full age– and trap–dependence simultaneously (see Pugesek et al., 1995; Frederiksen & Bregnballe, 2000). Age dependent survival was modelled in previous studies using a two–pseudo age class structure (first year after initial resighting vs. later; Kanyamibwa et al., 1990), which did not account for fine age variation in survival rate, and probably also resulted in underestimated adult survival rates (see also Tavecchia et al., 2001). A progressive increase of survival prospects early in life, as found here, has been shown in several long–lived species (Frederiksen & Bregnballe, 2000; Tavecchia et al., 2001), thus strongly encouraging complex modelling of age–

specific survival rate, using large data sets to provide sufficient power (see also Pugesek et al., 1995; Harris et al., 1997). Such an increase might be widespread among long–lived species, although it has only rarely been detected because the age structure modelled was too simple (Hafner et al., 1998; Prévot–Julliard et al., 1998; Forero et al., 2001; but see Bauchau et al., 1998). However, the use of large data sets collected at large spatial scales also imposes constraints and limitations, because their analysis involves (i) a large set of candidate models, and (ii) complex models with many parameters, as the influence of more factors are investigated. Here, we faced the problem of the upper limit of the number of parameters based on our available computer memory when using MARK (1,000). Because of this limit, we had to adopt a strategy to simplify the models investigated to reduce parameter number (see e.g. Pugesek et al., 1995), and chose a two–step procedure mixing a priori and a posteriori effects: (i) simplifying age and time modelling of survival and resighting probabilities, and (ii) using the a posteriori simplified model in a multi–state model to investigate the effect of the class of distance. We could not define a starting model including all effects of interest simultaneously in MARK. Moreover, we also reduced the number of models to be compared by simplifying resighting (and transition) probability before survival probability (i.e. we did not explore the whole model space). While this approach renders analyses tractable in MARK, it does not guaranty that the best model is retained. A posteriori modelling should normally be used to create new hypotheses to be tested with another data set, in an a priori way. Here we mixed both approaches, which decreases the strength of evidence for the selected model. A theoretical assessment of the influence of

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different strategies used to simplify model selection procedures on the model retained would be helpful to ensure that results are robust. It may turn out that, in the case of complex models on large data sets, MARK may not be the appropriate capture– recapture analysis software to perform model selection. The simplifying strategies could be based on a priori knowledge of the species biology and results of previous studies. For instance, a trap–happiness when individuals are identified without physical capture, as for the White stork, may be due to higher resighting probability of breeders compared to non– breeders, associated with delayed maturity (Lebreton, 1978; Kanyamibwa & Lebreton, 1992). In this case, trap–dependence may be modelled using two states, breeder and non–breeder, with state–specific resighting probabilities. Here, such a two–state approach again could not have been performed using MARK because of our limits in parameter numbers. However, Release tests on the data set restricted to resighting histories after the first resighting as a breeder indicated that trap– happiness occurred within adults. Furthermore, trap– happiness was retained when resightings of birds of unknown status, probably mostly non–breeders, were excluded from the data, a result differing from other studies (Lebreton et al., 1992; Cézilly et al., 1996). Thus, resighting heterogeneity was observed among breeders. Trap–happiness may rather be due to spatial heterogeneity in resighting effort associated with high philopatry (Prévot–Julliard et al., 1998). Birds may be expected to have a higher resighting rate when breeding close to the stations than far, because of time constraints in resighting effort. Further analyses including geographical estimates of resighting effort are required to investigate this mechanism of trap–dependence. Our detailed modelling of survival and resighting probabilities allowed us to start investigating the consequences of conservation measures, here supplementary feeding, on these traits. Providing food did not seem to strongly affect survival rate directly, but may affect it indirectly (see Schaub & Pradel, 2004; Schaub et al., 2004). The influence of supplementary feeding on the probability of migrating, and survival differences between residents and migrants is still in need of investigation. Eventually, a complete understanding of the White stork population dynamics in the Netherlands will also require the detailed characterisation of recruitment (Clobert et al., 1994; Pradel, 1996), reproductive success and dispersal, and of their variation with age, time (Pradel et al., 1997), breeding density (Barbraud et al., 1999), and supplementary feeding (Moritzi et al., 2001; Tortosa et al., 2002; Tryjanowski & Kuzniak, 2002). It will then become possible to build an integrated population model, incorporating the estimates obtained. Such a model would allow us to understand the mechanisms of population dynamics (see Thomson & Cotton, 2000) and thus to predict the consequences of future conservation actions for the White stork in the Netherlands.

Acknowledgements This study was funded by Vogelbescherming Nederland (partner of Birdlife International in the Netherlands), which initiated and coordinated the reintroduction programme of the White stork in the Netherlands. We thank in particular T. van der Have, who oversaw the contract from Vogelbescherming Nederland, and provided data, help and advice. This work could not have been conducted without the help of many volunteers, who, year after year, gathered data on White stork presence, breeding activities and success, and ringed nestlings in the whole of the Netherlands. We thank all the persons involved in data gathering and management, in particular G. Speek of the Nederlandse Ringcentrale (at CTO, NIOO, Heteren), and A. Enters, W. van der Nee, R. and I. Rietveld, who played a coordinating role with volunteers. We also thank R. Pradel for technical advices on goodness–of–fit tests and preparation of the data to account for trap–dependence, P. Warin for help in preparing the data, M. Schaub for advice in the model selection step, C. Deerenberg for preliminary work on the White stork data, and two anonymous referees for their constructive comments. References Alonso, J. C., Alonso, J. A. & Carrascal, L. M., 1991. Habitat selection by foraging White storks, Ciconia ciconia, during the breeding season. Can. J. Zool., 69: 1957–1962. Barbraud, C., Barbraud, J.–C. & Barbraud, M., 1999. Population dynamics of the White stork Ciconia ciconia in Western France. Ibis, 141: 469–479. Bairlein, F., 1991. Population studies of White storks (Ciconia ciconia) in Europe. In: Bird Population Studies. Relevance to Conservation and Management: 207–229 (G. J. M. Hirons, Eds.). Oxford Univ. Press, Oxford. Bauchau, V., Horn, H. & Overdijk, O., 1998. Survival of Spoonbills on Wadden Sea islands. J. Avian Biol., 29: 177–182. Berthold, P., van den Bossche, W., Fiedler, W., Kaatz, C., Kaatz, M., Leshem, Y., Nowak, E. & Querner, U., 2001. Detection of a new important staging and wintering area of the White stork Ciconia ciconia by satellite tracking. Ibis, 143: 450–455. Brittingham, M. & Temple, S. A., 1988. Impacts of supplemental feeding on survival rates of black– capped chickadees. Ecology, 69: 581–589. Burnham, K. P., Anderson, D. R., White, G. C., Brownie, C. & Pollock, K. H., 1987. Design and Analysis Methods for Ffish Survival Experiments Based on Release–Recapture. American Fisheries Society Monographs n°5, Bethesda, Maryland. Burnham, K. P., White, G. C. & Anderson, D. R., 1995. Model selection strategy in the analysis of capture–recapture data. Biometrics, 51: 888–898.

Animal Biodiversity and Conservation 27.1 (2004)

Cam, E. & Monnat, J.–Y., 2000. Apparent inferiority of first–time breeders in the kittiwake: the role of heterogeneity among age classes. J. Anim. Ecol., 69: 380–394. Carrascal, L. M., Alonso, J. C. & Alonso, J. A., 1990. Aggregation size and foraging behaviour of White storks Ciconia ciconia during the breeding season. Ardea, 78: 399–404. Caswell, H., 2001. Matrix Population Models. Sinauer Associates, Sunderland. Cézilly, F., Viallefont, A., Boy, V. & Johnson, A. R., 1996. Annual variation in survival and breeding probability in greater flamingo. Ecology, 77: 1143–1150. Choquet, R., Reboulet, A.–M., Pradel, R., Gimenez, O. & Lebreton, J.–D., 2003. U–Care user’s guide, Version 2.0 Mimeographed document, CEFE/ CNRS Montpellier. ftp://ftp.cefe.cnrs–mop.fr/Soft–CR. Clobert, J., 1995. Capture–recapture and evolutionary ecology: a difficult wedding? J.Applied Stat., 22: 989–1008. Clobert, J., Lebreton, J.–D., Allaine, D. & Gaillard, J. M., 1994. The estimation of age–specific breeding probabilities from recaptures or resightings in vertebrates populations: II. Longitudinal models. Biometrics, 50: 375–387. Dallinga, H. & Schoenmakers, M., 1984. Populatieveranderingenbij de ooievaar Ciconia ciconia ciconia in de periode 1850–1975 (Population changes of the White stork Ciconia ciconia ciconia during the period 1850–1975). Rijksuniversiteit Groningen, Groningen, The Netherlands. Fiedler, W., 1998. Joint VogelWarte Radolfzell– Euring migration project: a large–scale ringing recovery analysis of the migration of European bird species. EURING Newsletter, 2: 31–35. Forero, M. G., Tella, J. L. & Oro, D., 2001. Annual survival rates of adult Red–necked nightjars Caprimulgus ruficollis. Ibis, 143: 273–277. Frederiksen, M. & Bregnballe, T., 2000. Evidence for density–dependent survival in adult cormorants from a combined analysis of recoveries and resightings. J. Anim. Ecol., 69: 737–752. Hafner, H., Kayser, Y., Boy, V., Fasola, M., Julliard, A.–C., Pradel, R. & Cézilly, F., 1998. Local survival, natal dispersal, and recruitment in Little egrets Egretta garzetta. J. Avian Biol., 29: 216–227. Harris, M. P., Freeman, S. N., Wanless, S., Morgan, B. J. T. & Wernham, C. V., 1997. Factors influencing the survival of Puffins Fratercula arctica at a North Sea colony over a 20–year period. J. Avian Biol., 28: 287–295. Horak, P. & Lebreton, J.–D., 1998. Survival of adult Great tits Parus major in relation to sex and habitat; a comparison of urban and rural populations. Ibis, 140: 205–209. Hörnfeldt, B., Hipkiss, T., Fridolfsson, A.–K., Eklund, U. & Ellegren, H., 2000. Sex ratio and fledging success of supplementary–fed Tengmalm’s owl broods. Mol. Ecol., 9: 187–192. Johst, K., Brandl, R. & Pfeifer, R., 2001. Foraging

399

in a patchy and dynamic landscape: human land use and the White stork. Ecol. Appl., 11: 60–69. Kanyamibwa, S., Bairlein, F. & Schierer, A., 1993. Comparison of survival rates between populations of the White stork Ciconia ciconia in Central Europe. Ornis Scandinavica, 24: 297–302. Kanyamibwa, S. & Lebreton, J.–D., 1992. Variation des effectifs de cigogne blanche et facteurs du milieu: un modèle démographique. In: Les Cigognes d’Europe: 259–264 (J.–L. Mériaux, A. Schierer, C. Tombal & J.–C. Tombal, Eds.) Colloque International, Metz 1991, Inst. Européen d'Ecologie. Kanyamibwa, S., Schierer, A., Pradel, R. & Lebreton, J.–D., 1990. Changes in adult annual survival rates in a Western European population of the White stork Ciconia ciconia. Ibis, 132: 27–35. Lebreton, J.–D., 1978. Un modèle probabiliste de la dynamique des populations de Cigogne blanche Ciconia ciconia L. en Europe occidentale. In: Biométrie et Ecologie: 277–343 (R. Tomassone, Eds.). Société Française de Biométrie, Paris. Lebreton, J.–D., Burnham, K., Clobert, J. & Anderson, D. R., 1992. Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies. Ecol. Monogr., 62: 67–118. Lebreton, J.–D. & Clobert, J., 1991. Bird population dynamics, management, and conservation: the role of mathematical modelling. In: Bird Population Studies: Relevance to Conservation and Management: 105–125 (G. J. M. Hirons, Eds.). Oxford University Press, Oxford. Lindström, J., 1999. Early development and fitness in birds and mammals. TREE, 14: 343–348. Martin, T. E., Clobert, J. & Anderson, D. R., 1995. Return rates in studies of life history evolution: are biases large? J. Applied Stat., 22: 863–875. Moritzi, M., Maumary, L., Schmid, D., Steiner, I., Vallotton, L., Spaar, R. & Biber, O., 2001. Time budget, habitat use and breeding success of White storks Ciconia ciconia under variable foraging conditions during the breeding season in Switzerland. Ardea, 89: 457–470. 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. & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. J. Applied Stat., 22: 835–846. Pradel, R., 1993. Flexibility in survival analysis from recapture data: handling trap–dependence. In: Marked Individuals in the Study of Bird Populations: 29–37 (P. M. North, Eds.). Birkhäuser Verlag, Basel/Switzerland. – 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Pradel, R., Johnson, A. R., Viallefont, A., Nager, R. G. & Cézilly, F., 1997. Local recruitment in the

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Greater flamingo: a new approach using capture– mark–recapture data. Ecology, 78: 1431–1445. Prévot–Julliard, A.–C., Lebreton, J.–D. & Pradel, R., 1998. Re–evaluation of adult survival of Black– headed gulls (Larus ridibundus) in presence of recapture heterogeneity. Auk, 115: 85–95. Pugesek, B. H., Nations, C., Diem, K. L. & Pradel, R., 1995. Mark–resighting analysis of a California gull population. J. Applied Stat., 22: 625–639. Reid, J. M., Bignal, E. M., Bignal, S., McCracken, D. I. & Monaghan, P., 2003. Environmental variability, life–history covariation and cohort effects in the red–billed chough Pyrrhocorax pyrrhocorax. J. Anim. Ecol., 72: 36–46. Sarrazin, F., Bagnolini, C., Pinna, J.–L., Danchin, E. & Clobert, J., 1994. High survival estimates of Griffon vultures (Gyps fulvus fulvus) in a reintroduced population. Auk, 111: 853–862. Sarrazin, F. & Barbault, R., 1996. Reintroduction: challenges and lessons for basic ecology. TREE, 11: 474–478. Sasvari, L. & Hegyi, Z., 2001. Condition–dependent parental effort and reproductive performance in the White stork Ciconia ciconia. Ardea, 89: 281–291. 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? Biol. Cons., 119: 105–114. Senra, A. & Alés, E. E., 1992. The decline of the

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White stork Ciconia ciconia population of Western Andalousia between 1976 and 1988: causes and proposals for conservation. Biol. Cons., 61: 51–57. Stearns, S. C., 1992. The Evolution of Life Histories. Oxford University Press, Oxford. Tavecchia, G., Pradel, R., Boy, V., Johnson, A. R. & Cézilly, F., 2001. Sex– and age–related variation in survival and cost of first reproduction in greater flamingos. Ecology, 82: 165–174. Thomson, D. L. & Cotton, P. A., 2000. Understanding the decline of the British population of Song thrushes Turdus philomelos. In: Ecology and Conservation of Lowland Farmland Birds: 151– 155 (J. A. Vickery, Eds.). British Ornithologists Union, Tring. Tortosa, F. S., Caballero, J. M. & Reyes–Lopez, J., 2002. Effect of rubbish dumps on breeding success in the White stork in Southern Spain. Waterbirds, 25: 39–43. Tortosa, F. S., Manez, M. & Barcell, M., 1995. Wintering White storks (Ciconia ciconia) in South West Spain in the years 1991 and 1992. Die Vogelwarte, 38: 41–45. Tortosa, F. S., Perez, L. & Hillstrom, L., 2003. Effect of food abundance on laying date and clutch size in the White stork Ciconia ciconia. Bird Study, 50: 112–115. Tryjanowski, P. & Kuzniak, S., 2002. Population size and productivity of the White stork Ciconia ciconia in relation to Common vole Microtus arvalis density. Ardea, 90: 213–217. White, G. C. & Burnham, K. P., 1999. Program MARK: estimation from population of marked animals. Bird Study, 46, supplement: 120–138.

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Appendix. Simultaneous investigation of age– and trap–dependence on resighting rate. Apéndice. Investigación simultánea de la dependencia de la edad y de la trampa sobre la tasa de reavistaje.

The following capture–recapture history of an individual White stork during the period 1980–2000 is taken as an example. The individual considered here was born in 1983, first resighted when three years old (in 1986), and then resighted each year from 1987 to 1989, from 1991 to 1993, and in 1996 and 1997; it was not resighted again after 1997. Capture–recapture history 000100111101110011000 Trap–dependence can be accounted for by splitting the capture–recapture history after each capture, and considering the rest of the history as a new history. Using an artificial two–age structure for the parametrisation of resighting probabilities then allows us to distinguish resighting probability one year after a previous resighting from resighting probability later on (Pradel, 1993). To account for real age– dependence, one needs to specify the age at which each "new" history starts. Here, we consider 21 years, thus 21 age classes for full age–dependence. When re–injecting the individual after splitting the history, the age of the individual is specified by assigning it to an age group. The "– 1" values indicate when a history has been split, so the individual is considered to be removed ("loss on capture") from the analysis at this point (Choquet et al., 2003). Splitting the history and accounting for age at "release" 000100100000000000000 000000110000000000000 000000011000000000000 000000001100000000000 000000000101000000000 000000000001100000000 000000000000110000000 000000000000010010000 000000000000000011000

–1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 –1 0 0 0 0 0 –1 0 0 0 0 0 –1 0 0 0 0 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 –1 0 0 0 0 –1 0 0 0 0 –1 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

When parameterising resighting probabilities according to age, year and cohort, one then needs to set equal the probabilities for individuals of the same age and cohort in the same year, except just after a resighting (trap–dependence effect). In MARK, parameters are defined for each age group separately in matrices called Parameter Index Matrices (PIM). The full structure (age, time and cohort effects) is specified in the PIM of the first age group. Then, for each group i: (i) the parameters of the first i–1 rows of the PIM are set equal to 0 (because no individual of age i can be found before year 1980+i, nor belong to a cohort younger than 2000–i); (ii) the parameters of row j (j m i) are set equal to the 20–i+1 last parameters of row j–i+1 of the PIM of the first group; (iii) the first parameter of each row j (j m i) is set different from the corresponding parameter of the PIM of the first group, i.e. introducing an artificial two–age structure to account for trap–dependence. The PIMs for individuals of groups 1, 2 and 4 are shown below as an example. Only the effects of age and trap–dependence are considered (no cohort nor time effect). First group 1

2 1

3 2 1

4 3 2 1

5 4 3 2 1

6 5 4 3 2 1

7 6 5 4 3 2 ….

8 7 6 5 4 3

9 8 7 6 5 4

10 9 8 7 6 5

11 10 9 8 7 6

12 11 10 9 8 7

13 12 11 10 9 8

14 13 12 11 10 9

15 14 13 12 11 10

16 15 14 13 12 11

17 16 15 14 13 12

18 17 16 15 14 13

19 18 17 16 15 14

20 19 18 17 16 15

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Appendix. (Cont.)

Second group (birds released after a resighting at age 2) X

X 21

X 3 21

X 4 3 21

X 5 4 3 21

X 6 5 4 3 21

X 7 6 5 4 3 窶ｦ.

X 8 7 6 5 4

X 9 8 7 6 5

X 10 9 8 7 6

X 11 10 9 8 7

X 12 11 10 9 8

X 13 12 11 10 9

X 14 13 12 11 10

X 15 14 13 12 11

X X X 13 12 11

X X X 14 13 12

X X X 15 14 13

X 16 15 14 13 12

X 17 16 15 14 13

X 18 17 16 15 14

X 19 18 17 16 15

X 20 19 18 17 16

X X X 19 18 17

X X X 20 19 18

Fourth group (birds released after a resighting at age 4) X

X X

X X X

X X X 23

X X X 5 23

X X X X X X 6 7 5 6 23 5 窶ｦ.

X X X 8 7 6

X X X 9 8 7

X X X X X X 10 11 9 10 8 9

X X X 12 11 10

X X X 16 15 14

X X X 17 16 15

X X X 18 17 16

Parameters 21 and 23 here are set different from parameters 2 and 4 respectively to account for trap窶電ependence, and X is set to 0. In the analysis including class of distance as a state variable, a set of PIMs is defined for each state.

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

Saurola, P. & Francis, C. M., 2004. Estimating population dynamics and dispersal distances of owls from nationally coordinated ringing data in Finland. Animal Biodiversity and Conservation, 27.1: 403– Abstract Estimating population dynamics and dispersal distances of owls from nationally coordinated ringing data in Finland.— Amateur bird ringers can collect data at a geographic and temporal scale that is rarely possible with professional field crews, thus allowing truly national analyses of population dynamics and dispersal. Since the early 1970s, bird ringers in Finland have been strongly encouraged to focus on birds of prey, especially cavity– nesting owls. In addition to ringing nestlings and adults, ringers also provide data on population trends and breeding success. The resultant data indicate that numbers of breeding pairs fluctuated with the 3–4 year microtine cycle, but without any long–term trend. Mean productivity per nest varied from 2.18 to 3.33 fledglings per active nest in Tawny Owls, 1.56 to 2.87 in Ural Owls and 1.78 to 4.32 in Tengmalm’s Owls. Survival and breeding propensity also varied with the vole cycle and explained much of the observed variation in breeding populations. Observed median dispersal distances were 24 and 18 km for Ural and Tawny Owls respectively, but increased to 36 and 48 km, using a method presented here to adjust for uneven sampling effort, highlighting the importance of considering sampling effort when estimating dispersal. Key words: Amateur ringers, Natal dispersal, Population modelling, Mark–recapture analysis, Tawny Owl, Ural Owl, Tengmalm’s Owl. Resumen Estimación de la dinámica poblacional y de las distancias de dispersión en los búhos, efectuada utilizando datos de anillamiento de Finlandia coordinados a escala nacional.— Los anilladores aficionados pueden recopilar datos a una escala geográfica y temporal que rara vez está al alcance de los equipos de campo profesionales, lo que permite llevar a cabo análisis de dinámica poblacional y de dispersión de alcance verdaderamente a escala nacional. Desde principios de la década de 1970, se ha recomendado encarecidamente a los anilladores de Finlandia que se centren en las aves de presa, en concreto, en los búhos que anidan en cavidades. Además de anillar a los pollos nidífugos y a los ejemplares adultos, los anilladores también aportan datos acerca de las tendencias poblacionales y el éxito de reproducción. Los datos resultantes indican que el número de parejas reproductoras fluctuó con el ciclo microtino de 3–4 años, pero no pudo observarse ninguna tendencia a largo plazo. La productividad media por nido varió de 2,18 a 3,33 volantones por nido activo en el cárabo común, de 1,56 a 2,87 en el cárabo uralense, y de 1,78 a 4,32 en la lechuza de Tengmalm. La supervivencia y propensión a la reproducción también experimentaron cambios con el ciclo de los micrótidos, lo que explicaría, en gran parte, la variación apreciada en las poblaciones reproductoras. Las distancias de dispersión medias observadas fueron de 24 y 18 km para el cárabo uralense y el cárabo común, respectivamente, si bien, mediante el método descrito en el presente estudio, aumentaron hasta alcanzar los 36 y 48 km. Dicho método permite ajustar los esfuerzos de muestreo desiguales, al tiempo que resalta su importancia a la hora de estimar la dispersión. Palabras clave: Anilladores aficionados, Dispersión natal, Modelaje de la población, Análisis de marcajerecaptura, Cárabo común, Cárabo uralense, Lechuza de Tengmalm. Pertti Saurola, Finnish Museum of Natural History, Univ. of Helsinki, P. O. Box 17, FIN–00014, Finland.– Charles M. Francis, Canadian Wildlife Service, National Wildlife Research Centre, Ottawa, Ontario, K1A 0H3, Canada.

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Introduction In Finland, as in many other places in the world, bird ringers are an extremely dedicated and skilled group of birdwatchers. The majority of ringers are real amateurs with the best meaning of the word — lovers of their hobby, contributing many hundreds of hours as volunteers each year to collect data on their passion— birds. As such, they are an important resource for science and conservation, and professional ornithologists can benefit enormously by making good use of their data. This paper illustrates, through analysis of large– scale data sets on owls generated by ringers in Finland, the types of data that can be provided by ringers, some of the analyses that can be done with such data, and some of the statistical challenges that remain for working with these types of data, especially for analysis of dispersal. Specifically, this contribution has three main objectives: (1) to highlight to ringing schemes around the world, the value of coordinating amateur ringers on specialized projects, and of fully computerizing all of the data as precisely as possible; (2) to demonstrate how amateur ringing data can be used to study all aspects of the population dynamics of owls, providing background information for more detailed analyses presented elsewhere (Francis & Saurola, 2004); and (3) to encourage statisticians to develop new methods to estimate dispersal distances from data sets collected from large areas. In many countries, data produced by amateur ringers have been used primarily for studies of bird migration (e.g. Bairlein, 2001; Wernham et al., 2002). While such studies are obviously valuable, they tap only a small percentage of the potential uses of these data. Increasingly, efforts are being made to use amateurs in programs such as constant–effort mist–netting programs to monitor population trends (e.g. DeSante et al., 1999; Peach et al., 1996). Nevertheless, the vast majority of scientific research on detailed aspects of the population ecology and dispersal of birds has been based on data gathered by professional ornithologists (see e.g. references in Newton, 1986). In many cases, these studies are limited to relatively small study areas, although there are some notable exceptions. For example, a very impressive population study on owls has been carried out in north–western USA on the endangered Spotted Owl Strix occidentalis (e.g. Forsman et al., 1996, 2002), but this was realized with a remarkable funding base that can only be dreamed of by the majority of researchers. The Finnish owl data illustrate how, with dedicated volunteers, high quality data on all aspects of population biology can be collected by amateurs. Making maximum use of data from amateurs requires well organized, centralized computer files that incorporate all aspects of the data, as well as well– planned and coordinated data collection. Although most ringing schemes now have all of their recoveries computerized, many still do not store the details of the original ringing records, nor

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all of the recapture records in electronic form. Both of these types of data are required to estimate survival as well as dispersal, by providing information on the spatial and temporal distribution of ringing effort. Furthermore, in some ringing schemes, even if the data are computerized, the location data (ringing and encounters) are recorded only approximately (for example, in North America, they are recorded only to 10' of latitude and longitude), which severely limits the precision of many spatial analyses, including estimation of natal and breeding dispersal distances in birds. In Finland, since the start of ringing in 1913, all dead encounters reported by the general public have been computerized and are available in electronic form. In addition, since 1974 (larger ring sizes since 1973), all ringing as well as all live encounter data (recaptures and resightings) have also been stored, with the location recorded to within 100 m, in the central computer system of the Finnish Ringing Centre (Saurola, 1987a). Gaining maximum value from amateur ringers also requires a well coordinated research design. For both scientific and conservation reasons, Finnish ringers have been, since 1974, especially encouraged by the Finnish Ringing Centre to work on birds of prey (Saurola, 1987a). Now, after 30 years of coordinated voluntary work involving several hundred active ringers, very large data sets have been accumulated on a scale that would be impossible to achieve for a normal research team of professional ornithologists. Not only are they recording birds that they ringed, but they are also tracking numbers of breeding pairs, active nests, and young per nest. These data sets are particularly critical for raptors which, being at the top of the food chain, are intrinsically much less common than many other bird species, and are also particularly vulnerable to environmental threats such as the accumulation of pesticides or other toxic chemicals (e.g. Newton, 1979). The potential of these remarkable data sets for understanding the population dynamics of cavity– nesting owls is illustrated in this paper and a companion paper by Francis & Saurola (2004). These same data can also be used to estimate dispersal, but that presents a number of still unsolved statistical challenges. Walters (2000) suggested that "lack of information about dispersal has begun to limit progress on several biological fronts". These limitations arise not just because of limited data, but also because of limited statistical methods for their analysis. Saurola (2002) presented information on sexual and annual variation in natal dispersal of Finnish Owls, but did not take into account potential geographic variation in ringing and recapture efforts. Thomson et al. (2003) presented a method that uses data on the distribution of ringing effort, combined with population data from breeding bird atlases to adjust the observed dispersal distributions. Here, an alternative, though related, method is presented, and applied to two species of owls. Adjusting for effort substantially increases estimated dispersal distances,

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but also highlights a number of remaining challenges related to estimation of confidence limits and statistical comparisons among estimates.

ringers are asked to provide a measure of the effort on their plot. In general, effort within an individual plot has remained roughly the same from year to year (unpublished data).

Data collection methods

Raptor Questionnaire

Ringings and recoveries

In 1982, a Raptor Nest Card (in addition to the traditional nest card available for all species) was introduced and ringers were asked to complete a card for all nests found. In 1986, after a relatively poor initial response rate, an alternative report, called a Raptor Questionnaire was introduced, which must be completed as a requirement in the annual reports of all ringers working on raptors. The Raptor Questionnaire summarises the total numbers of (1) potential nest sites checked, (2) active nests and occupied territories found, and (3) nests of different clutch and brood sizes verified by ringers. In addition, measures of effort are reported. These results are summarized for reporting based on the "territories" of all local ornithological societies in different parts of the country. The main purpose of the Raptor Questionnaire is to collect data on annual breeding output. In addition, these data, although not precisely standardised from year to year, may be used with care to detect fluctuations and trends in population sizes, especially for areas where Raptor Grid data are too scanty.

Since 1974, the Finnish Ringing Centre has encouraged ringers to ring birds of prey. To ensure data quality, candidates interested in ringing in Finland must pass a fairly demanding test in which they demonstrate skills in identifying all species of birds encountered in Finland, in completing all required reporting and documentation (preferably in electronic form), and in techniques for safely handling birds without injury, etc. In addition, the ringer has to make a "ringing plan", that outlines the ringing activities that would be undertaken if the application is accepted. All new ringing licences are usually restricted to a limited number of species that meet this plan. Because birds of prey have had high priority, candidates applying for a ringing licence for birds of prey have been given a high priority as well. Further details on the ringing program were given by Saurola (1987a). Most ringers working on owls select a study area ranging from about 100 km2 to over 1,000 km2. These are chosen in consultation with other ringers to avoid overlap or confusion in responsibility. Ringers then place nest boxes throughout suitable habitat within their area, of an appropriate size and design for the species of holeâ&#x20AC;&#x201C;nesting owls expected to be within their area. These boxes are checked regularly during the breeding season and, if a nest is found, the ringer returns on the appropriate date to ring the nestlings. In addition, most ringers attempt to catch the adults, especially females, which in most species are much easier to catch. A relatively effective but still laborious method for catching adult males of the Tawny Owl (Strix aluco) and Ural Owl (Strix uralensis) was developed in the early 1970s (Saurola, 1987b), but so far only some ringers have regularly caught adult males of these species. Raptor Grid In 1982, the Finnish Ringing Centre, with some support for administration from the Ministry of Environment, started a monitoring project called the Raptor Grid to monitor diurnal and nocturnal birds of prey (Saurola, 1986). The Raptor Grid program is completely based on voluntary field work by raptor ringers. Ringers are asked (1) to establish a study group; (2) to select a 10x10 km study plot based on 10â&#x20AC;&#x201C;km squares of the Finnish National Grid; and (3) to try each year to find all the active nests or at least the occupied territories of all species of birds of prey in their study plot. The number of Raptor Grid study plots surveyed has averaged 120 per year (Saurola, 1997; BjĂśrklund et al., 2003). The amount of effort has varied considerably among study plots, but

Data analysis methods Changes in population size Data from the Raptor Grid were used for estimating changes in population size. While efforts have been made to retain the same set of study plots over time, in practice, some plots have become inactive and new ones have emerged, primarily because of changes in volunteers involved in the field work. Thus, analyses have to control for this potential variation in coverage among plots. To do this, for each year, population indices were calculated through pairwise comparisons of mean numbers in that year to those in a reference year for plots that were active in both years (cf. Baillie et al., 1986; Peach et al., 1996). For this analysis, 1997 was chosen as a reference year because it was a good year with many active plots and many data. Thus, only plots that were active in 1997 were included in this analysis. Two measures of abundance were examined: all occupied territories and active nests. Population parameters Productivity was estimated as the mean number of large nestlings (i.e. old enough to be ringed) produced per nest, for all active nests (i.e. nests in which eggs were laid) reported by ringers through the Raptor Questionnaire. Most of the owlets are ringed during the second half of the nestling period, when they were at least two weeks old, but prior to

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fledging. Age at ringing is an appropriate metric, because mortality within the nest after this age is low and, in any case, first–year survival estimates are based from the age of ringing onwards. Combined mark–recapture–recovery analyses, as implemented in program MARK (White & Burnham, 1999) were used to estimate age–specific survival rates (Francis & Saurola, 2002) as well as relative breeding propensity (based on annual and age– specific variation in capture probabilities). Details of the methods, as well as the results of those analyses are presented elsewhere (Francis & Saurola, 2004) and not repeated here. Natal dispersal distances In principle, the distribution of natal dispersal distances, defined as the distance from the natal site to the first breeding location, can be estimated by ringing birds as nestlings and later recapturing them as breeding adults. In practice, however, this procedure may produce biased estimates because only some of the marked nestlings that survive to breed are ever recaptured as breeders. If the dispersal distances of the observed recruits are not representative of those of recruits that were not observed, then estimates of dispersal distances may be biased unless appropriate correction factors can be developed (Barrowclough, 1978; Van Noordwijk, 1984). In many cases, the study area is smaller than the maximum dispersal distance, and distribution of potential dispersal distances is truncated, leading to an under–estimate of mean dispersal distances. Without any information on long–distance recruitment, no correction is possible. In the case of species such as owls in Finland, ringing takes place over a very large area, so that the distribution (at least for most species) is unlikely to be truncated. However, ringing effort may vary throughout the area, such that owls in some parts of the country may be more likely to be captured and ringed than owls in other parts. Thomson et al. (2003) presented a method for adjusting observed dispersal distances for effort, using data from breeding bird atlases combined with ringing totals. They estimated the relative probability of recapturing birds at different distances for each bird that was ringed. In this paper, an alternative, but related method is presented, first developed by Saurola & Taivalmäki (unpublished). This method involves studying dispersal in the reverse direction: for each first capture of a breeding adult originally ringed as a nestling (a recruit), calculating (a) the actual observed dispersal distance for that individual, and (b) the distribution of potential dispersal distances, which is defined as the distribution of distances from the recapture site to all birds ringed as nestlings in the same year as the observed recruit was first ringed. These represent birds that could have been detected if they had recruited to the site. These potential recruitment

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distances are then used to estimate the density of ringed individuals at different distances from the observed recruit by dividing by the land area at each distance. If the density of breeding owls is similar throughout the range of the species, the relative density of ringed owls at each distance provides an index of the proportion of owls that were ringed at each distance category. By repeating this process for all owls, and averaging across individuals, these relative densities can be used to adjust the estimated dispersal probabilities. Mathematically, this can be expressed as follows: observed recruit, a bird ringed as a nestling and caught for the first time as a breeding adult at the nest; potential recruits, birds ringed as nestlings in the same year as an observed recruit was ringed. Let us define the following symbols: Nbi. Number of observed recruits in distance category i in which the natal dispersal distance y is ri–1 < y < ri where ri is the radius of the outer and ri– the inner border of the distance category i. 1 Nqik. Number of potential recruits in distance category i for the kth capture. Distance is measured from the site of the capture to all the sites where nestlings were ringed in the same year as the observed recruit. Npi. Number of potential recruits in distance category i summed across all observed recruits. Npi can be calculated as follows: , where m is the total number of captures (= different individuals captured at their nest) in the distance category i. Let us then define the density of potential recruits in distance category i: , where Ai is the area of the distance category i. If the study area were large without boundaries, then Ai could be calculated as: Ai =

x (r2i – r2i–1)

However, in most landscapes, this will overestimate the true area, because some parts of the distance band may be in the ocean or otherwise in unsuitable habitat for owls. For this paper, a GIS analysis was used to estimate the land area of each distance band (excluding both ocean and inland water bodies) from each observed recruit. This was done using the "Digital Chart of the World" at a 1:1,000,000 scale. The map was rasterized at a 500m pixel resolution for the analysis, and then overlaid with 10 km distance bands from each observed recruit, using an Arc/Info script to determine the land area in each band. Furthermore, land areas north of the breeding range of the species were also excluded (for Tawny Owl, areas north of 63.9°N were excluded, as well as the inland area of northwestern Finland with few records —see fig. 1;

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for Ural Owl, areas north of 67.2°N were excluded). The area at each distance was then averaged across all recruits to generate Ai. This analysis was carried out separately considering only land areas within Finland, and also considering land areas in adjacent countries from which the species might also disperse. The corrected proportion of individuals dispersing from each distance category can then be estimated as:

Strix aluco Nestlings ringed by 10 x 10 km squares 1–5 (110) 6–25 (182) 26–100 (171) 101–200 (54) 201–526 (31)

100 km

where:

Strix uralensis

and Nc is the total number of distance categories. Results Ringing data During 1913–2002, altogether 223,981 owls were ringed in Finland, predominantly of 5 different species (table 1). Although the vast majority of these birds, for all species, were young birds, in the past 30 years most ringers attempted to catch the adults, at least the females, of hole nesting owls breeding in their nest– boxes. As a result of this effort, almost 10% of newly ringed birds for some species are of breeding adults (table 1). In addition, for these same species, a significant portion of the re–encounters represent birds recaptured alive at the nests in subsequent years. These data provide valuable information on the survival of older birds, and are also critical for capture–recapture and dispersal analyses, to determine what happens to the young birds. Unfortunately, only a few ringers have been systematically attempting to catch breeding males as well, owing to the substantially greater effort required. The spatial distribution of the ringing effort for the Tawny Owl, Ural Owl and Tengmalm’s Owl coincides fairly well with the distribution of each species that has been mapped using data from all available sources in the Finnish Breeding Bird Atlas (Väisänen et al., 1998). Both nestlings and breeding adults of the Tawny Owl have been ringed only in the southern quarter of Finland (fig. 1). For the Ural Owl this extends to the southern half of the country, while Tengmalm’s Owls have been ringed all over the country (fig. 2). However, in all these species, the spatial distribution of ringing effort is somewhat patchy: in some areas the work has been much more intensive than elsewhere. Demographic parameters In the process of ringing these owls, Finnish ringers have also inspected very large numbers of nest sites each year. For example, in 2002, 44,650

Nestlings ringed by 10 x 10 km squares 1–5 (311) 6–10 (149) 11–50 (411) 51–100 (114) 101–319 (58)

100 km

Fig. 1. Distribution of numbers of nestlings ringed for Tawny Owls (top) and Ural Owls (bottom) from 1973 to 2002 by 10 x 10 km squares of the Finnish National Grid. Fig. 1. Distribución del número de pollos anillados, correspondiente al cárabo común y al cárabo uralense, desde 1973 hasta 2002, por cuadrados de 10 x 10 km de la Red Nacional del Sistema de Coordenadas de Finlandia.

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Table 1. Accumulated ringing totals, since 1913, of five species of owls in Finland, along with the % of adults ringed of the total in 1993–2002 and the total number of recoveries and recaptures (the latter including only movements of at least 10 km or an elapsed time of at least 3 months): S. Species; RT. Ringing total; %A. Percentage of adults; R– R. Recaptures/recoveries; TeO. Tengmalm's Owl (Aegolius funereus); TaO. Tawny Owl (Strix aluco); UrO. Ural Owl (Strix uralensis); PyO. Pygmy Owl (Glaucidium passerinum); EaO. Eagle Owl (Bubo bubo).

Aegolius funereus

Nestlings ringed by 10 x 10 km squares 1–5 (337) 6–10 (249) 11–50 (591)

Tabla 1. Total de anillamientos acumulados, desde 1913, con respecto a cinco especies de búhos en Finlandia, junto con el porcentaje de adultos anillados sobre el total correspondiente a 1993–2002 y el número total de recuperaciones y recapturas (estas últimas sólo incluyen movimientos de por lo menos 10 km, o un tiempo transcurrido mínimo de tres meses): S. Especies; RT. Total de anillamientos; %A. Porcentage de adultos; R–R. Recapturas/ recuperaciones; TeO. Lechuza de Tengmalm (Aegolius funereus); TaO. Cárabo común (Strix aluco); UrO. Cárabo uralense (Strix uralensis); PyO. Mochuelo chico (Glaucidium passerinum); EaO. Buho real (Bubo bubo).

51–200 (272) 201–503 (31)

100 km

R–R

TeO

96,263

11.5%

4,962

TaO

37,067

8.7%

9.880

UrO

35,615

8.4%

9,779

PyO

20,305

9.6%

1,253

EaO

13,058

3.4%

2,884

potential nest sites of all species of birds of prey were inspected and reported by 236 ringers/groups (Björklund et al., 2003). These included the following numbers of nest–boxes for owls: Tengmalm’s Owl 9,264, Pygmy Owl 5,540, Ural Owl 4,338 and Tawny Owl 4,038. The corresponding numbers of active nests found in that year were 621, 578, 1,084 and 560. In addition to providing information on numbers of occupied territories, from most of these nests, data were also gathered on the number of nestlings produced per nest, an important measure of productivity. Analyses of data from the Raptor Grids show major fluctuations in the numbers of breeding owls and occupied territories each year for both the Tawny Owl and Ural Owl (fig. 3). These fluctuations coincide strongly with the 3–4 –year cycle of small mammals, especially voles, through the 1980s and early 1990s, although the correspondence does not

Fig. 2. Distribution of numbers of nestlings ringed for Tengmalm’s Owls (Aegolius funereus) from 1973 to 2002 by 10 x 10 km squares of the Finnish National Grid. Fig. 2. Distribución del número de pollos anillados, correspondiente a la lechuza de Tengmalm (Aegolius funereus), desde 1973 hasta 2002, por cuadrados de 10 x 10 km de la Red Nacional del Sistema de Coordenadas de Finlandia.

appear to be as strong during the past four years (see also Sundell et al. 2004). Furthermore, these data indicate that the populations of both species have remained at roughly the same mean levels during the last two decades, with no evidence of long–term trends. Sundell et al. (2004) used these data to demonstrate widespread spatial synchrony in the breeding patterns of the owls, apparently related to similar spatial synchrony in the cycle of microtine rodents Data from the Raptor Questionnaire show that the reproductive success of the Tawny Owl and

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100% 75%

Strix aluco

–02: 160 territ. / 46 plots –02: 120 nests / 41 plots

–02: 147 territ. / 59 plots –02: 119 nests / 49 plots

Strix uralensis

Index

50% 25% 0 –25% –50%

Year

2002

2000

1998

1996

1994

1992

1990

1988

1986

1984

1982

2002

2000

1998

1996

1994

1992

1990

1988

1986

1982

–100%

1984

–75%

Year

Fig. 3. Population indices of Tawny Owls (Strix aluco) and Ural Owls (Strix uralensis) during 1986–2002 in Finland, based on data from the Raptor Grid monitoring project. All indices represent the percent difference in population size relative to 1997 (index = 0%). Black dots, indices based on all occupied territories; white triangles, indices based on active nests. Fig. 3. Índices poblacionales del cárabo común (Strix aluco) y del cárabo uralense (Strix uralensis) durante 1986–2002 en Finlandia, elaborados mediante datos extraídos del Proyecto de Observación del Sistema de Coordenadas para Aves Rapaces. Todos los índices representan la diferencia porcentual en el tamaño poblacional correspondiente a 1997 (índice = 0%). Puntos negros, índices basados en todos los territorios ocupados; triángulos blancos, índices basados en nidos activos.

Ural Owl fluctuated from one year to the next (fig. 4). In these more generalist feeders the amplitude of fluctuations was not as large as might be expected for the most specialized vole feeders (Saurola, 1995) but was still substantial. For example during 1986–2002, the average annual production of large nestlings per active nest varied from 2.18 to 3.33 in the Tawny Owl and from 1.56 to 2.87 in the Ural Owl (fig. 4). In the more specialized Tengmalm’s Owl, production varied from 1.78 to 4.32 in the same time period. Francis & Saurola (2004) showed that Tawny Owl survival rates for both adults and young also varied considerably among years. The vole cycle explained some of the variation, with lowest survival in years when voles crashed and remained at very low levels over the following winter. Even more variation was explained by the severity of winter weather, with lower survival in very cold winters. Recapture probabilities, an index of breeding propensity, varied dramatically in response to the vole cycle, with lowest breeding in years of low vole abundance. The variation was most extreme for one–year old owls. Dispersal analyses For three species of Finnish hole–nesting owls, the Tawny Owl, Ural Owl and Tengmalm’s Owl, extensive data suitable for dispersal analysis are avail-

able for birds ringed as nestlings and later recaptured as breeding adults (figs. 5, 6). For Tawny and Ural Owls, adjusting the observed dispersal distribution for the density of ringed birds (potential recruits) leads to an outward shift of the estimated distribution of dispersal distances, with a substantial increase in the estimated proportion of birds that moved very long distances (fig. 7). Assuming that each species occurs throughout the land areas in both Finland and adjacent countries, south of the northern limit of its breeding range, the estimated median dispersal distance increased from 18 to 48 km for Tawny Owls, and 24 to 36 km for Ural Owls (table 2). If the presence of unsuitable areas (e.g. water, including sea) is ignored, then the estimated dispersal distances are substantially higher, while if the species are assumed only to disperse within Finland, then the estimated dispersal distances are somewhat lower, especially for Tawny Owls (table 2). The adjusted estimates suggest that both recaptures and recoveries may underestimate dispersal distances, despite the fact that recoveries of birds found dead during the breeding season are generated from a sampling method that is fairly independent of the ringing data, being found by the general public (table 2). Adjusting the dispersal distances also leads to changes in the estimated differences between the sexes (table 2), although these may be biased due to the much more limited areas where males were trapped (fig. 5).

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4 Strix uralensis

n = 6,147 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Year

n = 10,402 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Nestlings / active nest

Strix aluco

Year

Fig. 4. Mean breeding output (large young per active nest) of Tawny Owls (Strix aluco) and Ural Owls (Strix uralensis) during 1986–2002, based on data from the Raptor Questionnaires. Standard errors are indicated by vertical bars. Fig. 4. Promedio de resultados de reproducción (individuos jóvenes ya crecidos por nido activo) del cárabo común (Strix aluco) y del cárabo uralense (Strix uralensis) durante 1986–2002, obtenido a partir de datos extraídos de los Cuestionarios sobre Aves Rapaces. Los errores estándar se indican mediante barras verticales.

The Tengmalm’s Owl presents a challenge to this analysis method, because several very long–distance movements were observed into Sweden (fig. 6). Analyses would be enhanced with information on the numbers of nestlings ringed every year in Sweden, as well as information on the distribution within Sweden. Nevertheless, it is likely that adjusting these estimates would lead to a substantial increase in the estimated proportion of very long distance movements for that species as well. Discussion The role of Finnish ringers in monitoring birds of prey Ringers play a crucial role in monitoring birds of prey in Finland, for understanding all aspects of their population ecology, from changes in population size and range, to changes in demographic parameters. At one scale, field–work needed in special projects to monitor threatened species of birds of prey, is carried out by ringers. At another scale, monitoring of common birds of prey is totally based on voluntary work by ringers. The Finnish Ministry of Environment has provided only limited funding for this program, primarily for the administrative work needed to coordinate the field work, to file the data and to produce annual reports (see Björklund et al., 2003). Because every ringer who rings birds of prey has to report his observations through the Raptor Ques-

tionnaire, in addition to providing ringing data, ringers also provide an extensive overview of the annual variation of the average breeding performance (clutch size, brood size and breeding success) of different species of birds of prey. This is very important, because otherwise only a relatively small fraction of nests and territories of birds of prey found by ringers would be reported at all. Motivation to complete a summary form seems to be much higher than to complete more detailed nest records cards, especially from the nests of a common species such as the Tawny Owl. While this reduces the detailed information available on such things as nest site selection, habitat, nest height, etc., the most critical information on annual productivity is provided. Demographic analyses Both through ringing and through reporting nesting data and numbers of pairs, ringers in Finland provide information on all aspects of the life cycle of several species of owls. Survey data provide information on numbers of birds breeding and nesting success. Coordination of ringing effort by many ringers throughout Finland allows for much greater precision (through larger sample sizes) as well as a more representative national picture, than would be possible with only a few ringers, even though some of the most dedicated ringers have ringed over 1,000 owls each. Detailed analyses of survival rates for Tawny Owls (Francis & Saurola, 2004), indicate the

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Strix aluco Females

Males

100 km

Strix uralensis

Males

Females

100 km

Fig. 5. Natal dispersal patterns of Tawny Owls (Strix aluco) and Ural Owls (Strix uralensis) for females (left) and males (right) from 1951 to 2002. The locations of nests where ringed nestlings were first captured as breeding adults are indicated by dots, connected to the corresponding natal sites by lines. Fig. 5. Pautas de dispersión natal del cárabo común (Strix aluco) y del cárabo uralense (Strix uralensis) para hembras (izquierda) y machos (derecha), desde 1951 hasta 2002. La situación de los nidos en los que los pollos nidífugos fueron capturados por primera vez siendo adultos reproductores se indica a través de puntos, conectándose mediante líneas a los correspondientes lugares de nacimiento.

data are sufficient to estimate annual variation in survival rates for three age classes with good precision, and to model that variation in relation to prey abundance and winter weather conditions. Because the ringing effort has been consistently high, annual variation in capture probabilities can also be used as an index of age–specific variation in breeding probabilities. Francis & Saurola (2004) were able to use these parameter estimates to develop a simple matrix model of the demography of this population that, even in a deterministic fashion, captured some of

the key variation in population numbers in relation to vole abundance. The data should also be sufficient to estimate spatial variation in demographic parameters such as survival, but such analyses have not yet been undertaken. Natal dispersal distances The analyses presented here indicate that the dispersal estimates in Saurola (2002) may be a substantial underestimate of true dispersal distances

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Aegolius funereus

Females

Males

100 km

Fig. 6. Natal dispersal patterns of Tengmalm’s Owl (Aegolius funereus), females and males, ringed in Finland from 1951 to 2002. The locations of nests where ringed nestlings were first captured as breeding adults are indicated by dots, connected to the corresponding natal sites by lines. Fig. 6. Pautas de dispersión natal de las lechuzas de Tengmalm (Aegolius funereus), hembras y machos, anilladas en Finlandia entre 1951 y 2002. La situación de los nidos en los que los pollos fueron capturados por primera vez siendo adultos reproductores se indica con puntos, conectándose mediante líneas a los correspondientes lugares de nacimiento.

(fig. 7). The proportion of long–distance movements appears to be substantially underestimated based solely on observed recaptures, despite the fact that ringing took place throughout much of the range of the species in Finland. Even greater shifts can be anticipated for Tengmalm’s Owls, given the numbers of very long distance movements, and the likelihood of a similar drop–off in density of marked individuals with distance. These corrections can also potentially affect analyses of relative dispersal distance. For example, Saurola (2002) showed that if all recaptures at the nest of Ural Owls, Tawny Owls and Tengmalm’s Owls ringed as nestlings from the entire country were included in the analysis, statistically significant difference between males and females were detected (figs. 5, 6). However, if the analyses were restricted only to intensive study areas, where both sexes were captured at the nest, the difference remained significant only in the Tengmalm’s Owl. This suggests that at least some of the apparent differences between the sexes could be due to bias caused by geographic variation in dispersal distances: because most of the trapping of male owls took place within a few study areas (fig. 6), estimates of their dispersal distances may not have been comparable to those of females ringed throughout Finland. The analyses presented here indicate that the estimated differences between the sexes increase if the observed dispersal distances are adjusted for uneven sampling effort (table 2). However, these analyses did not consider the possibility of geographic variation in dispersal distances, and thus could still be biased. Further developments of the method are required to take that into account. Saurola (2002) showed that observed median natal dispersal distances of the Tawny Owl were about three times longer in Finland than in Britain (Paradis et al., 1998) and in southwest Sweden (Wallin et al., 1988). However, the average distances between the nests were 3.2 and 2.7 times longer, suggesting that natal dispersal distances of owls from these three areas may be closely related to territory densities, with birds moving farther when territories are large (Saurola, 2002). Again, these analyses might be affected by adjusting for sampling effort, which would likely vary among the three countries. This correction method depends on the assumption that the densities of owls are similar throughout their range. Furthermore, in order to apply this method, some assumptions are required about densities of each species outside of Finland. Comparison of the analysis methods (table 2) indicates that results for Tawny Owl are particularly sensitive to the assumptions, probably because its true breeding range is more restricted. An alternative assumption could be that the distribution of ringing records actually reflects the true distribution of owls, given the wide distribution of ringing activities. However, the estimated densities of ringed birds dropped off noticeably with distance, even over the first few

413

Animal Biodiversity and Conservation 27.1 (2004)

30%

20%

Observed distribution, median = 48 km

15% 10%

30%

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300

5% 0%

Percentage of owls

Observed distribution, median = 18 km (n = 1,451)

Strix uralensis

25% 20%

Observed distribution, median = 24 km (n = 1,315) Observed distribution, median = 36 km

15% 10% 5% 0%

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 222 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370

Percentage of owls

Strix aluco 25%

Dispersal distance (km) Fig. 7. Distribution of natal dispersal distances of Tawny Owls (Strix aluco) and Ural Owls (Strix uralensis) in Finland. Black columns, observed distribution of distances; grey columns, distribution corrected for density of ringed birds throughout their breeding range in Finland and adjacent countries (see Methods). Fig. 7. Distribución de las distancias de dispersión natal del cárabo común (Strix aluco) y del cárabo uralense (Strix uralensis) en Finlandia: Columnas negras, distribución de distancias observada; columnas grises, distribución corregida para la densidad de aves anilladas a lo largo de toda su zona reproductora en Finlandia y países adyacentes (véase Métodos).

distance bands, probably reflecting the fact that ringing activities are clumped into individual study areas. Most likely, the true dispersal distances are somewhere in between these approaches —densities do vary through Finland, but not exactly proportional to ringing densities. Thomson et al. (2003) proposed a method involving the use of atlas data to estimate both the distribution and relative abundance of the species, to provide a more accurate correction. Given the availability of atlas data from much of Finland, such an approach should be explored in future with the Finnish ringing data. An additional statistical challenge involves placing confidence limits on the estimates. These correction methods are particularly sensitive to long–distance movements, which are usually few in number, because the density of available ringed birds at longer distances is usually low. Much of the shift in the estimated median

dispersal distance for Tawny Owls arises because of a small number of observed movements > 250 km. Appropriate statistical methods, perhaps involving jack– knifing or randomization tests, are required to place confidence limits on these estimates. These must take into account not only the sampling characteristics, but also the uncertainty in estimates of the distribution of breeding owls. Furthermore, it is important to emphasize that ringing must take place over a wide enough area that at least some long–distance movements are observed —this method cannot adjust distance categories with no observed recoveries. Since the inception of the EURING technical meetings, huge advances have been made in statistical development of models for estimating survival and movement probabilities. Similar developments are now required for the estimation of dispersal distances (cf. Van Noordwijk, 1993).

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Table 2. Estimated median dispersal distances (km) from natal nest to site of first recapture during the breeding season for two species of owls in Finland, both unadjusted (Observed), and adjusted for variation in effort and area (see text); and distances between natal site and site of death for owls ringed as nestlings, and recovered dead in a subsequent year during the breeding season: 1 Adjusted based on the land area in the species’ breeding range within Finland; 2 Adjusted based on the land area south of the northern limits of the species range in Finland and adjoining countries; 3 Adjusted based solely on the calculated surface area of each distance band; 4 Derived from Saurola (2002). Tabla 2. Distancias de dispersión medias estimadas (km) desde el nido natal hasta el lugar donde se realizó la primera recaptura durante la estación reproductora para dos especies de búhos en Finlandia, indicándose tanto las no ajustadas (Observed), como aquéllas ajustadas para la variación con respecto a los esfuerzos y al área (véase el texto); y distancias entre el lugar del nacimiento y el lugar de la muerte correspondiente a búhos que fueron anillados siendo pollos nidífugas y que se recuperaron muertos en un año posterior durante la estación reproductora: 1 Ajuste basado en el área dentro del rango reproductor de la especie en Finlandia; 2 Ajuste basado en el área situada al sur del límite norte de distribución de la especie en Finlandia y paises cercanos; 3 Ajuste basado exclusivamente en la superficie del área calculada para cada banda de distancias; 4 Basado en Saurola (2002).

Recaptures Specie

Observed

Adjusted

Recoveries 2

Adjusted

Observed4

Tawny Owl Strix aluco Males only

439

Females only

999

1,451

Both

Ural Owl Strix uralensis Males only

213

Females only

1,084

Both

1,315

Conclusions

Acknowledgements

These analyses demonstrate the enormous potential for ecological research through coordinated, large– scale volunteer–based ringing projects involving ringing of both nestlings and older age classes. These data can be further enhanced through collecting relevant auxiliary data such as nesting productivity data. In addition, and most importantly, all of the data must be centrally and efficiently computerized. Unfortunately, the majority of the Ringing Centres around the world have not yet started to file data on original ringing records and/or local recaptures in electronic form. Furthermore, in many cases the location information is only recorded to within 10–15 km, which is not sufficiently precise to allow accurate estimation of dispersal for many species (median dispersal distance in many species may even be smaller than this limit). Nevertheless, with appropriate encouragement and guidance, similar coordinated volunteer–based projects could no doubt be developed through enthusiastic ringers in many parts of the world, leading to greatly enhanced understanding of population dynamics and thus more effective management and conservation planning.

We are grateful to Finnish ringers for their outstanding work, especially on birds of prey, to Jukka–Pekka Taivalmäki for fresh ideas in developing a new method to correct observed dispersal distances, to Heidi Björklund for producing ringing and recovery maps, and to Andrew Couturier of Bird Studies Canada for the GIS analyses required to help correct dispersal distances. References Baillie, S. R., Green, R. E., Boddy, M. & Buckland, S. T., 1986. An evaluation of the Constant Effort Sites Scheme. BTO Research Report 21, BTO. Thetford. Bairlein, F., 2001. Results of bird ringing in the study of migration routes. Ardea, 89(1) special issue: 7–19. Barrowclough, G. F., 1978. Sampling bias in dispersal studies based on finite area. Bird Banding, 49: 333–341. Björklund, H., Saurola, P. & Haapala, J., 2003.

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Breeding and population trends of common raptors and owls in 2002 –many new records saw the daylight. Linnut–vuosikirja, 2002: 18–40. (In Finnish with extensive English summary). DeSante, D. F., O’Grady, D. R., & Pyle, P., 1999. Measures of productivity and survival derived from standardized mist–netting are consistent with observed population changes. Bird Study, 46 (suppl.): 178–188. Forsman, E. D., Anthony, R. G., Reid, J. A., Loschl, P. J., Sovern, S. G., Taylor, M., Biswell, B. L., Ellingson, A., Meslow, E. C., Miller, G. S., Swindle, K. A., Thairkill, J. A., Wagner, F. F. & Seaman, D. E., 2002. Natal and breeding dispersal of Northern Spotted Owls. Wildlife Monographs, 149: 1–35. Forsman, E. D., DeStefano, S., Raphael, M. G. & Gutiérrez, R. J. (Eds.), 1996. Demography of the Northern Spotted Owl. Studies in Avian Biology No. 17. Cooper Ornithological Society. Francis, C. M. & Saurola, P., 2002. Estimating age– specific survival rates of tawny owls—recaptures versus recoveries. Journal of Applied Statistics, 29: 637–647. – 2004. Estimating components of variance in demographic parameters of Tawny Owls. Animal Biodiversity and Conservation, 27.1: 489–502. Newton, I.,1979. Population ecology of raptors. T. & A. D. Poyser. London. – 1986. The Sparrowhawk. T. & A. D. Poyser. London. Paradis, E., Baillie, S. R., Sutherland, W. J. & Gregory, R. D., 1998. Patterns of natal and breeding dispersal in birds. Journal of Animal Ecology, 67: 518–536. Peach, W. J., Buckland, S. T. & Baillie, S. R., 1996. The use of constant effort mist–netting to measure between–year changes in the abundance and productivity of common passerines. Bird Study, 43: 142–156. Saurola, P., 1986. The Raptor Grid: an attempt to monitor Finnish raptors and owls. Vår Fågelvärld. Suppl., 11: 187–190. – 1987a. Bird ringing in Finland: status and guide– lines. Acta Reg. Soc. Sci. Litt. Gothoburgensis. Zoologica, 14: 189–201. – 1987b. Mate and nest–site fidelity in Ural and Tawny Owls. In: Biology and conservation of northern forest owls. Symposium proceedings, February 3–7, 1987 Winnipeg, Manitoba: 81–86 (R. W. Nero, R. J. Clark, R. J. Knapton & R. H. Hamre, Eds.). USDA Forest Service Gen. Tech. Rep. RM–142.

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– 1995. Suomen pöllöt. Owls of Finland. Helsinki, Kirjayhtymä. (in Finnish with English summary). – 1997. Monitoring Finnish Owls 1982–1996: methods and results. In: Biology and conservation of owls of the northern hemisphere. 2nd International Symposium, February 5–9, 1997, Winnipeg, Manitoba, Canada: 363–380 (J. R. Duncan, D. H. Johnson & T. H. Nichols, Eds.). USDA Forest Service Gen. Tech. Rep. NC. – 2002. Natal dispersal distances of Finnish owls: results from ringing. In: Ecology and conservation of owls: 42–55 (I. Newton, R. Kavanagh, J. Olsen & I. Taylor, Eds.). CSIRO Publishing, Collingwood VIC, Australia. Sundell, J., Huitu, O., Henttonen, H., Kaikusalo, A., Korpimäki, E., Pietiäinen, H., Saurola, P., & Hanski, I., 2004. Large–scale spatial dynamics of vole populations in Finland revealed by the breeding success of vole–eating predators. Journal of Animal Ecology, 73: 167–178. Thomson, D. L., Van Noordwijk, A., & Hagemeijer, W., 2003. Estimating avian dispersal distances from data on ringed birds. Journal of Applied Statistics, 30: 1003–1008. Van Noordwijk, A. J., 1984. Problems in the analysis of dispersal and a critique on its "heritability" in the Great Tit. Journal of Animal Ecology, 53: 533–544. – 1993. On the role of ringing schemes in the measurement of dispersal. In: Marked Individuals in the Study of Bird Population: 323–328 (J.– D. Lebreton & P. M. North, Eds.). Birkhäuser Verlag, Basel. Väisänen, R. Lammi, E. & Koskimies, P., 1998. Distribution, numbers and population changes of Finnish breeding birds. Otava, Helsinki (In Finnish with English summary). Wallin, K., Carlsson, M. & Wall, J., 1988. Natal dispersal in female and male Tawny Owls (Strix aluco). In: Life history evolution and ecology in the Tawny Owls (Strix aluco) (Wallin, K.). Ph. D. Thesis, Univ. of Gothenburg. Walters, J. R., 2000. Dispersal behavior: an ornithological frontier. Condor, 102: 479–481. Wernham, C. V., Toms, M. F., Marchant, J. H., Clark, J. A., Siriwardena, G. M. & Baillie, S. R., 2002. The Migration Atlas: movements of the birds of Britain and Ireland. T. & A. D. Poyser, London. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46: S120–S139

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)

417

Estimating the contribution of survival and recruitment to large scale population dynamics R. Julliard

Julliard, R., 2004. Estimating the contribution of survival and recruitment to large scale population dynamics. Animal Biodiversity and Conservation, 27.1: 417–426. Abstract Estimating the contribution of survival and recruitment to large scale population dynamics.— At large spatial scales, variation in population abundance results from variation in the survival of reproducing adults and variation in the recruitment of new individuals. Which of these two parameters varies the most and how these parameters are correlated are fundamental questions if we want to understand the large–scale dynamics of such populations. I explore how Pradel’s seniority (complement of the proportion of new individuals in the population) may help to answer such questions. I show that the sign of the correlation between temporal variation in seniority and of an independent measure of population growth rate should indicate whether population growth rate is more influenced by variation in survival or by variation in recruitment. Various predictions are proposed for evaluating the degree of regulation in the population (i.e., the existence of a negative correlation between survival and recruitment). Data from the French integrated breeding bird monitoring programme, combining point count surveys, from which population growth rate is estimated, and standardized capture–recapture, allowing the estimation of survival and seniority variation, were used to evaluate the method. Patterns of variation were examined for the four most frequently captured species, using data from 32 trapping sites covering 13 years (1989–2001). For Blackcap and Chiffchaff, the pattern is consistent with population growth rate being under the additive influence of survival and recruitment. For the Reed Warbler, the population appears to be strongly regulated, but with recruitment unable to compensate entirely for survival variation. For the Blackbird, the pattern is more confused and may indicate complex population dynamics, with non–linear relationships between survival, recruitment and population growth rate. Altogether, the method appears extremely promising and is particularly suitable for large scale monitoring of breeding birds by means of ringing. Key words: Breediing bird survey, Constant effort site, Monitoring, Common bird, Seniority, Regulation. Resumen Estimación de la contribución de la supervivencia y del reclutamiento en la dinámica de poblaciones a gran escala.— A grandes escalas espaciales, la variación en la abundancia poblacional se produce como consecuencia de la variación en la supervivencia de los adultos en edad reproductora y la variación en el reclutamiento de nuevos individuos. Para comprender la dinámica a gran escala de estas poblaciones es fundamental conocer cuál de estos dos parámetros varía en mayor medida y de qué modo se correlacionan entre sí. En el presente estudio se analiza cómo el modelo de senescencia de Pradel (complemento del porcentaje de individuos nuevos en la población) puede contribuir a responder estas preguntas. Se demuestra que el signo de la correlación entre la variación temporal en la senescencia y el de una medición independiente de la tasa de crecimiento poblacional debería indicar si la tasa de crecimiento poblacional está más influenciada por la variación en la supervivencia o por la variación en el reclutamiento. Se proponen varias predicciones para evaluar el grado de regulación de la población (es decir, la existencia de una correlación negativa entre la supervivencia y el reclutamiento). Para evaluar este método se utilizaron datos del programa francés de seguimiento integral de aves en edad reproductora, en combinación con estudios de recuento de puntos, lo que permite estimar la tasa de crecimiento poblacional, así como la captura–recaptura estandarizada, que permite estimar la variación en la supervivencia y la senescencia. Se examinaron pautas ISSN: 1578–665X

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Julliard

de variación para las cuatro especies capturadas con mayor frecuencia, empleando datos de 32 localidades de trampeo a lo largo de un período de 13 años (1989–2001). Para la curruca capirotada y el mosquitero común, la pauta concuerda con una tasa poblacional sujeta a la influencia aditiva de la supervivencia y el reclutamiento. Para el carricero común, la población parece estar fuertemente controlada, si bien el reclutamiento no puede compensar por completo la variación en la supervivencia. Para el mirlo común, la pauta es más confusa, pudiendo indicar una dinámica poblacional compleja, con relaciones no lineales entre la supervivencia, el reclutamiento y la tasa de crecimiento poblacional. En conjunto, el método parece extremadamente prometedor y resulta muy adecuado para el seguimiento a gran escala de aves en edad reproductora por medio del anillamiento. Palabras clave: Censo de aves reproductoras, Estación de esfuerzo constante, Seguimiento, Passeriformes comunes, Senescencia, Regulación. Romain Julliard, Centre de Recherches sur la Biologie des Populations d’Oiseaux, Muséum National d’Histoire Naturelle, 55 rue Buffon, F–75005 Paris, France. E–mail: julliard@mnhn.fr

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Introduction There is a growing interest in the large–scale monitoring of populations (Yoccoz et al., 2001). One reason for this is global changes which impact biodiversity at large spatial scales. Another reason is the finding that the population dynamics of many species appears to be synchronized over very large geographical areas, leading to a series of emerging scientific questions on the underlying mechanisms. At such scales, and ignoring the age structure of the population, the dynamics of the population may be summarized by the basic equation: Nt+1 =

Nt = (St + Rt)Nt

(1)

where N is the population size (number of reproducing individuals), the population growth rate, S is the survival rate of reproducing individuals and R is the per capita recruitment rate (number of new reproducing individuals produced per reproducing individual per year). Because the biological process influencing survival and recruitment are generally under the influence of different constraints, it is important to determine their relative influence on population dynamics. In other words, which of survival or recruitment variation determine most of the variation in population growth rate (Saether & Bakke, 2000)? For such simple population dynamics, the theoretical sensitivities of these two parameters are equal to 1 (Caswell, 1989), that is variation of 1 unit of survival or 1 unit of recruitment will have the same effect on the population growth rate which will also vary by 1 unit. Hence, what really maters is the amount of natural variation in these two parameters (Gaillard et al., 1998). A source of such information may come from the estimation of the seniority rate using Pradel’s (1996) approach. Seniority rate is a direct measure of the relative contribution of survival and recruitment to the population growth rate, i.e., an analog of elasticity of these two parameters calculated for each time interval (Nichols et al., 2000). This paper explores the use of time–dependent seniority estimates to evaluate whether population growth rate variation is more likely to be due to survival variation, recruitment variation or both. The paper consists of two parts: one is a theoretical analysis of the components of seniority rate and the other is an analysis of data from large–scale ringing–based monitoring undertaken in France (13 years of data from more than 30 sites). Part 1. Preliminary considerations: temporal co– variation of survival and seniority rates The seniority rate as defined by Pradel (1996) is the estimated proportion of experienced individuals in the population at a given time step. It is obtained by performing a standard survival analysis on reverse capture–recapture histories. 1 – is thus the proportion of new individuals in the population and has been often called recruitment (following Pradel, 1996). This is a bit confusing because 1 – is

generally not equal to recruitment as usually defined by population biologist (i.e., Rt in [1]). Using the parameters from (1), the expected seniority rate may be expressed as a function of survival and recruitment rate: E( t) = StNt / Nt+1= StNt / (St+Rt)Nt = St / St+Rt (2) From (2), it can be shown that E( t) is always between St and 1 – Rt. The quantity 1 – t is equal to the "true" recruitment only for St + Rt = 1, i.e., when the population is stable (but then, recruitment can be directly inferred from survival as Rt = 1 – St). Hence, seniority is a function of recruitment and survival. In addition to this structural link between seniority and survival expectancies, sampling variation causes a further positive correlation between survival and seniority estimates. Indeed, sampling variation, i.e., the difference between estimated survival in the studied population and the actual survival, affects survival and seniority estimates similarly. In other words, in the absence of actual variation in survival and recruitment, survival and seniority estimates will still be positively correlated. A logit transformation shows that the actual variation in survival and recruitment contributes equally to variation in seniority rate: logit (E( t)) = log (St) – log (Rt)

(3)

It further shows that temporal variation in seniority should be positively correlated with survival rate variation and negatively correlated with recruitment rate variation. Interestingly, population growth rate as defined in (1) is positively correlated with survival and positively correlated with recruitment rate: t

= St + Rt

(4)

Hence, if population growth rate is more influenced by survival variation than by recruitment variation, then population growth rate and seniority rate should be positively correlated, and conversely, if population growth rate is more influenced by recruitment variation than by survival variation, then population growth rate and seniority rate should be negatively correlated. The sign of the correlation between population growth rate and seniority will thus indicate which of survival or recruitment most influence the population dynamics. When both survival and recruitment rates vary, the prediction depends on the sign of their co– variation: (1) survival and recruitment variation may be negatively correlated. This may occur when the population is strongly regulated, i.e., when high adult survival limits recruitment of new individuals or, when low adult survival is compensated by high recruitment. If the correlation is very high, then point estimates of seniority rate and survival rate for a given time step should be very similar, that is, . Furthermore, population growth rate variation should be much reduced.

420

(2) Survival and recruitment variation may be positively correlated. This may occur when the same environmental factor affects adult survival and recruitment of new individuals (for example, winter weather may strongly determine both adult and juvenile survival variation). This will result in low variation in the seniority rate (because variation in the number of experienced individual will be hidden by the same variation in the number of new individuals). Other tests may help to discriminate between these different situations. Three appear particularly useful: (i) the correlation between population growth rate and survival (predicted to be positive if survival variation affects population growth rate variation); (ii) comparison of the importance of temporal variation between survival and seniority (see prediction in table 1); (iii) the partial correlation between seniority rate and estimates of both survival rate and population growth rate (table 1). Part 2. Case study: estimating temporal variation in survival and recruitment from nation–wide capture–recapture monitoring In many countries, volunteer ornithologists monitor the abundance of common breeding birds by means of extensive surveys. Such monitoring is based on counts of individuals undertaken using a standardized method, at fixed points, transects or sites. The repetition of these counts year after year by the same observer at a given place makes it possible to estimate variation in abundance. These schemes, known as the Breeding Bird Survey in North America or the Common Birds Census in the U.K. were started as early as the 1960s. More recently, besides these traditional censuses using direct counts, amateur ringers have been involved in breeding bird population monitoring by a growing number of ringing schemes. Such programmes, known as the CES (Constant Effort Site, in Britain and Ireland, started in 1983, and subsequently in a growing number of countries in continental Europe; Peach et al., 1996) or MAPS (Monitoring Avian Populations for Productivity and Survival, in North America, started 1994; DeSante et al., 1999) are based on standardized mist–netting of breeding populations of small birds at a number of sites, repeated year after year. Date and number of capture occasions, as well as the numbers and locations of mist–nets are kept as constant as possible from one year to the next for a given site. The French breeding bird monitoring programme In France, both a BBS–like and a CES–like programme were initiated in 1989 (Julliard & Jiguet, 2002). For the BBS type of survey, individuals were counted on permanent plots during a fixed period of 5 minutes, counting both visible individuals and singers. To be valid, the counts must be repeated at approximately the same date (± 7 days within April

Julliard

to mid June), at the same time of day (± 15 minutes within 1 to 4 hours after sunrise), by the same observer. Between 1989 and 2001, 3,000 points were surveyed over an average six years per site. Survey plots usually comprised groups of 10–15 points, with at least 200 m between each point. Such groups of points were not spread homogenously over France. However, clusters of groups were found in the North, Normandy, Auvergne and Languedoc, i.e., regularly scattered on a north–south gradient. This data set makes it possible to estimate changes in the abundance of the various species considered, using a standard log–linear analysis with a site effect included (e.g., Julliard et al., 2004). The basis of the French CES protocol is the same as that used by other CES schemes: mist– nets are erected at the same place, year after year and capture sessions are conducted at nearly the same dates each year. However, compared to other CES schemes, more nets are erected per site (12– 50 12m–long nets), and there are usually only three trapping sessions per year between mid–May and mid–July, instead of 9–12 visits between May and August. Nets are concentrated on a small area to ensure a high density of nets (3–5 nets per ha) in order to maximize capture rates. As in other CES schemes, all individuals are ringed. Nets are visited every 30–60 minutes, and birds are immediately released near the net after data have been taken. All recaptures, even those from the same day, are recorded. For all species, plumage features provide good criteria for age determination, allowing accurate distinction between adults and young of the year (Svensson, 1992). The goal of this analysis was to compare temporal variation in survival and seniority, and to analyze how temporal variations in these parameters were correlated with variations in population growth rates estimated from the point count survey. Note that no point counts were run on CES sites. Data selection and parameter estimation Throughout the analysis, juveniles are ignored. The analysis was repeated for the four most abundant species: Blackcap Sylvia atricapilla (6,272 individuals), Blackbird Turdus merula (3,446 individuals), Reed Warbler Acrocephalus scirpaceus (3,205 individuals), and Chiffchaff Phylloscopus colybitta (2,722 individuals). All sites with at least three consecutive years of captures between 1989 and 2001 were selected. On average, each site was operated for eight years giving a total of 258 site–years. For three sites, the data set was split into two parts due to major changes in the number of nets. For any given site, only species with at least five individuals captured and at least one individual recaptured were selected. For the most widespread species, there were 31 data sets each of which included up to 13 years of data. The full time dependent model, including all possible combinations of years and sites, thus re-

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Table 1. Prediction of the effects of the actual temporal co–variation of survival (S) and recruitment (R) on the outcome of modeling temporal variation in estimated survival ( ) and seniority rates ( ). = St + Rt, is the population growth rate: (v). Variable; (c). Constant; + Positive correlation; – t Negative correlation; (1) Due to sampling correlation; (2) Adjusted effect more significant than main effect; NS. Test should be statistically not significant. Tabla 1. Predicción de los efectos de la covariación temporal real de la supervivencia (S) y el reclutamiento (R) en los resultados de la modelación de la variación temporal en la supervivencia estimada (v) y los índices de senescencia (c). kt = St + Rt, es la tasa de crecimiento poblacional: (v). Variable; (c). Constante; + Correlación positiva; – Correlación negativa; (1) Debido a la correlación de la muestra; (2) Efecto ajustado más significativo que el efecto principal; NS. El test debería ser estadísticamente no significativo. St(v) Rt(v) St(v), Rt(c)

St(c) Rt(v)

LRT time dependence

(NS) <

unpredictable

Sign of the correlation between t and t

–

unpredictable

not varying much

+ (1) (2) – t

+ (2) (2) – t

+ t NS or –

+ (1) t NS or –

Partial correlation between t and t and

+ NS

t t

quires 744 parameters to be defined. Such large numbers of parameters pose several problems. A minor one is that MARK (White & Burnham, 1999) is unusable if the analysis has to be done in a reasonable amount of time. I thus relied on SURGE (Reboulet et al., 1999). This may solve the issue of computing time but there are still problems of convergence. Local likelihood maxima were detected in many of the model runs. Being interested in models with common temporal variation in survival across sites, and not being willing to make unrealistic assumptions of constancy of capture probability over time or sites, I first planned to use the model [ site+year; psite*year], i.e., additive effect of year and site for survival and full time and site variation for capture probability, as a baseline model (notation as in Lebreton et al., 1992). A way to decrease the rate of presumably bad convergence was to constrain all capture parameters corresponding to year–site combinations that were not monitored to be equal. This represents about 1/3 of the capture parameters. This has the other advantage of checking that no mistakes occurred when the data set was being built: the estimated values for all of these parameters should be 0. Yet, there were still several cases of bad convergence. One of the symptoms was that the last survival (or seniority) estimate was equal to 1. Another sign of bad convergence in the case of models including external variables was that the ratio (estimated slope/standard error) differed substantially from the LRT for constancy of the parameter. Unfortunately, changing initial values was usually inefficient when the model included a large number of parameters. In order to reduce the number of parameters to be estimated, I used the mean number of within– season recaptures as a proxy for the efficiency of

uncorrelated

– correlated

+ correlated

capture rate for a given species in a given site. Many individuals are captured several times within a CES season. The average number of captures per captured individual calculated for each year, each species and each site was used as an external variable for modelling capture probability. For each data set, six models of capture rate variation were used [psite*year], [pc*site], [pc+site], [psite], [pc], [p·] where c represents the external variable as defined above. The AIC of model [pc] was generally far smaller than the AIC values of models with other parameterizations of capture rate, whatever the model for survival or seniority. LRT tests for the significance of the external variable [pc] <> [p·] varied between 15 and 30 (1 df) indicating that the variable was able to retrieve much of the variation in capture probability, both between and within sites. This result appears to be extremely useful because capture probability could be modelled with only two parameters without making assumptions of constancy over time or sites. Results For each species considered, I ran the various tests proposed in table 1. For all species, statistically significant temporal variation in survival and seniority were detected (table 2, fig. 1). Survival was clearly more variable than seniority in two cases (Blackbird & Chiffchaff, table 2). In the other two cases, the amount of temporal variation in survival and seniority were similar. For three cases out of four, the different tests fitted more or less with one of the scenarios described in table 1.

Julliard

422

Table 2. Modeling temporal variation in seniority ( ) and survival ( ). Changes in deviance between nested models (i.e., LRT) are presented (changes in AIC could be obtained as changes in deviance – 2df). For every species and model, capture probability was modeled with a site–, year– and species–specific capture efficiency index (see methods). (The sign is the sign of the slope of the external variable of interest.) Tabla 2. Modelación de la variación temporal en la senescencia ( ) y la supervivencia ( ). Se indican los cambios de desviación entre los modelos anidados (es decir, LRT) (los cambios de AIC podrían obtenerse como cambios de desviación – 2df). Para cada especie y modelo, la probabilidad de captura se modeló con un índice de eficiencia de captura dependiente del lugar, el año y la especie (ver métodos).

Test

Chiffchaff

LRT [ t] vs [ ·]

45.48

31.36

35.04

35.50

LRT [ t] vs [ ·]

19.78

36.36

31.97

19.99

LRT [ ] vs [ ·]

15.63 (+)

6.56 (+)

6.67 (+)

LRT [ ] vs [ ·]

2.25 (–)

0.99 (–)

14.32 (+)

0.01

LRT [ ] vs [ ·]

5.51 (+)

8.31 (+)

22.00 (+)

0.01

LRT [ LRT [

Blackcap

Reed Warbler

Blackbird

4.25 (+)

] vs [ ]

10.74 (–)

5.94 (–)

3.90 (+)

0.02

] vs [ ]

14.00 (+)

13.26 (+)

11.57 (+)

0.01

Chiffchaff and Blackcap For the Chiffchaff, survival variation was strongly positively correlated with to population growth rate variation (15.53 units of deviance from the 45.48 units for temporal variation; table 2). Seniority was slightly negatively correlated with population growth rate variation (2.25 units of deviance from the 19.78 units for temporal variation; table 2). However, after adding survival estimates as a second external variable, the test for the effect of population growth rate jumped to 10.74 units of deviance (table 2). The two variables explained almost all of the temporal variation in seniority (16.25 units of deviance explained with 2 df and 3.23 units of deviance unexplained with 9 df left; table 2). For the Blackcap, a similar tendency was detected, but the proportion of temporal variation explained was lower (table 2). Altogether, for these two species, the pattern of variation fitted well with population growth rate being determined concomitantly by variation in both survival and recruitment. Reed Warbler For the Reed Warbler, time–dependent seniority estimates were very similar to the time–dependent survival estimates (fig. 1). Both seniority and survival were positively correlated with population growth rate. Even after introducing the survival estimates as an external variable for modeling seniority variation, the estimated slope for population growth rate remained positive (table 2).

This pattern is close to the one predicted for a strongly regulated population with recruitment variation compensating for survival variation. In addition, the fact that seniority was rather strongly positively correlated with population growth rate suggested that population growth rate was more influenced by survival rate than by recruitment rate. Altogether, the observed pattern of variation suggests that variation in recruitment imperfectly compensated for variation in survival. Blackbird For the Blackbird, few correlations were found: survival was to a small extent correlated with population growth rate (4.25 units of deviance from the 35.50 units for temporal variation). Seniority was neither correlated with population growth rate nor with survival (0.03 units of deviance explained from the 19.99 units for temporal variation). Yet, inspection of figure 1 suggests that seniority and survival co–varied in some years but not in others. A plot of seniority estimates against population growth rate revealed a possible non– linear relationship (fig. 2). Indeed, the square of population growth rate ( ²) was significantly related to seniority (7.40 units of deviance). The main effect of population growth rate was still not significant indicating a symmetric relationship around zero. The slope for ² was negative. Hence, seniority tended to be positively correlated with when the population was decreasing and negatively related to when the population was increasing. This could indicate that variation in Blackbird sur-

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Demographic rate

0.7

Chiffchaff

0.6

0.5

0.4

0.3

0.2

0.1

0 1988 1990 1992 1994 1996 1998 2000 2002 0.7 Demographic rate

0.7

Reed Warbler

0 1988 1990 1992 1994 1996 1998 2000 2002 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1988 1990 1992 1994 1996 1998 2000 2002

Blackcap

Blackbird

0 1988 1990 1992 1994 1996 1998 2000 2002

Fig. 1. Temporal variation in survival (black diamonds) and seniority (open circles) for four common passerines: vertical bars ± 1 SE. (Data from the French Constant Effort Site scheme.) Fig. 1. Variación temporal en la supervivencia (rombos negros) y la senescencia (círculos blancos) para cuatro Paseriformes comunes: barras verticales ± 1 EE. (Datos extraídos del programa francés Constant Effort Site.)

vival is mainly responsible for population declines while population recovery is mostly under the control of recruitment. General discussion Data quality As almost everywhere else, the French point count survey was not based on an explicit sampling framework. Instead, observers were asked to choose their survey sites. Hence, there is no guarantee that population variation measured by such a scheme is representative of the actual population variation. Yet, previous analyses have shown fairly high similarities between population variation from this scheme and from comparable surveys in neighboring countries (Julliard et al., 2004). In addition to the lack of a sampling framework for the survey sites, using mist–nets added constraints to the choice of sites. The scrub layer has to be well developed to ensure efficient trapping. Hence, there is a further bias toward par-

ticular habitats amongst the sites used for mist– netting. Yet, population size variation (measured as the number of adults caught) estimated from capture data is very similar to population variation measured from point count surveys for many species (R. Julliard, unpublished analysis). This was the case in particular for Blackbird, Blackcap and Chiffchaff, but not for the Reed Warbler. Altogether, this means that finding relationships between population growth rates estimated with one survey and demographic parameters estimated with another, is in itself an important results. Indeed, few studies have demonstrated the usefulness of CES/MAPS ringing monitoring. This is partly due to the young age of such schemes (the French one is among the oldest), and partly due to the difficulty of the analysis (dealing with data from so many sites is rather time consuming). Robustness of the approach For Blackcap and Chiffchaff, the approach suggests that both survival and recruitment variation

Julliard

424

0.6

Seniority

0.5

–0.15

0.4

–0.10

0.3 –0.05 0 0.05 0.10 Population growth rate

0.15

Fig. 2. Relationship between seniority and population growth rate for the Blackbird showing a possible nonlinear relationship. Seniority estimated from the French Constant Effort Sites data; population growth rate estimated from the French Breeding Bird Survey. Fig. 2. Relación entre la senescencia y la tasa de crecimiento poblacional del mirlo común, indicando una posible relación no lineal. La senescencia se ha estimado a partir de los datos del programa francés Constant Effort Site; la tasa de crecimiento poblacional se ha estimado a partir de los datos del censo francés de aves reproductoras.

shape the population growth rate. Although such a claim may appear trivial, it is probably the first time it is supported by such convincing results at this spatial scale. The main limitation of previous studies of the contributions of different demographic parameters to population growth rates is that the population growth rate estimate and demographic parameter estimates came from the same data sets (e.g., Saether & Bakke, 2000; Saether et al., 2002). For the Reed Warbler, the results suggest that survival rather than recruitment is the main driver of population growth rate. In addition, there was evidence for some compensation between recruitment and survival. This suggests that Reed Warbler populations are strongly regulated (the level of recruitment is dependent of the level of survival), but that variation in survival is too large to be compensated by recruitment, leading to further population variation. It should be noted that the method described here makes it possible to clearly separate negatively correlated survival and recruitment (regulated populations) from positively correlated survival and recruitment (due for example to similar climatic effects). The adult survival of Reed Warblers may partly depend on rainfall in their African winter quarters, as has been found for the con–generic Sedge Warbler A. schoenobaenus (Peach et al., 1991). It is plausible that juvenile survival is also under the influence of the same climatic variable, in which case survival and recruitment should be positively correlated. Although this may be true, the results clearly indicate that regulation totally obscures any

such relationship, reversing the sign of the expected relationship between recruitment and survival. The results for the Reed Warbler highlight another aspect of the patterns for Chiffchaff and Blackcap. The fact that the effect of recruitment on population growth rate was detectable (negative relationship between seniority and population growth rate) suggests weak, if any, regulation by recruitment in these species. As noted in the preliminary analysis, perfect compensation between recruitment and survival should lead to equality between point estimates of seniority and survival for a given time step. As a consequence, the slope relating seniority and survival estimates should be equal to 1. Without such compensation, the relationship should be blurred by independent variation in recruitment, and the slope should be below 1. Hence, it might be possible to estimate the compensation between survival and recruitment by estimating the slope between seniority and survival. Further studies along these lines may be useful, although one has to keep in mind that sampling variation is likely to weaken the approach. Another interesting aspect would be to evaluate the relative contributions of survival and recruitment variation to population growth rates. The relevant information is reflected in the relative importance of temporal variation of survival and seniority and in the strength of the relationship between seniority and population growth rate. Again further studies are needed, and again, the sampling correlation between seniority and survival may reduce the chances of obtaining a clear answer. In this study we have already found contradictory results for the Chiffchaff

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where survival was found to be much more variable than seniority (suggesting more variation in survival than in recruitment) but the slope relating seniority to population growth rate was negative (suggesting a stronger influence of recruitment than survival). For the Blackbird, the method failed to identify a simple scenario. Although the possibility that the data set for this species was too biased cannot be ruled out, there is no reason why it should be more biased than those for the other species. In particular, Blackbirds have relatively high survival and capture rates. I suggested that the absence of simple patterns of variation in the parameters considered is due to non–linear effects of survival and recruitment on population growth rate, with survival variation being responsible for population decrease and recruitment variation being responsible for population increase. Although this interpretation is appealing at first glance, the biological mechanism that might bring about such a scenario is unclear. Furthermore, it is not entirely clear whether such a scenario would indeed result in the observed pattern. Again further work is needed. Yet, the fact that the method may allow the detection of non trivial patterns of population dynamics is particularly exciting. Suitability of data for such analyses It must be stressed that the method presented here depends on the availability of independent estimates of population growth rates. It is obvious that population growth rate estimated from the same CMR data using Pradel’s (1996) approach is unsuitable. Similarly, population growth rates estimated by other means, that are not independent of the number of captured individuals (e.g., number of nests in a study population for which the capture rate is high) are also likely to lead to spurious results, because of the strong sampling correlation between all of the estimated parameters. The other limit comes from the use of seniority. The general case where it is used here is noticeably different from the particular case where it has mostly been used previously, that is, to study age–specific accession to reproduction (Pradel & Lebreton, 1999). Indeed, in the case studied here, it is essential that the same technique is used to catch unmarked individuals and to recapture individuals: the capture rate, estimated from the sequence of recaptures, must apply to unmarked individuals. In particular, all data based on resightings do not satisfy this requirement, since the resighting probability cannot be assumed to be equal to the capture probability of unmarked individuals. Implication for CES/MAPS monitoring As stated above, this study considerably strengthens the case that CES/MAPS monitoring is capable of measuring temporal variation in recruitment and survival. In addition, it demonstrates the value of catching as many adults as possible instead of maximizing both adult and juvenile captures. This

has implications for the optimal number of trapping occasions. The 9–12 trapping occasions recommended by most schemes aims to cover the reproductive season as completely as possible. However, because the adult population is constant, the number of new adults caught is continuously decreasing from one trapping occasion to the next throughout the season. If there is a trade off at the scale of the country between the number of sites monitored and the number of trapping occasion within a site, then in order to maximize the number of adults caught, the optimal number of trapping occasions may be much lower than 10. In France, where the requested number of trapping sessions is 3 (ringers are allowed to do more), the number of sites reached 90 in 2003. In the U.K., there are about 120 CES, but about 10 times more ringers than in France. Acknowledgments Special thanks to Will Peach and an anonymous referee who managed to give very constructive comments on a rather rough version of the paper. The success of such long term, large scale surveys relies entirely on the continuous participation of voluntary observers to whom this paper is dedicated. The national coordination is financially supported by the Muséum National d’Histoire Naturelle, the CNRS and the French Ministry in charge of environment. References Caswell, H., 1989. Matrix population models. Sinauer Associates, Sunderland, Massachusetts, U.S.A. DeSante, D. F., O’Grady, D. R. & Pyle, P., 1999. Measures of productivity and survival derived from standardized mist–netting are consistent with observed population changes. Bird Study, 46: S178–188. Gaillard, J.–M., Festa–Bianchet, M. & Yoccoz, H. G., 1998. Population dynamics of large herbivores: variable recruitment with constant adult survival. Trends in Ecology and Evolution, 13: 58–63. Julliard, R. & Jiguet, J., 2002. Un suivis intégré des populations d’oiseaux communs en France. Alauda, 70: 137–147. Julliard, R., Jiguet, F. & Couvet, D., 2004. Common birds facing global changes: what makes a species at risk? Global Change Biology, 10: 148–154. 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. Nichols, J. D., Hines, J. E., Lebreton, J.–D. & Pradel, R., 2000. Estimation of contributions to population growth: A reverse–time capture–re-

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capture approach. Ecology, 81: 3362–3376. Peach, W., Baillie, S. & Underhill, L., 1991. Survival of British sedge warblers Acrocephalus schoenobaenus in relation to west African rainfall. Ibis, 133: 300–305. Peach, W. J., Buckland, S. T. & Baillie, S. R., 1996 The use of constant effort mist–netting to measure between–year changes in the abundance and productivity of common passerines. Bird Study, 43: 142–156. Pradel, R., 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Pradel, R. & Lebreton, J.–D., 1999. Comparison of different approaches to the study of local recruitment of breeders. Bird Study, 46 (suppl.): S74–81. Reboulet, A. M., Viallefont, A., Pradel, R. & Lebreton, J. D., 1999. Selection of survival and recruitment

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models with SURGE 5.0. Bird Study, 46 (suppl.): 148–156. Saether, B.–E. & Bakke, O., 2000. Avian life history variation and contribution of demographic traits to the population growth rate. Ecology, 81: 642–653. Saether, B.–E., Engen, S. & Matthysen, E., 2002. Demographic characteristics and population dynamical patterns of solitary birds. Science, 295: 2070–2073. Svensson, L., 1992. Identification Guide to European Passerines, 4th edition, Stockholm. White G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 (suppl.): 120–139. Yoccoz, N. G., Nichols, J. D. & Boulinier, T., 2001. Monitoring of biological diversity in space and time. Trends in Ecology and Evolution, 16: 446–453.

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

Yoccoz, N. G. & Ims, R. A., 2004. Spatial population dynamics of small mammals: some methodological and practical issues. Animal Biodiversity and Conservation, 27.1: 427–435. Abstract Spatial population dynamics of small mammals: some methodological and practical issues.— Small mammals have been widely used to further our understanding of spatial and temporal population dynamical patterns, because their dynamics exhibit large variations, both in time (multi–annual cycles vs. seasonal variation only) and space (regional synchrony, travelling waves). Small mammals have therefore been the focus of a large number of empirical and statistical (analysis of time–series) studies, mostly based on trapping indices. These studies did not take into account sampling variability associated with the use of counts or estimates of population size. In this paper, we use our field study focusing on population dynamics and demography of small mammals in North Norway at three spatial scales (0.1, 10 and 100 km) to illustrate some methodological and practical issues. We first investigate the empirical patterns of spatial population dynamics, focusing on correlation among time–series of population abundance at increasing spatial scales. We then assess using simulated data the bias of estimates of spatial correlation induced by using either population indices such as the number of individuals captured (i.e., raw counts) or estimates of population size derived from statistical modeling of capture–recapture data. The problems encountered are similar to those described when assessing density–dependence in time-series —a special case of the consequence of measurement error for estimates of regression coefficients— but are to our knowledge ignored in the ecological literature. We suggest some empirical solutions as well as more rigorous approaches. Key words: Spatial autocorrelation, Measurement error, Voles, Norway. Resumen Dinámica poblacional espacial de pequeños mamíferos: algunas cuestiones metodológicas y prácticas.— Los pequeños mamíferos se han utilizado ampliamente para ayudarnos a comprender mejor las pautas dinámicas espaciales y temporales de las poblaciones. Ello obedece a que sus dinámicas presentan importantes variaciones, tanto por lo que respecta al tiempo (ciclos multianuales frente a una única variación estacional) como al espacio (sincronía regional, ondas progresivas). Por consiguiente, los pequeños mamíferos han sido objeto de gran número de estudios empíricos y estadísticos (análisis de series temporales), basados principalmente en índices de capturas por trampa. Dichos estudios no tomaban en consideración la variabilidad del muestreo asociada al empleo de recuentos o de estimaciones del tamaño poblacional. En el presente artículo utilizamos nuestro estudio de campo para analizar la dinámica poblacional y la demografía de los pequeños mamíferos del norte de Noruega en tres escalas espaciales (0,1, 10 y 100 km), además de ilustrar algunas cuestiones prácticas y metodológicas. En primer lugar, investigamos las pautas empíricas de la dinámica poblacional espacial, centrándonos en la correlación existente entre las series temporales de la abundancia poblacional en escalas espaciales cada vez mayores. Seguidamente, utilizamos datos simulados para evaluar el sesgo de las estimaciones de la correlación espacial inducida mediante el empleo, bien de índices poblacionales como el número de individuos capturados (es decir, recuentos brutos) o estimaciones del tamaño poblacional derivadas de la modelación estadística de los datos de captura–recaptura. Los problemas encontrados son similares a los descritos cuando se evalúa la dependencia de la densidad en las series temporales —un caso especial de la consecuencia del error de medición con respecto a las estimaciones de coeficientes de regresión—, pese a que, según parece, se han ignorado en la literatura ecológica. Por último, sugerimos algunas soluciones empíricas, así como planteamientos más rigurosos. Palabras clave: Autocorrelación espacial, Error de medición, Ratón de campo, Noruega. Nigel G. Yoccoz(1) & Rolf A. Ims(2), Inst. of Biology, Univ. of Tromsø, 9037 Tromsø, Norway.– Norwegian Inst. for Nature Research, Polar Environmental Centre, N–9296 Tromsø, Norway. (1)

E–mail: nigel.yoccoz@ib.uit.no

ISSN: 1578–665X

(2)

E–mail: rolf.ims@ib.uit.no

428

Introduction Small mammals, and in particular voles and lemmings, are well known for their large multi–annual population fluctuations (Stenseth, 1999). Most interestingly, populations of the same species (e.g., the Field vole, Microtus agrestis) can exhibit different temporal and spatial dynamical patterns (Henttonen & Hansson, 1986). In Fennoscandia, southern populations show mainly seasonal changes, while northern populations are characterized by 3 to 5 year cycles. Spatial synchrony, meaning that populations in different locations exhibit correlated fluctuations, is also a variable phenomenon, with large–scale (over hundreds of km) synchrony as well as traveling waves having been described (Lambin et al., 1998; Steen et al., 1990). Populations of different species at the same location are often synchronous too (Henttonen & Hansson, 1986). However, most studies of synchrony have used population indices, and did not account for sampling error (e.g. Bjørnstad et al., 1999, Steen et al., 1996). Accurate estimation of the quantitative patterns of population synchrony, and in particular of the relationship between correlation and distance among populations, is of interest because it can shed light on the mechanisms driving the population dynamics (Bjørnstad et al., 2002; Bolker & Grenfell, 1996, Ranta et al., 1997; Ripa, 2000). The Moran effect (e.g., Bjørnstad et al., 1999) describes a situation where the correlation for the environmental noise is expected to be identical to the correlation observed for the population dynamics. Climate, predation and dispersal are three factors that are important for the population dynamics of small mammals (Hansson & Henttonen, 1985, 1988), but their consequences in terms of population synchrony are likely to be scale–dependent. Climate will generally result in large scale synchrony (e.g., over distances larger than 100 km Krebs et al., 2002), predation by small mustelids (weasel Mustela nivalis, stoat M. erminea) will be characterized by synchrony at intermediate (10 km) scale, whereas vole dispersal may influence small (1–5 km) scale synchrony (Lambin et al., 1998). Depending on the relative importance and extent of these factors, we would expect spatial correlation patterns with a characteristic scale (Engen et al., 2002). The consequences of ignoring the uncertainty in population indices or estimates have been emphasized for at least three decades when analyzing density–dependence (Ito, 1972; Lebreton, 1990; Solow, 2001). It is related to the more general issue of the effect of measurement error on regression coefficients estimates (Fuller, 1987): when the predictor variable is measured with some error (and assuming a simple measurement error model, i.e. additivity and independence), the regression coefficient is biased towards 0 (so–called attenuation). The issue has not been explicitly addressed when analyzing spatial correlation (but see e.g. Moilanen [2002] when one is interested in estimat-

Yoccoz & Ims

ing parameters of metapopulation models), even if it has been mentioned by Lande et al. (2003), and some statistical analytical models have incorporated measurement error (Viljugrein, 2000). The correlation between two time series of population indices or estimates and is given by: . If the measurement error process is additive and independent for the two populations, the covariance will not be affected, but the two standard deviations will be inflated by the measurement error (Link & Nichols, 1994), and therefore the correlation will be attenuated (biased towards 0). Viljugrein and her colleagues modeled the measurement error assuming a Poisson distribution. This is likely to underestimate the impact of measurement error as for example heterogeneity of capture, a common problem in most field studies of small mammals (Yoccoz et al., 1993), will lead to variability larger than Poisson. Assessing the size of the bias will therefore depend on having an accurate knowledge of the measurement error process. Dennis et al. (1998) considered a model for analyzing joint density–dependence and suggested using state–space models for taking into account sampling variance but did not implement such models to our knowledge. In this paper, we investigate how sampling variance due to the population size estimation process affects estimates of spatial autocorrelation. We first describe the empirical patterns of spatial autocorrelation in a multi–scale study of population dynamics of voles in north Norway. The sampling variance associated to raw counts or closed capture–recapture models incorporating heterogeneity of capture is not analytically known, and we therefore used simulated data including both process and sampling variance to investigate the effects on bias of correlation estimates. Finally, we suggest approaches to correct for the observed bias. Field study design and empirical estimation of spatial correlation patterns The empirical study is based on trapping 122 0.5 ha grids, twice a year (June and September) during 5 years (1998–2002). The sampling design was hierarchical, with three spatial scales: 1 km, 10 km and 100 km (fig. 1). Each grid covered an area of 0.5 ha, with 16 Ugglan live traps. Each primary trapping session included three secondary sessions (days). We had more than 12,000 captures of > 7,300 individuals belonging to four vole species, the Grey– sided Vole (Clethrionomys rufocanus; 3,300 ind., 5,600 capt.), the Red Vole (C. rutilus; 2,900 ind., 4,900 capt.), the Bank Vole (C. glareolus; 440 ind., 750 capt.) and the Field Vole (640 ind., 950 capt.). In this paper, we only consider the Grey–sided Vole. In addition, more than 5,000 shrews (mostly Common

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Porsanger

Porsanger Alta Nordreisa T5 MĂĽlselv Skjomen

T4 T3

40 km

T2 Birch forest Pine forest Taiga

Fig. 1. Map of study area (north Norway) with detailed study design for one fjord region (Porsanger). Each transect (T1 to T5) had five trapping grids from the sea level/valley bottom to the treeline. In two areas, a slightly different design was used: Skjomen (three transects with eight trapping grids each) and MĂĽlselv (four transects with six, six, six and five grids respectively). Fig. 1. Mapa del ĂĄrea de estudio (norte de Noruega), que incluye el diseĂąo de un estudio detallado para una regiĂłn de fiordos (Porsanger). Cada transecto (T1 a T5) presentaba cinco rejillas de captura, desde el nivel del mar/fondo del valle hasta la lĂnea de ĂĄrboles. En dos ĂĄreas se utilizĂł un diseĂąo ligeramente distinto: Skjomen (tres transectos con ocho rejillas de captura cada una) y MĂĽlselv (cuatro transectos con seis, seis, seis y cinco rejillas, respectivamente).

Shrews, Sorex araneus) were caught but most were killed by the trapping. Numbers of individuals captured per grid varied between 0 and 40. We estimated population size using models for closed populations (Pollock et al., 1990; Borchers et al., 2002). We considered two models: M0 (identical capture rate for all individuals) and Mh2 (2nd order jackknife estimator [Burnham & Overton, 1979], a nonâ&#x20AC;&#x201C;parametric estimator incorporating heterogeneity of capture). These two models were used to estimate population size independently for each grid and species. We investigated further the sources of variability in capture rates, both within and among grids using the Hugginsâ&#x20AC;&#x2122; conditional likelihood approach (Huggins, 1989; Alho, 1990; Borchers et al., 2002). We used grid, sex and body weight as covariates, and applying Horvitzâ&#x20AC;&#x201C; Thompsonâ&#x20AC;&#x201C;like estimator for each grid: ,

Simulations and estimation of spatial correlations Data were simulated according to two processes: a population dynamic process mimicking multiannual population cycles, and a sampling process describing the capture of individuals. Population dynamics differed among regions and among species. For the Greyâ&#x20AC;&#x201C;sided Vole, they were consistent with 5â&#x20AC;&#x201C;years cycles in the NE and somewhat shorter cycles in the SW (Hansson & Henttonen, 1985). For the Red Vole, some populations had only seasonal fluctuations with no indication of cycles. However, the timeâ&#x20AC;&#x201C; series are too short to estimate adequately the periodicity. We simulated the true population dynamics using a secondâ&#x20AC;&#x201C;order autoregressive model, with two caseâ&#x20AC;&#x201C;studies (cyclic populations and stable populations; see Stenseth, 1999 for general discussion of parameter values in autoregressive models):

is the estimated probability that the where are individual i is caught at least once. The given by: ,

AR2: log(Nt+1) = 4.5 + 0.2 * log(Nt) â&#x20AC;&#x201C; 0.5 * log(Ntâ&#x20AC;&#x201C;1) + t AR1: log(Nt+1) = 3.5 + 0.2 * log(Nt) + t,

where the probabilities pki of catching individual i in session k are modeled as function of the covariates using a logit link. We used the value of the conditional likelihood to calculate the AIC values and select an appropriate model.

where t, the stochastic component of population growth, is assumed to follow a normal distribution with mean 0 and variance " 2. The first caseâ&#x20AC;&#x201C;study, AR2, mimics a 5â&#x20AC;&#x201C;years cycle, whereas the second, AR1, mimics stable

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Fig. 2. Population dynamics of the Grey–sided Vole in Porsanger (A) and Alta (B), 1998–2002. The spring (left) and fall (right) population sizes are given for the five transects and the five trapping grids in each transect. Population size is estimated using the second–order jackknife estimator, M h2. The populations in Porsanger and Alta show multi–annual (5–years cycle) fluctuations, in addition to the seasonal fluctuations associated with the reproductive period (June to September). Even if the five transects are located in the same habitat type (birch forest), there is large variation in population abundance, within and among transects.

populations. The variance " 2 was set equal to 0.04 in order to get fluctuations with amplitude similar to the one observed. The capture–recapture data were simulated using WiSP (Borchers et al., 2002), assuming three secondary capture sessions, random variation in capture probabilities among the two populations and in time, and individual heterogeneity. The parameters describing capture prob-

abilities and heterogeneity were chosen in order to give capture probabilities similar to those observed in the actual vole data: secondary session capture probabilities were assumed to vary between the average values 0.18 and 0.5 (i.e., the probability of being caught at least once is then between 0.45 and 0.875: see fig. 3), with a random variation among sessions equal to 0.1 on the logit scale.

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Fig. 2. DinĂĄmica poblacional del ratĂłn de campo de flancos grisĂĄceos en Porsanger y Alta, 1998â&#x20AC;&#x201C;2002. Se indican los tamaĂąos poblacionales durante la primavera (izquierda) y el otoĂąo (derecha) para los cinco transectos y las cinco rejillas de cada transecto. El tamaĂąo poblacional se calcula utilizando el estimador jackknife de segundo orden, Mh2. Las poblaciones en Porsanger y Alta presentan fluctuaciones multianuales (ciclo de 5 aĂąos) y estacionales asociadas con el perĂodo reproductor (de junio a septiembre). Aunque los cinco transectos estĂŠn situados en el mismo tipo de hĂĄbitat, bosque de abedules, existe una amplia variaciĂłn en la abundancia poblacional, tanto en cada uno de los transectos como entre ellos.

Two synchronous populations were simulated by assuming that the stochastic components for the two population growth rates were correlated, that is the t were assumed to follow a binormal distribution with identical variance for the two populations and a correlation rĂ°. We estimated the correlations using the logarithm of number of animals caught (raw counts) as well as population size estimated using model M0 and Mh2.

Results Empirical correlation patterns The dynamics of populations differed among species and regions. In Porsanger (NE), the Greyâ&#x20AC;&#x201C; sided Vole experienced a synchronous decline in 1998â&#x20AC;&#x201C;99, followed by a synchronous increase and peak in 2002 (fig. 2). In Alta (ca. 150 km from

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Simulations Correlations based on estimated population size or number of animals caught were biased low, and the bias was relatively large (up to 30% for model Mo, tables 3, 4). The bias was due to the increased variance of population size, as the covariances were not biased. The worst case resulted from using the model Mo, whereas Mh2 and raw counts did give similar biases. The simple estimator (Mo) was both biased (because of the presence of heterogeneity of capture) and relatively imprecise, resulting in an overall larger measurement error than the jackknife estimator which was not biased (results not shown). The raw counts were of course biased, but did not include any estimation uncertainty. Using proper estimates for the process variance (Link & Nichols, 1994; Link et al., 1994; Burnham & White, 2002) would correct for most of

Probability of being captured only once

Porsanger), the Grey–sided Vole had very low population levels in 1998–1999, experienced a synchronous increase in 2000–01, but had already crashed in 2002 (fig. 2). The spatial synchrony at a large scale is therefore phase–dependent, being most pronounced in the increase phase 2000– 2001. Given this difference in the dynamics, we will restrict the discussion below to the intermediate (among transects: 10 km) and small (within transect: 1 km) scale spatial patterns in Porsanger. The correlations within and among transects were highly variable, for population size and population growth rates (table 1). They were in general higher within transects than among transects indicating stronger correlations at small spatial scales, but the pattern was not consistent. As population sizes were variable and sometimes quite low (e.g. transect 4; fig. 2), part of the variability results at least in part from sampling variability. One component of the sampling variability is linked to the population size estimation process. Given that we estimate population size of the same species, in the same habitat type, with the same trappers, and with trapping sessions occurring on the same days, we would assume that capture rates may vary little among trapping grids, even if this has not been assessed in most studies. To explore this possibility, we used the Huggins/Alho’s model with different sets of covariates (table 2). There was no evidence for variation among grids, but evidence for heterogeneity in capture rates linked to body weight: large, reproducing (adult) animals have a higher capture rate than small (juvenile/subadult) individuals (fig. 3). This implies that populations with different age structure would have different average probabilities of capture. The overall estimate for population size in the five grids was lower than the Mh2 estimate (table 2), either because body weight is only one source of heterogeneity of capture and/or the second–order jackknife estimator is biased high. Heterogeneity of capture is therefore likely to be one of the main sources of sampling variability in population size estimates.

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0.9

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Fig. 3. Probability of being captured at least once as a function of body weight for the Grey–sided Vole, transect 2, Porsanger, Fall 1998. Most breeding individuals weight more than 25 g, and their probability of being captured is higher than 0.8. Individuals with a body weight as low as 10 g have just been weaned and are caught only if the trap is close to their nest. Fig. 3. Probabilidad de ser capturado, como mínimo una vez, expresada como una función del peso corporal para el ratón de campo de flancos grisáceos, transecto 2, Porsanger, otoño de 1998. La mayoría de los individuos en edad reproductora pesan más de 25 g, y la probabilidad de que sean capturados es superior a 0,8. Los individuos con peso corporal de sólo 10 g han sido destetados recientemente y sólo resultan capturados si la trampa se halla cerca de su nido.

the bias, but assuming a Poisson distributed error would correct only for suggest that some population indices such as raw counts may result in similarly biased correlations compared to those obtained using estimated population size, and that using an inadequate model for capture rates is in fact worse than using raw counts. The main difficulty is that most trapping indices have unknown measurement error and therefore the biases are difficult to remove (i.e., by estimating the measurement error and the true process variance). This is indeed a general problem in

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Table 1. Empirical patterns for the correlations of population size and population growth rates within (average pairwise correlations among trapping grids within a transect) and among transects (average pairwise correlations among trapping grids between two transects) for the Grey–sided Vole in Porsanger regionin autumn. Transect 1 (T1) is south, transect 5 (T5) is north (see fig. 1). Tabla 1. Patrones empíricos para las correlaciones del tamaño poblacional y de las tasas de crecimiento poblacional en cada uno de los transectos (correlaciones promedio entre pares con relación a las rejillas de captura en un transecto) y entre varios transectos (correlaciones promedio entre pares con relación a las rejillas de captura entre dos transectos) para el ratón de campo de flancos grisáceos en la región de Porsanger. El transecto 1 (T1) está en el sur, mientras que el transecto 5 (T5) está en el norte (ver fig. 1).

Population size (Log(Nt)) T1

Table 2. Huggins/Alho’s model applied to the Grey–sided Vole and the five stations of transect 2, fall 1998, Porsanger: A. Conditional likelihood values (CL) and conditional AIC (CAIC) (N. Number of parameters; *Best model, include on the covariate body weight, see fig. 2); B. Parameter and population size estimates for model including only body weight as a covariate (E. Estimates; SE. Standard error). The resulting estimate is compared to the jackknife estimate (Model Mh2). Tabla 2. Valores de probabilidad condicional (CL), criterio de Información de Akaike Condicional (CAIC), estimaciones de parámetros y de tamaños poblacionales para el modelo de Huggins/Alho aplicado al ratón de campo de flancos grisáceos y las cinco estaciones del transecto 2, otoño de 1998, Porsanger: * Mejor modelo, incluye el peso corporal como covariante (ver fig. 2); B. Estimaciones del peso corporal y del parametro para el modelo que incluye sólo el peso corporal como covariante (E. Estima; SE. Error estandard). El valor de estima resultante se compara con el valor del transecto estimado mediante jackknife (Modelo Mh2).

0.54

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0.39

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0.12

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0.48 0.55

Grids + body weight + sex 7

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329.76 663.52

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335.04 672.08

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0.55

–

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–

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–

Total population size 229.4(13.8) 264.5(13.8)

most monitoring programs using field techniques that do not allow for uncertainty estimation (Yoccoz et al., 2001, 2003). Most of the recent discussion of spatial population dynamics is focusing on analyzing how correlations among populations change as a function of the distance and direction among populations (e.g., Bjørnstad et al., 2002), on comparing these correlations among species and comparing those correlations to those of, e.g., climatic variables such as temperature (in particular the so called "Moran effect" predicting identical spatial correlations for the environmental disturbance and the population abundance; Bjørnstad et al., 1999; Liebhold et al., 2004;

Peltonen et al., 2002). As we have shown here, correlations will in general be biased low and the extent of the bias will depend of the measurement error. Even in our case study with high capture rate, the bias was large (i.e., 20 to 30%), and we would expect much larger biases for studies based on simple indices. On the contrary, variables such as temperature are precisely measured and show little sampling variability and would result in low bias for estimates of spatial correlation. It means that further comparisons among species and variables should include measure-

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Table 3. Correlations and covariance matrices estimated using a 5,000 year time–series simulated using an AR1 process (stable population). All covariances and correlations are calculated using log–transformed data: A. Covariance matrix of the true population sizes; B. Covariance matrix of the estimated population size using M h2; C. Covariance matrix of the population indices (number of animals caught Nc); D. Bias in the estimated correlations (true value for ! = 0.7)

Table 4. Correlations and covariance matrices estimated using a 1,000 year time–series and a AR2 process (cyclic population). All covariances and correlations are calculated using log–transformed data: A. Covariance matrix of the true population sizes; B. Covariance matrix of the estimated population size using Mh2; C. Covariance matrix of the population indices (number of animals caught Nc); D. Bias in the estimated correlation (true value for correlation = 0.7).

Tabla 3. Correlaciones y matrices de covarianza estimadas utilizando una serie temporal de 5.000 años y un proceso AR1 (población estable). Todas las covariantes y correlaciones se calculan mediante datos transformados logarítmicamente: A. Matriz de covarianza de tamaños poblacionales verdaderos; Matriz de covarianza de tamaños poblacionales estimados con Mh2.; C. Matriz de covarianza de índices poblacionales (número de animales capturados Nc); D. Diagonal en las correlaciones estimadas (valor verdadero para ! = 0,7)

Tabla 4. Correlaciones y matrices de covariantes estimadas utilizando una serie temporal de 1.000 años y un proceso AR2 (población cíclica). Todas las covariantes y correlaciones se calculan mediante datos transformados logarítmicamente: A. Matriz de covarianza de tamaños poblacionales verdaderos; Matriz de covarianza de tamaños poblacionales estimados con Mh2.; C. Matriz de covarianza de índices poblacionales (número de animales capturados Nc); D. Diagonal en las correlaciones estimadas (valor verdadero para correlación = 0,7).

Pop1

Pop2

Pop1

Pop2

A Pop1

0.213

0.149

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0.268

0.165

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0.149

0.206

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0.235

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0.312

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0.279

B Pop1

0.249

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C Pop1

0.255

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Estimated population size using model M0: 0.54

Estimated population size using model M0: 0.50

Estimated population size using model Mh2: 0.60

Estimated population size using model Mh2: 0.58

Population counts (Nc):

0.59

ment error since the extent of measurement error and therefore the bias is likely to differ among species and variables. Acknowledgements We thank the Norwegian Research Council (Biodiversity programme) and the Norwegian Institute for Nature Research for financial support, and two anonymous referees for constructive comments.

0.57

References Alho, J. M., 1990. Logistic regression in capture– recapture models. Biometrics, 46: 623–635. Bjørnstad, O. N., Peltonen, M., Liebhold, A. M. & Baltensweiler, W., 2002. Waves of larch budmoth outbreaks in the European Alps. Science, 298: 1020–1023. Bjørnstad, O. N., Stenseth, N. C. & Saitoh, T., 1999. Synchrony and scaling in dynamics of voles and mice in northern Japan. Ecology, 80: 622–637. Bolker, B. M. & Grenfell, B. T., 1996. Impact of

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vaccination on the spatial correlation and persistence of measles dynamics. Proceedings of the National Academy of Sciences USA, 93: 12648–12653. Borchers, D. L., Buckland, S. T. & Zucchini, W., 2002. Estimating animal abundance. Closed populations. Springer–Verlag, London. Buckland, S. T., Newman, K. B., Thomas, L. & Koesters, N. B., 2004. State–space models for the dynamics of wild animal populations. Ecological Modelling, 171: 157–175. Burnham, K. P. & Overton W. S., 1979. Robust estimation of population size when capture probabilities vary among animals. Ecology, 60: 927–936. 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. Dennis, B., Kemp, W. P. & Taper, M. L., 1998. Joint density dependence. Ecology, 79: 426–441. Engen, S., Lande, R. & Sæther B. E., 2002. The spatial scale of population fluctuations and quasi–extinction risk. American Naturalist, 160: 439–451. Fuller, W. A., 1987. Measurement error models. John Wiley and Sons, New York. Hansson, L., & Henttonen, H., 1985. Gradients in density variations of small rodents: the importance of latitude and snow cover. Oecologia, 67: 394–402. – 1988. Rodent dynamics as community processes. Trends in Ecology and Evolution, 3: 195–200. Henttonen, H. & Hansson, L., 1986. Synchrony and asynchrony between sympatric rodent species with special reference to Clethrionomys. In: Causes and geographic patterns of microtine cycles (H. Henttonen, Ed.). Ph. D. Thesis, Univ. of Helsinki, Helsinki. Huggins, R. M., 1989. On the statistical analysis of capture experiments. Biometrika, 76: 133–140. Ito, Y., 1972. On the methods for determining density–dependence by means of regression. Oecologia, 10: 347–372. Krebs, C. J., Kenney, A. J., Gilbert, S., Danell, K., Angerbjörn, A., Erlinge, S., Bromley, R. G., Shank, C. & Carriere, S., 2002. Synchrony in lemming and vole populations in the Canadian Arctic. Canadian Journal of Zoology, 80: 1323–1333. Lambin, X., Elston, D. A., Petty, S. J. & MacKinnon, J. L., 1998. Spatial asynchrony and periodic travelling waves in cyclic populations of field voles. Proceedings of the Royal Society, London B, 265: 1491–1496. Lande, R., Engen, S. & Sæther, B.–E., 2003. Stochastic population dynamics in ecology and conservation. Oxford University Press, Oxford. Lebreton, J.–D., 1990. Modelling density dependence, environmental variability, and demographic stochasticity from population counts: an example using Wytham Wood great tits. In:

435

Population biology of passerine birds: 89–102 (J. Blondel, A. Gosler, J.–D. Lebreton, & R. H. McCleery, Eds.). Springer–Verlag, Berlin. Liebhold, A., Sork, V., Peltonen, M., Koenig, W., Bjørnstad, O. N., Westfall, R., Elkinton, J., & Knops, J. M. H., 2004. Within–population spatial synchrony in mast seeding of North American oaks. Oikos, 104: 156–164. Link, W. A. & Nichols, J. D., 1994. On the importance of sampling variance to investigations of temporal variation in animal population size. Oikos, 69: 539–544. Link, W. A., Barker, R. J., Sauer, J. R. & Droege, S., 1994. Within–site variability in surveys of wildlife populations. Ecology, 75: 1097–1108. Moilanen, A., 2002. Implications of empirical data quality to metapopulation model parameter estimation and application. Oikos, 96: 516–530. Peltonen, M., Liebhold, A. M., Bjørnstad, O. N. & Williams, D. W., 2002. Spatial synchrony in forest insect outbreaks: Roles of regional stochasticity and dispersal. Ecology, 83: 3120–3129. Pollock, K. H., Nichols, J. D. , Brownie, C. & Hines, J. E., 1990. Statistical inference for capture–recapture experiments. Wildlife Monographs, 107: 1–97. Ranta, E., Kaitala, V. & Lundberg, P., 1997. The spatial dimension in population fluctuations. Science, 278: 1621–1623. Ripa, J., 2000. Analysing the Moran effect and dispersal: their significance and interaction in synchronous population dynamics. Oikos, 89: 175–187. Solow, A. R., 2001. Observation error and the detection of delayed density dependence. Ecology, 82: 3263–3264. Steen, H., Ims, R. A. & Sonerud, G. A., 1996. Spatial and temporal patterns of small–rodent population dynamics at a regional scale. Ecology, 77: 2365–2372. Steen, H., Yoccoz, N. G. & Ims, R. A., 1990. Predators and small rodent cycles: an analysis of a 79–year time series of small rodent population fluctuations. Oikos, 59: 115–120. Stenseth, N. C., 1999. Population cycles in voles and lemmings: density dependence and phase dependence in a stochastic world. Oikos, 87: 427–461. Viljugrein, H., 2000. Spatio–temporal patterns of population fluctuations in waterfowl and small mammals. Ph. D. Thesis, Oslo Univ. Yoccoz, N. G., Nichols, J. D. & Boulinier, T., 2001. Monitoring of biological diversity in space and time. Trends in Ecology and Evolution, 16: 446–453. – 2003. Monitoring of biological diversity – a response to Danielsen et al. Oryx, 37: 410. Yoccoz, N. G., Steen, H., Ims, R. A. & Stenseth, N. C., 1993. Estimating demographic parameters and the population size: an updated methodological survey. In: The biology of lemmings: 565–587 (N. C. Stenseth & R. A. Ims, Eds.). Academic Press, London.

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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Abundance estimation and Conservation Biology J. D. Nichols & D. I. MacKenzie

Nichols, J. D. & MacKenzie, D. I., 2004. Abundance estimation and Conservation Biology. Animal Biodiversity and Conservation, 27.1: 437–439. Abundance is the state variable of interest in most population–level ecological research and in most programs involving management and conservation of animal populations. Abundance is the single parameter of interest in capture–recapture models for closed populations (e.g., Darroch, 1958; Otis et al., 1978; Chao, 2001). The initial capture–recapture models developed for partially (Darroch, 1959) and completely (Jolly, 1965; Seber, 1965) open populations represented efforts to relax the restrictive assumption of population closure for the purpose of estimating abundance. Subsequent emphases in capture–recapture work were on survival rate estimation in the 1970’s and 1980’s (e.g., Burnham et al., 1987; Lebreton et al., 1992), and on movement estimation in the 1990’s (Brownie et al., 1993; Schwarz et al., 1993). However, from the mid–1990’s until the present time, capture–recapture investigators have expressed a renewed interest in abundance and related parameters (Pradel, 1996; Schwarz & Arnason, 1996; Schwarz, 2001). The focus of this session was abundance, and presentations covered topics ranging from estimation of abundance and rate of change in abundance, to inferences about the demographic processes underlying changes in abundance, to occupancy as a surrogate of abundance. The plenary paper by Link & Barker (2004) is provocative and very interesting, and it contains a number of important messages and suggestions. Link & Barker (2004) emphasize that the increasing complexity of capture–recapture models has resulted in large numbers of parameters and that a challenge to ecologists is to extract ecological signals from this complexity. They offer hierarchical models as a natural approach to inference in which traditional parameters are viewed as realizations of stochastic processes. These processes are governed by hyperparameters, and the inferential approach focuses on these hyperparameters. Link & Barker (2004) also suggest that our attention should be focused on relationships between demographic processes such as survival and recruitment, the two quantities responsible for changes in abundance, rather than simply on the magnitudes of these quantities. They describe a type of Jolly–Seber capture–recapture model that permits inference about the underlying relationship between per capita recruitment rates and survival rates (Link & Barker, this volume). Implementation used Bayesian Markov Chain Monte Carlo methods and appeared to work well, yielding inferences about the relationship between recruitment and survival that were robust to selection of prior distribution. We believe that readers will find their arguments compelling, and we expect to see increased use of hierarchical modeling approaches in capture–recapture and related fields. Otto (presentation without paper) also recommended use of hierarchical models in analysis of multiple data sources dealing with population dynamics of North American mallards. He integrated survival inferences from ringing data, abundance information from aerial survey data, and recruitment information based on age ratios from a harvest survey. He used a Leslie matrix population projection model as an integrating framework and obtained estimates of breeding population size using all data.

James D. Nichols, USGS Patuxent Wildlife Research Center, 11510 American Holly Drive, Laurel, MD 20708–4017, U.S.A. E–mail: jim_nichols@usgs.gov Darryl I. MacKenzie, Proteus Research & Consulting Ltd., PO Box 5193, Dunedin, New Zealand. E– mail: darryl@proteus.co.nz ISSN: 1578–665X

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Ottoâ&#x20AC;&#x2122;s approach also permitted inference about biases in estimated quantities. As with the work of Link & Barker (2004), we find Ottoâ&#x20AC;&#x2122;s recommendation to use hierarchical models to integrate data from multiple sources to be very compelling. Alisauskas et al. (2004) report results of an analysis of captureâ&#x20AC;&#x201C;recapture data for a Saskatchewan population of whiteâ&#x20AC;&#x201C;winged scoters. They used the approach of Pradel (1996) to estimate population growth rate ( ) directly. Estimates for 1975â&#x20AC;&#x201C;1985 were quite low, but estimates for the recent period, 2000â&#x20AC;&#x201C;2003, increased to values > 1. Parameter estimates for seniority, survival and per capita recruitment (Pradel, 1996) led to the inference that increased recruitment was largely responsible for the improvements in population status and growth. However, various data sources also indicated that this increase in recruitment was likely a result of increased immigration rather than improved reproduction on the area. This latter inference is important from a conservation perspective in indicating the importance of birds in other locations to growth and health of the study population. Lukacs and Burnham presented material to be published elsewhere that dealt with the use of genetic markers in captureâ&#x20AC;&#x201C;recapture studies. The data sources for such studies are samples of hair or feces, which are then analyzed using molecular genetic techniques in order to determine individual genotypes with respect to a usually small number of loci. Two types of classification error can arise in such analyses. First, if only a small number of loci is examined, then there may be nonnegligible probabilities that multiple individual animals will have the same genotypes. The second type of error arises during the polymerase chain reaction (PCR) process and can result from failure of alleles to amplify (allelic dropout) or from PCR inhibitors in hair and feces that produce the appearance of false alleles or misprinting (Creel et al., 2003). Lukacs and Burnham developed models that formally incorporate possible misclassification of samples resulting from these errors. These models permit estimation of parameters such as abundance and survival in a manner that properly incorporates this uncertainty of individual identity. We anticipate that noninvasive sampling based on molecular genetic analyses of hair or feces will become extremely important for some species, and that the models of Lukacs and Burnham will become very popular for such analyses. MacKenzie & Nichols (2004) discuss the use of occupancy (proportion of patches or habitat area that is occupied) as a surrogate for abundance. In cases of territorial species and where birds occur at low densities, the number of occupied patches may provide a reasonable estimate of abundance. In other cases, occupancy can be viewed as providing information about one tail of the abundance distribution, P (N = 0). The motivation for considering occupancy as a surrogate for abundance is that occupancy is based on soâ&#x20AC;&#x201C;called presenceâ&#x20AC;&#x201C;absence surveys that are frequently less expensive of time and effort than methods that estimate abundance directly. We describe one set of models that can be used to estimate occupancy for a single season and another that can be used to estimate parameters such as local probabilities of extinction and colonization that are associated with occupancy dynamics. We outline a possible hybrid approach that combines occupancy data with data on marked individuals in order to better explore the mechanisms underlying occupancy dynamics. These five presentations made for an interesting session containing useful information and recommendations for future work. A number of themes connecting these presentations could be emphasized. For example, two of the presentations considered alternatives to standard captureâ&#x20AC;&#x201C;recapture sampling that can be used to draw inferences about abundance, or a portion of the abundance distribution, with field methods that should be less expensive than usual captureâ&#x20AC;&#x201C;recapture approaches of handling animals. We believe that the most important theme of the session was the emphasis on the processes responsible for changes in abundance. In particular, we are excited by the potential for using hierarchical models as a means of investigating relationships among vital rates and as a means of combining multiple sources of data relevant to system dynamics. Indeed, we expect the importance of this session theme to be reflected in the content and presentations of the next EURING meeting. References Alisauskas, R. T., Traylor, J. J., Swoboda, C. J. & Kehoe, F. P., 2004. Components of population growth rate for whiteâ&#x20AC;&#x201C;winged scoters in Saskatchewan, Canada. Animal Biodiversity and Conservation, 27.1: 451â&#x20AC;&#x201C;460. Brownie, C., Hines, J. E., Nichols, Pollock, K. H., & Hestbeck, J. B., 1993. Captureâ&#x20AC;&#x201C;recapture studies for multiple strata including nonâ&#x20AC;&#x201C;Markovian transition probabilities. Biometrics, 49: 1173â&#x20AC;&#x201C;1187. Burnham, K. P., Anderson, D. R., White, G. C., Brownie, C. & Pollock, K. P., 1987. Design and analysis of methods for fish survival experiments based on releaseâ&#x20AC;&#x201C;recapture. American Fisheries Society Monograph, 5: 1â&#x20AC;&#x201C; 437. Chao, A., 2001. An overview of closed captureâ&#x20AC;&#x201C;recapture models. Journal of Agricultural, Biological, and Environmental Statistics, 6: 158â&#x20AC;&#x201C;175. Creel, S., Spong, G., Sands, J. L., Rotella, J., Zeigle, J., Joe, L., Murphy, K. M. & Smith, D., 2003. Population size estimation in Yellowstone wolves with errorâ&#x20AC;&#x201C;prone noninvasive microsatellite genotypes. Molecular Ecology, 12: 2003â&#x20AC;&#x201C;2009.

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Darroch, J. N., 1958. The multiple–recapture census: I. Estimation of a closed population. Biometrika, 45: 343–359. – 1959. The multiple–recapture census: II. Estimation when there is immigration or death. Biometrika, 46: 336–351. Jolly, G. M., 1965. Explicit estimates from capture–recapture data with both death and immigration – stochastic model. Biometrika, 52: 225–247. 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. Link, W. A. & Barker, R. J., 2004. Hierarchical mark–recapture models: a framework for inference about demographic processes. Animal Biodiversity and Conservation, 27.1: 441–449. MacKenzie, D. I. & Nichols, J. D., 2004. Occupancy as a surrogate for abundance estimation. Animal Biodiversity and Conservation, 27.1: 461–467. 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: 1–135. Pradel, R., 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Schwarz, C. J., 2001. The Jolly–Seber model: more than just abundance. Journal of Agricultural, Biological, and Environmental Statistics, 6: 195–205. 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. Seber, G. A. F., 1965. A note on the multiple–recapture census. Biometrika, 52: 249–259.

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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Hierarchical mark–recapture models: a framework for inference about demographic processes W. A. Link & R. J. Barker

Link, W. A. & Barker, R. J., 2004. Hierarchial mark–recapture models: a framework for inference about demographic processes. Animal Biodiversity and Conservation, 27.1: 441– 449. Abstract Hierarchial mark–recapture models: a framework for inference about demographic processes.— The development of sophisticated mark–recapture models over the last four decades has provided fundamental tools for the study of wildlife populations, allowing reliable inference about population sizes and demographic rates based on clearly formulated models for the sampling processes. Mark–recapture models are now routinely described by large numbers of parameters. These large models provide the next challenge to wildlife modelers: the extraction of signal from noise in large collections of parameters. Pattern among parameters can be described by strong, deterministic relations (as in ultrastructural models) but is more flexibly and credibly modeled using weaker, stochastic relations. Trend in survival rates is not likely to be manifest by a sequence of values falling precisely on a given parametric curve; rather, if we could somehow know the true values, we might anticipate a regression relation between parameters and explanatory variables, in which true value equals signal plus noise. Hierarchical models provide a useful framework for inference about collections of related parameters. Instead of regarding parameters as fixed but unknown quantities, we regard them as realizations of stochastic processes governed by hyperparameters. Inference about demographic processes is based on investigation of these hyperparameters. We advocate the Bayesian paradigm as a natural, mathematically and scientifically sound basis for inference about hierarchical models. We describe analysis of capture–recapture data from an open population based on hierarchical extensions of the Cormack–Jolly–Seber model. In addition to recaptures of marked animals, we model first captures of animals and losses on capture, and are thus able to estimate survival probabilities (i.e., the complement of death or permanent emigration) and per capita growth rates f (i.e., the sum of recruitment and immigration rates). Covariation in these rates, a feature of demographic interest, is explicitly described in the model. Key words: Bayesian hierarchical analysis, Capture–recapture, Demographic analysis, Jolly–Seber Model, Open population estimation. Resumen Modelos jerárquicos de marcaje–recaptura: un marco para la inferencia de procesos demográficos.— El desarrollo de sofisticados modelos de marcaje–recaptura a lo largo de las últimas cuatro décadas ha proporcionado herramientas fundamentales para el estudio de poblaciones de fauna silvestre, lo que ha permitido inferir con fiabilidad los tamaños poblacionales y las tasas demográficas a partir de modelos claramente formulados para procesos estocásticos. En la actualidad, los modelos de marcaje–recaptura se describen de forma rutinaria mediante una extensa serie de parámetros. Dichos modelos representan el siguiente reto al que deberán enfrentarse los modeladores de fauna silvestre: discriminar las señales del ruido en amplias series de parámetros. La pauta que encontramos en los parámetros puede describirse mediante sólidas relaciones deterministas (como en los modelos ultraestructurales), pero resulta más flexible y creíble si se modela utilizando relaciones estocásticas más débiles. No es probable que la tendencia en las tasas de supervivencia se manifieste por una secuencia de valores hallados concretamente en una curva paramétrica dada; por ello, si pudiéramos llegar a conocer los valores reales, podríamos prever una relación de regresión entre parámetros y variables explicativas, de forma que el valor verdadero ISSN: 1578–665X

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equivaldría a la señal más el ruido. Los modelos jerárquicos proporcionan un marco útil para la inferencia acerca de series de parámetros relacionados. Así, en lugar de interpretar los parámetros como cantidades fijas, pero desconocidas, los interpretamos como realizaciones de procesos estocásticos regidos por hiperparámetros. La inferencia acerca de los procesos demográficos se basa en la investigación de dichos hiperparámetros. Por este motivo, defendemos el paradigma bayesiano como una base natural, matemática y científicamente sólida para la inferencia acerca de modelos jerárquicos. En el presente estudio describimos el análisis de datos de captura–recaptura obtenidos a partir de una población abierta basada en ampliaciones jerárquicas del modelo de Cormack–Jolly–Seber. Además de las recapturas de animales marcados, también modelamos las primeras capturas de animales y de pérdidas durante la captura, lo que nos permitió estimar las probabilidades de supervivencia de (es decir, el complemento de la muerte o la emigración permanente) y las tasas de crecimiento per cápita f (es decir, la suma de las tasas de reclutamiento y de migración). En el modelo se describe explícitamente la covariación en estas tasas, que constituye una característica de interés demográfico. Palabras clave: Análisis jerárquico bayesiano, Captura–recaptura, Análisis demográfico, Modelo de Jolly– Seber, Estimación de población abierta. William A. Link, Patuxent Wildlife Research Center, Laurel, MD, 20708, U.S.A.– Richard J. Barker, Univ. of Otago, Dunedin, New Zealand.

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Introduction Observations of biological systems are variable. Mathematical models describing this variability incorporate two sources of variation: variation relating to the biological system itself (birth, death, and behavioral processes), and variation relating to the collection of data. Thus there are two categories of parameters in models for biological data. We will let denote the collection of demographic parameters describing biological processes, and denote the collection of parameters governing collection of data; we will assume that these are unrelated, though this need not be the case. The parameters comprising are the parameters of interest. Those comprising are often referred to as "nuisance parameters". It is an appropriate name: their inclusion in the modeling effort is a necessary evil, and one which must be handled with care, because incorrect assumptions about can lead to profound biases in estimation of . The field of capture–recapture has developed over the last forty years with its first priority being on accounting for biologically irrelevant sources of variation in data. Increases in computational efficiency have allowed for analysis of larger data sets which not only include many nuisance parameters, but also many parameters of interest. The vector may have hundreds of components, including survival rates, recruitment rates, population sizes, movement parameters. We believe that the next priority in capture–recapture analysis should be the development of statistically sound methods for analysis, not of data, but of the parameters comprising . In this paper, we describe hierarchical models useful in examining pattern in parameters, making note of various ad hoc methods that have been used for examining them, and arguing for the usefulness and appeal of the Bayesian paradigm in this context. We illustrate our discussion with an hierarchical extension of the Cormack–Jolly–Seber model which allows for efficient, statistically sound analysis of covariation among demographic rates. Hierarchical models defined; ad hoc analytic methods Suppose that we did not have to deal with nuisance parameters, indeed, that we did not need to estimate , but knew its components without error. What would we do? Why, we’d do some statistical analyses, regressing survival rates against time, examining whether recruitment rates were related to population size, and performing similar analyses. After all, the entire purpose of collecting biological data is to make inference about . We would be mightily surprised if there were perfect, deterministic relations among the actual parameters. That is, if t is a survival rate for time period [t, t+1), we would not expect log ( t)= A + Bt

(1)

though we might anticipate acceptable fit of a model with log ( t)= a + bt + t (2) where the values t are a sample from a mean–zero normal distribution. The first of these specifies strong, deterministic relations among parameters; the survival rates are perfectly predictable, changing in lockstep fashion through time. The second specifies weaker, stochastic relations among parameters, general tendencies rather than predetermined patterns. The role of the error term t is to account for sources of variation that are not explained by known covariates, but that nevertheless are an important source of variation to the survival probabilities; modeling thus, we assume that the collective effects of unknown covariates amount to uncorrelated random noise. All statistical analyses, even nonparametric ones, begin with the specification of a family of distributions from which data have been drawn. Parametric analyses restrict the family to a general form, known except for certain unknown parameters, typically regarded (in the Frequentist paradigm) as "fixed but unknown constants". Hierarchical models treat these parameters as though they also were sampled from parametric families of distributions. Thus, many data are described by model specifications and fewer parameters; these parameters are described by further model specifications and even fewer parameters. The simplest hierarchical models are familiar as random effects models. Indeed, the notion of finding and evaluating pattern among parameters is no new one. What is changing is the manner in which such models are fit to data. Over the years, the approach to investigating stochastic relations among demographic parameters has been one of two sorts (Link, 1999). First, we might fit an unconstrained model, treating components of as completely unrelated, then attempting to uncover the stochastic relations among parameters by examination of their estimates . Thus, for example, if we were interested in examining temporal pattern of change in survival rates, we might fit the model log ( )= a* + b*t +

(3)

instead of (2), in the hope that inference about b* in (3) could inform us about the parameter b in (2). This approach could be called a two–stage modeling approach: first estimate the parameters without specifying relations among them; then, look for those relations among the parameter estimates obtained. The other commonly used approach to examining pattern among parameters has been ultrastructural modeling, in which the pattern among parameters is treated as deterministic. To examine temporal patterns of change in survival, we estimate the parameters t subject to the constraint given by equation (1). Thus the parameter set is reduced from { 1, 2,...} to {A, B}. We attempt to model {a, b} using the pair {A, B}.

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The two–stage and ultrastructural approach are approximate and unsatisfactory: in two–stage analyses, there may be problems of attenuation bias when random regressors are used. There may be inefficiencies, even gross ones, due to heterogeneity of variances among the estimators. Failure to adequately account for the sampling variability of parameter estimates, when using these as surrogates for actual parameter values, can be spectacularly misleading. Ultrastructural analyses often (but not inevitably) produce reasonable estimates, though usually with overstated precision. Various methods for improving ultrastructural and two–stage analyses have been developed, but may generally be said to only work asymptotically. In using them, we ought not to forget Le Cam’s 7th principle: "If you need to use asymptotic arguments, do not forget to let your number of observations tend to infinity" (Le Cam, 1990). Hierarchical models and Bayesian analysis Instead of using ad hoc methods of analysis for hierarchical models, we recommend the use of Bayesian methods. The first advantage of these is the naturalness of their application to hierarchical models. All quantities in Bayesian inference are random variables; the only distinction being whether quantities are known or not. Thus the hierarchical modeling view of parameters as random variables is completely natural to Bayesian analysis. Second, Bayesian analysis, properly conducted, requires no fussing over optimality criteria, choice of estimation technique, or asymptotics. There is nothing but a calculation, the calculation of a posterior distribution from a prior and a likelihood. All inference is based on features of the posterior distribution. But what of the prior? Choice of the prior distribution is the classically trained statistician’s Bayesian bugaboo. Choice of the prior is thought by some to introduce an irremediable and unacceptable subjectivity into the analysis. Attempts to define noninformative priors for objective analysis are dismissed by some critics on the grounds that the quality of noninformativeness is not transformation invariant. For instance, a noninformative prior for 2 is not a noninformative prior for . It is a marvelous thing that such critics will frequently choose the (n – 1) weighted variance estimator S2 as an estimator of 2 because it is unbiased, giving no thought to the fact that S, the value they will actually use in discussion, is a biased estimator of . Unbiasedness, like noninformativeness of priors, is not transformation invariant. Lack of transformation invariance is a problem for Frequentists as well as for Bayesians. The simple solution to the problem of choosing priors is to try several, and to see whether and how the choice influences posterior inference. In our view, this is a virtue rather than a vice of Bayesian inference. What is more, the Bernstein–Von Mises Theorem (also known as the Bayesian Central Limit

theorem) implies that, subject to minor constraints, the influence of the prior diminishes as the sample size increases, so that the choice will not matter if sample sizes are reasonably large. Putting aside all of the overheated and overblown Bayesian/Frequentist polemics, we find that the differences of inference generally prove to be slight, provided we have adequate data. If we are willing to adopt the Bayesian paradigm, we may avail ourselves of the powerful computational techniques known as Markov chain Monte Carlo (MCMC) to examine features of the posterior distribution. And even if we simply cannot shake misgivings about the Bayesian paradigm, MCMC is still useful: by specifying uniform prior distributions, the posterior distribution is simply the scaled likelihood, and its mode the maximum likelihood estimator. We thus obtain a tool for formal examination of complex demographic structures in data sets; specification of stochastic relations among demographic parameters becomes part of data modeling, rather than something done after the fact using dubious ad hoc methods. An anal ysis of association demographic parameters

among

We illustrate the use of hierarchical models with an analysis of temporal relations between demographic parameters governing rates of population change. These are survival rates (i.e., the complement of death or permanent emigration) and per capita growth rates f (i.e., the sum of recruitment and immigration rates); the analysis will allow formal evaluation and testing of associations between these demographic parameters. We analyze the Gonodontis data set of Bishop et al. (1978), which has been evaluated in several subsequent papers developing methods for open–population survival analysis (Crosbie, 1979; Crosbie & Manly, 1985; Link & Barker, in press); the model we describe is an extension of the Cormack–Jolly–Seber (CJS) model (Cormack, 1964; Jolly, 1965; Seber, 1965). The CJS model uses likelihood proportional to the joint distribution of sufficient statistics r and m, given statistics u and R. These statistics are vector–valued; Ri from R is the number of individuals released after the ith sampling occasion, and ri from r is the number of these subsequently recaptured. Components of m and u are the numbers of marked and unmarked animals, respectively, at each of the sampling occasions. The conditional distribution corresponding to the CJS model can be written in self–explanatory notation as [r, m | R, u]

(4)

The dependence of this distribution on demographic and nuisance parameters is suppressed in this notation. We note however that the only demographic parameters are survival rates i. Neither population size nor population growth rates are

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included in the model. An extension of the CJS model is needed; we propose one subsequently. Step one: obtaining a likelihood based on parameters of interest The first step in conducting a Bayesian hierarchical analysis is the specification of a likelihood for the data in terms of the parameters of interest. This is generally straightforward, but can sometimes require some thought about reparameterizing and extending models. The first attempt to extend the CJS model so as to allow examination of population change was the Jolly–Seber (JS) model. This extension had two components. The first, relatively minor change, is the addition of a parametric description of loss on capture, thus removing the conditioning on R, so that the likelihood used is proportional to [r, m, R | u]

(5)

Of greater importance is the modeling of first captures ui as binomial with index Ui. Identifiability of Ui, the number of unmarked animals in the population just prior to the ith sampling occasion, is obtained by assuming that the nuisance parameters (detection probabilities) are the same for marked and unmarked animals. The JS model thus uses a likelihood proportional to the joint distribution [r, m, R | u] [u | U]

(6)

The population numbers of unmarked animals, Ui, are not of much interest per se, for investigating relations between survival and other parameters governing population growth rates. The model is not expressed in terms of quantities in which we are directly interested. If we want to use the JS model for our hierarchical investigations, we are forced to carry out a two–stage analysis, first obtaining estimates of demographic parameters (first, we estimate the population number of marked animals, and add this to the estimated number of unmarked animals to obtain an estimate of population size; more precisely, we should say, "predict" the population size, since population size is not part of the JS model; with the additional assumption that survival rates are the same for marked and unmarked animals, the resulting estimates of population change can be partitioned among changes due, on the one hand, to mortality, and to nonmortality sources, on the other), and then looking for associations among the estimates. Instead, we consider an alternative extension of the CJS model, following work by Crosbie & Manly (1985), and later developed by Schwarz & Arnason (1996). Schwarz & Arnason describe the model in terms of a likelihood proportional to [r, m, R | u] [u | N*]

where N* is the number of distinct animals available for capture on at least one of the sampling occasions. Their model includes the additional assumption that survival and detection rates are the same for marked and unmarked animals. We demonstrate elsewhere that [u | N*] = [u | u.] [u. | N*] where u. = i ui, and that there is very little information in the likelihood component proportional to [u. | N*] (Link & Barker, in press). We thus eliminate that from the likelihood, and base our analysis on [r, m, R | u] [u | u.]

(7)

It can be shown that the distribution [u | u.] is multinomial with index u., and cell probabilities i determined by detection and survival rates from the CJS model and t – 1 additional estimable parameters, namely 1,f2 ,f3,...,ft–2 and t. Here, parameters fi (slightly different from the growth rate parameters of Pradel et al., 1996; for details, see Link & Barker, in press) are per capita growth rates (recruitment plus immigration). Parameters and t are confounded combinations of demo1 graphic and nuisance parameters. Details on these and the functional form of i are given in the Appendix. We use (7) as the basis for hierarchical analysis of relations between survival and growth rates rather than (5), because it is expressed entirely in terms of the demographic parameters of interest, and a clearly identified set of nuisance parameters. We have = { ; f} = {

,...,

;f ,f ,...,ft–2}

t–2 2 3

and = { 2, 3,...,

t–1

;v2,v3,...,vt–1; 1, t,

}

t–1

here, t is the number of sampling occasions, i is the detection probability at sampling occasion i, v i is the probability of successfully releasing an animal captured at sampling occasion i, and = t–1 t. t–1 Step two: describing stochastic relations among parameters of interest The next step in a hierarchical analysis is a description of stochastic relations among parameters of interest. From the Bayesian perspective, as we shall see subsequently, this amounts to a partial specification of the prior distributions of parameters. Our goal is to examine stochastic relations between i's and fi's. Since 0 < i < 1 and fi > 0, it is natural to transform the parameters in order to remove the range restrictions. We thus suppose that pairs = {logit ( i) log (fi)} i

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follow a bivariate normal distribution with mean vector and variance matrix 3. The correlation parameter in the variance matrix is of primary interest, as determining the strength of association between i's and fi's . We will refer to = { , 3} as the hyperparameters. The process of specifying this part of the model is similar to the corresponding process for modeling stochastic relations among data. Ideally, there should be some basis in first principles for deciding on whether data are normally distributed, or whether a Poisson distribution, or Binomial is appropriate. Alternatively, distributional choices are often made on the basis of convenience. In observational studies, these choices are usually made after informal inspection of the data; similar informal evaluations of parameters might be based on the ad hoc methods described at the outset. In any case, the same sort of model checking used for evaluating distributional assumptions for data should be used to evaluate distributional assumptions about parameters. Step three: selection of prior distributions So far, we have suppressed in our notation the dependence of the likelihoods on the parameters. In order to complete the specification of a Bayesian hierarchical model, we make this dependence explicit, rewriting (7) as [r, m, R | u; ,

0.8 0.6 0.4 0.2 0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 1. Annual survival vs. nonsurvival growth rates (symbol size proportional to precision). Fig. 1. Tasas de crecimiento de supervivencia anual respecto a no supervivencia (el tamaño de los símbolos es proporcional a la precisión).

as the distribution of

] [u | u.; , ]

All that remains for a fully Bayesian specification of the model, is a description of prior distributions for parameters and hyperparameters, i.e. [ , , ]. Here, we assume prior independence of the nuisance and demographic parameters, so that [ , , ] = [ , ] [ ] = [ |

] [ ] [ ]

Our study of association among demographic parameters will be based on the partial specification of the prior structure, namely [ | ]. The posterior distribution upon which we shall base inference is proportional to the joint distribution [r, m, R | u; ,

f 1.0

] [u | u.; , ] [ |

][ ][ ]

(8)

All that remains is specification of the prior distributions [ ] and [ ]. We chose flat priors [ ] ≠ c, for the nuisance parameters. We chose a Normal–Inverse Wishart prior distribution for . This is the distribution obtained by supposing that [ , 3 ] is a bivariate normal distribution with mean 0 and variance matrix 0, and that 3 /n0 has the inverse Wishart distribution with parameters V and df. It is an appealing choice for a prior distribution for parameters of the multivariate normal, because the resulting posterior distributions for are easily calculated, also being members of the inverse Wishart family of distributions. The inverse Wishart distribution can be thought of

where Xi are independent and identically distributed bivariate normal random variables with variance matrix V–1. Choice of the particular Normal– Inverse Wishart prior requires specification of 0, n0 m 0, V, and df m 2. For the Gonodontis data set t = 17. Recognizing that there are only t – 3 = 14 estimable pairs ( i, fi) informing our inference about the covariation, we anticipated some sensitivity to the choice of parameters governing the choice of NIW priors for ( , 3), hence decided to repeat the analysis for four choices of prior. We set n0 = 0 in all analyses, inducing vague priors on . Choice of parameters for the inverse Wishart distribution for 3 was guided by the observations that if df m 1, 1) The diagonal elements of 3 have inverse Gamma distributions:

where Vi,i is the ith diagonal element of V, so that i2 has the same distribution as Vi,i / A, where A i 2df–1, and 2) That given V is a diagonal matrix, the marginal distribution of the correlation parameter p is such that

The four priors we considered were: (1) df = 2,

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V = diag (1, 1); (2) df = 2, V = diag (2.5, 1.25); (3) df = 3, V = diag (1, 1); (4) df = 3, V = diag (5.0, 2.5). The first two priors maximize the coefficient of variation of the diagonal elements of 3 , subject to the requirement that [3 ] be proper; the third and fourth induce a uniform prior on the correlation parameter. The second and fourth fix the prior means of the precision (inverse variance) for logit survival rates and log birth rates at values of 0.40 and 0.80, respectively; these values were chosen as representing large levels of variability in demographic parameters. The first and third priors were chosen as representative of moderate levels of variability in demographic parameters.

these questions can be dismissed, what about computing a confidence interval for the estimated correlation parameter, ? Are we confident that the estimator has a normal distribution, even though there are only 14 values (estimates, at that) from which it is calculated? Far more satisfactory, to our mind, is the Bayesian analysis we have developed in the foregoing sections. We find relatively minor differences among the results based on four distinct priors; these are summarized in plots of the posterior distributions for p in figure 2, and on summaries of these distributions in table 1. The posterior probability that p > 0 is roughly 84% for all of the priors considered (0.843, 0.857, 0.816 and 0.843, for priors 1, 2, 3, and 4); the posterior odds are 5:1 in favor of a positive correlation.

Analysis of correlation in Gonodontis data Figure 1 is a scatter plot of maximum likelihood estimates based on the Pradel (1996) model, as implemented in program MARK (White & Burnham, 1999); note that the size of the plotting image reflects the relative precision of estimates. Three points, with imprecise estimates on the edge of the parameter space, were excluded. The plot suggests a positive association between the vital rates. We wish to study the association between log(f) and logit( ). However, uncovering this association by "doing statistics on statistics" is problematic. First, how are we to transform estimates = 1 to the logit scale? How are we to account for sampling variation, and covariation? Are the asymptotic variance estimates obtained from the estimated Fisher information matrix reliable? And supposing that all of the uncertainties raised by

Summary and comments Associations among demographic parameters are very naturally modeled by treating the parameters as random variables, the associations arising because the parameters have been sampled from related distributions. Such hierarchical models posit the existence of weak, stochastic relations among parameters, rather than unrealistic deterministic relations. The Bayesian paradigm, in which all quantities are treated as stochastic, is particularly appropriate for consideration of hierarchical models. We therefore encourage the following view of capture–recapture models. Data Y are described in terms of their dependence on demographic parameters and nuisance parameters , through distri-

Wishart priors on variance matrix df df df df

= = = =

2, 3, 2, 3,

S S S S

= = = =

diag diag diag diag

(1,1) (1,1) (2.5,1.25) (5.0,2.5)

Pr( > 0 |Y) = 0.84

–0.8

–0.6 –0.4 –0.2

Fig. 2. Posterior distributions of . Fig. 2. Distribución posterior de .

0.2

0.4

0.6

0.8

1.0

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Table 1. Summaries of posterior distributions for p in analysis of Gonodontis data. First column indicates prior as numbered in text. Tabla 1. Resúmenes de distribuciones posteriores de p en el análisis de datos de Gonodontis. En la primera columna se indican las distribuciones anteriores según constan numeradas en el texto.

Quantiles of posterior distribution SD

0.025

0.100

0.250

0.500

0.750

0.900 0.975

0.383

Mean

0.353

–0.425

–0.131

0.157

0.447

0.664

0.793

0.880

0.372

0.323

–0.356

–0.089

0.164

0.420

0.623

0.754

0.850

0.345

0.359

–0.450

–0.174

0.108

0.403

0.632

0.772

0.868

0.314

0.296

–0.330

–0.098

0.118

0.346

0.54

0.677

0.788

butional assumptions specifying [Y | , ]. The prior distribution [ , ] is factored as [ , ] ≠ [ ] [ |

] [ ]

or possibly as [ , ] ≠ [ | ] [ |

] [ ]

if it is thought that the nuisance and demographic parameters are related. Associations among demographic parameters are modeled in the partial specification of the prior, [ | ]. Given the specification of likelihood and prior, there is no need for asymptotic approximations, no need for selection among multitudinous optimality criteria, no need to "do statistics on statistics". There remains only a calculation, for the likelihood and prior determine the posterior distribution, upon which all inference is based. Software such as WinBUGS (Spiegelhalter et al., 2003) provides an easy entree into Bayesian analysis through implementation of Markov chain Monte Carlo (simulation based) evaluation of the posterior distribution. References Bishop, J. A., Cook, L. M. & Muggleton, J., 1978. The response of two species of moths to industrialization in northwest England. II Relative fitness of morphs and populations size. Phil. Trans. R. Soc. Lond. B., 281: 517–540. Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438.

Crosbie, S. F., 1979. The mathematical modelling of capture mark recapture experiments on animal populations. Ph. D. Thesis, Univ. Otago, Dunedin, New Zealand. Crosbie, S. F. & Manly, B. F. J., 1985. Parsimonious modelling of capture mark recapture studies. Biometrics, 41: 385–398. Jolly, G. M., 1965. Explicit estimates from capture recapture data with both death and immigration stochastic model. Biometrika, 52: 225–247. Le Cam, L. M., 1990. Maximum likelihood: an introduction. International Statistical Review, 58: 153–171. Link, W. A., 1999. Modeling pattern in collections of parameters. Journal of Wildlife Management, 63: 1017–1027. Link, W. A. & Barker, R. J. (in press). Modeling association among demographic parameters in analysis of open population capture recapture data, Biometrics. Pradel, R., 1996. Utilization of capture mark recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Schwarz, C. J. & Arnason, A. N., 1996. A general methodology for the analysis of capture recapture experiments in open populations. Biometrics, 52: 860–873. Seber, G. A. F., 1965. A note on the multiple recapture census. Biometrika, 52: 249–259. Spiegelhalter, D., Thomas, A., Best, N. & Lunn, D., 2003. WinBUGS User Manual, version 1.4. http://www.mrc–bsu.cam.ac.uk/bugs White, G. C. & Burnham, K. P., 1999. Program MARK: Survival estimation from populations of marked animals. Bird Study, 46 (Supplement): 120–139.

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Appendix / Apéndice

Cell probabilities of the multinomial distribution [u | u.] are 1, 2,..., t, defined in terms of t – 1 estimable parameters 1,f2,f3,...,ft–2 and t. Parameters 1 and t, like t–1 = t–1 pt in the CJS model, are confounded combinations of demographic and nuisance parameters. Specifically, these are =(

Cell probabilities

+ f 1 ) / p1

and

are defined as follows: Let

and for i = 3,4,...,t – 1.

Let

h 1,

–

) p2 ,

for i = 2,3,...,t–2 and

Then

= ft–1 pt.

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)

451

Components of population growth rate for White–winged Scoters in Saskatchewan, Canada R. T. Alisauskas, J. J. Traylor, C. J. Swoboda & F. P. Kehoe

Alisauskas, R. T., Traylor, J. J., Swoboda, C. J. & Kehoe, F. P., 2004. Components of population growth rate for White–winged Scoters in Saskatchewan, Canada. Animal Biodiversity and Conservation, 27.1: 451–460. Abstract Components of population growth rate for White–winged Scoters in Saskatchewan, Canada.— Breeding range and abundance of White–winged Scoters (Melanitta fusca deglandi) have declined in northwestern North America. Hypotheses proposed to account for this trend are that survival and/or recruitment of females had declined. Thus, we used a reverse–time capture–recapture approach to directly estimate survival, seniority and capture probabilities for females of breeding age at Redberry Lake, Saskatchewan, Canada for 1975–1980 and 2000–2003. We also estimated population size of breeding females for 1975–1985 and 2000–2003 using capture–recapture data. Initially, this local population was in serious decline [95% CL ( ) = 0.89 ± 0.09], but has since stabilized and may be slowly increasing [95% CL ( ) = 1.07 ± 0.11]. This reversal in trajectory apparently resulted from increased recruitment rather than increased apparent survival. Importantly, recent recruitment of adult females appeared to be driven solely by immigration of adult females with no detectable in situ recruitment, suggesting a hypothesis that the local population is being rescued by females produced elsewhere. Key words: Melanitta fusca deglandii, Population growth, Saskatchewan, Survival, Recruitment, White–winged Scoter. Resumen Componentes de la tasa de crecimiento poblacional en el negrón especulado de Saskatchewan, Canadá.— El rango reproductivo y la abundancia del negrón especulado (Melanitta fusca deglandi) han disminuido en la zona noroeste de América del Norte. La hipótesis propuesta para explicar esta tendencia es que se ha producido una disminución en la supervivencia y/o el reclutamiento de hembras. Por consiguiente, utilizamos un enfoque de captura–recaptura con el tiempo invertido para estimar directamente la supervivencia, la jerarquía y las probabilidades de captura de las hembras en edad reproductora del lago Redberry, Saskatchewan, Canadá, durante los periodos 1975–1980 y 2000– 2003. También estimamos el tamaño poblacional de las hembras reproductoras durante los periodos 1975–1985 y 2000–2003, mediante el empleo de datos de captura–recaptura. En un principio, esta población local experimentó una importante disminución [95% CL ( ) = 0,89 ± 0,09], pero posteriormente se estabilizó, y es posible que poco a poco vaya aumentando [95% CL ( ) = 1,07 ± 0,11]. Por lo visto, la inversión de esta trayectoria se produjo como consecuencia de un mayor reclutamiento, en lugar de una mayor supervivencia aparente. Es importante destacar que el reclutamiento reciente de hembras adultas parece haber obedecido exclusivamente a la migración de hembras adultas sin un reclutamiento detectable in situ, lo que sugiere la hipótesis de que la población local está siendo rescatada por hembras procedentes de otros lugares. Palabras clave: Melanitta fusca deglandi, Crecimiento poblacional, Saskatchewan, Supervivencia, Reclutamiento, Negrón especulado. Ray T. Alisauskas, Canadian Wildlife Service, Prairie and Northern Wildlife Research Centre, 115 Perimeter Road, Saskatoon, SK S7N 0X4, Canada; and Dept. of Biology, Univ. of Saskatchewan, 112 Science Place, Saskatoon, Saskatchewan, S7N 5E2, Canada.– Joshua J. Traylor & Cindy J. Swoboda, Dept. of Biology, Univ. of Saskatchewan, 112 Science Place, Saskatoon, Saskatchewan, S7N 5E2, Canada.– F. Patrick Kehoe, Ducks Unlimited Canada, #200 10720–178 St., Edmonton, Alberta, T5S 1J3, Canada. Corresponding author: R. T. Alisauskas. E–mail: ray.alisauskas@ec.gc.ca

ISSN: 1578–665X

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Introduction

Material and methods

Surveys done on the breeding grounds of White– winged Scoters in northwestern North America since 1955 demonstrate declines in continental population size, as well as northward retraction of breeding range (Trost, 1998; fig. 1). The contribution of changes in survival and recruitment to such population declines remains unknown. For instance, there have only been 142 recoveries of 4,006 White–winged Scoters banded in North America from 1923 to 2001. Krementz et al. (1997) noted that such low recovery rates led to imprecise parameter estimates using band recovery models (e.g., Brownie et al., 1985). However, Krementz et al. (1997) were able to estimate probabilities of apparent survival, , and capture, p, for a well–defined population of female White–winged Scoters captured during nesting from 1975 to 1985 on islands in Redberry Lake, Saskatchewan (52.66° N, 107.17° W). They drew inferences about temporal dynamics of productivity from range–wide harvest data and applied those inferences to the dynamics of the population breeding locally at Redberry Lake. Since then, there have been additional developments in the direct estimation of population growth rate using reverse–time, capture–recapture models (Pradel, 1996; Nichols et al., 2000; Nichols & Hines, 2002). These approaches also permit inference in a more tractable fashion about local components of population growth. Seniority probability, , is a useful metric for understanding the proportional contribution of survival to population growth rate, i.e., = / , and is analogous to elasticity for survival (Nichols et al., 2002). Values of seniority approaching 1.0 suggest that there is very little contribution to population growth rate by recruitment. If seniority and survival probabilities are estimable, then can be estimated by substitution. Similarly, local population size, , can be estimated from number of captures, n, and capture probability, . Finally, local recruitment, , defined as the per capita addition of individuals to the local population, can be calculated from the difference in local estimates of and . We analyzed the same capture histories used by Krementz et al. (1997) to directly estimate survival, seniority and capture probabilities. We also initiated a capture–recapture study of nesting females and ducklings on the same population in 2000 because of: (1) declines in scoters over much of the continent (fig. 1); (2) availability of historical data (Krementz et al., 1997); and (3) the location of Redberry Lake in an area of the continent with continuing declines (fig. 1). We reasoned that information gathered during 1975–1985 could serve as a useful historical benchmark for understanding current demography of the Redberry Lake population. While all of our conclusions apply to the Redberry Lake population, some of our inferences likely have range–wide relevance for North American White–winged Scoters (hereafter, scoters).

All scoters used in this analysis were captured during 1975–1985 or 2000–2003 on islands of Redberry Lake in the aspen parkland biome of west–central Saskatchewan, Canada. Redberry Lake is within a Migratory Bird Sanctuary, so there is no mortality of White–winged Scoters from hunting there. Scoters were captured by different sets of researchers in 1975–1980 (Brown, 1981), 1981– 1983 (C. S. Houston, pers. comm.), and 1984– 1985 (Kehoe, 1989). Details about 1975–1985 data representing 520 encounters of 280 nesting females were reported by Krementz et al. (1997); their analysis of capture histories for 1981–1985 did not include captures of unmarked scoters, ui, and only included recaptures of previously marked birds, mi, but this nevertheless permits estimation of survival and detection probabilities. Although not used in their analyses, Krementz et al. (1997) reported numbers of previously unmarked captures, ui, in their table 1, from which we calculated total captures, ni = ui + mi. Assuming equal capture probabilities of new captures and recaptures, we applied capture probabilities from 1981–1985 to ni for respective years to estimate local population size, (see below). Data from 1975 to 1980 only were used for estimation of other vital rates (see below) before 2000–2003 field work. The 2000– 2003 data represent 347 encounters of 184 nesting females. We analyzed both time series independently of one another, as there were no individuals common to both. During 1975–1985, scoters were marked only with legbands, and so reading marks required capture of females. From 2000–2003, all scoters were marked with legbands and nasal markers (Lokemoen & Sharp, 1985). Thus, the 2000– 2003 dataset included captures as well as resights without capture. Hence, all inferences refer to the population of female scoters that nested on the study area during the course of the study; temporary emigrants were considered as part of the local population unless and until they permanently emigrated. We also applied plasticine–filled legbands (Blums et al., 1994), to 265 one–day–old ducklings of both sexes in 2000, 399 in 2001, 273 in 2002, and 622 in 2003. We used Program MARK with capture–recapture/resight data to estimate probabilities of apparent survival, , capture, , and seniority, , based on Pradel’s (1996) model for survival and seniority (White & Burnham, 1999) for 1975–1980 and 2000–2003, using only adult females, i.e., those that had nested at least once. We also used the Cormack–Jolly–Seber (CJS) model to estimate for 1975–1985 data to derive estimates of population size, (see below), but used estimates of from Pradel’s (1996) model mentioned above for estimation of for 2000–2003. We used an information–theoretic approach for selection of models from a candidate set (Burnham & Anderson, 1998) using QAICc based on time dependence, (t), and time independence, (.), of

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Alaska / Yukon

Northern Alberta / B.C. / N.W.T.

700,000 600,000 500,000 400,000 300,000 200,000 100,000 1950 1960 1970 1980 1990 2000

1,800,000 1,600,000 1,400,000 1,200,000 1,000,000 800,000 600,000 400,000 200,000 0 1950 1960 1970 1980 1990 2000

Northern Saskatchevan 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 1950 1960 1970 1980 1990 2000

stable

Northern Manitoba 140,000

Southern Alberta

120,000

40,000 35,000

declining

100,000 80,000

30,000 25,000

no trend

20,000 15,000

60,000 40,000

gone

20,000

gone

10,000 5,000

0 1950

declining

0 1950 1960 1970 1980 1990 2000

Southern Saskatchevan

1960 1970 1980 1990 2000

Southern Manitoba 12,000

14,000 12,000

10,000

8,000

6,000

4,000

4,000 2,000 01950 1960 1970 1980 1990 2000

2,000 0 1950 1960 1970 1980 1990 2000

Fig. 1. Combined population estimates of White–winged (Melanitta fusca deglandi), Surf (Melanitta perspicillata), and Black (Melanitta nigra americana) Scoters from surveys conducted in Canada and Alaska, 1955 to 1998 (Trost, 1998). Although northern strata represent trends in all 3 species, strata in Southern Alberta, Saskatchewan and Manitoba are south of the main breeding distributions of Surf and Black Scoters, and contain mostly White–winged Scoters. Shown are annual estimates for different strata in the survey area with corresponding 5–year running averages. Approximate location of Redberry Lake, Saskatchewan, Canada, is indicated by the star. Fig. 1. Estimaciones poblacionales combinadas del negrón especulado (Melanitta fusca deglandi), negrón careto (Melanitta perspicillata) y negrón especulado (Melanitta nigra americana) a partir de estudios realizados en Canadá y Alaska, entre 1955 y 1998 (Trost, 1998). Aunque los estratos del norte representan tendencias en la totalidad de las tres especies, los estratos de Alberta del Sur, Saskatchewan y Manitoba se encuentran al sur de las principales distribuciones reproductivas del negrón careto y del negrón común americano, y contienen principalmente negretas de alas blancas. Se indican las estimaciones anuales de distintos estratos del área de estudio, junto con los correspondientes promedios de cinco años consecutivos. La estrella indica el emplazamiento aproximado del lago Redberry, Saskatchewan, Canadá.

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Table 1. Models for evaluating annual variation in estimates of probabilities for apparent survival ( ) capture (p) and seniority ( ) for female White–winged Scoters nesting at Redberry Lake, Saskatchewan, Canada, 1975–1980. Temporal variation was either constant (.), annual (t), or linear trend on a logit scale (T). AICc for the best model was 1737.78 with : w. Model weight; L. Model likelihood; K. Estimable parameters; D. Deviance. Tabla 1. Modelos para evaluar la variación anual en las estimaciones de probabilidades de supervivencia aparente ( ), captura (p) , y experiencia ( ) para negretas de alas blancas hembras que anidaron en el lago Redberry, Saskatchewan, Canadá, entre 1975 y 1980. La variación temporal fue constante (.), anual (t), o una tendencia lineal en una escala logit (T). El AICc correspondiente al mejor modelo fue 1737,78 con : w. Peso modelo; L. Modelo de probabilidad; K. Parámetros estimables; D. Desviación.

Model

{ (.) p(t) (.)}

0.00

AICc

0.80

1.00

1039.39

{ (t) p(t) (.)}

4.73

0.08

0.09

1037.84

{ (.) p(T) (.)}

4.74

0.07

0.09

1052.38

{ (.) p(t) (t)}

5.69

0.05

0.06

1038.81

{ (t) p(t) (t)} global

10.45

0.00

0.01

1037.20

{ (t) p(.) (t)}

19.71

0.00

1052.83

{ (.) p(.) (t)}

21.28

0.00

1062.75

{ (.) p(.) (.)}

30.20

0.00

1079.87

{ (t) p(.) (.)}

32.35

0.00

1073.82

each of the parameters. A linear time trend (T) was considered for estimated from Pradel’s (1996) model for both 1975–1980 and 2000–2003 data, because we reasoned that capture efficiency by field researchers may have improved with increased experience at finding and capturing nesting scoters even over only 4–6 years. We adjusted for goodness–of–fit by calculating variance inflation factors, , for estimates of and from CJS models with 1975–1985 data by performing 100 bootstrap simulations of expected deviance for the global model (White & Burnham, 1999); the quotient of observed deviance/simulated deviance for the global model was = 1.1168. from 1975–1980 and Similarly, we calculated 2000–2003 data sets used for variance estimation of , , and by performing bootstrap simulation on respective global models using forward capture histories { (t) p(t)}, and applied respective adjustments to variance in estimates from Pradel’s (1996) models: = 1.0, = 1.1168, and = 1.2457. Hence, we used AICc for model selection involving 1975–1980 data, but applied the corresponding variance inflation factor for calculation of QAICc and adjustment of variance in parameter estimates for 2000–2003 data. Direct estimates of , , and were used to calculate derived estimates, and . We estimated population size in year i as

where ni equals the number of scoters captured in year i, and

is the associated approximate variance suggested by Williams et al. (2002: 503). We estimated population growth rate in year i as a parameter derived by Program MARK, i.e.,

We estimated recruitment probability, as

, ourselves

and the complement to seniority, referred to as entry probability, is the proportional contribution of recruitment to population growth rate related to these as

(Nichols et al., 2000). Approximate 95% confidence limits for derived parameters ( ) were constructed as

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Table 2. Models for evaluating temporal variation in estimates of probabilities for apparent survival ( ), capture (p), and seniority ( ), for female White–winged Scoters nesting at Redberry Lake, Saskatchewan, Canada, 2000–2003. Temporal variation was either constant (.), annual (t), or linear trend on a logit scale (T). QAICc for the best model was 703.49 with : w. Model weight; L. Model likelihood; K. Estimable parameters; QD. Quasi deviance. Tabla 2. Modelos para evaluar la variación temporal en las estimaciones de probabilidades de supervivencia aparente ( ), captura (p), y experiencia ( ), para negrones especulares que anidaron en el lago Redberry, Saskatchewan, Canadá, entre 2000 y 2003. La variación temporal fue constante (.), anual (t), o una tendencia lineal en una escala logit (T). El QAIC correspondiente al mejor modelo fue 703,49 con : w. Modelo de peso; L. Modelo de probabilidad; K. Parámetros estimables; QD. Quasi desviación.

Model

QAICc

{ (.) p(T) (.)}

0.00

0.36

1.00

408.707

{ (.) p(t) (.)}

0.86

0.23

0.65

405.436

{ (t) p(.) (t)}

1.43

0.17

0.49

406.004

{ (t) p(t) (.)}

2.66

0.09

0.26

405.156

{ (.) p(t) (t)}

2.79

0.09

0.25

405.287

{ (t) p(t) (t)} global

4.66

0.03

0.10

405.056

{ (.) p(.) (t)}

6.27

0.02

0.04

412.922

{ (t) p(.) (.)}

9.09

0.00

0.01

417.799

{ (.) p(.) (.)}

12.06

0.00

422.812

Results Of 4 CJS models considered for 1975–1985 data, { (.) p(t)} had w = 0.997, so we estimated using numbers of scoters captured, ni, and capture probability, , in turn estimated from this model. Numbers captured or resighted, ni, were highly variable, ranging from only 2 to 114/yr during 1975–1985, but increasing from 47 to 127 during 2000–2003. Most annual variation in ni (fig. 2A) appeared to be strongly related to capture probability (fig. 2B). We considered the same 9 candidate Pradel models for analysis of 1975–1980 (table 1) and 2000–2003 data (table 2). There was very little uncertainty about the best model { (.) p(t) (.)} for 1975–1980 data with model weight, w = 0.80. There was less certainty for 2000–2003 data, but the two best models were merely variants of temporal variation in (table 2), with cumulative w = 0.59. Hence we used the best models from analysis of each time series for parameter estimation (table 3). On a logit scale, slope between capture probability and year for 1975–80 was 0.44 ± 0.16 (95% CL) estimated from { (.) p(T) (.)}, and 0.73 ± 0.34 for 2000–03 estimated from { (.) p(T) (.)}. These results support the notion of increased efficiency of capture with experience by respective teams of researchers during 1975–1980 (Brown, 1981) and 2000–2003 (this study).

Estimated population size of females that had bred at least once showed a strong decline from ~450 in 1975 to ~100 in 1985 (fig. 2). There was no evidence of substantial recovery between 1985 and 2000–2003 as shown by overlapping 95% CL ( ) for these years. Most recent estimates of population size remained below estimates from 1975–1979. Annual rate of population change for 1975–1980 indicated a population decline with 95% CL ( ) < 1 (table 3). Most recently, suggests absolute population growth with 95% CL ( ) non–overlapping with those for 1975–1980, but with inclusion of unity. The 20% increase in between 1975–1980 and 2000–2003 was accounted for more by a 92% increase in recruitment, than it was by a 9% increase in survival probability (fig. 3). Apparent survival was constant in each time series, but had increased from 1975–1980 to 2000–2003; however 95%CL( ) overlapped between the two time series (table 3, fig. 3). During 1975–1985, survival probability constituted a higher proportion of for which upper 95% CL ( ) < 1.0 (table 3, fig. 3). When the population was declining in 1975–1980, only ~0.12 new females, on average, entered the breeding population each year for every female that had bred the previous year at Redberry Lake; the annual recruitment rate of breeding females had almost doubled by 2000–2003 (table 3).

456

We marked 1559 male and female ducklings marked with permanent plasticine bands during 2000–2003, but could only have recaptured the females because our sampling did not include males. However, no females had been recaptured up to 3 years later. Thus, contrary to expectation, all recruitment at Redberry Lake apparently was through immigration of adults from elsewhere, or possibly of some unmarked ducklings > 4–years–old hatched at Redberry Lake before 2000. Discussion Our retrospective analysis indicated that the population of White–winged Scoters nesting at Redberry Lake was in serious decline during 1975–1985, shrinking from at least 283 [i.e., lower 95% ( )] females to, at most, 136 [upper 95% ( )] females capable of breeding (fig. 2C). Since then, findings from our own mark–recapture study during 2000–2003 suggested that the population had increased somewhat, and most recently was at least stable or continuing to increase slowly. Precision of some of our recent estimates was poor compared to 1975–1985, but we anticipate that this will improve with additional years of study. The acute local population decline during 1975–1985 was accompanied by stable survival rates (table 3, fig. 3), as also found by Krementz et al. (1997). The striking change in population trajectory between 1975–1985 and 2000–2003 was largely the result of improved recruitment, although estimates of apparent survival of female scoters that had bred at least once had also increased from 0.77 to 0.84. Vital rates that contribute to in situ recruitment include age of first reproduction, breeding propensity, clutch size, nest success, duckling survival, annual survival after fledging before breeding age, and duckling philopatry. Additionally, immigration rate, i+1 (expressed as the ratio of immigrants in year i + 1, Ii+1, to adults from the previous year, Ni ), added to in situ recruitment equals total recruitment. We have not yet estimated breeding probability, which may have declined. However, Traylor et al. (in press) found that White–winged Scoters at Redberry Lake nested later in 2000–2001 and had protracted nesting of 41 days between nest initiation and hatch, compared to 1977–1980 when mean nesting duration was 36–39 days. They suggested that later and protracted nesting may have resulted from poor nutrition of breeding females which may impinge on their ability to successfully complete incubation and to raise offspring. Nest success and egg hatchability in 2000–2001 (Traylor et al., 2004) was comparable to that estimated by Brown (1981). However, Traylor (2003) estimated that 95% CL ( ) was only 0.016 ± 0.015 and 0.021 ± 0.021 in 2000 and 2001, respectively, for the first 28 days after hatch at Redberry Lake; to our knowledge, these are the lowest estimates ever published for any North American duck species.

Alisauskas et al.

Our findings are consistent with Traylor’s (2003) and indicate that improved recruitment during 2000–2003 was not the result of in situ production of ducklings. White–winged Scoter females are physiologically capable of nesting as 2–year–olds; of 1,740 ducklings marked with webtags over 4 years (1977–1980) by Brown (1981), 3 were detected nesting at Redberry Lake as 2–year–olds, and 7 as 3 years olds. We marked a comparable number of 1,559 ducklings over the same number of years (2000–2003). Despite (1) our use of more permanent plasticine–filled legbands (Blums et al., 1994) compared to webtags with lower retention used by Brown (1981), and (2) generally higher capture probabilities in 2000–2003 compared to 1977–1980 (fig. 2B), we had not detected any as breeding adults originally marked as ducklings. Although probability of duckling survival to 28 days of age was very low at Redberry Lake in 2000–2001 (Traylor, 2003), the discrepancy in return rates between 1975–1985 and 2000–2003 may also have resulted from delays in age of first nesting. If some recruitment during 2000–2003 resulted from in situ production of ducklings, these could only have been produced before 2000 and so would be at least 4–years–old. If true, then age of first reproduction may have increased in this population. Permanent emigration of ducklings from Redberry Lake may have increased recently. However, we suspect that low survival rate of ducklings is the most important factor behind low in situ production and, instead, recruitment into the Redberry Lake population appeared to be entirely from immigration of adults produced elsewhere. While the scope of our inferences is most pertinent to the Redberry Lake population, we also suggest that attention be focused on ecological factors that influence recruitment to better understand dynamics of White–winged Scoter populations in southern Saskatchewan. Estimates of apparent survival provide a minimal estimate of true survival, and may be representative continentally. If true, our results suggest that long– term changes in survival played only a minimal role, at best, in declines in Prairie Canada and possibly in the boreal forest (fig. 1). Kehoe (1994) suggested that disappearance of White–winged Scoters from some historic breeding sites may have been related to local harvest pressure, although this seems unlikely for widespread declines that have occurred until recently. There is no hunting on Redberry Lake, a migratory bird sanctuary, so local harvest pressure was not a factor during this study. Afton & Anderson (2001) similarly suggested that harvest likely was not related to recent continental declines in Greater (Aythya marila) and Lesser (A. affinis) Scaup. White–winged Scoters (Dobush, 1986) and waterfowl in general (Alisauskas & Ankney, 1992) rely on nutrient reserves, often stored great distances from nesting areas, for use during breeding. Annual variation in levels of prebreeding nutrient reserves can influence continental production

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130 120 110 100 90 80 70 60 50 40 30 20 10 0 1974 1.0

1979

1984

1989

1994

1999

2004

0.9

)

0.7

95% CL (

0.8

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1974

1979

1984

1989

1994

1999

2004

700

95% CL (

)

600 500 400 300 200 100 0 1974

1979

1984

1989 Year, i

1994

1999

2004

Fig. 2. A. Numbers of nesting scoters captured or sighted, n; B. Detection probabilities, 95% CL ( ); and C. Annual estimates of population size, 95% CL ( ), for female White–winged Scoters nesting at Redberry Lake, Saskatchewan, 1975–1985 and 2000–2003. Open circles denote numbers of annual captures, ni, range 4–15, otherwise closed circles denote ni > 45. Fig. 2. A. Número de negrones comunes anidados que fueron capturados o avistados, n; B. Probabilidades de detección, 95% CL ( ); C. Estimaciones anuales del tamaño poblacional, 95% CL ( ), para los negrones especulares que anidaron en el lago Redberry, Saskatchewan, entre 1975 y 1985, y entre 2000 y 2003. Los círculos blancos indican el número de capturas anuales, ni, rango 4–15, mientras que los círculos negros indican ni > 45.

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1.4 1.2

Recruitment Survival

)

1.0 95% CL(

of young waterfowl subsequent to the storage of such reserves (Alisauskas, 2002). Several hypotheses related to the connection between events during winter or spring migration and subsequent recruitment have been proposed to explain the decline in population estimates of scoters and other duck species nesting in the boreal forest and prairie biomes of northwestern North America. For example, Afton & Anderson (2001) suggested that declines in estimates of continental scaup (Aythya spp.) populations may stem from females arriving on breeding areas with lower nutritional reserves than in the past, possibly due to diminished food resources. As noted above, Traylor et al. (2004) speculated that depleted body reserves may have protracted incubation duration, thereby delaying hatch date in scoters at Redberry Lake. A general delay in nesting often lowers nest success in waterfowl (Flint & Grand, 1996) reducing the number of recruits to local populations (Dzus & Clark, 1998; Blums et al., 2002). Large–scale changes (i.e., degradation) in quality of nesting habitat, and human alteration of lakes used by breeding waterfowl through development for recreational use or agriculture may have affected recruitment (Turner et al., 1987). Agriculture, commercial forestry, and mineral extraction may favor increased abundance and foraging efficiency of generalist predators (Krasowski & Nudds, 1986; Turner et al., 1987). The main predators of ducklings at Redberry Lake were California (Larus californicus) and Ring–billed (L. delawarensis) Gulls (Traylor, 2003). Thus, increased predation pressure from growing populations of gulls, in response to the growth in size and number of garbage dumps associated with human activity, may impinge directly on recruitment of White–winged Scoters.

0.8 0.6 0.4 0.2 0.0 1975–1980

2000–2003

Fig. 3. Estimates of annual rate of population change, 95% CL ( ), for female White–winged Scoters breeding at Redberry Lake, Saskatchewan, Canada, 1975–1985 and 2000–2003. Also shown are estimates of survival ( ) and recruitment ( ) the components of . Fig. 3. Estimaciones de la tasa anual de cambio poblacional, 95%CL ( ) correspondientes al nedrón especular que se reprodujeron en el ago Redberry, Saskatchewan, Canadá, entre 1975 y 1985, y entre 2000 y 2003. También se indican las estimaciones de supervivencia ( ) y el reclutamiento ( ) así como los componentes de .

Table 3. Ninety–five % confidence limits of estimates of population parameters for female White– winged Scoters nesting at Redberry Lake, Saskatchewan, Canada, 1975–1980 and 2000–2003. Estimates shown are from the best of Pradel’s (1996) models considered from separate analysis of 1975–1980 data { (.) p(t) (.)} and 2000–2003 data { (.) p(T) (.)}: . Apparent survival; . Seniority; . Annual rate of population change; . Recruitment; . Entry. Tabla 3. Límites de intervalos de confianza del 95% en las estimaciones de parámetros poblacionales para los negrones especulares que anidaron en el lago Redberry, Saskatchewan, Canadá, entre 1975 y 1980, y 2000 y 2003. Las estimaciones indicadas corresponden al mejor de los modelos de Pradel (1996) considerados en análisis independientes de datos obtenidos entre 1975 y 1980 { (.) p(t) (.)} y entre 2000 y 2003 { (.) p(T) (.)}: . Supervivencia aparente; . Precedencia; . Tasa anual de cambio poblacional; . Reclutamiento; . Entrada.

Year 1975–1980

0.77 ± 0.08

0.87 ± 0.09

0.89 ± 0.09

0.12

0.13 ± 0.09

2000–2003

0.84 ± 0.07

0.78 ± 0.09

1.07 ± 0.11

0.23

0.22 ± 0.09

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Heavy metals acquired on winter or migration areas (White & Finley, 1978) and subsequent biomagnification of other contaminants (Di Giulio & Scanlon, 1984) may disrupt reproductive potential by reducing breeding propensity, egg size, egg hatchability, clutch size, and/or duckling survival. Contaminants also reduce adult survival (DeKock & Bowmer, 1993; Perkins & Barclay, 1997), although this has never been tested directly in free–ranging waterfowl, in either a design–based (i.e., experimental) or model–based fashion other than by Grand et al. (1998) to our knowledge. Thus, identification of wintering/migration areas of breeding females and associated levels of contaminants may provide further insights on reduced breeding success and survival of scoter nests and ducklings. Changing food resources, varying harvest pressures and contaminants on wintering areas all have potential to limit reproductive success and local recruitment on breeding areas in a multifactorial fashion. As events on these areas are not mutually exclusive, it is important to link breeding and wintering areas to address conservation issues and to understand factors that influence population dynamics. Consequently, we have expanded our research scope with a view toward a better understanding of ecological processes behind variation in survival and recruitment in White–winged Scoters. In conjunction with continued mark–recapture of nesting females, our objectives have expanded to include development of methodology based on stable isotope analyses for assigning breeding females from this population to winter areas, as was done successfully for King Eiders (Somateria spectabilis; Mehl et al., in press). As well, we have taken blood samples during 2001– 2003 to evaluate concentrations of Cadmium, Mercury, Lead and Selenium on annual survival and on components of recruitment. Linkage of individuals nesting at Redberry Lake to respective wintering areas may uncover differences in contaminant loads, nest success, and egg hatchability in relation to winter origin, and identify areas responsible for limiting recruitment. Inferences from those findings could have relevance to much of the continental population. Finally, as have others (e.g., Nichols, 1992; Anderson, 2001), we caution against drawing inferences about population change strictly from counts or number of animal captures. Numbers of White– winged Scoters captured annually (fig. 2A) increased regardless of whether the local population was in a precipitous decline ( l 0.89) during 1975–1980, or was increasing ( l 1.07) during 2000–2003 (fig. 2C). Instead, annual number of captures was related largely to increasing capture efficiency of field researchers in each phase of study (fig. 2B). Inferences about size or dynamics of free–ranging populations should be based on methods that account for variation in capture or detection probabilities, and that provide unbiased estimates. This seems particularly critical in situations for which there may be great conservation concerns.

Acknowledgments We thank David G. Krementz for graciously supplying capture histories from 1975–1985, and Patrick W. Brown and C. Stuart Houston for their efforts capturing and marking scoters at Redberry Lake. Jason Traylor, Hollie Remenda, Credence Wood, and Mike Hill and associates from Ducks Unlimited (Canada), assisted with field work during 2000–2003. Research during 2000–2003 was supported by the Institute for Wetlands and Waterfowl Research, Ducks Unlimited (Canada), Redberry Lake World Biosphere Reserve, Canadian Wildlife Service, and the University of Saskatchewan. Earlier versions of this paper were improved greatly with suggestions from Bob Clark, Steve Dinsmore, Jim Nichols and Eric Reed. References Afton, A. D. & Anderson, M. G., 2001. Declining scaup populations: a retrospective analysis of long–term population and harvest survey data. Journal of Wildlife Management, 65: 781–796. Alisauskas, R. T. & Ankney, C. D., 1992. The cost of egg laying and its relationship to nutrient reserves in waterfowl. In: Ecology and Management of Breeding Waterfowl: 30–61 (B. D. J. Batt, A. D. Afton, M. G. Anderson, C. D. Ankney, D. H. Johnson, J. A. Kadlec & G. L. Krapu, Eds.). Univ. of Minnesota Press, Minneapolis. Alisauskas, R. T., 2002. Arctic climate, spring nutrition, and recruitment in mid–continent Lesser Snow Geese. Journal of Wildlife Management, 66: 181–193. Anderson, D. R., 2001., The need to get basics right in wildlife field studies. Wildlife Society Bulletin, 29: 1294–1297. Blums, P., Mednis, A. & Nichols, J. D., 1994. Retention of web tags and plasticine–filled leg bands applied to day–old ducklings. Journal of Wildlife Management, 58: 76–81. Blums, P., Clark, R. G. & Mednis, A., 2002. Patterns of reproductive effort and success in birds: path analyses of long–term data from European ducks. Journal of Animal Ecology, 71: 280–295. Brown, P. W., 1981. Reproductive ecology of the White–winged Scoter. Ph. D. Thesis, Univ. of Missouri. Brownie, C., Anderson, D. R., Burnham, K. P. & Robson, D. S., 1985. Statistical inference from band recovery data – a handbook. Resource publication / United States Department of the Interior, Fish and Wildlife Service; number 156. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference: a practical information– theoretic approach. Springer–Verlag, New York. Dekock, W. C. & Bowmer, C. T., 1993. Bioaccumulation, biological effects, and food chain transfer of contaminants in the zebra mussel (Dreissena polymorpha). In: Zebra Mussel Biology, Impacts, and Control: 503–533 (T. F. Nalepa

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& D. W. Schoesser, Eds.). Lewis Publishers, Boca Raton. Di Giulio, R. T. & Scanlon, P. F., 1984. Heavy metals in tissues of waterfowl from Chesapeake Bay, U.S.A. Environmental Pollution Service A., 35: 29–48. Dobush, G. R., 1986. The accumulation of nutrient reserves and their contribution to reproductive success in the White–winged Scoter. M. Sc. Thesis, Univ. of Guelph. Dzus, E. & Clark, R. G., 1998. Brood survival and recruitment of mallards in relation to wetland density and hatching date. Auk, 115: 311–318. Flint, P. L. & Grand, J. B., 1996. Nesting success of Northern Pintails on the coastal Yukon– Kuskokwim Delta, Alaska. Condor, 98: 54–60. Grand, J. B., Flint, P. L., Peterson, M. R. & Moran, C. L., 1998. Effect of lead poisoning on Spectacled Eider survival rates. Journal of Wildlife Management, 62: 1103–1109. Kehoe, F. P., 1989. The adaptive significance of crèching behavior in the White–winged Scoter (Melanitta fusca deglandi). Canadian Journal of Zoology, 67: 406–411. – 1994. Status of sea ducks in the Atlantic Flyway with strategies toward improved management. Ad hoc Sea Duck Committee, Atlantic Flyway Technical Section. Krasowski, T. P. & Nudds, T. D., 1986. Microhabitat structure of nest sites and nesting success of diving ducks. Journal of Wildlife Management, 50: 203–208. Krementz, D. G., Brown, P. W., Kehoe, F. P. & Houston, C. S., 1997. Population dynamics of White–winged Scoters. Journal of Wildlife Management, 61: 222–227. Lokemoen, J. T. & Sharp, D. E., 1985. Assessment of nasal marker materials and designs used on dabbling ducks. Wildlife Society Bulletin, 13: 53–56. Mehl, K. R., Alisauskas, R. T., Hobson, K. A. & Merkel, F. (in press). Linking breeding and wintering grounds of King Eiders: making use of polar isotopic gradients. Journal of Wildlife Management. Nichols, J. D., 1992. Capture–recapture models:

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using marked animals to study population dynamics. Bioscience, 42: 94–102 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. & Hines, J. E., 2002. Approaches for the direct estimation of , and demographic contributions to , using capture–recapture data. Journal of Applied Statistics, 29: 539–568. Perkins, C. R. & Barclay, J. S., 1997. Accumulation and mobilization of organochlorine contaminants in wintering greater scaup. Journal of Wildlife Management, 61: 444–449. Pradel, R., 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Traylor, J. J., 2003. Nesting and duckling ecology of White–winged Scoters (Melanitta fusca deglandii) at Redberry Lake, Saskatchewan. M. Sc. Thesis, Univ. of Saskatchewan. Traylor, J. J., Alisauskas, R. T. & Kehoe, F. P. (2004). Nesting ecology of White–winged Scoters (Melanitta fusca deglandi) at Redberry Lake, Saskatchewan. Auk, 121: 950–962. Trost, R., 1998. 1998 Pacific Flyway data book. U.S. Fish and Wildlife Service. Turner, B. C., Hochbaum, G. S., Caswell, F. P. & Nieman, D. J., 1987. Agricultural impacts on wetland habitats on the Canadian Prairies, 1981– 85. Transactions of the North American Wildlife and Natural Resource Conference, 52: 206–215. White, D. H. & Finley, M. T., 1978. Uptake and retention of dietary cadmium in mallard ducks. Environmental Research, 17: 53–59. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 (suppl.): S120– 139. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations: modeling, estimation, and decision making. Academic Press, San Diego.

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Occupancy as a surrogate for abundance estimation D. I. MacKenzie & J. D. Nichols

MacKenzie, D. I. & Nichols, J. D., 2004. Occupancy as a surrogate for abundance estimation. Animal Biodiversity and Conservation, 27.1: 461–467. Abstract Occupancy as a surrogate for abundance estimation.— In many monitoring programmes it may be prohibitively expensive to estimate the actual abundance of a bird species in a defined area, particularly at large spatial scales, or where birds occur at very low densities. Often it may be appropriate to consider the proportion of area occupied by the species as an alternative state variable. However, as with abundance estimation, issues of detectability must be taken into account in order to make accurate inferences: the non–detection of the species does not imply the species is genuinely absent. Here we review some recent modelling developments that permit unbiased estimation of the proportion of area occupied, colonization and local extinction probabilities. These methods allow for unequal sampling effort and enable covariate information on sampling locations to be incorporated. We also describe how these models could be extended to incorporate information from marked individuals, which would enable finer questions of population dynamics (such as turnover rate of nest sites by specific breeding pairs) to be addressed. We believe these models may be applicable to a wide range of bird species and may be useful for investigating various questions of ecological interest. For example, with respect to habitat quality, we might predict that a species is more likely to have higher local extinction probabilities, or higher turnover rates of specific breeding pairs, in poor quality habitats. Key words: Occupancy, Species distribution, Abundance, Metapopulation, Monitoring. Resumen La ocupación como sustituto de la estimación de la abundancia.— En muchos programas de monitorización puede resultar extremadamente caro estimar la abundancia real de una especie de ave en un área definida, especialmente a grandes escalas espaciales, o donde las aves se dan a densidades muy bajas. A menudo, es posible que resulte conveniente considerar la proporción del área ocupada por la especie como una variable de estado alternativa. Sin embargo, al igual que sucede con la estimación de la abundancia, para poder realizar deducciones exactas es preciso tener en cuenta ciertas cuestiones de detectabilidad: el hecho de que una especie no pueda detectarse no significa que realmente esté ausente. En este estudio analizamos algunos modelos de reciente desarrollo que permiten una estimación no sesgada de la proporción del área ocupada, de la colonización y de las probabilidades de extinción local. Estos métodos permiten un esfuerzo de muestreo desigual, así como la posibilidad de incorporar información sobre covariantes en los emplazamientos de muestreo. También describimos el procedimiento para ampliarlos a fin de incorporar información acerca de individuos marcados, lo que permitiría abordar con mayor detalle cuestiones acerca de la dinámica poblacional (como el índice de rotación de los emplazamientos de los nidos por parte de parejas de reproducción específicas). Consideramos que estos modelos podrían aplicarse a una amplia gama de especies de aves, pudiendo resultar útiles para investigar diversas cuestiones de interés ecológico. Por ejemplo, respecto a la calidad del hábitat, podríamos predecir que una especie presenta más probabilidades de extinción local, o índices de rotación más elevados de determinadas parejas de reproducción, en hábitats de baja calidad. Palabras clave: Ocupación, Distribución de especies, Abundancia, Metapoblación, Control. Darryl I. MacKenzie, Proteus Wildlife Research Consultants, P.O. Box 5193, Dunedin, New Zealand.– James D. Nichols, USGS Patuxent Wildlife Research Center, 11510 American Holly Drive, Laurel, MD 20708–4017, U.S.A. Corresponding author: D. I. MacKenzie. E–mail: darryl@proteus.co.nz

ISSN: 1578–665X

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Introduction One of the primary roles of a wildlife monitoring program should be to track the status of populations so that substantial changes can be identified and appropriate management actions taken. Abundance (the number of individuals in a population) is one measure that can be used to characterise the state of a population for a single species, with changes in abundance reflecting changes in the population’s status. However, in order to make accurate conclusions about changes in abundance, it is important that the probability of observing an individual is incorporated into our inferential process (e.g., Yoccoz et al., 2001; MacKenzie & Kendall, 2002; Williams et al., 2002; Schmidt, 2003). This often requires that individuals are identifiable (either by natural patternings such as colour patterns, or by applying unique marks such as rings) to keep accurate records of the number of encounters for each animal. For some bird species, especially those that are difficult to capture, this may require a level of effort that is infeasible to sustain as part of a long–term monitoring program, particularly at a reasonably large spatial scale. An alternative state variable that could be considered in many situations is the proportion of area occupied by a species (which we refer to henceforth as occupancy). Determining whether a target species is present at a sampling location may be much less costly than collecting the relevant information (if possible at all) for estimating the number of individuals in an area. Such an approach has been considered in the past for a number of bird species including the northern spotted owl (Azuma et al., 1990), marbled murrelet (e.g., see Stauffer et al., 2002) and goshawks (P. Kennedy, pers. comm.). The reasoning behind using occupancy rather than abundance is that at an appropriate scale the two state variables should be positively correlated (i.e., occupancy may increase with increasing abundance), although it should be noted that the two state variables are addressing distinctly different aspects of the population dynamics. While intuitively the questions "What fraction of the landscape does the species occupy?" and "How many individuals of this species are in the landscape?" are similar, it must be recognised that some changes in the size of the population may not be identified using an occupancy approach to monitoring (e.g., changes in animal density) and that some changes in range and occupancy may not be reflected by changes in abundance. However, for certain species the discrepancies between the two state variables may be minimal if the size of sampling unit is chosen appropriately. For example, the number of breeding pairs of a territorial bird species (such as many raptors) may be closely related to occupancy if the sampling unit is chosen to be approximately the same size as a nesting territory. There may also be situations where occupancy is actually the state variable of direct interest, such as when investigating changes in species range and metapopulation incidence functions.

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Few species are likely to be so conspicuous that they will always be detected at a sampling unit (site) when present. Dependent upon the survey methods being used, there may be a reasonable chance that the species goes undetected and is declared to be “falsely absent”. By not correcting for the fact that the species may go undetected, a naïve count of the number of sites where the species is detected will underestimate the true level of occupancy. Furthermore, inferences about changes in occupancy based upon an observed difference between two (or more) naïve counts should be made with caution, as the difference may be the result of a change in our ability to detect the species rather than a change in occupancy. The arguments against using a naïve count for occupancy are very similar to those given for not using a simple count as an index of abundance (e.g., Yoccoz et al., 2001, MacKenzie & Kendall, 2002; Williams et al., 2002; Schmidt, 2003). Recently there have been a number of methodological advances for modelling occupancy data while explicitly allowing for the fact that the species may go undetected at a site when present. These can be classified into single season models with homogenous detection probabilities (MacKenzie et al., 2002; Tyre et al., 2003); a single season model with heterogeneous detection probabilities caused by variation in abundance (Royle & Nichols, 2003); and a multiple season model without heterogeneity (Barbraud et al., 2003; MacKenzie et al., 2003). These new likelihood–based approaches provide a statistically robust framework for modelling occupancy data, enabling occupancy to be seriously considered as a surrogate for abundance in monitoring programs. There are strong similarities between these methods and mark–recapture models for individual animals, but there are also some subtle differences in their application. In this paper we briefly review the multiple season model of MacKenzie et al. (2003). This is very similar to the approach of Barbraud et al. (2003) although the latter approach only models the detection histories following the season in which the species was first detected at the site. The differences between the approaches of MacKenzie et al. (2003) and Barbraud et al. (2003) are in some ways analogous to the differences between the Jolly–Seber and Cormack–Jolly–Seber mark–recapture models (e.g., Seber, 1982; Williams et al., 2002). We also outline how information from marked individuals may be incorporated into the model. These approaches to the modelling of occupancy dynamics may be very useful for identifying the underlying processes that generate patterns in occupancy (e.g., metapopulations). In particular, we believe that such modelling is likely to be more useful than the common approach of attempting to draw inferences about such processes by observations of occupancy pattern over space at a single point in time (e.g., Hanski, 1992, 1994, 1997). Indeed, there are often many different bio-

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logically reasonable processes that can result in the same pattern of occupancy (e.g., Tyre et al., 2001). This should not be surprising. As an analogy, suppose that you are given a randomly selected photograph from a stack of photographs taken throughout a football game. You are then asked to comment on the current state of the game, and how the game has progressed up to that point. It would be possible to tell the current state of play such as which team has the ball and possibly the score; however it would be impossible to make further comment on how the game has progressed. Not until you are able to go through the entire stack of photographs (in order) would you be able to get some idea of how the game progressed. It is the same situation in ecological studies where processes of population dynamics can only be fully understood by observing the population at systematic points in time, noting how the patterns change and modelling these changes in terms of relevant rate parameters. Basic sampling scheme Suppose we wish to estimate the level of occupancy for a target species in some arbitrarily defined "area". The term "area" is used ambiguously here, and may consist of a continuous region such as a forest or national park, or it may be a collection of discrete habitat patches such as ponds or fragmented forest stands. The area can be considered as a collection of subunits that we shall generically refer to as sites. Depending upon the situation and target species, a site may constitute a suitably sized quadrat, potential nesting territory or an individual habitat patch. At n chosen sites, multiple presence/absence (or more correctly detection/nondetection) surveys are conducted for the target species over a relatively short timeframe: a season. During the season all sites are closed to changes in occupancy so that sites are either always occupied or always unoccupied (this may be relaxed as long as the changes are completely random, although it alters the interpretation of the parameters, e.g. proportion of area occupied becomes proportion of area used). Careful consideration needs to be given to the exact method for selecting the n sites from the area of interest. One of the fundamental rules for statistical inference states that in order to be able to generalize the results from the study sites to the larger area, the sites must be selected from the larger area using a valid probability sampling scheme (e.g., random sampling). This is sometimes overlooked in ecological studies. Failure to select the sites appropriately may lead to estimates that do not correspond to the desired characteristic of the population. We do not give further consideration to the issue of site selection here as the best advice is often situation specific, but we wish to highlight that it is an important issue that is often not given adequate deliberation.

The series of detections and nondetections from the repeated surveys of a generic site i can be recorded as a sequence of 1’s and 0’s (respectively), which we refer to as a detection history (Hi). For example Hi = {10 00 11} would denote that the site has been surveyed for three seasons, with two surveys per season. In this case the species was detected in the first survey of season one; not detected at all in season two; then detected in both surveys during season three. By modelling the underlying stochastic processes that may have caused the observed detection history (just as in much of mark–recapture modelling), we can build a model that will enable us to estimate the quantities of interest. A multiple season model Let 1 be the probability a site is occupied by the species in the first season (t =1) and pt,j be the probability of detecting the species, given presence, in survey j within season t. Further, let t denote the probability an unoccupied site becomes occupied by the species between seasons t and t+1 (colonization), and let t denote the probability a site that was occupied by the species in season t, is unoccupied in t+1 (local extinction). These dynamic parameters enable the modelling of changes in occupancy that may occur between seasons. For any given detection history, these parameters can be used to describe the process that may have resulted in the observed data. For example, consider the history Hi = {01 00} indicating that the species was detected in the second survey of the site in season 1, and undetected otherwise. Obviously the site was occupied in the first season with the species being detected, hence the probability of observing the first season’s data would be 1(1 – p1,1)p1,2, but in the second season there are two options. Either the species did not go locally extinct and was not detected in either survey, with probability

or the species did go locally extinct between seasons so it was not there to be detected (with probability 1). The probability of observing the complete history would therefore be:

. A slightly more complicated second example would be for the history Hi = {00 11}. Now there are two options for the occupancy state of the site in the first season; therefore we must consider the possible processes that could have resulted in the site being occupied immediately before the start of the surveys for the second season. Either the spe-

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cies was present, but undetected, in the first season then did not go locally extinct, or the species was not present at the site in the first season and colonized the site between seasons. The probability of observing this history could therefore be expressed as:

Generally, however, there could be a large number of possible pathways that would result in the same detection history. It is therefore useful to define a transition probability matrix that details how sites can transition between an occupied and unoccupied state between seasons t and t + 1 (1). A row vector must also be defined to indicate which occupancy state the site is in the first season (2). (1)

Once the likelihood has been defined then estimation may proceed using either maximum likelihood or Markov chain Monte Carlo. It is worth noting that based upon the probabilities defined above it is possible to derive two additional biologically relevant quantities, the probability of occupancy in any given year (5) and the rate of change in occupancy between successive years (6). = t–1(1 – t–1) + (1 – t–1) t–1 (5) t (6) The model may even be reparameterized so that these quantities are estimated (or modelled) directly. However, experience to date suggests that it can be difficult to obtain convergence on the estimates for reparameterized models. Extensions Missing observations

(2) A detection probability column vector needs to be defined that indicates the probability of observing the portion of the detection history relating to season t, pHt, conditional upon each state. Whenever the species is detected at least once during a season, then the second element must be zero as clearly the site cannot be in the unoccupied state (for example see equation (3). Conversely, when the species is not detected within a season, then there is some probability associated with the occupied state, and if the site is unoccupied then not detecting the species is the only possible history that could be observed that season (4). (3)

(4) The probability of observing any given detection history can now be easily calculated by using the following expression,

where D(pH,t) is a diagonal matrix with the elements of pH,t along the main diagonal (top left to bottom right), zero otherwise, and T is the number of seasons of data collection. The model likelihood is then calculated in the usual manner assuming that the detection histories from the n sites are independent. L (

, , ,pxH1,...,Hn) =

A likely feature of many ecological studies is the existence of missing observations. In some instances it might not be possible to collect the required data: weather conditions may prevent access to some sites; vehicles may breakdown en route; or logistically it may not be possible to sample all sites within a suitably small time frame. MacKenzie et al. (2002) and MacKenzie et al. (2003) show that missing observations can easily be incorporated into the models described above. In effect, the detection probability for the respective survey of a site is set to zero, which fairly reflects the fact that the species could not be detected (even if present) as no survey was conducted at that time. Essentially, this removes the detection probability parameter from the model likelihood (with respect to the site and time in question). The ability of the model to handle missing observations has important ramifications for study designs, as it enables different sites to have different sampling intensities. Incorporating covariates Often researchers may be interested in potential relationships between the model parameters (occupancy, colonization, local extinction and detection probabilities) and characteristics of the sites or generalized weather patterns (e.g., drought years). Further, the surveyor’s ability to detect the species during any given survey may also be affected by localized conditions at the sampling site (e.g. weather conditions or intensity of nearby traffic noise). Using the logistic model (7), MacKenzie et al. (2003) detail how such covariate information can be incorporated. The logistic model allows the relationship between the probability of interest for site i ( i) and the respective covariate (or covariates; Yi) to be modelled, where (which may be a vector) is the magnitude or coefficient

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for the covariate(s). Analyses of this type could be considered as generalised logistic regression analyses, where allowance has been made for uncertainty in the binary observation of occupancy state (due to imperfect detectability). (7) It should be noted that the logistic model is not the only possible method for including covariate information, and that other functional forms may be used if desired. Including information from marked individuals An important question that often arises in various ecological studies is whether sites that are continuously occupied are occupied by the same individuals or whether there is instead turnover of animals at some sites (the "rescue effect" of Brown & Kodric– Brown, 1977). In our opinion, it is not possible to reliably differentiate between the two possibilities from detection/nondetection data, and auxiliary information is required. Such information may be obtained from having uniquely marked individuals in the study population. Below we conceptualize how the above modelling approach could be extended to include this type of information into our inference. We imagine that such an approach would be useful for species where a site is only occupied by a single unit such as a single individual or where small groups effectively exist as a single unit (e.g., breeding pairs). We could now consider that a site may be in one of three possible mutually exclusive states; i) occupied by the same individual as the previous season (state S); ii) occupied by a different individual from the previous season (D); or iii) not occupied by the species (N). However in the first season, there is no information regarding which sites were occupied by which individuals in the previous year as the sites were not previously being monitored. Therefore, in the first season there are only two states that can be considered; occupied and unoccupied. The transition probability matrices for t m 1 can then be redefined as (see table 1 for parameter definitions); S

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element of the transition matrix will always be zero. The probabilities in each row of the transition matrix should sum to 1.0, hence not all of the parameters can be independently estimated (i.e., the third probability could be obtained by subtraction), although we have presented the concepts here in terms of a very general model. In practice not all of the parameters may be identifiable; this is a continuing area of research. However, various constraints could be imposed upon the parameters to express (and compare competing) plausible biological hypotheses. For example, is the probability that a new individual occurs at a site different for sites that had an established individual last season (transition: StD) from those sites that had a new individual last season (DtD). Such a hypothesis could be investigated by comparing sets of models where the constraint S = tD is imposed against models without such a t constraint. In any given season, however, there are four types of observations that could be made. The same or a different individual may be detected at the site, the species may not be detected at all during the season (which could mean the site is truly in any of the three states because of imperfect detectability), and fourthly, because not all of the individuals may be marked, the species may be observed there but it is unknown whether it is the same or a different individual occupying the site (U). However these can be easily accommodated within redefined capture probability vectors. In the case where the site’s state is known with certainty, there is only one non–zero element in the vector, i.e.,

where {D} denotes some detection history within season t that indicates the site is occupied by a different individual, and p't is the probability of observing the specific sequence of detections within a season. For seasons where the species was not detected, all three elements will be non–zero indicating that the site may have been in any state, i.e.,

for t m 2.

Rows of t denote the occupancy state of sites in season t, and columns denote the state in season t + 1. Between any two seasons, then all possible transitions are possible except that a site can not go from an unoccupied state to being occupied by the same individual the following season (as no individual was there previously), hence the bottom–left

For the final situation where the site is known to be occupied, but it unknown whether it is a new or previous occupant, then the first two elements will be non–zero, i.e.,

Deriving the probability for a given detection history and calculating the model likelihood can then proceed as above.

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Table 1. Definition of parameters (P) used to describe the transitions between states for occupancy studies with information from marked individuals. Tabla 1. Definición de los parámetros (P) utilizados en los estudios de ocupación para describir transiciones entre estados, con información proporcionada por individuos marcados.

Definition of parameters

Probability that a site unoccupied in season t is occupied in t + 1

Probability that an occupied site becomes unoccupied between seasons 1 and 2

Probability that a site occupied by an established individual in season t, is unoccupied in t + 1

D t

Probability that a site occupied by a new individual in season t, is unoccupied in t + 1

Probability that the same individual occupies the site in season 2 as in season 1

Probability that a site occupied by an established individual in season t is occupied by the same individual in t + 1

Probability that a site occupied by a new individual in season t is occupied by the same individual in t + 1

Probability that a site occupied by an established individual in season t is occupied by a different individual in t + 1

Probability that a site occupied by a new individual in season t is occupied by a different individual in t + 1

Probability that a different individual occupies the site in season 2 than in season 1

Discussion Abundance or population size has been the traditional state variable used in animal population studies and monitoring programmes. Here we propose the use of an alternative state variable, occupancy or the proportion of area occupied by a species. For some questions involving geographic distribution, range size, and metapopulation dynamics, this is the state variable of primary interest. In other situations, the reduced effort required to estimate occupancy, relative to that required to estimate abundance, may warrant consideration of occupancy as a surrogate for abundance. Our initial work on occupancy estimation focused on estimation within a single season or other short time period and on a single species (MacKenzie et al., 2002; MacKenzie & Bailey, in press). We have recently extended this single–season work to incorporate multiple species with possible dependencies in both occupancy and detection (MacKenzie et al., 2004). Heterogeneity of detection probabilities among different sites or sampling units beyond that associated with identified covariates is a topic of special interest. Promising approaches developed by Royle (in press) and Royle & Nichols (2002) deal with heterogeneity associated with site–specific variation in abundance and even permit inference about abundances.

Above, we describe our more recent work extending estimation to deal with occupancy dynamics over longer time scales (also see MacKenzie et al., 2003) and outline a proposed approach to building more mechanistic models for the sampling situation in which uniquely marked individuals can sometimes be identified. Our current models permit estimation of rate of change in occupancy, as well as local rates of extinction and colonization, the vital rates of occupancy dynamics. In addition, these rate parameters can be modelled as functions of potentially relevant covariates including site–specific habitat, site isolation or proximity to source locations, etc. We thus believe that this framework permits investigation of a number of interesting ecological hypotheses. References Barbraud, C., Nichols, J. D., Hines, J. E. & Hafner, H., 2003. Estimating rates of extinction and colonization in colonial species and an extension to the metapopulation and community levels. Oikos, 101: 113–126. Bradford, D. F., Neale, A. C., Nash, M. S., Sada, D. W. & Jaeger, J. R., 2003. Habitat patch occupancy by toads (Bufo punctatus) in a naturally fragmented desert landscape. Ecology,

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84: 1012–1023. Brown, J. H. & Kodric–Brown, A., 1977. Turnover rates in insular biogeography: effect of immigration on extinction. Ecology, 58: 445–449. Hanski, I. , 1992. Inferences from ecological incidence functions. American Naturalist, 139: 657– 662. – 1994. A practical model of metapopulation dynamics. Journal of Animal Ecology, 63: 151–162. – 1997. Metapopulation dynamics: from concepts and observations to predictive models. In: Metapopulation biology: ecology, genetics, and evolution: 69–91 (I. A. Hanski & M. E. Gilpin, Eds.). Academic Press, New York, USA. 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–occurrence when species are detected imperfectly. Journal of Animal Ecology, 73: 546–555. MacKenzie, D. I., Nichols, J. D. Hines, J. E., Knutson, M. G. & Franklin, A. D., 2003. Estimating site occupancy, colonization and local extinction probabilities when a species is not detected with certainty. Ecology, 84: 2200–2207. 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. Royle, J. A. (in press). N–mixture models for estimating population size from spatially replicated

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counts. Biometrics. Royle, J. A. & Nichols, J. D., 2003. Estimating abundance from repeated presence–absence data or point counts. Ecology, 84: 777–790. Schmidt, B. R., 2003. Count data, detection probabilities, and the demography, dynamics, distribution, and decline of amphibians. Comptes Rendus Biologies, 326: S119–S124 Stauffer, H. B., Ralph, C. J. & Miller, S. L., 2002. Incorporating detection uncertainty into presence– absence surveys for marbled murrelet. In: Predicting species occurances: 357–365 (J. M. Scott, P. J. Heglund, M. L. Morrison, J. B. Haufler, M. G. Raphael, W. A. Wall & F. B. Samson, Eds.). Island Press, Washington, District of Columbia, U.S.A. Tyre, A. J., Possingham, H. P. & Lindenmayer, D. B., 2001. Inferring process from pattern: can territory occupancy provide information about life history parameters? Ecological Applications, 11: 1722–1737. Tyre, A. J., Tenhumberg, B., Field, S. A., Niejalke, D., Parris, K. & Possingham, H. P., 2003. Improving precision and reducing bias in biological surveys by estimating false negative error rates in presence–absence data. Ecological Applications, 13: 1790–1801. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and management of animal populations. Academic Press, New York. Yoccoz, N. G., Nichols, J. D. &. Boulinier, T., 2001. Monitoring of biological diversity in space and time. Trends in Ecology and Evolution, 16: 446–453.

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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Population dynamics E. G. Cooch & A. A. Dhondt

Cooch, E. G. & Dhondt, A. A., 2004. Population dynamics. Animal Biodiversity and Conservation, 27.1: 469–470. Increases or decreases in the size of populations over space and time are, arguably, the motivation for much of pure and applied ecological research. The fundamental model for the dynamics of any population is straightforward: the net change over time in the abundance of some population is the simple difference between the number of additions (individuals entering the population) minus the number of subtractions (individuals leaving the population). Of course, the precise nature of the pattern and process of these additions and subtractions is often complex, and population biology is often replete with fairly dense mathematical representations of both processes. While there is no doubt that analysis of such abstract descriptions of populations has been of considerable value in advancing our, there has often existed a palpable discomfort when the "beautiful math" is faced with the often "ugly realities" of empirical data. In some cases, this attempted merger is abandoned altogether, because of the paucity of "good empirical data" with which the theoretician can modify and evaluate more conceptually–based models. In some cases, the lack of "data" is more accurately represented as a lack of robust estimates of one or more parameters. It is in this arena that methods developed to analyze multiple encounter data from individually marked organisms has seen perhaps the greatest advances. These methods have rapidly evolved to facilitate not only estimation of one or more vital rates, critical to population modeling and analysis, but also to allow for direct estimation of both the dynamics of populations (e.g., Pradel, 1996), and factors influencing those dynamics (e.g., Nichols et al., 2000). The interconnections between the various vital rates, their estimation, and incorporation into models, was the general subject of our plenary presentation by Hal Caswell (Caswell & Fujiwara, 2004). Caswell notes that although interest has traditionally focused on estimation of survival rate (arguably, use of data from marked individuals has been used for estimation of survival more than any other parameter, save perhaps abundance), it is only one of many transitions in the life cycle. Others discussed include transitions between age or size classes, breeding states, and physical locations. The demographic consequences of these transitions can be captured by matrix population models, and such models provide a natural link connecting multi–stage mark–recapture methods and population dynamics. The utility of the matrix approach for both prospective, and retrospective, analysis of variation in the dynamics of populations is well–known; such comparisons of results of prospective and retrospective analysis is fundamental to considerations of conservation management (sensu Caswell, 2000). What is intriguing is the degree to which these methods can be combined, or contrasted, with more direct estimation of one or more measures of the trajectory of a population (e.g., Sandercock & Beissinger, 2002). The five additional papers presented in the population dynamics session clearly reflected these considerations. In particular, the three papers submitted for this volume indicate the various ways in which complex empirical data can be analyzed, and often combined with more classical modeling approaches, to provide more robust insights to the dynamics of the study population. The paper by Francis & Saurola

Evan G. Cooch, Dept. of Natural Resources, Cornell Univ., Ithaca, NY 14853 U.S.A. E–mail: evan.cooch@cornell.edu André A. Dhondt, Laboratory of Ornithology, Cornell Univ., Ithaca, NY 14853, U. S . A . E–mail: aad4@cornell.edu ISSN: 1578–665X

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(2004) is an example of rigorous analysis and modeling applied to a large, carefully collected dataset from a long–term study of the biology of the Tawny Owl. Using a combination of live encounters and dead recoveries, the authors were able to separate the relative contributions of various processes (emigration, mortality) on variation in survival rates. These analyses were combined with periodic matrix models to explore comparisons of direct estimation of changes in population size (based on both census and mark– recapture analysis) with model estimates. The utility of combining sources of information into analysis of populations was the explicit subject of the other two papers. Gauthier & Lebreton (2004) draw on a long–term study of an Arctic–breeding Goose population, where both extensive mark–recapture, ring recovery, and census data are available. The primary goal is to use these various sources of information to to evaluate the effect of increased harvests on dynamics of the population. A number of methods are compared; most notably they describe an approach based on the Kalman filter which allows for different sources of information to be used in the same model, that is demographic data (i.e. transition matrix) and census data (i.e. annual survey). They note that one advantage of this approach is that it attempts to minimize both uncertainties associated with the survey and demographic parameters based on the variance of each estimate. The final paper, by Brooks, King and Morgan (Brooks et al., 2004) extends the notion of the combining information in a common model further. They present a Bayesian analysis of joint ring–recovery and census data using a state–space model allowing for the fact that not all members of the population are directly observable. They then impose a Leslie–matrix–based model on the true population counts describing the natural birth–death and age transition processes. Using a Markov Chain Monte Carlo (MCMC) approach (which eliminates the need for some of the standard assumption often invoked in use of a Kalman filter), Brooks and colleagues describe methods to combine information, including potentially relevant covariates that might explain some of the variation, within a larger framework that allows for discrimination (selection) amongst alternative models. We submit that all of the papers presented in this session indicate clearly significant interest in approaches for combining data and modeling approaches. The Bayesian framework appears a natural framework for this effort, since it is able to not only provide a rigorous way to evaluate and integrate multiple sources of information, but provides an explicit mechanism to accommodate various sources of uncertainty about the system. With the advent of numerical approaches to addressing some of the traditionally "tricky" parts of Bayesian inference (e.g., MCMC), and relatively user–friendly software, we suspect that there will be a marked increase in the application of Bayesian inference to the analysis of population dynamics. We believe that the papers presented in this, and other sessions, are harbingers of this trend. References

Brooks, S. P., King, R. & Morgan, B. J. T., 2004. A Bayesian approach to combining animal abundance and demographic data. Animal Biodiversity and Conservation, 27.1: 515–529. Caswell, H., 2000. Prospective and retrospective perturbation analyses: Their roles in conservation biology. Ecology, 81: 619–627. Caswell, H. & Fujiwara, M., 2004. Beyond survival estimation: mark–recapture, matrix population models, and population dynamics. Animal Biodiversity and Conservation, 27.1: 471–488. Francis, C. M. & Saurola, P., 2004. Estimating components of variance in demographic parameters of Tawny Owls, Strix aluco. Animal Biodiversity and Conservation, 27.1: 489–502. Gauthier, G. & Lebreton, J.–D., 2004. Population models for Greater Snow Geese: a comparison of different approaches to assess potential impacts of harvest. Animal Biodiversity and Conservation, 27.1: 503–514. Pradel, R., 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Nichols, J. D., Hines, J. E., Lebreton, J. D. & Pradel, R., 2000. Estimation of contributions to population growth: A reverse–time capture–recapture approach. Ecology, 81: 3362–3376. Sandercock, B. K. & Beissinger, S. R., 2002. Estimating rates of population change for a neotropical parrot with ratio, mark–recapture and matrix methods. Journal of Applied Statistics, 29: 589–607.

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

Caswell, H. & Fujiwara, M., 2004. Beyond survival estimation: mark–recapture, matrix population models, and population dynamics. Animal Biodiversity and Conservation, 27.1: 471–488. Abstract Beyond survival estimation: mark–recapture, matrix population models, and population dynamics.— Survival probability is of interest primarily as a component of population dynamics. Only when survival estimates are included in a demographic model are their population implications apparent. Survival describes the transition between living and dead. Biologically important as this transition is, it is only one of many transitions in the life cycle. Others include transitions between immature and mature, unmated and mated, breeding and non– breeding, larva and adult, small and large, and location x and location y. The demographic consequences of these transitions can be captured by matrix population models, and such models provide a natural link connecting multi–stage mark–recapture methods and population dynamics. This paper explores some of those connections, with examples taken from an ongoing analysis of the endangered North Atlantic right whale (Eubalaena glacialis). Formulating problems in terms of a matrix population model provides an easy way to compute the likelihood of capture histories. It extends the list of demographic parameters for which maximum likelihood estimates can be obtained to include population growth rate, the sensitivity and elasticity of population growth rate, the net reproductive rate, generation time, measures of transient dynamics. In the future, multi–stage mark–recapture methods, linked to matrix population models, will become an increasingly important part of demography. Key words: Matrix population models, Right whale, Eubalaena glacialis, Sensitivity, Elasticity. Resumen Más allá de la estimación de supervivencia: marcaje–recaptura, modelos matriciales de poblaciones y dinámica de poblaciones.— La probabilidad de supervivencia resulta especialmente interesante como componente de la dinámica poblacional. Sólo cuando las estimaciones de supervivencia se incluyen en un modelo demográfico, puede apreciarse su repercusión en la población. La supervivencia describe la transición entre la vida y la muerte. Pese a su importancia biológica, dicha transición sólo constituye una más de las que componen el ciclo vital, debiendo destacarse, entre otras, la que se produce entre la inmadurez y la madurez, la ausencia de apareamiento y el apareamiento, la reproducción y la ausencia de reproducción, el estado larval y el de adulto, pequeño y grande, y entre el emplazamiento x y el emplazamiento y. Las consecuencias demográficas de dichas transiciones pueden determinarse mediante modelos matriciales de poblaciones, que proporcionan un enlace natural capaz de vincular los métodos de marcaje–recaptura de fases multiestados con la dinámica poblacional. El presente estudio analiza algunas de dichas conexiones, incluyendo ejemplos extraídos de un análisis que sigue en marcha de la ballena franca (Eubalaena glacialis), en peligro de extinción. La formulación de problemas, considerados desde la perspectiva del modelo matricial de poblaciones, permite calcular fácilmente las probabilidades de las historias de captura, al tiempo que amplía la lista de parámetros demográficos con respecto a los que pueden obtenerse estimaciones por máxima verosimilitud, incluyendo la tasa de crecimiento poblacional, la sensibilidad y elasticidad de dicha tasa, la tasa neta de reproducción, el tiempo generacional y las mediciones de la dinámica transitoria. En el futuro, los métodos de marcaje–recaptura de multiestados, en combinación con los modelos matriciales de poblaciones, constituirán una parte cada vez más importante de la demografía. Palabras clave: Modelos matriciales de poblaciones, Ballena franca, Eubalaena glacialis, Sensibilidad, Elasticidad. Hal Caswell, Biology Dept. MS–34, Woods Hole Oceanographic Institution, Woods Hole MA 02543 U.S.A.– Masami Fujiwara, Dept. of Ecology, Evolution, and Marine Biology, Univ. of California, Santa Barbara, CA 93106– 9610, U.S.A.

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Introduction Throughout its history, and certainly since the 1960s, the field of mark–recapture (MR) statistics has emphasized the estimation of survival probability. Lebreton et al. (1992) provided a state–of–the art review of the field, emphasizing tests of hypotheses about, and selection of models describing, survival probability. But survival probability is, in itself, of limited interest. After all, every individual dies eventually; why should we care about the short–term probability of this ultimately certain event? The answer, of course, is that survival is vitally important as a component of population growth. In the simple population growth model N(t + 1) = [P (survival) + E (reproduction)] N( t ) (1) = N(t ) the rate of increase is the sum of the survival probability and the birth rate. A model like (2) makes the demographic analog of the homogeneity assumption of MR theory: that all individuals are identical. Since all individuals are obviously not identical, there is a long history (a century or more) of demographic population models —models that disaggregate individuals on the basis of age, physiological condition, size, developmental stage, spatial location, etc. (Lotka, 1907, 1924, 1934– 1939; Kermack & McKendrick, 1927; Leslie, 1945; Keyfitz, 1968; Metz & Diekmann, 1986; Caswell, 1989, 2001; Tuljapurkar & Caswell, 1997). Structured demographic models do two things. The most obvious but least important is to provide more accurate descriptions of population dynamics, by incorporating more biological differences among individuals and the way that those differences affect individuals’ fates in a given environment. The more important thing has nothing to do with accuracy. Structured demographic models are valuable because they provide explanations of population dynamics in terms of the fates of individuals. Calculating the population growth rate from age– or stage–specific data may or may not produce more accurate predictions, but it shows how the life cycle influences population dynamics in ways that unstructured models cannot do. Structured models can be written as partial differential equations, delay–differential equations, integrodifference equations, or matrix population models, depending whether individuals are divided into discrete classes or measured by a continuous variable, and whether time is discrete or continuous (Tuljapurkar & Caswell, 1997; Caswell, 2001): Discrete–state Continuous–state Discrete–time Matrix Integrodifference population equations models Continuous–time Delay–differential Partial differential equations equations

Regardless of the mathematical framework, demographic models link the fates of individuals to the dynamics of populations. This individual– population link is the key to using MR methods to estimate the parameters in such models, because MR data are essentially individual data; putting a tag on an animal (or plant) distinguishes that individual from all others in the population. Repeated observations on that individual record its history, distinct from the history of any other individual. Here, we will focus on matrix population models, because they correspond most closely to the structure of typical MR data, but it is worth noting that connecting MR methods and demographic models in the other frameworks is an important problem (e.g., Fujiwara, 2002). We will describe, briefly, the structure of a matrix population model and some of the ways that such a model characterizes population dynamics (for a much more complete description, see Caswell, 2001). We will emphasize perturbation analysis as an integral part of demographic analysis. This will lead to a discussion of the link of such models to MR analysis, and to ways in which matrix population models naturally encapsulate the fundamental notion of the "likelihood of a capture history". Finally, we will illustrate some of these points with aspects of a demographic analysis of the North Atlantic right whale. Matrix population models A matrix population model requires, first, a choice of a projection interval, or time step, over which to project the population, and a set of life cycle stages into which to classify individuals. The stages are chosen by a homogeneity criterion: knowing the stage of an individual must suffice to predict, at least probabilistically, the response of an individual to the environment (Metz & Diekmann, 1986, Caswell, 2001). Choosing stages requires a knowledge of, and an ability to balance, the biology of the organism and the limitations of the available data. Given the projection interval and the stages, the model can be written as (2) Here n(t) is a vector whose entries give the numbers of individuals in each stage, and A is a square population projection matrix whose entries may depend on time, the environment, and/or population density. Characterizing population dynamics The dynamics induced by (2), and the quantities calculated to describe those dynamics, depend on the nature of the projection matrix. It is these quantities that a demographic analysis sets out to estimate.

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Linear time–invariant models The solution to (2) in this case can be written as a sum of exponentials of the eigenvalues of A. Because A is inherently non–negative, the Perron– Frobenius theorem guarantees that one of these eigenvalues is real, non–negative, and as great as or greater than the magnitude of any of the others. If A is also primitive, then this eigenvalue 1 (or, where the subscript is not needed for the context, ) is strictly greater than any other, and the population eventually grows at this rate with a stable structure proportional to the corresponding right eigenvalue w. The left eigenvector v gives the distribution of reproductive values. Before this asymptotic growth rate is realized, the population will exhibit transient fluctuations in both abundance and structure, which can be analyzed in terms of the subdominant eigenvalues and eigenvectors of A.

where ||x|| = 3i |xi| is the 1–norm (Furstenberg & Kesten, 1960; Oseledec, 1968; Cohen, 1976, 1977a, 1977b; Tuljapurkar & Orzack, 1980; Tuljapurkar, 1989, 1990). Density–dependent models The dynamics of a density–dependent population described by the nonlinear model n(t + 1) = An(t) n(t)

(7)

are not characterized by exponential growth. Instead, trajectories typically converge to an attracting invariant set (equilibrium point, cycle, invariant loop, or strange attractor) on which the long–term average growth rate is 1. Often, these attracting invariant sets exhibit bifurcations as any parameter in the model is varied. Figure 1 shows an example from a simple two–stage model (juveniles and adults) with density–dependent fertility, in which

Periodic models Periodic models are useful for describing effects of seasonal variation in the vital rates (e.g., Hunter, 2001; Hunter & Caswell, 2004a; Smith et al., 2004). They can also be used to describe inter–annual variation as an approximation to other kinds of variation. Population growth over an annual cycle of p phases, starting at phase 1, is given by the product A1 = Bp···B2B1

(3)

Note that the seasonal matrices are multiplied from right to left, in order. The population growth rate is given by the dominant eigenvalue of A1. The stable stage distribution (at phase 1) is given by the corresponding right eigenvector w. The reproductive value distribution is given by the left eigenvector v. The stable stage distribution and reproductive value at other phases of the cycle are given by the eigenvectors of the appropriate cyclic permutations of the Bi ; e.g., those at phase 2 would be obtained from A2 = B1Bp···B2

(4)

Stochastic models In a stochastic environment, population growth is described by the time–varying model (5) where the matrices At are generated by a stochastic model for the environment. Given some reasonable assumptions about the environmental process and the matrices, asymptotic population growth is, with probability 1, characterized by the stochastic growth rate (6)

(8)

where 1 and 2 are juvenile and adult survival probabilities, is the maturation rate, and f is the fertility at low densities. As f is increased, the equilibrium population increases from zero (at any value of f below the critical value at which the population is capable of growing at all at low densities) to higher and higher values. As f increases, a flip bifurcation occurs and the stable equilibrium is replaced by a stable 2– cycle. As f is increased further, the 2–cycle is replaced by a 4–cycle, which in turn is replaced by cycles of period 8, 16, etc. Eventually the dynamics become chaotic. For more details on such bifurcations, see Caswell (1997, 2001) and Neubert & Caswell (2000a). For an account of an outstanding experimental investigation of such bifurcations in laboratory populations of Tribolium flour beetles, see Cushing et al. (2003). The point of this demonstration is not concern over whether a particular 2–stage population might exhibit chaotic dynamics, but to make the point that the characterization of a density–dependent model —the answer to the question "what are the implications of this set of parameters?"— is the entire bifurcation sequence. It is the result of the parameter values and the functional forms in the matrix An, which are exactly what would be estimated by a MR analysis using density as a covariate. Because the long–term performance of a density–dependent population involves its attractor(s), two different measures of population performance have attracted attention. The first is the long–term population composition or some function of it. Choosing the function has not received much careful thought. There is a tendency to think of total density (for equilibria) or time–averaged total density (for cycles, etc.) without considering the bio-

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logical justification for the choice. Since total density involves adding together individuals of very different properties (tiny seedlings and enormous trees, etc.), it is unlikely to capture much of relevance, but alternatives are not totally clear (for one, see (34) below)

Environment–dependent models

The invasion exponent

The dynamics of such a population depend on what the environment does, and there is surprisingly little to say in general about such models. If (t) represents a particular fixed environmental condition (habitat type, or level of pollution, say), interest might focus on potential population growth under different but fixed environmental conditions. Then life table response experiment (LTRE) analyses can be used to decompose the effects of the environment on into contributions from effects on each of the vital rates. For example, suppose that two environments ( 1 and 2) are being compared. The matrices and yield population growth rates and . To first order

An alternative measure of population performance is the invasion exponent. It can be motivated by comparing the growth rates of two density–independent populations, each with its own vital rates, and thus with population growth rates (1) and (2). If Ni is any measure of population size of type i, then asymptotically (9) Type 2 will increase in frequency relative to type 1 if and only if log (2) > log (1). Since log determines the ability of a type to invade, it is referred to as the invasion exponent. The sensitivity and elasticity of show the effect of parameter changes on the invasion exponent. Invasion calculations can be applied to linear and nonlinear, deterministic and stochastic models (Metz et al., 1992; Ferriére & Gatto, 1995; Rand et al., 1994; Grant, 1997; Grant & Benton, 2000). Consider two types, each defined by vector i of parameters; suppose that type 2 is trying to invade type 1. Its dynamics during this invasion will depend on its parameters, on the density of the resident, and on its own density. To make this dependence clear, we will write the projection matrix for type 2 as (10) The invasion exponent describes the dynamics of type 2 when it invades at such a low density that n2(t) is negligible for a very long time. It is given by the long–term average growth rate of type 2 while type 1 is on its attractor ; (11)

The notation I (Caswell, 2001) emphasizes the relation between the invasion exponent and the growth rate in a constant environment ( ) or a stochastic environment ( s). The superscript (2 t1) indicates that type 2 is invading type 1. In the special case in which is an equilibrium point, the invasion exponent is just the log of the dominant eigenvalue of the constant projection matrix for type 2, evaluated at the equilibrium density of type 1: (12) For examples of the use of the invasion exponent in calculations regarding the evolution of dispersal, see Khaladi et al. (2000) and Lebreton et al. (2000).

Suppose that the vital rates are functions of some environmental variable (t); then we could write (13)

(14) where the derivative of is calculated according to (15) below. For details see Caswell (1989a, 2000, 2001); for examples see Levin et al. (1996), Cooch et al. (2001). When (t) represents an observed temporal trend, asymptotic dynamics may be of little relevance, and attention would focus on short–term population projections. Those projections will depend on the ability to forecast the trend accurately. In other cases, as when (t) represents a climatic variable like rainfall, the asymptotic growth rate under a specified rainfall regime (characterized, say, by a time–series model or a stochastic model for disturbances like fire; Silva et al., 1991; Caswell & Kaye, 2001) would be of great interest. In each of these cases, the environment–dependent model is reduced to a simpler model (short–term transient dynamics, long–term stochastic dynamics, or long–term linear dynamics in different environments). But this kind of model has received little general attention, perhaps because it has not been easy to estimate vital rates as functions of environmental variables in the field; that task becomes much easier with mark–recapture approaches. Perturbation analyses The demographic analyses in the preceding (non– exhaustive) list all take as given a set of demographic parameter values. But almost never are we interested in only one specific set of parameter values. There is always the possibility that the values could change, because of natural environmental change, human activity (including management actions), evolutionary change, or because the parameters are estimated with error. Perturbation analyses evaluate the effect of such changes. They are available for many of the demographic indices just described (Caswell, 2001).

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Linear time–invariant models of a

(15) where the eigenvectors are scaled so that their scalar product <v,w> = 1 (Caswell, 1978, 2001) and vT is the trasposse of v. In the important case where = aij, this reduces to: (16) The proportional sensitivity, or elasticity, measures both changes and their effects on a logarithmic scale, and is given by: (17) See Link & Doherty (2002) for explorations of some interesting alternatives to considering the effect of proportional changes. Their paper is a response to concerns raised by the unfortunate tendency to think that elasticities (or sensitivities) measure the "importance" of a parameter, and then to be puzzled when the importance of survival, say, is not the same as the importance of mortality. The confusion can be alleviated by repeating, over and over, that sensitivities and elasticities are derivatives, nothing more. That the derivative of a quantity with respect to one variable is not the same as its derivative with respect to another variable is not particularly surprising. Because is generally a nonlinear function of the parameters, sensitivities and elasticities can only predict approximate results of large perturbations in . Experience has shown that, in most cases, they give usefully accurate predictions even for quite large changes. Since the elasticities of to changes in the aij sum to 1, they can (with care) be interpreted as showing the proportional contribution of the aij to population growth rate. In evolutionary contexts, the sensitivities of measure the selection gradients on the aij. (The elasticities, in contrast, do not measure selection gradients, and are of only limited use in evolutionary calculations; see Caswell 2001, Section 11.2)

12 Total population

The sensitivity of population growth rate change in a parameter is

10 8 6 4 2 0

200

400 Fertility

800

1,000

Fig. 1. A bifurcation diagram for the densitydependent model (8). For each value of fertility f, the asymptotic attractor is plotted. Equilibria appear as single lines, cycles as multiple lines, and chaotic dynamics as a cloud of points. Fig. 1. Diagrama de bifurcación para el modelo dependiente de densidades (8). Para cada valor de fertilidad f, se representa gráficamente el atractor asintótico. Los equilibrios se muestran como líneas simples, los ciclos mediante líneas múltiples y la dinámica caótica como una nube de puntos.

where F(m) =

G(m) =

{ {

Bp···Bm+1 I

m!p m=1

(19)

Bm–1···B1 m !1 I m=p

The sensitivity matrix whose entries are the sensitivities of to bij(m) is (20) The elasticity matrix

is (21)

Periodic models In a periodic model, we want to compute the sensitivities and elasticities of to changes in the entries of each of the seasonal matrices Bi, using the approach of (Caswell & Trevisan, 1994; Lesnoff et al., 2003). Let SA be the sensitivity matrix for A (i.e., the matrix whose (i,j) entry is ∑ /∑aij). In general, suppose that there are p matrices in a seasonal cycle, B1,...,Bp. To calculate the sensitivity of to the entries of Bm, let A = F(m) Bm G(m)

600

m = 1,...,p

(18)

where ° denotes the Hadamard, or element–by– element, matrix product. The elasticities eij(m) sum to 1 for each m. Thus they can be interpreted as proportional contribution of the vital rates bij(m) to population growth, exactly is done for non–seasonal models. Stochastic models Tuljapurkar (1990) derived the sensitivity of log s and the elasticity of s to changes in aij(t). The

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matrices At are generated by the stochastic environment. Assume that At is subject to a small perturbation, so that

Substituting (29) into (26) gives the elasticity of to aij:

(30) (22) The entries of Ct determine which elements of At are perturbed, and the relative magnitudes of those perturbations. To calculate the effect of the perturbation on log s, the stochastic model for the environment is used to generate a sequence of matrices A0,..., AT–1, where T is a large number. Starting from an arbitrary nonnegative initial vector w(0), with ||w(0)|| = 1, use the sequence of matrices to generate a sequence of stage distribution vectors

Density–dependent models The perturbation analysis of a density–dependent model can be carried out in terms of the invasion exponent I or the equilibrium density . There are intimate connections between the two measures of performance (Takada & Nakajima, 1992, 1998; Caswell et al., 2004). Let us write the projection matrix as: (31)

(23)

and one–step growth rates (24) where ||·|| denotes the 1–norm. Similarly, starting with an arbitrary nonnegative terminal vector v(T) with ||v(T)|| = 1, generate a backwards sequence of reproductive value vectors (25)

where is a parameter and the fi are functions of density that appear in the model. There will be more than one such function if different vital rates are affected by different sets of stages in the life cycle. Let (32) be the population growth rate calculated from the matrix at equilibrium ( = 1). Then it can be shown (Caswell et al., 2004) that (33)

In terms of these quantities, Tuljapurkar (1990) showed that the stochastic growth rate after the perturbation, log s( ), is:

where Ñ is an effective equilibrium density, which is a linear combination of stage densities (34)

(26) The coefficient of on the right hand side of (26) is the sensitivity of the stochastic growth rate to the perturbations imposed by the sequence of perturbation matrices Ct. If only a single element of At is perturbed (say, aij (t)) and the perturbation is the same at each time, Ct is a constant matrix with a 1 in the (i,j) position and zeros elsewhere; C t = ei ej T

(27)

where ei is a vector with a 1 in the i th entry and zeros elsewhere. Substituting (27) into (26) leads to Tuljapurkar’s formula for the sensitivity of log s to aij: (28)

with the weight on stage i given by (35)

The biologically effective density Ñ weights the density of the stages by their importance to density–dependent effects (∑fh/∑ni) and the importance of those effects to demography (∑ /∑fh). This combination of density–dependence and demography is exactly the content of the nonlinear model, so Ñ has ample biological justification as an interesting quantity. This result provides a valuable link between (and thus all kinds of evolutionary invasion questions) and equilibrium population (and thus all kinds of questions related to population management); see Grant & Benton (2003) for a discussion of the need for such relationships. Estimation: beyond survival

The elasticity of s to aij is calculated by assuming that the perturbation cij(t) is proportional to aij(t), so that Ct is a matrix with aij(t) in the (i,j) position and zeros elsewhere: C t = e i e iT A t e j e jT

(29)

The point of this quick tour through the types of matrix population models is to emphasize that the construction of such a model creates a whole suite of population parameters, particularly those describing perturbations, that can be estimated.

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Estimation with mark–recapture analyses

The probability of capture histories

The population projection matrix and its dependence on time or the environment is the central entity in population dynamics. Mark–recapture methods–in particular, the multi–state versions–are an extraordinarily powerful tool for constructing such models. Demographers have been slow to realize this fact. For example, Caswell (1989) wrote an entire book on matrix population models without mentioning multistate MR methods, although they date back to Arnason (1973). He knew better by 2001, partly because Nichols et al. (1992) presented the first explicit application of multistate MR methods to matrix population models. The link is simple. A matrix population model contains a complete description of the probabilities of transition among all the identified stages plus death. The history of any individual consists of a sequence of such transitions, beginning with birth and ending with death. MR data consists of observations of such sequences, with the possible extra complication of failing to observe or capture the individual at each time; in other words, of capture histories. The key to estimating parameters (survival or more complex patterns of transition), is to use the probability of a set of capture histories as the likelihood of the parameters generating the probability. A matrix population model makes it possible to do this in a remarkably straightforward way.

For our present purposes, the most important use of is that it makes it easy to write down the probability of a multi–state capture history. Let

Decomposing the projection matrix In most (all?) cases, the projection matrix can be decomposed into a part T that represents transitions of individuals already present in the population and a part F that represents the creation of new individuals by reproduction: A=T+F

(36)

(Feichtinger, 1971; Cochran & Ellner, 1992; Cushing & Yicang, 1994). We will focus on T, and return to F later. Let s be the number of stages in the model. Create the (s + 1) × (s + 1) matrix by adding death as a stage

(38)

be a diagonal matrix of capture probabilities. If ps+1 = 0, recoveries of dead individuals are not considered, but the life cycle graph can be extended to include categories representing newly dead and dead–and–gone, and both recaptures and recoveries considered together. For what follows, it will be useful to define ei as a column vector with a 1 in the i th entry and zeros elsewhere, e as a column vector of ones, and Ei = eieiT as a matrix with a 1 in the (i,i) position and zeros elsewhere. The columns of a matrix can be summed by multiplying on the left by eT. An individual is first marked at time t = 1 and then recaptured (or not) at times t = 2,...,T. The capture history consists of a sequence of numbers, h = X1,X2, ...,XT

(39)

where Xt indicates the stage of the individual or the fact that it was not seen at time t. If the individual is not seen, let Xt = 0. Suppose an individual was marked in state X 1 at t = 1. Then the vector gives the probability distribution of its state at t = 1. The probability distribution of its state after the transition from t = 1 to t = 2 is the vector including the probability of death. The entries of the vector give the probability of capturing the individual in each of the stages at t = 2. Similarly, the entries of the vector give the probabilities of failing to capture the individual in each of the stages at t = 2. Continuing this process leads to the following simple formula for the probability of any capture history (Caswell, 2001; Fujiwara & Caswell, 2002).

(37) (40) where m is a row vector of stage–specific probabilities of death, mj = 1– 3i tij. The matrix is the transition matrix of an absorbing Markov chain, and can be used to calculate many demographically useful quantities, including the distribution of ages at death (even though age may not appear in the model), the net reproductive rate, the generation time, age–specific survivorship, the stable age distribution within each stage, and the probability of any event that can be expressed in terms of stages (Caswell, 2001). Because is derived from A, it may vary with time, density, or environmental factors.

(41) where if Xt+1 g 0 Qt =

(42) if Xt+1 = 0

This provides a simple matrix extension of the familiar formula for the probability of a capture history in the CJS model; it is possible because includes death as a state.

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For example, suppose that an individual is marked in stage 1, captured in stage 3, captured in stage 5, and then not captured for two time intervals, so that

eT(I – P) E1P z = (1 – p1)t11p1t12

(50)

+ (1 – p 2)t21p1t12

(51)

+ (1 – p 3)m1p 1t12

(52)

h = 1,3,5,0,0 Applying (42) gives (43) P[h] = eT[(I – P5)

][(I – P4)

][E5P3

][E3P2

]e1

Because death is included as an absorbing state in , this formula automatically accounts for all the possibilities of survival and state transition during the terminal string of (two) zeros in this sighting history. To see how this works, consider a simple example with only two stages and time–invariant transitions

The first term in this sum corresponds to (reading from right to left) transition from 2 to 1, capture in 1, transition from 1 to 1 and failure to capture in 1. The second term corresponds to transition from 2 to 1, capture in 1, transition from 1 to 2, and failure to capture in 2. The third term corresponds to transition from 2 to 1, capture in 1, death fro stage 1, and failure to recover as dead. Adding yet another zero to the capture history and multiplying (49) by (I – P) will give the somewhat larger set of possibilities at t = 4. Making the customary assumption of independent and identical individuals, the log likelihood for the entire collection of capture histories is

(44) (53) and sighting probabilities. Consider the capture history h = 2,1,0. The individual is known to be in stage 2 at t = 1, so the probability distribution of its initial state is Z = (0 1 0) T. At t = 2 the probability distribution of its state is (45)

The probability of being captured in each of those stages is

(46)

But we know it was captured in state 1, so only the first entry in this vector is relevant:

Maximum likelihood estimates of the parameters can be found by maximizing log L (e.g., using Matlab optimization routines). Nested models can be compared using log–likelihood tests, and model selection using information–theoretic methods (Akiake’s Information Criterion, or AIC; see Burnham & Anderson, 1998) can be carried out directly using log L. Models in which and P depend on time, external covariates, or individual covariates can be defined by making the appropriate matrix entries functions of the covariates and maximizing the likelihood with respect to the resulting parameters. The m–array We note that it is also possible to construct the multi–state version of the m–array from and P, in calculations essentially identical to those of Brownie et al. (1993), and to use it to derive the likelihood function.

(47) More things to estimate Applying the transition matrix to this vector gives the probability, at t = 3, of the three states

(48)

The individual was not captured at t = 3, the probability of which is

(49)

Finally, we add all these probabilities to obtain the probability of all possible transition and sighting sequences compatible with the capture history.

Fertility The estimate of provides estimates of the entries in T, the transition portion of the projection matrix A. Fertility, however, appaears in F, and estimation of F requires extra information on reproduction. One way to obtain some of this information is to include a state in the life cycle corresponding to reproduction or breeding. Every time an individual enters this state, she produces some number of offspring. If the expectation of that number is known, or can be estimated, then the estimate of also provides an estimate of F, which will contain positive values only in entries corresponding to transitions to the reproductive state. The fertilities in F will depend on the reproductive biology of the species, with care taken to account for when in the

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annual cycle censuses occur (Caswell, 2001). In the example we consider here, the right whale produces only a single calf in a reproductive event, so construction of F is relatively easy. Demographic indices If the parameters defining the model are estimated by maximizing the likelihood log L, then any invertible function of those parameters is also a maximum likelihood estimate. That means that the matrix population model provides maximum likelihood estimates of, inter alia, the population growth rate, stable stage distribution, reproductive value distribution, damping ratio, period of oscillation, and the sensitivity and elasticity of population growth rate to all parameters. And that’s just for the linear time invariant case. If a stochastic model is estimated, then the analysis provides maximum likelihood estimates of the stochastic growth rate and its sensitivity and elasticity. And a density–dependent model yields maximum likelihood estimates of the attractor(s) to which the population will converge (equilibria, cycles, invariant loops, or strange attractors) and the sensitivity of the invasion exponent and the effective equilibrium population to changes in the parameters. Standard errors for all these parameters can be obtained from the results of maximizing the likelihood, either by the Taylor series expansion method for those quantities (like ) whose sensitivity to parameters can be written down, or by a parametric bootstrap approach using the information matrix. For example, if C is the covariance matrix of the parameter vector , then to first order (54) where the vector of derivatives ∑ /∑ is obtained from (15). In addition, model uncertainty can be analyzed using information–theoretic methods (Burnham & Anderson, 1998). It is hard to overstate the potential of this for demographic studies. An example: the North Atlantic right whale The North Atlantic right whale (Eubalaena glacialis) was once abundant in the northwestern Atlantic, but as an early preferred target of commercial whalers by 1900 it had been hunted to near extinction. The remaining population (only about 300 individuals) is distributed along the Atlantic coast of North America, from summer feeding grounds in the Gulf of Maine and Bay of Fundy to winter calving grounds off the Southeastern U.S. In the more than 50 years since the end of commercial whaling, the population has recovered only slowly. Right whales are killed by ship collisions and entanglement in fishing gear, and may be affected by pollution of coastal waters. . Individual right whales are photographically identifiable by scars and callosity patterns. Since 1980,

the New England Aquarium has surveyed the population, accumulating a database of over 10,000 sightings. Treating the first year of identification of an individual as marking, and each year of resighting as a recapture, we have used MR statistics to estimate demographic parameters of this endangered population (Caswell et al., 1999; Fujiwara & Caswell, 2001, 2002; Fujiwara, 2002). From a conservation point of view, the most alarming finding has been a declining trend in the survival of reproducing females, which in turn has driven a decline in population growth rate so that is now less than 1. Life cycle structure Figure 2 shows a transition graph for the right whale. It distinguishes males and females, and divides each sex into developmental stages. Females are classified as calves, immature, mature, mothers (reproducing females), and post–mothers (i.e., mature females in an interbirth interval). This graph differs from that in Fujiwara and Caswell (2001) by incorporating the post–mother stage, which enforces a 2–year minimum for the interbirth interval, which is supported by the data. The inclusion of a stage representing breeding females captures the biologically important act of breeding as an explicit transition in the life cycle, which is particularly critical when mature females do not breed every year. It permits calculation of the fertility part of the projection matrix, because in this case we know that only a single calf is produced by a reproducing female. Modelling transitions A model in which all transition and sighting probabilities were free to vary independently would, for this data set, have about 300 parameters. We chose a much more parsimonious universe of models to investigate. Because previous analyses (Fujiwara & Caswell, 2001) had shown that variation in calf survival, mother survival, and reproductive rate were the most variable, we permitted variation only in 21, 54, and the birth probability conditional on survival, , defined by 43 43

(55)

where 3 is the survival probability of stage 3. We examined models in which each of these three parameters was constant or a logistic function of time and/or of the North Atlantic Oscillation (NAO). The NAO is a major climatic and oceanographic oscillation, defined in terms of the barometric pressure difference between Iceland and the Azores (e.g., Hurrell, 1995); the NAO is know to have effects on a variety of ecological systems (Ottersen et al., 2001), including plankton in the western North Atlantic, where right whales feed.

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Fig. 2. Transition graph for the right whale (Fujiwara, 2002). Stages N1–N5 are females: N1. Calf; N2. Immature; N3. Mature; N4. Mother; N5. Post–breeding. Stages N6–N8 are males: N6.Calf; N7. Immature; N8. Mature; N9. Dead. Fig. 2. Gráfico de transición correspondiente a la ballena franca (Fujiwara, 2002). Las fases N1–N5 son hembras: N1. Ballenato; N2. Inmadura; N3. Madura; N4. Madre; N5. Post reproducción. Las fases N6– N8 son machos: N6. Ballenato; N7. Inmaduro; N8. Maduro; N9. Muerto.

(t) = f (time, NAO)

(56) (57)

(t) = f (time, NAO)

Models including only a temporal trend or only NAO dependence are also not supported. The best model has time dependence of mother’s survival and birth probability, and NAO dependence of all three parameters.

(58) The F matrix (59)

(t) = f (time, NAO)

(60) (61)

This yields 43 = 64 different models for the transition probabilities, with 37–42 parameters. Previous analyses had shown that sighting probability varied greatly over time, but with considerable correlation among stages. Thus we modelled sighting by letting p3 (sighting probability of mature females) vary freely over time, and setting (62) Each model was fit by maximizing the likelihood L in (53). AIC values were calculated as 2 log L – 2 np, where n p is the number of param e t e r s . The spectrum of AIC values (AIC relative to the minimum) is shown in figure 3. The time– invariant model has the highest AIC value; the data clearly do not support constant vital rates.

Each female that becomes a mother (which happens with probability 43(t)) produces a single calf which, is female with probability 0.5. To be counted as reproduction, the calf must survive long enough to be catalogued. We assume that this requires 6 months, and that the calf will die during this time if its mother dies (which happens with probability 54(t + 1)). Thus F13(t) = 0.5

(t) (

(t + 1) )0.5

(63)

Some demographic results Combining Tt and Ft gives us a series of population projection matrices At for each model. As an example of the kind of results available, consider the following. A time–invariant model Even though the data do not support a time– invariant model, it is worth examining the resulting projection matrix as the best single image of the overall demography of the right whale during the 1980s and 1990s (64):

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Constant

25 NAO 20 AIC

Trend

10 Trend + NAO 5

0 0

Fig. 3.

30 40 Model rank

AIC values for 64 models fit to the demographic transitions of the right whale.

Fig. 3. Los valores AIC correspondientes a 64 modelos se ajustan a las transiciones demográficas de la ballena franca.

of N give the life expectancies of individuals in different stages (66):

The fertility elements of A are in bold face; the transition matrix T has zeros in those entries and the fertility matrix F is zero except for those entries. The upper left submatrix describes production of females by females, the lower right submatrix the production of males by males, the lower left submatrix the production of males by females, and the upper right submatrix the production of females by males. T is the transient portion of an absorbing Markov chain, with eventual absorbtion by death. Thus, the (i,j) entry of the fundamental matrix N = ( I – T )–1

(65)

gives the expected number of time intervals spent in stage j before absorbtion (i.e., death) by an individual starting in stage i. Thus the column sums

Thus the life expectancy for a female calf is 32 years (the sum of column 1); that for male calf is 18.4 years (the sum of column 6). This pattern, with male life expectancy shorter than that of females, appears to be not unusual in cetaceans. A female in stage 4 has just reproduced. The inter–birth interval is the time before she reproduces again. This interval is infinite if she dies before reproducing, so a meaningful average can be calculated only from the distribution of interval lengths conditional on reproducing again. The conditional distribution is calculated from T by creating a new absorbing state ("reproduced before dying"), calculating the probability of absorbtion in this state rather than its competitor ("died before

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0.4 Elasticity

0.3 0.2 0.1 0 1 2 To

4 5 1

Fig. 4. The elasticity of

3 m Fro

to changes in the elements of A for the time–invariant right whale model.

Fig. 4. La elasticidad de frente a la presencia de cambios en los elementos de A para el modelo de ballena franca invariable con el tiempo.

reproducing"), and creating a conditional transition matrix T(c), conditional on absorbtion in this state (Caswell, 2001, Chapter 5). This estimate of T yields a median inter–birth interval of 4.0 years. Combining F and T, we calculate the net reproductive rate (the expected lifetime reproductive output of a newborn female) as R0 = dominant eigenvalue of FN = 2.18

(67) (68)

The population growth rate, calculated from the dominant eigenvalue of A, is = 1.025

(69)

that is, a growth of about 2.5% per year. The elasticity of to changes in the aij is shown in figure 4. is most elastic to changes in the transitions representing survival and growth of mature females, mothers, and inter–birth females. Changes in fertility would have little effect on . Beyond time–invariance The matrix A gives the best possible time–invariant model, given the mark–recapture data from 1980– 1997. But the AIC values show it placing a dismal last among all 64 models examined. Clearly the data do not support a time–invariant model. A detailed analysis of the whole family of models is not possible here (Fujiwara et al., in prep.), but it is worth considering the implications of some of the models.

Given a time–varying model, it is possible to calculate at each time, as a measure of the quality of the environment at that time. This is a hypothetical calculation, giving the rate at which the population would be capable of growing if the environment was fixed in the state it was in when the vital rates were measured. As in previous analyses (Fujiwara & Caswell, 2001), a model with a temporal trend in survival and breeding probability was a great improvement over the time–invariant model. The best model ( AIC = 0) included the temporal trend in mother’s survival and breeding probability, and NAO effects on calf survival, mother’s survival, and breeding probability. Figure 5 shows calf survival, mother’s survival, and breeding probability as functions of time for these two models. There has been a slight decline in the first, and a dramatic decline in the second and third of these quantities. The two models agree in the rate and amount of decline; the best model adds some NAO–driven fluctuations around the smooth trend of the time model. These survival and transition estimates are only the beginning. Figures 6 and 7 translate them into trajectories of female life expectancy and net reproductive rate, both of which have declined. The birth interval has increased; figure 8 shows the mean inter–birth interval implied by the two models; it has increased from about 4 to about 7 years. Similar patterns are shown by the median interval and others measures of the time between births. Figure 9 shows the population growth rates cal-

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Calf survival

0.9

0.85

0.8 1980

1985

1990

1995

1985

1990

1995

1985

1990

1995

0.9

0.8

0.7

0.6 1980

Conclusion

0.5 Birth probability

These fragments of right whale demography only begin to suggest the power of combining multi–state mark–recapture analysis and matrix population models. All that is needed to take advantage of this power is a model that includes a description of the life cycle. We say "a" description advisedly, many such descriptions are possible. Demographers may be more accustomed to flexibility in defining stages and structuring life cycles than are mark–recapture practitioners. Given such a description of the entire life cycle, all becomes possible. For example, in the simplest case of a constant matrix: 1. From an estimate of the matrix , one can estimate life expectancy, age–specific survival (even if the model is stage– rather than age–classified), and the distribution of inter–event times (e.g., inter–breeding intervals). 2. The fertility matrix F, combined with , can provide an estimate of the parameters in a multi– type branching process, which permits a detailed analysis of demographic stochasticity, including variability in population growth and probability of extinction (Caswell, 2001, Chapter 15). 3. In another direction, F and together provide estimates of the net reproductive rate R0, age– specific fertility (even though model is stage–classified), generation time, and the stable age–within– stage distribution. 4. Extracting the transition matrix T from and combining it with F gives an estimate of the population projection matrix A from which both transient and asymptotic dynamics can be estimated, including population growth rate, stable stage distribu-

Trend Best

0.95

Mom survival

culated from these two models, compared with the value from the time–invariant model. Conditions for the right whale appear have deteriorated since 1980. If conditions typical of the late 1990s were to be maintained the population would be doomed to extinction, since < 1. It can be shown that the decline in is due mainly to the reduction in survival of mothers. The best model, which includes NAO effects, creates variability around this trend, but does not obscure it. The time–invariant model, unsupported though it may be, yields a value of comfortably in the middle of the range spanned by the time–varying models. Although the vital rates and population growth rates have changed dramatically, the results of perturbation analysis have not. Figure 10 shows one way of looking at this, presenting the elasticity of to changes in the survival probability of each stage. While there has been a gradual decline in all of them, the elasticity to survival of mature females is consistently highest, followed by that to immature survival and survival of mothers or post–mothers. The elasticity to to calf survival is consistently the lowest. It is encouraging that the all these models point to the same target —improvement in survival— for management actions.

0.4

0.3

0.2

0.1 1980

Fig. 5. Temporal variation in calf survival, mother survival, and birth probability in the temporal trend model and the best model. Fig. 5. Variación temporal en la supervivencia de los ballenatos, de las madres y de la probabilidad de nacimiento en el modelo de tendencias temporales y en el mejor modelo.

tion, reproductive value, and the sensitivity and elasticity of those quantities. 5. If the model is density–dependent, the projection matrix provides the machinery for a complete nonlinear analysis, including estimates of equilibria, stability, resilience, reactivity, bifurcations, invasion

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Trend Best

50 40 30 20

Net reproductive rate R0

Female life expectancy

Trend Best

5 4 3 2 1 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998

10 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998

Fig. 6. Temporal variation in female life expectancy at birth in the temporal trend model and the best model.

Fig. 7. Temporal variation in net reproductive rate R0 in the temporal trend model and the best model.

Fig. 6. Variaci贸n temporal en la esperanza de vida al nacer en las hembras en el modelo de tendencias temporales y en el mejor modelo.

Fig. 7. Variaci贸n temporal en la tasa neta de reproducci贸n R0 en el modelo de tendencias temporales y en el mejor modelo.

10 9

1.07 Trend Best

8 7 6 5 4 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998

Fig. 8. Temporal variation in the mean interbirth interval, conditional on survival, in the temporal trend model and the best model. Fig. 8. Variaci贸n temporal en el intervalo medio entre nacimientos, condicionada a la supervivencia, en el modelo de tendencias temporales y en el mejor modelo.

exponents, and the sensitivity and elasticity analysis of both equilibria and invasion exponents. 6. We have not emphasized spatial models, but there are two directions in which matrix models

1.06 Population growth rate

Mean interbirth interval

1.05

Trend Best Constant

1.04 1.03 1.02 1.01 1 0.99 0.98 1980 1982 1984 1986 1988 1990 1992 1994 19961998

Fig. 9. The population growth rate produced by three models for the right whale. Fig. 9. Tasa de crecimiento poblacional generada por tres modelos para la ballena franca.

provide powerful analyses of demography and dispersal. If a population projection matrix (linear or nonlinear) is combined with a distribution of dispersal distances, it is possible to estimate the invasion wave speed as a measure of the ability of the population to expand into previously unoccupied

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Fig. 10. The elasticity of population growth rate to changes in survival of each stage, in the constant, time trend, and best models. Elasticities to changes in mother survival and post–mother survival are identical. Fig. 10. Elasticidad de la tasa de crecimiento poblacional frente a la presencia de cambios en la supervivencia de cada fase, en el modelo de efectos fijos, en el de tendencias temporales y en el mejor modelo. Las elasticidades frente a la presencia de cambios en la supervivencia de las madres y en la supervivencia tras haber sido madres son idénticas.

territory (Neubert & Caswell, 2002b). If matrices are available for several sites and location is considered as a state along with life cycle stage, then the resulting multiregional model will describe both population growth and distribution, including sensitivity and elasticity of population growth rate to both demographic and dispersal parameters (e.g., Rogers, 1995; Lebreton, 1996; Hunter & Caswell, 2003b). Although some of these methods can be applied to even if it does not include the entire life cycle, doing so provides only a small fraction of the information that a complete demographic analysis can provide. There is much to gain in our understanding of population dynamics by making the

estimation of demographic models a goal at the outset of a mark–recapture study. Open problems We close by pointing out some unsolved problems. We anticipate that this section will rapidly become obsolete. Because MR methods make it so natural to estimate projection matrices as functions of environmental covariates, it would be good to have a more coherent theory for environment–dependent models. The lack of theory is as much a problem of not knowing the questions to ask as of not knowing how

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to answer them, especially when the the environmental dependence cannot be reduced to either time–invariance or a stationary stochastic process. Pradel’s (1996) approach to estimating the observed rate of growth of a population (he originally used for this quantity, but it has since regrettably been denoted by ) is an important advance. Understanding the relation between a fully stage– specific version of and the demographic , both obtained from multi–state mark–recapture analysis, would be extremely useful (Nichols et al., 2000 have made a start). There are many reasons why a population might fail to grow at its potential rate (e.g., a non–stable stage–distribution), but there are few documented cases to generalize about (e.g. Sandercock & Beissinger, 2002). The task of estimating F deserves more attention. Including reproductive states in the life cycle graph (as in the right whale) helps, but requires information on fecundity to go with it. Perhaps reverse–time mark–recapture analyses (Pradel, 1996) can help. Integrated modelling approaches, in which mark–recapture and census data are combined (Besbeas et al., 2002, 2003) could also provide estimates of F. The estimation of density dependence is another important problem. In principle, density should be no different from any other external covariate. We know of only one statistical analysis of bifurcation patterns (in the flour beetle Tribolium (Cushing et al., 2003), and it used inverse methods rather than MR). There should be more. Stochastic models are essential for understanding the effect of environmental fluctuations on population dynamics and persistence, but the estimation of stochastic models by MR is in its infancy. Bayesian methods for hierarchical random effects models may contribute to solving this problem. One challenge will be to identify appropriate distributions for the necessary matrix–valued random variates. Finally, methods that integrate mark–recapture and count information open up exciting possibilities. Taken by itself, the inverse problem of determining a model from a time–series of population estimates is usually poorly conditioned (because there are many sets of parameters that can generate the same or nearly the same time–series). Methods include transforming the matrix population model into a nonlinear autoregressive model with lognormally distributed errors (Dennis et al., 1995), quadratic programming methods that minimize the squared deviations between observed and predicted time–series (Wood, 1997), methods based on the Kalman filter and other state–space approaches (Besbeas et al., 2002, 2003; De Valpine & Hastings, 2002), and Bayesian methods (Gross et al., 2002). Although in this venue we view count data as strengthening the analysis of mark–recapture data, it is just as legitimate to think of the problem the other way around: mark–recapture data can render an ill–conditioned inverse problem soluble.

Acknowledgments We thank the National Science Foundation (grants DEB–9973518 and DEB–0235692), the National Oceanic and Atmospheric Administration (grant NA03NMF4720491), the Environmental Protection Agency (grant R–82908901), the David and Lucille Packard Foundation (grant 2000–01740), and the Robert W. Morse Chair at the Woods Hole Oceanographic Institution for financial support. Conversations with Christine Hunter, Jean–Dominique Lebreton, Mike Neubert, and Jim Nichols have shaped the ideas here, although those individuals should not be blamed. Comments by Evan Cooch helped improve the manuscript. The right whale analyses were based on data kindly provided by the New England Aquarium and the North Atlantic Right Whale Consortium. Crucial logistical support was provided by Le Soulcie. References Arnason, A., 1973. The estimation of population size, migration rates, and survival in a stratified population. Researches in Population Ecology, 15: 1–8. Besbeas, P., Freeman, 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. Besbeas, P., Lebreton, J.–D. & Morgan, B. J. T., 2003. The efficient integration of abundance and demographic data. Journal of the Royal Statistical Society C, 52: 95–102. 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. & Anderson, D. R., 1998. Model selection and inference: a practical information– theoretic approach. Springer–Verlag, New York, New York, U.S.A. Caswell, H., 1978. A general formula for the sensitivity of population growth rate to changes in life history parameters. Theoretical Population Biology, 14: 215–230. – 1989. Matrix population models. Sinauer Associates, Sunderland, Massachusetts, U.S.A. – 1997. Methods of matrix population analysis. In: Structured population models in marine, terrestrial and freshwater systems: 19–58 (S. Tuljapurkar & H. Caswell, Ed.). Chapman and Hall, New York, U.S.A. – 2001. Matrix population models. Second edition. Sinauer Associates, Sunderland, Massachusetts, U.S.A. Caswell, H., Fujiwara, M. & Brault, S., 1999. Declining survival probability threatens the North Atlantic right whale. Proceedings of the National Academy of Sciences, U.S.A., 96: 3308–3313. Caswell, H. & Kaye, T., 2001. Stochastic demography and conservation of Lomatium bradshawii in

Animal Biodiversity and Conservation 27.1 (2004)

a dynamic fire regime. Advances in Ecological Research, 32: 1–51. Caswell, H., Takada, T. & Hunter, C. M., 2004. Sensitivity analysis of equilibrium in density– dependent matrix population models. Ecology Letters, 7: 380–387. Caswell, H. & Trevisan, M. C., 1994. The sensitivity analysis of periodic matrix models. Ecology, 75: 1299–1303. Cochran, M. E. & Ellner, S., 1992. Simple methods for calculating age–based life history parameters for stage–structured populations. Ecological Monographs, 62: 345–364. Cohen, J. E., 1976. Ergodicity of age structure in populations with Markovian vital rates, I: countable states. Journal of the American Statistical Association, 71: 335–339. – 1977a. Ergodicity of age structure in populations with Markovian vital rates, II: general states. Advances in Applied Probability, 9: 18–37. – 1977b. Ergodicity of age structure in populations with Markovian vital rates, III: finite–state moments and growth rate; an illustration. Advances in Applied Probability, 9: 462–475. Cooch, E. G., Rockwell, R. F. & Brault, S., 2001. Retrospective analysis of demographic responses to environmental change: an example in the lesser snow goose. Ecological Monographs, 71: 377–400. Cushing, J. M., Costantino, R. F., Dennis, B., Desharnais, R. A. & Henson, S. M., 2003. Chaos in ecology. Academic Press, New York, U.S.A. Cushing, J. M. & Yicang, Z., 1994. The net reproductive value and stability in matrix population models. Natural Resources Modeling, 8: 297–333. Dennis, B., Desharnais, R. A., Cushing, J. M. & Costantino, R. F., 1995. Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecological Monographs, 65: 261–281. De Valpine, P. & Hastings, A., 2002. Fitting population models incorporating process noise and observation error. Ecological Monographs, 72: 57–76. Feichtinger, G., 1971. Stochastische modelle demographischer prozesse. Lecture notes in operations research and mathematical systems. Springer–Verlag, Berlin. Ferriére, R. & Gatto, M., 1995. Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theoretical Population Biology, 48: 126–171. Fujiwara, M., 2002. Mark–recapture statistics and demographic analysis. Ph. D. Dissertation, MIT/ WHOI Joint Program in Oceanography/Applied Ocean Science and Engineering. Fujiwara, M. & Caswell, H., 2001. Demography of the endangered North Atlantic right whale. Nature, 414: 537–541. – 2002. Estimating population projection matrices from multi–stage mark–recapture data. Ecology, 83: 3257–3265 Furstenberg, H. & Kesten, H., 1960. Products of

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random matrices. Annals of Mathematical Statistics, 31: 457–469. Grant, A., 1997. Selection pressures on vital rates in density dependent populations. Proceedings of the Royal Society B, 264: 303–306. Grant, A. & Benton, T. G., 2000. Elasticity analysis for density dependent populations in stochastic environments. Ecology, 81: 680–693. – 2003. Density–dependent populations require density–dependent elasticity analysis: an illustration using the LPA model of Tribolium. Journal of Animal Ecology, 72: 94–105. Gross, K., Craig, B. A. & Hutchinson, W. D., 2002. Bayesian estimation of a demographic matrix model from stage–frequency data. Ecology, 83: 3285–3298. Hunter, C. M., 2001. Demography of procellariids: model complexity, chick quality, and harvesting. Ph. D. Dissertation, Otago University, Dunedin, New Zealand. Hunter, C. M. & Caswell, H., 2004a (in press). Selective harvest of sooty shearwater chicks: effects on population dynamics and sustainability. Journal of Animal Ecology. – 2004b (in press). The use of the vec–permutation matrix in spatial matrix population models. Ecological modelling. Hurrell, J. W., 1995. Decadal trends in the North Atlantic oscillation: regional temperatures and precipitation. Science, 269: 676–679. Kermack, W. O. & McKendrick, A. G., 1927. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Statistical Society of London, A 115: 700–721. Keyfitz, N., 1968. Introduction to the mathematics of population. Addison–Wesley, Reading, Massachusetts, U.S.A. Khaladi, M., Grosbois, V., Lebreton, J. D., 2000. An explicit approach to evolutionarily stable strategies with a cost of dispersal. Non Linear Analysis, Real world applications, 1: 137–144. Lebreton, J. D., 1996. Demographic models for subdivided populations: the renewal equation approach. Theoretical Population Biology, 49: 291–313. 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., Khaladi, M. & Grosbois, V., 2000. An explicit approach to evolutionarily stable strategies: no cost of dispersal. Mathematical Biosciences, 165: 163–176. Leslie, P. H., 1945. On the use of matrices in certain population mathematics. Biometrika, 33: 183–212. Lesnoff, M., Ezanno, P. & Caswell, H., 2003. Sensitivity analysis in periodic matrix models: A postscript to Caswell and Trevisan. Mathematical and Computer Modelling, 37: 945–948. Levin, L. A., Caswell, H., Bridges, T., DiBacco, C., Cabrera, D. & Plaia, G., 1996. Demographic

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response of estuarine polychaetes to pollutants: Life table response experiments. Ecological Applications, 6: 1295–1313. Link, W. A. & Doherty Jr, P. F., 2002. Scaling in sensitivity analysis. Ecology, 83: 3299–3305. Lotka, A. J., 1907. Studies on the mode of growth of material aggregates. American Journal of Science, 24: 199–216, 375–376. – 1924. Elements of physical biology. Williams and Wilkins, Baltimore. Maryland, USA. (Reprinted 1956 by Dover Publications, New York, New York, U.S.A, as Elements of mathematical biology.) – 1934–1939. Théorie analytique des associations biologiques. Part I (1934), Part II (1939). Actualités Scientifiques et Industrielles, Hermann, Paris. (Published in English translation as A. J. Lotka. 1998. Analytical theory of biological populations. Translated by D. P. Smith & H. Rossert. Plenum Press, New York, U.S.A.) Metz, J. A. J. & Diekmann, O., 1986. The dynamics of physiologically structured populations. Springer–Verlag, Berlin, Germany. Metz, J. A. J., Nisbet, R. M. & Geritz, S. A. H., 1992. How should we define "fitness" for general ecological scenarios? Trends in Ecology and Evolution, 7: 198–202. Neubert, M. G. & Caswell, H., 2000a. Density– dependent vital rates and their population dynamic consequences. Journal of Mathematical Biology, 43: 103–121. – 2000b. Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology, 81: 1613–1628. 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. Nichols, J. D., Hines, J. E., Lebreton, J. D. & Pradel, R., 2000. Estimation of contributions to population growth: a reverse–time capture–recapture approach. Ecology, 81: 3362–3376. Oseledec, V. I., 1968. A multiplicative ergodic theorem: Ljapunov characteristic numbers for dynamical systems. Transaction of the Moscow Mathematical Society, 19: 197–231. Ottersen, G., Planque, B., Belgrano, A., Post, E., Reid, P. C. & Stenseth, N. C., 2001. Ecological effects of the North Atlantic Oscillation. Oecologia, 128: 1–14.

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Pradel, R. 1996. Utilization of capture–mark–recapture for the study of recruitment and population growth rate. Biometrics, 52: 703–709. Rand, D. A., Wilson, H. B. & McGlade, J. M., 1994. Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philosophical Transactions of the Royal Society, London B, 343: 261–283. Rogers, A., 1995. Multiregional demography: principles, methods, and extensions. Wiley, New York, U.S.A. Sandercock, B. K. & Beissinger, S. R., 2002. Estimating rates of population change for a neotropical parrot with ratio, mark–recapture and matrix methods. Journal of Applied Statistics, 29: 589–607. Silva, J. F., Raventos, J., Caswell, H. & Trevisan, M. C., 1991. Population responses to fire in a tropical savanna grass Andropogon semiberbis: a matrix model approach. Journal of Ecology, 79: 345–356. Smith, M., Caswell, H. & Mettler, P., 200a (in press). Stochastic flood and precipitation regimes and the population dynamics of a threatened floodplain plant. Ecological Applications. Takada, T. & Nakajima, H., 1992. An analysis of life history evolution in terms of the density–dependent Lefkovitch matrix model. Mathematical Biosciences, 112: 155–176. – 1998. Theorems on the invasion process in stage– structured populations with density–dependent dynamics. Journal of Mathematical Biology, 36: 497–514. Tuljapurkar, S. D., 1989. An uncertain life: demography in random environments. Theoretical Population Biology, 35: 227–294. – 1990. Population dynamics in variable environments. Springer–Verlag, New York. Tuljapurkar, S. & Caswell, H., 1997. Structured population models in marine, terrestrial and freshwater systems. Chapman and Hall, New York, U.S.A. Tuljapurkar, S. D. & Orzack, S. H., 1980. Population dynamics in variable environments I. Long– run growth rates and extinction. Theoretical Population Biology, 18: 314–342. Wood, S. N., 1997. Inverse problems and structured– population dynamics. In: Structured–population models in marine, terrestrial and freshwater systems: 555–586 (S. Tuljapurkar & H. Caswell, Ed.). Chapman and Hall, New York, U.S.A.

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

Francis, C. M. & Saurola, P., 2004. Estimating components of variance in demographic parameters of Tawny Owls, Strix aluco. Animal Biodiversity and Conservation, 27.1: 489–502. Abstract Estimating components of variance in demographic parameters of Tawny Owls, Strix aluco.— Survival rates of Tawny Owls (Strix aluco) were estimated using recapture and recovery data from approximately 20,000 nestling and adult owls ringed between 1980 and 1999 in southern Finland. Survival rates averaged 33% in the first year of life, 64% in the second, and 73% in subsequent years, but varied dramatically among years. Approximately 50% of annual variation in survival could be explained by stage of the vole cycle and severity of winter weather. Capture probabilities, an index of breeding propensity, varied dramatically among years, and could almost entirely be explained by the vole cycle, superimposed on a long–term increase in capture effort. Matrix models based on mean values in each year of the vole cycle, predict that in 2 out of 3 years, the population would decline by 13%–15% per year, offset by a large increase in the 3rd year. Numbers of nesting pairs are predicted to be low in one of three years, with no long–term trend, consistent with observed estimates of active nests. Key words: Population fluctuations, Voles, Winter weather, Survival estimation, Matrix models. Resumen Estimación de los componentes de la varianza en los parámetros demográficos del cárabo común, Strix aluco.— Se calcularon las tasas de supervivencia del cárabo común (Strix aluco) utilizando datos de recaptura y recuperación correspondientes a unos 20.000 cárabos comunes —entre polluelos y adultos—, anillados entre 1980 y 1999 en el sur de Finlandia. Las tasas de supervivencia alcanzaron un promedio del 33% en el primer año de vida, del 64% en el segundo y del 73% en los años subsiguientes, variando de forma espectacular entre los distintos años. Alrededor del 50% de la variación anual en la supervivencia pudo ser explicada por el estadio en que se encontraba el ciclo poblacional de los micrótidos y el rigor del clima invernal. Las probabilidades de captura —que representan un índice de la propensión a la reproducción— variaron considerablemente entre los distintos años, pudiendo explicarse en su práctica totalidad por el ciclo de los micrótidos, superpuesto a un aumento a largo plazo del esfuerzo de captura. Según los modelos matriciales basados en los valores promedio correspondientes a cada año del ciclo de los micrótidos, predicen que en dos de cada tres años la población disminuirá entre un 13% y un 15% anual, aunque ello se verá compensado por un considerable aumento durante el tercer año; asimismo, se calcula que el número de parejas nidificantes será bajo uno de cada tres años, sin ninguna tendencia a largo plazo, lo que concuerda con las estimaciones observadas acerca de los nidos activos. Palabras clave: Fluctuaciones poblacionales, Ratones de campo, Clima invernal, Estimación de supervivencia, Modelos matriciales. Charles M. Francis, Canadian Wildlife Service, National Wildlife Research Centre, Ottawa, Ontario, K1A 0H3, Canada.- Pertti Saurola, Finnish Museum of Natural History, P.O. Box 17, FIN–00014 University of Helsinki, Finland.

ISSN: 1578–665X

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Introduction Populations of many animal species fluctuate in response to changing environmental conditions. For example, breeding populations of some seabirds are affected by multi–annual fluctuations in oceanographic patterns such as the El Niño Southern Oscillation, which affect food supply and hence both breeding success and survival (Nur & Sydeman, 1999). In other species, irregular severe weather events may cause substantial mortality at a particular time of year, such as for Grey Herons Ardea herodias overwintering in Britain (Freeman & North, 1990). Of particular interest, from the perspective of variability, are the multi–year population cycles of small mammals and their associated predator communities. These have attracted considerable research attention, both studying the impact of small mammals on predators (e.g. Brand & Keith, 1979) and the impact of predators on small mammals (e.g., Korpimäki et al., 2003). Intensive long–term studies of the 10–year cycle of the Snowshoe Hare (Lepus americanus) and its associated predators in northern Canada suggest that this system is driven largely intrinsically, by predator–prey dynamics, with dispersal of predators, and possibly large–scale climatic factors, helping to synchronize the cycle (Krebs et al., 2001). In contrast, there is still considerable controversy about the factors driving the three to four year cycles of small rodents, particularly voles, in northern Europe (e.g., Oli, 2003; Korpimäki et al., 2003). Ruesink et al. (2002) emphasized the importance of accurate demographic data for understanding population cycles. Among the predators involved in these northern forest systems, are several species of owls. Breeding propensity, clutch size, nesting success, movement patterns and survival rates of Great Horned Owls (Bubo virginianus) have all been shown to vary in relation to the 10– year cycle in Snowshoe Hare (Houston & Francis, 1995; Rohner et al., 2001). Breeding success and emigration patterns of several European owl species have also been shown to fluctuate in response to the 3–4 year cycle of many microtine rodents (Saurola, 1997). Dispersal distances of many Finnish owls are greater in years of low vole abundance (Saurola, 2002). Survival rates of male Tengmalm’s Owls (Aegolius funereus) vary dramatically in response to vole abundance with estimated survival decreasing from 50–75% in peak years to only 30– 35% in poor years (Hakkarainen et al., 2002). In addition, surviving males were less likely to breed in low vole years, as indicated by lower recapture rates. Breeding success and survival also vary dramatically among years in the Ural Owl (Strix uralensis) (Brommer et al., 2002). Tawny Owls (Strix aluco) breed throughout much of Europe. In Finland, the Tawny Owl is a relative newcomer, with the first record in 1875, and the distribution still largely restricted to the south (Saurola, 1995). The Tawny Owl may be restricted from expanding farther north by winter tempera-

tures and snow conditions. Like most other owls in Finland, Tawny Owls feed on voles when these are abundant, but as generalist feeders, they can also feed on other prey when voles are scarce (Saurola, 1995; Sunde et al., 2001; Solonen & Karhunen, 2002). Nevertheless, numbers of occupied territories, numbers of nesting pairs, and numbers of young per nesting attempt have been found to vary in response to vole abundance (Saurola, 1997). Changes in population numbers with the vole cycle could be influenced by variation in breeding success (productivity), variation in survival rates, or a combination of both. To determine the relative importance of these factors requires estimates of annual variation in each of these parameters. Data from large–scale, coordinated ringing in Finland provide an opportunity to examine the impact of vole abundance, as well as other environmental factors such as winter weather, on demographic parameters of this species, and to estimate the impact of these factors on population dynamics of the owls. In this paper, joint analyses of recapture and recovery data of Tawny Owls ringed throughout southern Finland (Francis & Saurola, 2002) are used to estimate annual variation in age–specific survival probabilities of this species and the proportion of variation that can be explained by the vole cycle and by winter weather conditions. Annual variation in recapture parameters of yearlings and adults is also estimated. Because most adults are captured as breeders, capture probabilities can be considered an index of the proportion of pairs that are breeding (breeding propensity), after allowing for any variation in capture effort. Finally, the results of these analyses are put into a simple demographic model to estimate the overall impact of fluctuations in vole abundance on population size of these owls. Methods Field data Especially since the early 1970s, bird ringers in Finland have been strongly encouraged to ring nestlings and capture adults of several species of birds of prey (Saurola, 1997; Saurola & Francis, 2004). As a consequence of this activity, a very large data base of ringing, recapture and recovery data has been accumulated for several species of owls. For the Tawny Owl, ringing has taken place during the breeding season throughout much of the species breeding range in Finland. Many individuals now breed in nest boxes, mostly installed by ringers and inspected regularly during the breeding season. At successful nests, the nestlings are ringed when they are of sufficient age. In addition, most ringers also attempt to trap and ring the adult female and many also trap the male at each nest. In this paper, data from the 20 year period from 1980 to 1999 were selected for analysis. The sample sizes during this period (including only breeding

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season captures) were 18,166 nestlings ringed, leading to 1,737 recaptures and 1,655 recoveries, along with 1,742 adults ringed which generated 1,278 recaptures and 190 recoveries. Small mammal abundance varies dramatically among years, with variation in abundance of voles between high and low points in the population cycle up to two orders of magnitude (Brommer et al., 2002; Hanski et al., 1991). Quantitative measures of vole abundance were only available from some areas of Finland. Previous analyses have shown that these are generally well–correlated at broad spatial scales with a 3–year cycle in most of southern Finland (Sundell et al., 2004). By combining these quantitative observations with casual observations in the field, years were readily classified into one of three categories, here labelled based on the potential impact for survival of owls over the coming year: Poor – voles are at their peak at the beginning of the year, but crash over the course of the year, such that owls have very few voles available for food during the year, especially in the following winter; Medium – voles are low at the start of the year, but gradually increase over the course of the year; Good – voles are moderately abundant at the beginning of the year and increase to a peak over the summer and following winter. Winter severity was indexed by two measures: mean daily temperature over the winter, derived from the mean monthly temperatures between December and March, and mean monthly snow depth during the same period. Both measures were taken as the mean of data from five weather stations of the Finnish Meteorological Institute in southern Finland: 1,201 Jokioinen (60° 49’ N, 23° 30’ E); 1,303 Hattula Leteensuo (61° 04’ N, 24° 14’ E); 1,304 Hattula Lepaa (61° 08’ N, 24° 20’ E); 1,306 Pälkäne Myttäälä (61° 20’ N, 24° 13’ E); 1,403 Lammi, Biological station (61° 03’ N, 25° 03’ E). Statistical analyses Survival probabilities were estimated using joint recapture and recovery models (Burnham, 1993). Previous analyses have shown that these combined models allow estimation of age–specific survival rates with little apparent bias due to emigration from the study area (Francis & Saurola, 2002). These models estimate four classes of parameters: survival ( ) – the probability that an animal alive at the beginning of the year (here defined as 1 June) will be alive the following year; recapture (p) – the probability that a marked individual alive and present in the breeding population will be captured in a particular year; recovery (r) – the probability that an individual that dies in a particular year will be found and its ring number reported to the ringing office; and "fidelity" (F) – the probability that a marked surviving individual that was in the local population the previous year is still in the population available for recapture. This "fidelity" parameter is often considered as the probability that an individual will not emigrate from the study area (Burnham, 1993), but this interpreta-

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tion is dependent on several assumptions: recapture effort is concentrated in a particular study area; all animals present in the study area are equally likely to be captured; and all individuals that have left the study area are equally like to be recovered. For Tawny Owls, even individuals that emigrate from a particular area can potentially be recaptured, because capture effort occurred throughout the species range in Finland. Also, although individuals from anywhere in the country can potentially be recovered, the probability may vary geographically with human population densities. Finally, in some areas, capture probabilities of males were probably lower than those of females, with the result that males may appear to have "emigrated" from the study area (note that Tawny Owls were not sexed as nestlings nor on recovery, so this is not easily modelled). For these reasons "fidelity" parameters can not reliably be interpreted in any biological context, and we do not present the estimates. Initially, a series of models was tested with 2 or 3 age classes for each parameter, and annual variation in all parameters except “fidelity” parameters, which were constrained to be constant within each age class. For models with more than 2 age classes, birds ringed as "adults" were treated as if they were all in the highest age class. In practice, these likely included some birds in their second year which were not distinguished from adults. While these are unlikely to be a large percentage of birds captured (because of the lower nesting probability of yearlings), this may cause a slight downwards bias in the adult estimates. Goodness of fit was tested for the most general model examined (with three age classes for survival and recapture probabilities, two age classes for recovery, and three for "fidelity", the first three all varying among years) using a generalized bootstrap goodness of fit within MARK (White & Burnham, 1999). This indicated some residual lack of fit, which was dealt with through quasi–likelihood approaches, using an overdispersion parameter = 1.2. QAICc was then used for selection among these models. All model selection and parameter estimation was performed using MARK; however, because of the complexity involved with editing the large Parameter Index Matrices and the Design Matrix, the input files for the analyses were created using custom–written SAS programs (SAS Institute, 2001). Models were also checked, and in many cases rerun, from within the MARK interface. This was particularly necessary for some constrained parameter models that converged on a local maximum of the likelihood and had to be rerun with alternative starting values. Two approaches were selected to estimate annual variation in survival and recapture parameters (the latter being related to breeding propensity because only breeding birds were captured). The first approach was to use the Variance Components option within MARK, modelling annual variation in age–specific survival as a random effect. This es-

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sentially involves modelling residual variation in the annual estimates, after correcting for sampling error, using the variance–covariance matrix of the estimates (White & Burnham, 1999). This was done with no covariates except the mean (to measure total variance), then with voles as a covariate (treated as a class variable with 3 unranked levels to avoid imposing any assumptions about the nature of the response of different parameters to the vole cycle), with weather as covariates (including both snow depth and temperature, each standardized to a range between +1 and –1), and with both together. The second approach involved using ultrastructural models to incorporate covariates directly into the estimation procedure. First, a model constraining survival to be constant over time was fitted to the data and the difference in deviance (–2 * Likelihood) between this model and the year–specific model was taken as an index of total variation in annual survival rates. Then models were fitted constraining survival rates to vary in relation to voles, weather, or both, using the same coding as above. The percent reduction in the deviance, relative to the difference between constant and year–specific models, was used as a measure of the total variance in annual survival rates that could be explained by the covariates. This procedure was carried out separately for first–year survival (allowing adult and yearling survival to vary among years) and for survival of older age classes (which varied in parallel, while allowing first–year survival to vary independently among years). The variance components approach was also used to estimate the percentage of variance in capture probabilities that could be explained by the vole cycle. In addition, as there appears to have been an increase in trapping effort for adults over time, a model incorporating year as a linear covariate, as well as voles, was also fitted. Mean productivity estimates (nestlings per active nest) were derived from the Raptor Questionnaire (Saurola & Francis, 2004) that is completed by all active ringers in Finland. Population modelling A complete analysis of the impacts of both voles and winter weather on population dynamics of Tawny Owls, through stochastic simulation models, is beyond the scope of this paper. However, as a preliminary evaluation of the potential impact of the vole cycle, a deterministic, 2–stage, pre–breeding matrix model was developed, based on mean parameter values during each of the three stages of the vole cycle. The two stages were yearlings and adults, with transition probabilities from yearlings to adults and adults to adults represented by second year and adult survival probabilities respectively, while transitions to yearlings were estimated by the product of age–specific breeding propensity (probability a bird will breed), mean nestlings per nest (divided by 2 to allow for the fact that two adults are associated with each nest), and first–year survival probabilities (fig.

1). Breeding propensity for yearlings and adults at each stage of the vole cycle was estimated assuming that it was directly proportional to estimated mean recapture rates at each stage of the cycle and that 90% of adults breed during peak years of vole abundance. This 90% estimate was based on general impressions of the numbers of non–breeders during field work ringing owls, and is similar to peak estimates of percentage breeders of Ural Owls (Brommer et al., 2002). Three separate matrices were developed, one for each year of the three stages of the vole cycle. The dominant eigenvalue of each matrix was used to estimate what population growth rate would have been, if population parameters had remained unchanged at those values. Population growth over the vole cycle, assuming that it continues as a regular three year cycle, can be estimated by analysing the matrix generated by multiplying the three matrices together in reverse order (Caswell, 2001). Age ratios at each stage of the cycle, as well as realized population growth rates in each year of the cycle, were determined by iteratively multiplying the matrices until the age distributions stabilized. Results Model selection indicated that survival probabilities varied for at least three age classes, whereas capture parameters could be adequately modelled with two age classes (table 1). This apparent discrepancy is, in fact, consistent with a model that allows different parameters for nestlings, yearlings and adults, because the first capture parameter actually refers to capture at one year of age (capture probabilities of nestlings are not included in the models). For the survival parameters, a model in which adult and yearling survival parameters are constrained to vary in parallel (on a logit scale) was found to be an adequate fit, and hence was used for all further analyses. Survival rates varied considerably among years in all age classes (fig. 2), but the variance of first– year survival rates was about 40% higher than that of adults (table 2). Variance components analysis indicated that stage of the vole cycle could explain about 39% of annual variation in first–year survival, but only 11% of variation in adult survival (table 2). In contrast, winter severity, as measured by snow depth and temperature, explained only 35% of variation in first–year survival, but about 58% of variation in adult survival. Collectively, voles and winter weather explained approximately 50% and 58% of total variation in survival for young and adults respectively (table 2, fig. 2). Estimates of variance components based upon changes in the deviance of ultra–structural models produced very similar results, with overall variance explained by all covariates estimated at 54% and 63% for young and adults respectively (table 3). Annual survival rates for both age classes increased with warmer winter temperatures (fig. 3)

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BAD· (N/2) · S1 BY · (N/2) · S1

SAD S2

Yearlings

Adults

Fig. 1. Life cycle diagram for a pre–breeding, stage–based matrix model of Tawny Owls in Finland: BY. Proportion of yearlings (one–year old birds) that breed; BAD. Proportion of adults that breed; N. Mean number of nestlings fledged per nesting attempt (assumed the same for yearlings and adults); S1. Survival probability of owls in their first year, from fledging to the following breeding season; S2. Survival probability of owls over their second year; SAD. Annual survival probability of adults. Fig. 1. Diagrama del ciclo vital según un modelo de matrices por etapas para la pre–reproducción del cárabo común en Finlandia: BY. Proporción de aves de un año de vida que se reproducen; BAD. Proporción de adultos que se reproducen; N. Promedio de polluelos volantones por cada intento de anidación (presuponiendo lo mismo para aves de un año de vida y para adultos); S1. Probabilidad de supervivencia del cárabo común durante su primer año de vida, desde la fase de volantones hasta la siguiente estación de reproducción; S2. Probabilidad de supervivencia del cárabo común durante su segundo año de vida; SAD. Probabilidad de supervivencia anual de los adultos.

and with reduced winter snow depth (fig. 4). However, both of these weather variables were strongly correlated, and after taking into account temperature, snow depth explained little residual variation. Mean winter temperatures varied by nearly 10°C among years over the study period. The predicted

impact on the owls was most dramatic for first– year survival, with a predicted increase from only 20% annual survival in the coldest years, up to 40% in warmer years. Adult survival was predicted to vary from 60% up to 80% over the same temperature range (fig. 3).

Table 1. Results of initial model selection process to estimate annual variation in age–specific survival and recapture probabilities: Model parameters represent probabilities for survival ( ), recapture (p), recovery (r) and "fidelity" (F) (see text for definitions). In each column, letters refer to variation modeled in first, second, and third age classes (if present) (Y. Year–specific; C. Constant; Y+Y. Different intercepts, but parallel —on a logit scale— annual variation in the 2nd and oldest age classes). was estimated at 1.2 from a parametric bootstrap procedure. Tabla 1. Resultados del proceso de selección inicial de modelos para estimar la variación anual en la supervivencia específica dependiente de la edad y las probabilidades de recaptura: "Model parameters" representan la probabilidad de supervivencia ( ), recaptura (p), recuperación (r) y fidelidad (F) (ver el texto para las definiciones). En cada columna las letras se refieren a la variación modelada en la primera, segunda y tercera clases de edad (Y. Anual; C. Constante; Y+Y. Distintas intercepciones, pero variación anual paralela —en escala logit— en la segunda y superiores clases de edad). se calcula a 1,2 a partir de un proceso bootstrap paramétrico.

Model parameters

Model fit F

AIC( )

# Parm.

Deviance

Y,Y+Y

Y,Y

C,C,C

0.00

120

3270.65

Y,Y+Y

Y,Y,Y

Y,Y

C,C,C

7.43

137

3243.70

Y,Y,Y

Y,Y

C,C,C

14.04

136

3252.33

Y,Y,Y

Y,Y

C,C,C

21.67

151

3229.58

Y,Y

C,C

34.84

117

3311.57

Y,Y

56.70

141

3284.87

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Survival probability

1.0 0.9 0.8

First–year

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 80

Survival probability

1.0 0.9

88 90 Year

96 98

Adult

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 80

Fig. 2. Estimated annual survival probabilities with 95% confidence intervals for Tawny Owls in their first year of life (top) and as adults (bottom), from 1980 to 1997. Note that second–year survival probabilities were modelled as fluctuating in parallel (on a logit scale) with adult survival, but averaged about 13% lower. Dashed lines show predicted survival rates based on a regression model incorporating stage of the vole cycle and two measures of severity of winter weather. Fig. 2. Estimaciones de las probabilidades de supervivencia anual, con intervalos de confianza del 95%, del cárabo común durante su primer año de vida (parte superior) y como adulto (parte inferior), desde 1980 hasta 1997. Obsérvese que las probabilidades de supervivencia durante el segundo año se modelaron como si fluctuaran en paralelo (en una escala logit) con la supervivencia de los adultos, pero el promedio fue alrededor de un 13% inferior. Las líneas discontinuas indican las tasas de supervivencia previstas basadas en un modelo de regresión que incorpora la etapa del ciclo de los micrótidos y dos medidas de rigor invernal.

Capture probabilities also varied dramatically among years, especially for yearlings (fig. 5). In this case, stage of the vole cycle explained nearly all of the variation in capture probabilities for yearlings, with only minimal additional variation explained by a linear trend variable (table 4). For adults, stage of the vole cycle, on its own, explained only 64% of variation in capture probabilities, but addition of a trend variable, possibly reflecting increased effort in recent years, brought this up to nearly 95% as well. Weather variables explained no additional variation. Considering only the impact of the vole cycle, mean survival rates in poor and good years ranged

from 26% to 43% in first–year, 57% to 71% in second year, and 67% to 79% in older owls (table 5). Capture probabilities, which can be considered an index of breeding propensity as only nesting birds were captured, were lowest in years following poor survival, dropping to only 4% for yearlings and 29% for adults. For yearlings, the next lowest years were in periods when voles crashed, presumably because the crash starts just before or during the breeding cycle. In contrast, for adults, there was little difference in breeding propensity between good years and years when voles crashed. Data on nesting productivity from 1986–1997 (Saurola & Francis, 2004) indicate that nest suc-

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Table 2. Components of variance in annual survival probabilities for first year birds and older birds, based on using the variance components option within program MARK for survival estimates from model (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C) from table 1. Residual variance in this model is estimated after adjusting for variance due to sampling error. Voles were modelled as a class variable with three states, and weather variables (snow depth and temperature) were modelled as continuous covariates. Tabla 2. Componentes de varianza en las probabilidades de supervivencia anual para las aves de un año de vida y las de más edad, a partir de la opción de componentes de varianza en el marco del programa MARK para estimaciones de supervivencia a partir del modelo (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C) de la tabla 1. En este modelo la varianza residual se estima tras haber ajustado la varianza como consecuencia del error de muestreo. Los ratones de campo se modelaron como una variable de clases con tres estados, mientras que las variables climáticas (profundidad de la nieve y temperatura) se modelaron como covariantes continuas.

First–Year

Yearling/adults

Covariates

Residual variance

% Reduction in variance

Residual variance

%Reduction in variance

No covariates

0.01130

0.0

0.00802

0.0

Voles alone

0.00691

38.8

0.00712

11.2

Weather variables alone

0.00730

35.4

0.00335

58.3

Voles and weather variables

0.00566

50.0

0.00340

57.6

Table 3. Estimates of the percent of total variance in annual survival probabilities for first year birds and older birds, based on fitting ultra–structural models with various constraints placed on first year survival, starting with model (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C) from table 1. All constraints were based on logit–link models, with voles modelled as a class variable with three states, and weather variables (snow depth and temperature) modelled as continuous covariates. In all models, the other age classes were allowed to vary among years. Tabla 3. Estimaciones del porcentaje de la varianza total en las probabilidades de supervivencia anual para las aves de un año de vida (parte superior) y las aves de más edad (parte inferior), tras haber ajustado modelos ultraestructurales con varias restricciones en la supervivencia del primer año, empezando con el modelo (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C) de la tabla 1. Todas las restricciones se basaron en modelos logit-link, con los ratones de campo modelados como una variable de clases con tres estados, mientras que las variables climáticas (la profundidad de la nieve y la temperatura) se modelaron como covariantes continuas. En todos los modelos se permitió que las otras clases de edad variaran entre los distintos años.

Model

Deviance

3270.653

0.00

% Variance explained

First–year survival Year–specific

100

Voles and weather variables

3311.691

41.04

Weather variables alone

3328.888

58.23

Voles alone

3334.364

63.71

No covariates (constant survival)

3360.361

89.71

Yearling and adult survival Year–specific

3270.653

0.00

100

Voles and weather variables

3302.467

31.81

Weather variables alone

3309.923

39.27

Voles alone

3349.046

78.39

No covariates (constant survival)

3357.027

86.37

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Survival probability

1.0 0.9

First–year

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Temperature (ºC)

Survival probability

1.0 0.9

Adult

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Temperature (ºC)

Fig. 3. Estimated annual survival probabilities with 95% confidence intervals for Tawny Owls in their first year of life (top) and as adults (bottom), from 1980 to 1997 plotted in relation to mean winter temperature. Dashed line shows a simple linear regression. Fig. 3. Estimaciones de las probabilidades de supervivencia anual, con intervalos de confianza del 95%, del cárabo común durante su primer año de vida (parte superior) y como adulto (parte inferior), desde 1980 hasta 1997, representadas gráficamente con relación a la temperatura media invernal. La línea discontinua indica una regresión lineal simple.

cess, among active nests, parallels breeding propensity of yearlings, with lowest productivity (2.30 young per active nest) in years following a crash, medium productivity (2.76 young/nest) in years when the crash actually occurred, and highest productivity (3.03 young/nest) in years when voles were abundant and increasing. These values were used to construct population projection matrices based on the model in figure 1 for each of the three phases of the vole cycle (table 6). The lower entries in each of the component 2 x 2 matrices represent survival probabilities for yearlings and adults (table 5). The upper values represent the product of the number of young produced and the first year survival rate, calculated as Bx · (N/2) · S1, where Bx is estimated by

Px · (0.90/PAD–peak). For example, the number of new yearlings produced by yearlings in poor years is estimated at: 0.189 · (0.90/0.469) · (2.76/2) · 0.26 = 0.130. The dominant eigenvalues of these matrices suggest that, if parameters were to remain stable at each of those levels, conditions of poor and medium years of the vole cycle would lead to a 10% to 12% decline per year, whereas conditions for good years would lead to an increase of about 26% (table 6). Such analyses do not, however, reflect actual population changes over the cycle, because the age ratio changes through the cycle. The projection matrix over the complete vole cycle indicates that, over a 3–year period, the popu-

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1.0 Survival probability

0.9

First–year

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

20 30 Snow depth (cm)

1.0 Survival probability

0.9

Adult

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

Fig. 4. Estimated annual survival probabilities with 95% confidence intervals for Tawny Owls in their first year of life (top) and as adults (bottom), from 1980 to 1997 plotted in relation to mean snow depth. Dashed line shows a simple linear regression. Fig. 4. Estimaciones de las probabilidades de supervivencia anual, con intervalos de confianza del 95%, del cárabo común durante su primer año de vida (parte superior) y como adulto (parte inferior), desde 1980 hasta 1997, representadas gráficamente con relación a la profundidad media de la nieve. La línea discontinua indica una regresión lineal simple.

lation would be approximately stable (at least within the uncertainty of the parameter estimates), but fluctuations in numbers of birds are greater than suggested by the eigenvalues. The overall population drops by 13% in the first year, another 15% in the second year, but then increases by 32% in the third year (table 6). Much of this variation arises as a result of changes in the age ratio, with a large surge in production of young in the third year. Population changes for adults lag one year behind, and are less extreme: the number of adults in the population increases by 7.4% over the poor year (as the large number of yearlings produced in the previous good year matures), then decreases by 1.7% in the medium year, and 6.3% in the good year. Based on these models, and taking into account the breeding probability of each age class, the model predicts that

the number of active nests should be lowest at the beginning of a medium year, increase by 74% by the start of a good year, then increase by a further 5% at the start of a poor year before dropping again. Discussion These analyses indicate that all aspects of the demography of Tawny Owls in Finland vary substantially among years, including survival probabilities, capture probabilities (an index of breeding propensity) and nesting success. The variance components analysis suggests that more than half of the variation in survival rates for all age classes can be explained by a combination of vole abundance and winter weather.

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Capture probability

1.0 Yearling 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 80

282

88 90 Year

Capture probability

1.0 0.9 Adult 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 80

Fig. 5. Estimated capture probabilities with 95% confidence intervals for yearling (top) and adult (bottom) Tawny Owls in Finland from 1981 to 1998. Dashed lines show predicted capture probabilities based on a regression model incorporating stage of the vole cycle and year as a linear covariate. Fig. 5. Estimaciones de las probabilidades de captura, con intervalos de confianza del 95%, del cárabo común durante su primer año de vida (parte superior) y como adulto (parte inferior), en Finlandia, desde 1981 hasta 1998. Las líneas discontinuas indican las probabilidades de captura previstas a partir de un modelo de regresión que incorpora la etapa del ciclo de los micrótidos y el año como una covariante lineal.

Both approaches to estimating variance components produced similar estimates, though there were some slight differences, presumably reflecting the fact that both methods involve approximations and asymptotic assumptions. The variance components approach (White & Burnham, 1999) produces the apparent anomaly of a poorer fit when both weather and voles are considered together, than with just weather effects. This may be due to the small number of years for estimating variance (18) relative to the number of parameters (4), and changes in the estimated error variance with reduced degrees of freedom as more dependent variables are added. The approach using ultrastructural models is robust to this particular limitation, but has other limitations, including the

fact that only one set of parameters can be modelled at a time without introducing bias. Bayesian approaches, using Markov Chain Monte Carlo (MCMC) would allow more robust estimates of variance components (Link & Barker, 2004), for both strictly random effects models as well as for mixed effects models such as this one, and should be considered for future analyses. An examination of the individual survival estimates (fig. 2) suggests that a substantial amount of the residual variation, after accounting for voles and weather, is due to particularly high survival, especially for first–year birds, in two peak vole years in 1985 and 1988. Anecdotal evidence suggests that voles were exceptionally abundant in those two years, despite the fact that capture prob-

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Table 4. Components of variance in annual capture probabilities for yearling and older birds, based on using the variance components option within program MARK for recapture estimates from model (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C). Residual variance in this model is estimated after adjusting for variance due to sampling error. Voles were modelled as a class variable with three states, while year was modelled as a linear covariate. Tabla 4. Componentes de varianza en las probabilidades de captura anual para las aves de un año de vida y de más edad, a partir de la opción de componentes de varianza en el marco del programa MARK para estimaciones de recaptura a partir del modelo (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C). En este modelo, la varianza residual se estima tras haber ajustado la varianza como consecuencia del error de muestreo. Los ratones de campo se modelaron como una variable de clases con tres estados, mientras que el año se modeló como una covariante lineal. Yearlings

Adults

Covariates

Residual variance

No covariates

0.01324

0.0

0.01048

0.0

Voles alone

0.00061

95.4

0.00375

64.2

Voles and year

0.00061

95.4

0.00068

93.5

abilities, which fluctuate almost perfectly in synchrony with the vole cycle (fig. 5) show nothing exceptional about those years. Different responses of survival and breeding propensity to quantitative variation in vole abundance might be expected, because the limiting factors are at different times of

% reduction in variance

Residual variance

% reduction in variance

year. Breeding activity is influenced mainly by vole abundance early in the season and may plateau at moderate levels of vole abundance. Survival, in contrast, is likely to be limited later in the year, especially in winter, and may not asymptote at the same level. Unfortunately, the quantitative data re-

Table 5. Estimates of mean survival rates (S) and capture probabilities (p) with their standard errors in relation to the vole cycle, estimated from the model (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C). Stages of the vole cycle are labelled with respect to their predicted impact on survival probabilities. Because the main impact of the crash is after the breeding season, the relative impacts on capture probabilities (assumed proportional to breeding propensity) are different from those on survival. Tabla 5. Estimaciones de las tasas de supervivencia media (S) y probabilidades de captura (p), con sus errores estándar con relación al ciclo de los micrótidos, estimadas a partir del modelo (Y,Y+Y) p(Y,Y) r(Y,Y) F(C,C,C). Los estadios del ciclo de los micrótidos se determinan en función del impacto esperado sobre la probabilidad de supervivencia. Dado que el principal impacto se da después de la estación de reproducción, los impactos relativos sobre las probabilidades de captura (asumidos proporcionales a la propensión de reporducirse) son diferentes a los de la supervivencia.

Stage of the vole cycle Poor

Medium

Good

Parameter

Mean

26.0

4.2

30.1

4.4

42.7

4.8

57.3

5.6

64.6

5.7

70.6

5.4

SAD

67.1

4.5

73.8

4.5

79.2

4.3

PYearling

18.9

2.2

3.9

1.4

30.5

3.7

PAD

45.8

3.0

28.9

3.0

46.9

3.1

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Table 6. Projection matrices, their dominant eigenvalues ( ), and estimated population change ( N) for Tawny Owls in Finland during each year of the vole cycle, based on the model in figure 1. The "whole cycle" matrix represents the projection from one good year to the next good year (see text), and as such and N for this matrix refers to a 3â&#x20AC;&#x201C;year period. Change in population size from the beginning of a particular phase of the vole cycle to the start of the next phase ( N) and the percentage of yearlings at the end of the cycle (% Yearlings), calculated through iterative deterministic modelling. Tabla 6. Matrices de proyecciĂłn, sus valores propios dominantes ( ), y cambio poblacional estimado ( N) para el cĂĄrabo comĂşn en Finlandia durante cada aĂąo del ciclo de los micrĂłtidos, a partir del modelo de la figura 1. La matriz del "ciclo completo" representa la proyecciĂłn de un aĂąo bueno hasta el siguiente aĂąo bueno (vĂŠase el texto) y, por consiguiente, en esta matriz, y N se refieren a un perĂodo de 3 aĂąos. Cambio en el tamaĂąo de poblaciĂłn desde el inicio de una determinada fase del ciclo de los micrĂłtidos hasta el inicio de la siguiente fase ( N) y el porcentaje de individuos de un aĂąo al final del ciclo (% Yearlings) calculado mediante un modelo determinĂstico itarativo.

Projection matrix

Stage of vole cycle

Yearling

Adults

Yearling

0.130

0.315

0.904

0.869

27.5

Adults

0.573

0.671

Medium

Yearling

0.026

0.192

0.883

0.859

17.0

Adults

0.646

0.738

Good

Yearling

0.378

0.582

1.259

1.324

41.3

Adults

0.706

0.792

Yearling

0.338

0.458

0.989

â&#x20AC;&#x201D;

Adults

0.482

0.650

Poor

Whole cycle

quired to test this hypothesis, on variation among cycles both in vole abundance and in how this changes across seasons, are lacking. The relationship between capture probabilities and the vole cycle, after allowing for the longâ&#x20AC;&#x201C;term increase, is almost unbelievably close â&#x20AC;&#x201D;it is important to note that all of the estimates in figure 5 were derived from unconstrained yearâ&#x20AC;&#x201C;specific parameter estimates. This relationship is unlikely to be affected by ringing effort, which is believed to be quite independent of the stage of the vole cycle. Most ringers check every nest box in their territory every year, and make a concerted effort to trap nesting birds at all occupied nest boxes. However, the longâ&#x20AC;&#x201C;term increase in capture probabilities suggests that efforts by the ringing office to encourage ringers to capture more adults, especially males, have been successful. In this analysis, we did not consider differences in capture probabilities between the sexes, which are likely to differ based on the fact that nearly three times as many adult females as males are ringed (unpublished data). Unfortunately, sex can only be determined for adults, and thus is unknown for nearly all nestlings that were not subsequently recaptured, including most of those that were later recovered without being recaptured. It would be possible to

%Yearlings

develop models that allow sexâ&#x20AC;&#x201C;specificity in some parameters such as capture probabilities, but only by making explicit assumptions about other parameters that are not estimable because of missing information. In any case, it is quite likely that the main difference between the sexes is in capture probability, rather than either breeding propensity or survival. Matrix models indicate that the abundance of voles has a dramatic impact on the demography of Tawny Owls, with populations declining for two years out of every three, and then bounding back in the 3rd year, such that the longâ&#x20AC;&#x201C;term trend is for a roughly stable population (table 6). Although total population size is predicted to show two years of decline followed by an increase, the numbers of nesting pairs are predicted to show low breeding numbers in medium years followed by 2 years of increases. This latter pattern agrees at least qualitatively with observed numbers of nesting pairs estimated from the Raptor Census from 1982â&#x20AC;&#x201C;1997 (Saurola, 1997; Saurola & Francis, 2004 â&#x20AC;&#x201D;note that 1984 was a medium year). The matrix model suggests a slight longâ&#x20AC;&#x201C;term decline, of 1% every 3 years, but a slight change in some of the parameter estimates, such as the percentage of adults that attempt to breed in a good year,

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could shift this to a stable or slightly increasing population. In any case, a change this small would not be detectable over a 20 year period. Observed population data show no sign of any long–term trend over the study period. The impact of the vole cycle on this species is surprisingly great, considering that the Tawny Owl is a generalist feeder and can switch to other species of prey when voles are scarce (Saurola, 1995; Sunde et al., 2001; Solonen & Karhunen, 2002). The limited effect of vole abundance on adult survival (table 5) suggests that adults are able to use alternative prey to survive, although their breeding propensity is influenced by vole abundance. In contrast, young birds appear to be strongly dependent on small rodents both for survival over their first year, and for breeding in the subsequent year, if they do survive. Previous analyses of data from Tengmalm’s Owl (Hakkarainen et al., 2002), which is relatively specialized on voles, indicate a more dramatic effect on adult survival, with nearly a 50% reduction in survival during years of low vole abundance, in addition to reduced breeding propensity. Survival probabilities of Ural Owls also vary considerably among years for both young and adults. Brommer et al. (1998, 2002) found that adult survival of Ural Owls ranged from a low of 60% to as high as 90%, while first–year survival estimates ranged from a low of 0% to as high as 75%, but only some of that variation could be explained by vole abundance. However, sample sizes were fairly small for some years, such that a substantial portion of that variance may be due to sampling variance. The strong correlation between survival, of all age classes, and mean winter temperature supports the hypothesis that the northward spread of this species may be limited by severe winter weather. This suggests several directions for further research. One is to develop stochastic matrix models that consider the interaction between the vole cycle and winter weather patterns and their impact on projected population growth rates. These models would benefit from improved estimates of the variance components, derived from Bayesian approaches (Link & Barker, 2004). Such models could be used to determine whether the observed relationships between survival and temperature are consistent with the current northern limits of this species in southern Finland based on winter isoclines of temperature and snow depth. Predictions from such a model could be tested by looking for geographic variation in survival rates of this species within Finland because the geographic location of each territory is available as an individual covariate. Finally, such models could also be used to predict the potential impacts of various climate change scenarios on this species. Such detailed demographic models, especially if similar models can be developed for some of the other owl species such as Ural Owl, can be used to enhance understanding of the extent to which these owls may actually be driving the population cycles of the voles, as opposed to merely responding to them.

Acknowledgements We are particularly grateful to all of the bird–ringers in Finland for their outstanding work in gathering the data used in this analysis. References

Brand, C. J. & Keith, L. L. B., 1979. Lynx demography during a snowshoe hare decline in Alberta. Journal of Wildlife Manage., 43: 827–849. Brommer, J. E., Pietiäinen, H, & Kolunen, H., 1998. The effect of age at first breeding on Ural owl lifetime reproductive success and fitness under cyclic food conditions. Journal of Animal Ecology, 67: 359–369. – 2002. Reproduction and survival in a variable environment: Ural Owls (Strix uralensis) and the three–year vole cycle. The Auk, 119: 544–550. Burnham, K. P., 1993. A theory for joint 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.). Birkhäuser, Basel. Caswell, H., 2001. Matrix Population Models: construction, analysis and interpretation, 2nd Edition. Sinauer Associates, Sunderland, Massachusetts. Francis, C. M. & Saurola, P., 2002. Estimating age– specific survival rates of tawny owls—recaptures versus recoveries. Journal of Applied Statistics, 29: 637–647. Freeman, S. N. & North, P. M., 1990. Estimation of survival rates of British, Irish and French Grey Herons. The Ring, 13: 139–166. Hakkarainen, H., Korpimäki, E., Koivunen, V. & Ydenberg, R., 2002. Survival of male Tengmalm’s owls under temporally varying food conditions. Oecologia, 131: 83–88. Hanski, I., Hansson, L. & Henttonen, H., 1991. Specialist predators, generalist predators, and the microtine rodent cycle. Journal of Animal Ecology, 60: 353–367. Houston, C. S. & Francis, C. M., 1995. Survival of Great Horned Owls in relation to the snowshoe hare cycle. The Auk, 112: 44–59. Korpimäki, E., Klemola, T., Nordahl, K., Oksanen, L., Oksanen, T., Banks, P. B., Batzli, G. O. & Henttonen, H., 2003. Vole cycles and Predation. Trends in Ecology and Evolution, 18: 494–495. Krebs, C. J., Boonstra, R., Boutin, S., & Sinclair, A. R. E., 2001. What drives the 10–year cycle of Snowshoe Hares? Bioscience, 51: 25–35. Link, W. & Barker, R., 2004. Hierarchical mark– recapture models: a framework for inference about demographic processes. Animal Biodiversity and Conservation, 27.1: Nur, N. & Sydeman, W. J., 1999. Survival, breeding probability and reproductive success in relation to population dynamics of Brandt’s Cormorants Phalacrocorax penicillatus. Bird Study, 46: S92– S103.

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Oli, M. K., 2003. Population cycles of small rodents are caused by specialist predators: or are they? Trends in Ecology and Evolution, 18: 105–107. Rohner, C., Doyle, R. I. & Smith, J. N. M., 2001. Great Horned Owls. In: Ecosystem dynamics of the boreal forest (C. J. Krebs, S. Boutin & R. Boonstra, Eds.). Oxford Univ. Press, Oxford. Ruesink, J. L., Hodges, K. E. & Krebs, C. J., 2002. Mass–balance analyses of boreal forest population cycles: merging demographic and ecosystem approaches. Ecosystems, 5: 138–158. SAS Institute, 2001. The SAS System for Windows, Release 8.02. SAS Institute Inc., Cary, NC, USA. Saurola, P., 1987. Mate and nest–site fidelity in Ural and Tawny Owls. In: Biology and conservation in northern forest owls: symposium proceedings; 1987 February 3–7; Winnipeg, Manitoba: 81–86 (R. W. Nero, R. J. Clark, R. J. Knapton & R. H. Hamre, Eds.). Gen. Tech. Rep. RM–142. U.S. Department of Agriculture, Forest Service, Rocky Mountain Forest and Range Experiment Station, Fort Collins, Co. – 1995. Suomen pöllöt. Owls of Finland. Helsinki, Kirjayhtymä. (in Finnish with English Summary). – 1997. Monitoring Finnish Owls 1982–1996: methods and results. In: Biology and conservation of owls of the northern hemisphere. 2nd International Symposium, February 5–9, 1997, Winnipeg, Manitoba, Canada: 363–380 (J. R. Duncan,

Francis & Saurola

D. H. Johnson & T. H. Nichols, Eds.). USDA Forest Service Gen. Tech. Rep. NC. – 2002. Natal dispersal distances of Finnish owls: results from ringing. In: Ecology and conservation of owls: 42–55 (I. Newton, R. Kavanagh, J. Olsen & I. Taylor, Eds.). CSIRO Publishing. Collingwood VIC, Australia. Saurola, P., & Francis, C. M., 2004. Estimating population parameters of owls from nationally coordinated ringing data in Finland. Animal Biodiversity and Conservation, 27.1: Solonen, T. & Karhunen, J., 2002. Effects of variable feeding conditions on the Tawny Owl Strix aluco near the northern limit of its range. Ornis Fennica, 79: 121–131. Sunde, P., Overskaug, K., Bolstad, J. P. & Øien, I. J., 2001. Living at the limit: ecology and behaviour of Tawny Owls Strix aluco in a northern edge population in Central Norway. Ardea, 89: 495–508. Sundell, J., Huitu, O., Henttonen, H., Kaikusalo, A., Korpimäki, E., Pietiäinen, H., Saurola, P. & Hanski, I., 2004. Large–scale spatial dynamics of vole populations in Finland revealed by the breeding success of vole–eating predators. Journal of Animal Ecology, 73: 167–178. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46: S120–S139.

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

Gauthier, G. & Lebreton, J.–D., 2004. Population models for Greater Snow Geese: a comparison of different approaches to assess potential impacts of harvest. Animal Biodiversity and Conservation, 27.1: 503–514. Abstract Population models for Greater Snow Geese: a comparison of different approaches to assess potential impacts of harvest.— Demographic models, which are a natural extension of capture–recapture (CR) methodology, are a powerful tool to guide decisions when managing wildlife populations. We compare three different modelling approaches to evaluate the effect of increased harvest on the population growth of Greater Snow Geese (Chen caerulescens atlantica). Our first approach is a traditional matrix model where survival was reduced to simulate increased harvest. We included environmental stochasticity in the matrix projection model by simulating good, average, and bad years to account for the large inter–annual variation in fecundity and first–year survival, a common feature of birds nesting in the Arctic. Our second approach is based on the elasticity (or relative sensitivity) of population growth rate (lambda) to changes in survival as simple functions of generation time. Generation time was obtained from the mean transition matrix based on the observed proportion of good, average and bad years between 1985 and 1998. If we assume that hunting mortality is additive to natural mortality, then a simple formula predicts changes in lambda as a function of changes in harvest rate. This second approach can be viewed as a simplification of the matrix model because it uses formal sensitivity results derived from population projection. Our third, and potentially more powerful approach, uses the Kalman Filter to combine information on demographic parameters, i.e. the population mechanisms summarized in a transition matrix model, and the census information (i.e. annual survey) within an overall Gaussian likelihood. The advantage of this approach is that it minimizes process and measured uncertainties associated with both the census and demographic parameters based on the variance of each estimate. This third approach, in contrast to the second, can be viewed as an extension of the matrix model, by combining its results with the independent census information. Key words: Greater Snow Geese, Population model, Transition matrix, Generation time, Hunting mortality, Kalman Filter. Resumen Modelos poblacionales del gran ánsar nival: comparación entre distintos enfoques empleados para evaluar los impactos potenciales de la cosecha.— Los modelos demográficos, que son una ampliación natural de la metodología de captura–recaptura (CR), constituyen un excelente instrumento orientativo a la hora de decidir cómo gestionar las poblaciones de flora y fauna. Comparamos tres enfoques de modelos distintos para evaluar los efectos de una mayor cosecha en el crecimiento poblacional del ánsar nival (Chen caerulescens atlantica). Nuestro primer enfoque consiste en un modelo de matrices tradicional en el que se redujo la supervivencia a efectos de simular una mayor cosecha. Incluimos estocasticidad medioambiental en el modelo de proyección matricial simulando años buenos, medios y malos a efectos justificar la significativa variación interanual en la fecundidad y en la supervivencia durante el primer año, dado que constituyen una característica común de las aves que nidifican en el Ártico. Nuestro segundo enfoque se basa en la elasticidad (o sensibilidad relativa) de la tasa de crecimiento poblacional (lambda) con respecto a los cambios en la supervivencia como funciones simples del tiempo generacional. El tiempo generacional se obtuvo a partir de la matriz de transición ISSN: 1578–665X

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media basada en la proporción observada de años buenos, medios y malos entre 1985 y 1998. Si suponemos que la mortalidad por caza se suma a la mortalidad natural, una fórmula simple predice cambios en la lambda como una función de cambios en la tasa de cosechas. El segundo enfoque puede considerarse como una simplificación del modelo de matrices, puesto que emplea resultados de sensibilidad formal derivados de la proyección poblacional. Nuestro tercer enfoque, de mayor alcance potencial, utiliza el filtro de Kalman para combinar información sobre parámetros demográficos; es decir, los mecanismos poblacionales resumidos en un modelo de matrices de transición, y la información censal (es decir, la inspección anual) en una probabilidad gaussiana general. La ventaja de este enfoque es que minimiza los procesos y las incertidumbres medidas que se asocian, tanto con el censo como con los parámetros demográficos basados en la varianza de cada estimación. El tercer enfoque, a diferencia del segundo, puede considerarse como una ampliación del modelo de matrices, combinando sus resultados con la información censal independiente. Palabras clave: Ánsar nival, Modelo poblacional, Matriz de transición, Tiempo generacional, Mortalidad por caza, Filtro de Kalman. Gilles Gauthier, Dépt. de biologie and Centre d’études nordiques, Pavillon Vachon, Univ. Laval, Québec, Qc, G1K 7P4, Canada.– Jean–Dominique Lebreton, Centre d’écologie fonctionnelle et évolutive, CNRS, 1919 Route de Mende, 34 293 Montpellier cedex 5, France. Corresponding author: G. Gauthier. E–mail: gilles.gauthier@bio.ulaval.ca

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Introduction Demographic models based on transition matrices are a natural extension of standard capture–recapture (CR) methodology because age or stage–specific survival and fecundity parameters essential to build transition matrices are often estimated using standard CR analyses. More sophisticated CR analyses (e.g. multi–state models) are also well suited for estimating parameters such as dispersal, which are required for more complex transition matrix models like those built for metapopulations (Caswell, 2001; Lebreton & Pradel, 2002; Lebreton et al., 2003). In an applied context, population models are especially useful for the conservation of endangered species or the management of exploited species, and they can be instrumental in recommending sustainable harvest levels. In structured populations, transition matrices are widely used to model growth rate (Caswell, 2001). In this paper, we focus on a harvested species through a case study, the Greater Snow Goose (Chen caerulescens atlantica) population. The Greater Snow Goose is a long–distant migrant that breeds in the eastern high Arctic of Canada and winters in temperate areas of eastern North America (Reed et al., 1998). Even though the species is hunted, its population has increased considerably over the past three decades (Menu et al., 2002). The high population growth rate (9%/yr) and the apparent lack of density– dependence have generated great concerns over the potential negative impact of overgrazing from high densities of geese on their breeding and staging habitats. Even though density–dependence may be locally important in some goose populations (Cooch & Cooke, 1991; Pettifor et al., 1998), Menu et al. (2002) failed to find any evidence for it at the population level in Greater Snow Geese. Use of food subsidy in farmlands in winter and spring may be an important factor to explain the absence of population–wide density–dependence effect in this population. This situation led to recommendations to take actions to stop the population growth as soon as possible (Giroux et al., 1998). Population models were thus developed to determine the harvest levels required to achieve management goals. Our objective was to compare three different modelling approaches that were applied to the case of the Greater Snow Goose, and discuss their advantages and disadvantages. We show how models incorporating a functional relationship between survival and hunting mortality can be built and used to explore various harvest scenarios. Finally, we will show how the Kalman Filter, a technique rarely used in wildlife biology, can be used to improve parameter estimates of the model, and thus model projections. Data set Data for the Greater Snow Goose population were available from several sources. First, fecundity and survival data come from a long–term capture–recap-

ture study conducted since 1990 at the breeding colony of Bylot Island in the Canadian Arctic (see Lepage et al., 2000; Reed et al., 2003 for details). Adult survival came from a detailed mark–resight studies conducted on both the breeding and southern staging grounds, and young survival came from band recovery analyses (Gauthier et al., 2001; Menu et al., 2002). Thus, our survival estimates were not confounded by permanent emigration. Second, fall age–ratio counts have been conducted by the Canadian Wildlife Service since the early 1970s in southern Quebec. Third, accurate estimates of population size came from a spring photo inventory conducted annually since 1970 in southern Quebec by the Canadian Wildlife Service (Reed et al., 1998). Finally, harvest data was obtained from the annual National Hunter Surveys conducted by the Canadian Wildlife Service and the US Fish and Wildlife Service. We divided the total number of young or adults harvested by the fall population size to obtain an index of harvest rate in this population as explained by Menu et al. (2002). The alternative method of using band recovery rates to estimate harvest rates was not possible because band reporting rate was confounded by several factors in this population. These factors include the occurrence of band solicitation at some periods, the introduction of toll–free number bands in the course of the study, and language difference between Quebec and the US which likely affected reporting rate. Recently, however, Calvert (2004) showed a very good correlation between our harvest rate estimates and adult band recovery rates in this population, which suggests that our harvest rate index tracked fairly well annual changes in harvest. Annual reproductive output of Greater Snow Geese is highly variable and is characterized by boom and bust years because of the short nesting season and strong environmental variability of the high Arctic. Late onset of spring is associated with a reduced probability of breeding, delayed nest initiation, and poor gosling growth and survival rates. Annual production, as indexed by fall age:ratio count, can vary by more than one order of magnitude (Gauthier et al., 1996). Environmental stochasticity is thus an important aspect of the demography of this population. Transition matrix model Transition matrices are relatively easy to use when assessing the effects of various harvest scenarios on population growth rates. For instance, one can empirically modify survival rates to evaluate the effects of changes in harvest rates on population growth rates (e.g. Rockwell et al., 1997). Gauthier & Brault (1998) developed a first population model for Greater Snow Geese based on a four age–class transition matrix because the age when all females have started to breed is four years in this species. They used a post–breeding census formulation (sensu Caswell, 2001) because survival rates were estimated with birds banded in late summer, shortly

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before the fledging of young (i.e. the time interval is from summer to summer). The four age–classes were thus fledging young and adults 1, 2 and 3+ year old:

(1)

where F = fecundity, SA = adult survival and SY = young (i.e. first–year) survival. Fecundity was obtained as follows: F = BP·(TCL/2)·P1·NS·P2·P3

(2)

where BP = breeding propensity, TCL = total clutch size, NS = nesting success, P1 = egg survival in successful nests, P2 = hatching success, and P3 = gosling survival from leaving the nest to fledging (Lepage et al., 2000). Instead of using a single transition matrix, Gauthier & Brault (1998) used three transition matrices to characterize three different states: good (G), average (A) and bad (B) years of reproduction (hereafter called quality of reproduction). Although this categorization is an oversimplification of reality, it recognizes the large environmental stochasticity encountered on the breeding ground. The annual quality of reproduction was categorized as bad (B) when the proportion of young in the fall age–ratio counts was < 10%, average (A) when it was 10–30%, and good (G) when it was m 30%. Parameters that differed between the three matrices were fecundity and young survival in their first–year, SY (most fecundity components in equation 2 and young survival were moderately or considerably reduced in average or bad years, respectively), but not adult survival, SA (see Gauthier & Brault, 1998 for details). The three matrices yielded different asymptotic growth rate ( =1.17 for G years, 1.01 for A years and 0.84 for B years). Gauthier & Brault (1998) used Monte Carlo simulations to combine the three transition matrices in various proportions. At each yearly iteration, one matrix was randomly selected based on probabilities equal to a chosen ratio of good:average:bad years. A ratio of 6G:3A:1B years yielded a growth rate similar to the rate observed from the spring population survey over a 16–year period when the observed ratio of G:A:B years was 5:3:2. To evaluate the impact of increased harvest on population growth, they reduced the survival of adults, young, or both in each matrix by various proportions. They also explored the effect of reduced survival under various combinations of good, average or bad years. For instance, under the scenario of 6:3:1, adult survival had to be reduced from 0.83 to 0.76 to stop population growth ( = 1) if only the harvest of adults was increased; in contrast, if only the harvest of young was increased, their survival had to be reduced from 0.42 to 0.24 to stop population growth. Using an ad hoc procedure, Gauthier & Brault (1998) estimated that stability could be achieved if harvest rates were in-

creased 1.6 times in both adults and young, 2.0 times if the harvest was increased in young only, and 2.3 times if it was in adults only (fig. 1A). A model based on the relationship between generation time and elasticity The approach of Gauthier & Brault (1998) implicitly assumed a direct, inverse relationship between survival and hunting mortality, i.e. that hunting mortality is additive to natural mortality. However, hunting mortality can be compensatory or additive to natural mortality, or somewhere in between (Nichols et al., 1984). Hunting mortality is compensatory to natural mortality when the risk of dying from natural causes decreases in response to increase in hunting mortality (Boyce et al., 1999). In contrast, hunting mortality is additive when the risk of dying from natural causes is independent from hunting mortality. Although these two concepts are quite simple, their analytical treatment becomes complex when both forms of mortality (i.e. natural and hunting) occur simultaneously, as it is commonly the case. Analysis of the relationship between hunting mortality and survival then involves the theory of competing risk (Anderson & Burnham, 1976; Lebreton, 2005). However, for high values of survival and moderate value of harvest, Burnham & Anderson (1984) showed that the interaction between survival (S) and harvest (H) can be approximated by the relationship: S = S0(1 – bH)

(3)

where S0 = survival in absence of hunting (this is analogous to the equation for discrete time scale, i.e. when harvest and natural mortality do not overlap in time; Lebreton, 2005). When hunting mortality is fully compensatory, b = 0 and thus S = S0. In contrast, when hunting mortality is fully additive, b = 1 and thus can be ignored. The assumption of additive mortality provides a starting point for modelling the impact of harvest on population growth in age–structured populations. The variation in survival ( S) that is induced by a variation in harvest rate ( H) can be expressed as: S = S0(1 – (H + H) – S0(1–H) = –S0 H

(4)

The relative change in survival can be expressed as: (5) Ultimately, we want to determine the impact of a change in survival S induced by a change in harvest H on the growth rate of the population ( ). In a transition matrix population model, the sensitivity (s) of to a change in the value of element a of the matrix is given by the formula (Caswell, 2001): (6)

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1.0

0.9

0.8

0.7 1.0

Population growth rate ( )

1.1

1.5

2.0

2.5

3.0 B

1.1

1.0

0.9

0.8

0.7 1.0

Adult Young Adult + young 1.5 2.0 2.5 3.0 Increase in hunting mortality (x)

Fig. 1. Change in Greater Snow Goose population growth rate as function of various proportional increases in harvest (hunting mortality) of adults only, young only, or both: A. Model based on stochastic simulations using three different transition matrices for good (G), average (A) and bad (B) year of reproduction combined in a ratio 6G:3A:1B (approach 1; mean with 1 SE); B. Model based on the relationship between generation time of the mean transition matrix and elasticity (from fig. 2; approach 2). Fig. 1. Cambio en la tasa de crecimiento poblacional del ánsar nival como función de varios aumentos proporcionales en la cosecha (mortalidad por caza) de adultos, jóvenes o ambos: A. Modelo basado en simulaciones estocásticas utilizando tres matrices de transición distintas para un año de reproducción bueno (G), medio (A) y malo (B) combinadas en una ratio de 6G:3A:1B (enfoque 1; media con 1 EE). B. Modelo basado en la relación entre el tiempo generacional de la matriz de transición media y la elasticidad (de la fig. 2; enfoque 2).

The change in induced by a change in the element a of the matrix (say, survival) is thus the product of the change in survival and the sensitivity of to this parameter (Caswell, 2001). Given that we have an age–structured population, we further want to separate the effects of harvesting young (i.e. first–year birds) or adults on the population. We are thus interested in changes induced by harvest on the survival of adults (SA) and young (SY). We can write:

(7) The relative change in

is given by: (8)

in the last equation, we note that S/ d /dS is the

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elasticity (e), i.e. the proportional change in induced by a proportional change in survival (Caswell, 2001). Lebreton & Clobert (1991) showed that there is a simple and direct link between generation time (T) and the elasticity of with respect to fecundity and adult survival. Several definitions of generation time exist but the most meaningful one here is the mean generation time ( ), which can be defined as the mean age of the parents of all offspring produced at the stable age distribution. This statistic can be easily calculated from a transition matrix model. The elasticity of with respect to fecundity and first–year survival is equal to 1/ and the one of adult (i.e. after–first year) survival is equal to 1 – 1/ (fig. 2). Finally, we can substitute the expression S/S by its equivalent in term of harvest (equation 5) and thus obtain:

over a post–breeding one (e.g. see equation 1) in order to have all "first–year" elements (i.e. fecundity and young survival) in the first line of the matrix as it is the elasticity of to those elements that is related to 1/ (fig. 2). This mean matrix yielded an asymptotic of 1.096, which was similar to the realized growth rate of the population over the same period based on the spring census (1.094), and a mean generation time of 6.446. Using the mean harvest rate observed on adults (0.06) and young (0.30) during 1985–1998, we increased the harvest of young and adult by various factors (fig. 1B). According to this model, population growth could be stopped if the current harvest level was increased by 1.75 times on both adults and young, 2.3 times on young only, and 2.7 times on adults only. Improving the parameter estimates: integrated modelling using the Kalman Filter

(9) Equation (9) provides a simple and straightforward way to assess the impact of harvest rates on population growth. The equation can be generalized by including a term for fecundity (F). This generalization can be useful when harvest affects fecundity in addition to survival (e.g. due to egg harvesting). Equation (9) then becomes: (10) If F varies across age classes, one can use a mean fecundity weighed by the stable age distribution. To apply this approach to Greater Snow Geese, we first calculated a mean transition matrix for the period 1985–1998 to account for the annual environmental stochasticity. For each year, we had a known Qi quality of reproduction (Qi, good, average or bad) and the realized harvest rate (Hi, from Menu et al., 2002). Each value of Qi had a corresponding set of age–specific fecundity parameters. Adult survival was defined as a function of annual harvest rate, i.e. SA = f(Hi). We thus generated 14 different matrices for the period 1985–1998, filling up the elements of each matrix with the elements of vectors Qi and Hi. The mean transition matrix was:

(11)

A pre–breeding formulation is here preferred

In the two previous approaches, models were validated by comparing the projected growth rates with the realized rates estimated by the annual spring survey. Although this is a standard procedure in the literature, it is an ad hoc one that has no formal basis (Lebreton & Clobert, 1991). This procedure also ignores the variance associated with both the demographic parameters and the survey. An alternative is the Kalman Filter (Kalman, 1960; Kalman & Bucy, 1961), a more robust approach that combines both sources of information, that is demographic data (i.e. transition matrix) and census data (i.e. annual survey). For each time step, the filter calculates the best estimate using both the prediction generated by the model (called a state equation) and the actual population measure (Harvey, 1989). Thus, when model prediction differs from the survey estimate, the filter updates the model parameters to find the best likelihood–based compromise between the prediction and the survey estimate based on the measurement errors of model parameters and survey estimates. The overall likelihood is the product of the Kalman Filter likelihood, as a function of the parameter values, and the capture–recapture likelihood. The former contains the information on parameters brought by the census, the second that brought by capture–recapture data. In that sense, the Kalman Filter does not privilege one source of information over the others and gives them weights directly linked to the amount of information they contain. This approach was proposed by Besbeas et al. (2002) for integrating matrix–based population model and census information. Besbeas et al. (2003) further described a normal approximation that maximized the likelihood of the Kalman Filter (see also Morgan et al., in prep.). In our case, the state equation was the transition matrix model:

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Elasticity (e)

Adult survival (SA)

Fecundity (F)

0 1

4 5 6 7 Generation time (T)

Fig. 2. Elasticity of the asymptotic growth rate ( ) with respect to adult (i.e. after–first year) survival and fecundity as a function of generation time (Lebreton & Clobert, 1991). Fig. 2. Elasticidad de la tasa de crecimiento asintótico ( ) respecto a la supervivencia de los adultos (i.e. despues del primer año) y fecundidad como función del tiempo de generación (Lebreton & Clobert, 1991).

(12) where t is a random term for departure from the model, incorporating in particular demographic stochasticity (see Besbeas et al., 2002), distinct form uncertainty in parameters. The observation equation was the census of the total population, (t) (i.e. spring inventory of the population, from Reed et al., 1998 and unpublished data): (13) where (t) is a random variable for census uncertainty. We assumed a constant coefficient of variation for the census, i.e. Var ( (t)) = y(t)2 c2 where c is the coefficient of variation of the census. The key relationship in the state equation relates adult survival to harvest rate. Gauthier et al. (2001) empirically estimated the parameters of this relationship using a complex CR analysis of live recaptures of adult Greater Snow Geese throughout the year. In their analysis, they modeled the relationship S = (a – bH)/r where r is a parameter to account for band loss (in this equation, b has a slightly different meaning than

in the equation S = S0(1 – bH) as here it is the product bS0). We used this equation to define adult survival in the state equation, i.e. we substituted SA by (a–bH)/r in the transition matrix. The other elements of the matrix (fecundity and young survival) were similar to those used in the previous modelling approach (equation 11) and considered to be constant (i.e. we assumed no measurement error as a first approximation). The measurement error associated with the census information, y(t), was the coefficient of variation (c) associated with the population survey, which is a further parameter to be estimated, brought into the overall likelihood by the combination of the state and census equations inherent in the Kalman Filter. We used the approach described by Besbeas et al. (2003) to maximize a combined likelihood function with four parameters, L(a, b, r, c). Complete results are developed in Gauthier, Besbeas, Lebreton and Morgan (in prep.). In particular, the results were quite insensitive to various reasonable choices for the initial population size and structure and its standard error. When we applied the Kalman Filter to the data for the period 1985–2002, the ML estimates of r, a, and b only changed slightly compared to the initial values taken from Gauthier et al. (2001), with a slight improvement in precision (table 1). The similarity between the updated estimates and the initial values indicates that the census information was compatible with the demographic information as built into the transition matrix model. The estimated CV of the censused population size, which is a further parameter estimated by the Kalman Filter besides the CR parameters, was equal to 0.199.

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This value was larger than the estimated CV of the survey (i0.10, Béchet et al., 2004) but compatible with it because of its large standard error. The model predictions generally tracked the observed changes in population size much better than the transition matrix model alone based on the initial parameter values (fig. 3). The change in survival estimates, although small, has nevertheless a strong effect on the quality of population projections. The weakness of the increase in precision is not surprising for two reasons. First, the overall good compatibility between the census and the capture–recapture information on survival induces only a slight change in estimates within their capture–recapture based confidence interval. Second, the estimated CV of the survey clearly limits the amount of information on survival processes brought in by the census data. Discussion The modelling approaches we described are prospective, i.e. they attempt to evaluate the impact of changes in survival rate induced by variations in harvest rates on population growth rate (Caswell, 2000). The standard approach of empirically varying survival rates in a transition matrix is simple and straightforward. In this case, the impact on is directly related to the elasticity of the parameter (as shown above) and thus can easily be derived analytically. The model of Gauthier & Brault (1998) also attempted to capture in the model the very high variance in annual fecundity and survival of young typical of species living in highly unpredictable environments like the Arctic. Their model recognized three environmental states and at each time t a state was randomly chosen, assuming independence between each state. More complex models with dependence between environmental states (i.e. state at time t + 1 depends of state at time t) are possible (Caswell, 2001) but were not considered for Greater Snow Geese given the low correlation found between environmental states (Gauthier, unpubl. data). In stochastic models, the effect of changes in survival on can not be obtained directly from the elasticity values but must be evaluated through simulations using the stochastic growth rate estimator (Tuljapurkar, 1990; Caswell, 2001). However, care must be used when calculating the growth rate in stochastic simulations. For instance, we later found that the stochastic simulations of Gauthier & Brault (1998) were slightly biased. This bias was uncovered when the three matrices were recast into a pre–breeding census using the same demographic parameters. The asymptotic growth rate of each matrix remained the same (as it should be), but we obtained a different stochastic growth rate. For instance, when running 10,000 simulations of 10,000 time steps with a 6:3:1 ratio of G, A and B years, the stochastic growth rate for a pre–breeding formulation was 1.094 (1.092–1.096, 95% CI) compared to 1.071 (1.068–1.074) for a post–breeding formulation (calculated with ULM, Legendre & Clobert, 1995;

Table 1. Initial parameter values for the relationship S = (a – bH)/r estimated by Gauthier et al. (2001) and Kalman Filter values obtained from the approximate combined likelihood analysis (Mean ± SE): P. Parameter; I. Initial values; KF. Kalman Filter values. Tabla 1. Valores de parámetros iniciales para la relación S = (a – bH)/r estimada por Gauthier et al. (2001) y valores del filtro de Kalman obtenidos a partir del análisis aproximado de probabilidades combinadas (Media ± EE): P. Parametro; I. Valores iniciales; KF. Valores del filtro de Kalman.

0.950 ± 0.008

0.943 ± 0.007

0.926 ± 0.022

0.935 ± 0.018

1.207 ± 0.560

1.100 ± 0.524

0.199 ± 0.190

Gauthier, unpubl. data). Cooch et al. (2003) showed that this difference was due to significant covariation among matrix elements, i.e. that years of low fecundity also have low survival of young. In a typical post–breeding formulation, this covariation is broken because the survival of young born in year i is found in the matrix selected for the next time step. Seasonal matrices must then be used to solve this problem (Cooch et al., 2003). Our modelling based on the relationship between generation time and elasticity conveniently summarized the link between harvest rate and in a simple equation. This is advantageous because, for managers who want to set harvest levels to reach specific management goals, harvest is the variable of primary interest, not survival. Our model assumes that hunting mortality is additive to natural mortality. In adult Greater Snow Geese, Gauthier et al. (2001) provided evidence, based on live recaptures of marked birds throughout the year, that hunting mortality is additive. This is probably a robust result for long–lived species like geese because their low natural mortality rate ([ 10%, Gauthier et al., 2001) does not allow much room to compensate for additional mortality due to harvest (see also Francis et al., 1992; Rexstad, 1992). In cases where some compensation in hunting mortality occurs (i.e. b < 1 in the equation S = S0 (1 – bH)), the model can be modified to accommodate a different value of b (see Lebreton, 2005). However, this assumes that a precise estimate of b is available, which is rarely the case. Hence, in many circumstances, one may be forced to assume the default value of 1 for b and use the equation presented here as an approximation. The model based on generation time cannot account for

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1,000

Population size (x1,000)

900 800 700

Census Census (modified) Kalman Filter Matrix model Matrix model (SA*1.05)

600 500 400 300 200 100 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 Year

Fig. 3. Trajectory of population size over time predicted by the Kalman Filter, the matrix model alone, the matrix model when adult survival has been increased by 1.05x, and measured by the spring photographic inventory (census). From 1998 onward, the census method changed (see text). Fig. 3. Trayectoria del tamaño poblacional a lo largo del tiempo, prevista por el filtro de Kalman, por sólo el modelo de matrices, y por el modelo de matrices cuando la supervivencia de los adultos se ha visto incrementada en 1,05x y cuando ha sido medida por el inventario fotográfico primaveral (censo). A partir de 1998, el método censal cambió (véase el texto).

environmental stochasticity in the same way that Gauthier & Brault (1998) did in their model. As an alternative, we used a mean matrix for Greater Snow Geese, which provided a good approximation of the observed growth rate for the population over the period considered. One counterintuitive result that came out from this modelling approach and the previous one is that the same proportional increase in the harvest rates of young has slightly greater impact on population growth rate than when applied to adults. This result is surprising because adult survival has the highest elasticity (0.84). The reason is that the actual harvest rate is much higher in young than in adults. For the period of reference (1985–1998), the mean harvest rate of adults was only 0.06 compared to 0.33 in young, more than a five–fold difference (Menu et al., 2002). This reflects the fact that young are much more vulnerable to hunting than adults in geese due to their inexperience (Menu et al., 2002; Calvert, 2004). Hence, a doubling of harvest in young means a far greater increase in absolute number of birds killed than in adults. Our approach based on generation time can be viewed as a simplification of the matrix model approach because it uses formal sensitivity results derived from population projection. In contrast, the approach based on the Kalman Filter can be viewed as an extension of the matrix model that combined

model results with the independent census information. The greatest advantage of the Kalman Filter is that it attempts to incorporate uncertainties associated with both the census and demographic parameters based on the variance of each estimate. Our application of the Kalman Filter was centered on the parameters of the equation relating adult survival to harvest, S = (a–bH), with an additional parameter (r) accounting for marker loss. The parameter values updated by the filter differed only slightly compared to the initial values estimated by Gauthier et al. (2001) and the difference was greatest for b, as expected given that this parameter had the lowest precision. The updated value of b (1.10) in the relation S = (a–bH)/r is closer to the theoretical slope of 0.93 (i.e. the value of parameter a) expected for complete additivity of hunting mortality than the initial value of 1.21. This reaffirms the original conclusion of Gauthier et al. (2001). The Kalman Filter may be especially useful in situations where the slope parameter (b) is poorly known or in situations where hunting mortality can be partly compensatory (i.e. 0 < b < 1). The updated parameter values generated by the filter can thus provide information on the extent of compensatory mortality occurring in the population. The Kalman Filter dramatically improved the prediction of the model compared to the transition matrix alone as judged by the correspondence be-

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tween the prediction and the observed population size. Although the fit appears poorer for the most recent years, part of the discrepancy results from a change in the survey method starting in 1998. A telemetry study conducted in 1998–2000 showed that the survey underestimated the true population (Béchet et al., 2004) because it increasingly missed some flocks as the population expanded. The survey method was thus definitively modified in 2001 to reduce this negative bias. The reduction of the value of parameter b by the Kalman Filter indicates that the impact of hunting on survival may have been slightly overestimated, and thus adult survival underestimated. Even though this underestimation was slight, and quite within the confidence interval of the parameter estimated by the CR analysis, it nonetheless had a large impact on the model prediction. This is not surprising given the very high sensitivity of to adult survival. An ad–hoc increase of 5% in adult survival in the transition matrix model alone yielded a prediction almost identical to the one obtained by the Kalman Filter (fig. 3), further suggesting that the impact of hunting on survival was slightly overestimated. In our application of the Kalman Filter, we ignored the error associated with the fecundity and young survival rates for simplicity. We expected that the bias would be slight given the very low elasticity of these parameters compared to adult survival (the combined elasticity of fecundity and young survival is only 0.16 compared to 0.84 for adult survival). Subsequent analyses suggested that inclusion of the error associated with fecundity had a negligible effect on the model results (Gauthier et al, in prep.). Concluding remarks We believe that our second modelling approach based on the relationship between generation time and elasticity provides a simple mean to directly model the impact of harvest rate on population growth rate and we recommend its use. This minimally requires some demographic parameters and harvest rate data. Projections from the model can easily be used by managers to evaluate the effect of various harvest scenarios on population growth rate. In the case of the Greater Snow Goose where the initial goal was to stop population growth, one can directly estimate the harvest rate of adults, young, or both needed to reach this goal (fig. 1). Even though we used a mean matrix and thus ignored the stochastic component of fecundity modeled by Gauthier & Brault (1998), predictions from this model were similar to those of their more complex stochastic model. Implementation of changes in harvest regulations will of course result in perturbations that will affect generation time and thus model predictions; hence, updated parameter values will eventually be required in the transition matrix. However, this is inherent to any prospective analyses based on elasticity (Caswell, 2001). When census information is available, the Kalman

Filter allows a formal integration of independent information from the survey into the model, and this improved model predictions. The updated parameter values of the relationship between survival and harvest (a, b and r) generated by the filter could, in turn, be used to improve projections of the impact of variations in harvest rate on population growth rate using our second approach. For the case of the Greater Snow Goose, doing that did not change markedly the projections shown in fig. 2B, probably because parameter values were relatively well estimated to start with. However, in an adaptive management framework (Walters, 1986), a more sensible way to use the Kalman Filter would be to generate projections one–step at a time; e.g., based on the current year production and the latest survey figure, one can project the next year population for various harvest scenarios and choose the scenario that matches most closely the population goal. By doing that on an annual basis, the model constantly updates its parameters, and hence should improve the quality of predictions over time (see Fonnesbeck & Conroy, 2004 for a similar approach). We are currently implementing this predictive component in the Greater Snow Goose model. Acknowledgments We thank S. Brault for her assistance in developing the first Greater Snow Goose model, and B. J. T. Morgan and P. Besbeas for their help in developing the Kalman Filter model for the Greater Snow Goose. This work was funded by grants to G. Gauthier from the Natural Sciences and Engineering Research Council of Canada and the Fonds pour la Formation de Chercheurs et l’Aide à la Recherche (Ministère de l’Éducation du Québec). The comments of Sue Sheaffer and Barry D. Smith greatly improved an earlier version of this manuscript. References Anderson, D. R. & Burnham, K. P., 1976. Population ecology of the mallard. VI. The effects of exploitation on survival. US Fish and Wildlife Service Resource Publication 128. Béchet, A., Reed, A., Plante, N., Giroux, J.–F. & Gauthier, G., 2004. Estimating the size of large bird populations: the case of the greater snow goose. J. Wildl. Manage.,68: 639–649. Besbeas, P., Freeman, 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. Besbeas, P., Lebreton, J.–D. & Morgan, B. J. T., 2003. The efficient integration of abundance and demographic data. Appl. Stat., 52: 95–102. Boyce, M. S., Sinclair, A. R. E. & White, G. C., 1999. Seasonal compensation of predation and

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harvesting. Oikos, 87: 419–426. Burnham, K. P. & Anderson, D. R., 1984. Tests of compensatory vs additive hypotheses of mortality in mallards. Ecology, 65: 105–112. Calvert, A., 2004. Variations spatiales et temporelle de la mortalité due à la chasse et les effets des mesures de gestion chez la grande oie des neiges (Chen caerulescens atlantica). M. Sc. Thesis, Univ. Laval, Ste–Foy, Québec, Canada. Caswell, H., 2000. Prospective and retrospective perturbation analyses: Their roles in conservation biology. Ecology, 81: 619–627. – 2001. Matrix population models – Construction, analysis, and interpretation. 2nd ed. Sinauer, Sunderland. Cooch, E. G. & Cooke, F., 1991. Demographic changes in a snow goose population: biological and management implications. In: Bird population studies: 168–189 (C. M. Perrins, J.–D. Lebreton & G. J. M. Hirons, Eds). Oxford Univ. Press, Oxford. Cooch, E. G., Gauthier, G. & Rockwell, R. F., 2003. Apparent differences in stochastic growth rates based on timing of census: a cautionary note. Ecol. Model., 159: 133–143. Fonnesbeck, C. & Conroy, M., 2004. Application of Bayesian decision making and MCMC to the conservation of a harvested species. Animal Biodiversity and Conservation, 27.1: 267–281. Francis, C. M., Richards, M. H., Cooke, F. & Rockwell, R. F., 1992. Long–term changes in survival rates of lesser snow geese. Ecology, 73: 1346–1362. 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 and Canadian Wildlife Service, Ottawa. 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. Gauthier, G., Rochefort, L. & Reed, A., 1996. The exploitation of wetland ecosystems by herbivores on Bylot Island. Geosc. Can., 23: 253–259. Giroux, J.–F., Batt, B. D. J., Brault, S., Costanzo, G., Filion, B., Gauthier, G., Luszcz, D. & Reed, A., 1998. Conclusions and management recommendations. In: The greater snow goose: report of the Arctic Goose Habitat Working Group: 81–88 (B. D. J. Batt, Ed). Arctic Goose Joint Venture Special Publication. U.S. Fish and Wildlife Service, Washington and Canadian Wildlife Service, Ottawa. Harvey, A. C., 1989. Forecasting, structural time series models and the Kalman Filter. Cambridge Univ. Press, Cambridge. Kalman, R. E., 1960. A new approach to linear fitting and prediction problems. A.S.M.E. J. Bas. Eng. Ser. D, 82: 35–45.

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Kalman, R. E. & Bucy, R. S., 1961. New results in linear filtering and prediction theory. A.S.M.E. J. Bas. Eng. Ser. D, 83: 85–108. Lebreton, J.–D., 2005. Dynamical and statistical models for exploited populations. Aust. N. Z. J. Stat., 47: 901–915. Lebreton, J.–D. & Clobert, J., 1991. Bird population dynamics, management, and conservation: the role of mathematical modelling. In: Bird population studies: 105–125 (C. M. Perrins, J.–D. Lebreton & G. J. M. Hirons, Eds.). Oxford Univ. Press, Oxford. 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. Oikos, 101: 253–264. Lebreton, J.–D. & Pradel, R., 2002. Multistate recapture models: modelling incomplete individual histories. J. Appl. Stat., 29: 353–369. Legendre, S. & Clobert, J., 1995. ULM, a software for conservation and evolutionary biologists. J. Appl. Stat., 22: 817–834. Lepage, D., Gauthier, G. & Menu, S., 2000. Reproductive consequences of egg–laying decisions in snow geese. J. Anim. Ecol., 69: 414–427. Menu, S., Gauthier, G. & Reed, A., 2002. Changes in survival rates and population dynamics of greater snow geese over a 30–year period: Implications for hunting regulations. J. Appl. Ecol., 39: 91–102. Morgan, B. J. T., Besbeas, P., Thomas, L., Buckland, S., Harwood, J., Duck, C. & Pomeroy, P. (in prep.). Integrated analysis of wildlife population dynamics. 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. Trans. North Am. Wildl. Nat. Res. Conf., 49: 535–554. Pettifor, R. A., Black, J. M., Owen, M., Rowcliffe, J. M. & Patterson, D., 1998. Growth of the Svalbard barnacle goose Branta leucopsis winter population 1958–1996: An initial review of temporal demographic changes. In: Research on Arctic Geese, Proceedings of the Svalbard Goose Symposium, Oslo, Norway: 147–164 (F. Mehlum, J. M. Black & J. Madsen, Eds). Norsk Polarinstitutt Skrifter 200, Oslo, Norway. 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 and Canadian Wildlife Service, Ottawa. Reed, E. T., Gauthier, G., Pradel, R. & Lebreton J.–D., 2003. Age and environmental conditions affect recruitment in greater snow geese. Ecology, 84: 219–230. Rexstad, E. A., 1992. Effects of hunting on annual survival of Canada geese in Utah. J. Wildl. Manag., 56: 297–305.

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Rockwell, R. F., Brault, S. & Cooch. E. G., 1997. Dynamics of the mid–continent population of lesser snow geese – projected impacts of reduction in survival and fertility on population growth rates. In: Arctic ecosystems in peril: report of the Arctic Goose Habitat Working Group: 73–100 (B. D. J. Batt, Ed.). Arctic Goose Joint Venture Special Publication. U.S.

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Fish and Wildlife Service, Washington and Canadian Wildlife Service, Ottawa. Tuljapurkar, S. D., 1990. Population dynamics in variable environments. Springer Verlag, New York. Walters, C. J., 1986. Adaptive management of renewable resources. MacMillan Publishing Company, New York.

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

Brooks, S. P., King, R. & Morgan, B. J. T., 2004. A Bayesian approach to combining animal abundance and demographic data. Animal Biodiversity and Conservation, 27.1: 515–529. Abstract A Bayesian approach to combining animal abundance and demographic data.— In studies of wild animals, one frequently encounters both count and mark–recapture–recovery data. Here, we consider an integrated Bayesian analysis of ring–recovery and count data using a state–space model. We then impose a Leslie– matrix–based model on the true population counts describing the natural birth–death and age transition processes. We focus upon the analysis of both count and recovery data collected on British lapwings (Vanellus vanellus) combined with records of the number of frost days each winter. We demonstrate how the combined analysis of these data provides a more robust inferential framework and discuss how the Bayesian approach using MCMC allows us to remove the potentially restrictive normality assumptions commonly assumed for analyses of this sort. It is shown how WinBUGS may be used to perform the Bayesian analysis. WinBUGS code is provided and its performance is critically discussed. Key words: Census data, Integrated analysis, Kalman filter, Logistic regression, Ring–recovery data, State– space model, WinBUGS. Resumen Aproximación bayesiana para combinar abundancia y datos demográficos.— En estudios de animales salvajes, es frecuente encontrarse tanto con datos de recuento como datos de marcaje–recaptura– recuperación. En el presente estudio consideramos un análisis integrado bayesiano de recuperación de anillas y datos de recuento utilizando un modelo de estado–espacio. Seguidamente aplicamos un modelo basado en las matrices de Leslie en los recuentos de población verdadera para describir los procesos naturales de nacimiento–muerte y de transición de edades. Nos centramos en el análisis de los datos de recuento y de recuperación recopilados en avefrías europeas (Vanellus vanellus) en combinación con los registros del número de días de helada de cada invierno. Demostramos cómo el análisis combinado de estos datos proporciona un marco inferencial más sólido, y discutimos cómo el enfoque bayesiano usando MCMC nos permite eliminar los supuestos de normalidad potencialmente restrictivos que suelen adoptarse en análisis de este tipo. Se demuestra cómo puede utilizarse WinBUGS para realizar el análisis bayesiano. Se facilita el código WinBUGS, y se discute su funcionamiento. Palabras clave: Datos de censo, Análisis integrado, Filtro de Kalman, Regresión logística, Datos de recuperación de anillas, Modelo de estado–espacio, WinBUGS. S. P. Brooks, The Statistical Laboratory–CMS, Univ. of Cambridge, Wilberforce Road, Cambridge CB3 0WB, U.K.– R. King, CREEM, Univ. of St. Andrews, Buchanan Gardens, St. Andrews KY16 9LZ, U.K.– B. J. T. Morgan, Inst. of Matemathics, Statistics and Actuarial Science, Univ. of Kent, Canterbury CT2 7NF, U.K.

ISSN: 1578–665X

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Introduction Studies of wildlife populations often result in different forms of data being collected from different sources. Useful data comprise capture–recapture data (of live animals), ring–recovery data (of dead animals), radio–tagging (where the state of each animal is known at all times), data on productivity (as in nest–record data, for example), location data and/or count data (estimates of total population size). By combining data from different sources, we obtain more robust (and self–consistent) parameter estimates that fully reflect the information available. Previous studies of combined data of this sort include the analysis of joint capture–recapture and ring–recovery data (Catchpole et al., 1998; King & Brooks, 2002b), multi–site data (King & Brooks, 2002a; King & Brooks, 2003) and joint ring–recovery and either census data or population indices (Besbeas et al., 2002). In this paper, we shall consider a Bayesian analysis of joint ring–recovery and population index data, revisiting the analysis of Besbeas et al., 2002. We demonstrate how the state space model used to describe the index data can be easily fitted using Markov chain Monte Carlo (MCMC; Gamerman, 1995; Gilks et al., 1996; Brooks, 1998) and implemented via WinBUGS (Spiegelhalter et al., 2002b; Gentleman, 1997; Link et al., 2002). An appendix provides code to analyse the dataset described here. MCMC methods provide an alternative to the Kalman filter based approaches typically applied to problems of this sort. They also permit more general modelling frameworks for cases where the usual normality and linearity assumptions are not appropriate. We begin in "Data and modelling" section with an introduction to the data and of the models we will use here. In "Analysis and results" section we describe the Bayesian analysis of these data using WinBUGS and provide estimates for key parameters of interest. In "Non–mortality" section we provide an example where the Bayesian analysis is more appropriate due to the small count values. Finally, in "Discussion" section we discuss the use of WinBUGS, both for the application of this paper, and more generally. Data and modelling The British lapwing (Vanellus vanellus) population has been declining over recent years and has been placed on the "amber" list of species of conservation concern in Britain. As such, it has received a great deal of attention over recent years (Tucker et al., 1994) not least because it can be regarded as an "indicator" species in that by understanding the reasons for its decline, we might gain insight into the dynamics of similar farmland birds. We have two distinct sources of data, both of which are provided by the British Trust for Ornithology (BTO): index data providing annual population size estimates and recovery data from birds ringed as chicks and subsequently reported dead. Note that

in the case of lapwings, the index data do not provide a formal census of a national population, but may be regarded as estimating the population size for the set of sites at which observations take place. We also introduce the number of days each year that a Central England temperature fell below zero, as a covariate to help describe the variation in survival over time. We begin with a description of the population index data and associated model. Population index data The population index data are derived from the Common Birds Census (CBC) which has been the main source of information on population levels for common British birds since it was established in 1962. More recently it has been replaced by the Breeding Bird Survey. Annual counts are made at a number of sites around the UK and from these an index value is calculated based upon a statistical analysis of the data collected (Ter Braak et al., 1994). The raw data are not used, but the index provides a measure of the population level, taking account of the fact that each year only a small proportion of sites are actually surveyed. We shall consider the analysis of single–site data in Section 4. Here, we analyse the index values collected for adult females from 1965 to 1998 inclusive and we omit data from earlier years of the study during which the index protocol was being standardised. The data are plotted in (Besbeas et al., 2002). We denote the index value for year t by yt and, for consistency with the ring–recovery data described later, we associate the year 1963 with the value t = 1, so that we actually observe the values y3,...,y36. Since these yt are only estimates, we first try to estimate the true underlying population levels that we will subsequently use as input into our system model. Here we shall assume that yt

- N ( Na,t,

2 y

)

(1)

where Na,t represents the true underlying numbers of adult females aged $ 2 years at time t. Here y2 is taken as a constant variance, although other assumptions could also be made. For an index, a constant variance seems reasonable; we do not have access to the estimated standard errors resulting from the separate statistical analysis that has resulted in the population index. Note that we estimate y2 from the index data, and not from the raw survey data. This then describes the observation process by which the estimates yt are derived from the true underlying process Na,t. We next need to describe the underlying system process which provides a model for the evolution of the true underlying population size over time. We follow the notation of Besbeas et al. (2002), rather than Durbin & Koopman (1997). A natural model would be to assume that Na,t - Bin (N1, t–1 + Na,t–1,

a,t–1

)

517

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where N1,t denotes the number of females of age 1 in year t and a,t denotes the adult survival rate in year t. We note here that lapwings are considered adult after year 1 of life. Thus, the number of adults aged $ 2 years in year t is derived directly from the number of adults and birds in their first year of life in the previous year which survive from t – 1 to t. In a similar manner, we might model the number of 1– year old females in year t by N1,t - Po (Na,t–1

t–1

1,t–1

)

where 1,t denotes the first–year survival rate in year t and t denotes the productivity rate in year t i.e., the average number of female offspring per adult female. We therefore assume that breeding begins at age 2. Thus, the number of birds aged 1 in year t stems directly from the number of chicks produced the year before which then survive from t – 1 to t. Traditionally, this model is difficult to fit classically as it falls beyond the standard normal framework (Durbin & Koopman, 1997). Thus, we adopt instead the common normal approximation in which we take (2) where the 1,t and a,t are assumed to be independent and Normally distributed, each with mean zero and variance 21,t and 2a,t, respectively. To approximate the Poisson/Binomial model above, we take 2 1,t 2 a,t

= Na,t–1

= (N1,t–1 + Na,t–1)

t–1

1,t–1

a,t–1

(1 –

a,t–1

)

See Sullivan (1992), Newman (1998) and Besbeas et al. (2002) for example. It is worth noting here that though the model depends upon the survival rates, there is typically very little information in the data with which to estimate them. In order to provide additional information, we can combine these data with those from a recovery study which provides far greater information on the survival rates. Recovery data To augment the index data, we also have recovery data from lapwings ringed as chicks between 1963 and 1997 and later found dead and reported between 1964 and 1998. Adult birds were also ringed as part of the study, but they make up a very small proportion of the total dataset and are ignored. The data are reproduced in Besbeas et al. (2001). Here, we denote the observed recovery data by , t 1 = 1,...,35, t 2 = t 1 + 1,...,37, where denotes the number of animals released at the beginning of year t1 and subsequently recovered (dead) in the year up to the end of year t2 for t2 [ 36 and denotes the number of animals ringed in year t1 and never subsequently returned. We then

assume that for each t 1 , the values , t2 = t1 + 1,...,37 follow a multinomial distribution with proportions which denote the probability that a chick ringed in year t1 is subsequently returned in year t2. Here we shall assume, as for the index data, that adults and first years have different time– varying survival rates, but common time–varying recovery rate t denoting the probability that a bird which dies in year t is recovered. See Besbeas et al. (2002) for further details and assumptions underpinning this model. Under this model, we have that:

t2 = t1 + 2,...,36

(3)

and . Throughout this paper, we follow the convention that a null sequence has sum 0 and product 1. Thus, in the formula (3) for , the product term is 1 when t2 = t1 + 2. Incorporating covariates As well as the index and recovery data, we also have a variety of weather covariates that we can use to try to explain the variation of our model parameters over time. Of particular relevance are the number of frost days (i.e., the number of days during which a Central England temperature went below freezing) each year. For year t, ft denotes the number of days below freezing between April of year t and March of year (t + 1), inclusive. This covariate was used by Besbeas et al., 2002. The survival probability of wild birds is likely to be more affected by lengthy cold periods rather than by low average temperatures, which might result from relatively short cold spells. Thus, we take

and

logit logit

= =

1,t a,t

(4) (5)

We expect to encounter a decline over time of the reporting probability of dead animals (see e.g., Baillie & Green, 1987) and in addition we are interested in seeing whether productivity might change over time. It should be noted that we base our models on the model selected by Besbeas et al. (2002), and do not carry out a model comparison. That will be the subject of further work (King et al., 2004). Thus, here, we set logit

(6)

and, since productivity is constrained only to be positive and not simply to values in [0,1], we have log

(7)

Thus the model components (index, recovery and covariates) can then be combined to provide a single comprehensive analysis.

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The integrated model The model for the population indices described in "Population index data" section depends upon parameters t, 1,t, a,t, 2y and the underlying population levels N1 and Na, which we treat as missing values to be estimated. This model is described as a joint probability density for the observed data y = (y3,...,y36) in terms of these parameters as follows. ƒ(y*N1, Na, , = ƒ(y*Na,

2 ) y

ƒ(N1, Na, ,

2 ) y

)

where ƒ(y*Na, 2y) is the density corresponding to Equation (1) and ƒ(N1, Na* , 1, a) is derived from Equation (2). The recovery model described in "Recovery data" section depends upon parameters t, 1,t and a,t and has corresponding joint density ƒ(m* , 1, a) under the multinomial model with probabilities given in Equation (3). It is clear that both of these models have parameters in common ( 1 and a). Thus, combining the two datasets and analysing them together pools the information regarding these parameters and this filters into the estimation of the remaining parameters. The combination of these two models is most clearly demonstrated in the Directed Acyclic Graph (DAG) given in figure 1. In the DAG, known quantities (i.e., data) are represented by squares and unknown quantities (parameters to be estimated) by circles. Arrows between nodes in the graph represent dependencies within the model between the corresponding nodes. Continuous arrows denote stochastic dependencies such as those given in Equations (1)–(3), whilst dashed arrows denote deterministic dependencies such as those described in Equations (4)–(7). Analysing the combined data simply involves merging the two individual DAG’s. Similarly, and under the assumption of independence between the two data sources, we now obtain a corresponding joint probability distribution for the combined data as follows ƒ (y, m* , , = ƒ(y* ,

, Na, N1)

, Na, N1) ƒ(m* ,

From the Bayesian perspective, we elicit priors for the model parameters and combine these with the joint probability density above to obtain a posterior density via Bayes’ theorem. The nuisance parameters are then integrated out using MCMC. Several recent papers (Brooks et al., 2002; Dupuis et al., 2002; He et al., 2001; McAllister et al., 1994) discuss the application of Bayesian statistical methods to parameter estimation for ecological models and many use the WinBUGS package (see e.g., Link et al., 2002; Meyer & Millar, 1999) to carry out their analyses. We provide the corresponding WinBUGS code for our analyses in the appendix.

)

This is our basis for inference. From the classical perspective, we treat this as a likelihood function for the model parameters given the data and seek to maximise it with respect to those parameters. The underlying population levels N1 and Na are essentially nuisance parameters which, ideally, we would like to integrate out of the likelihood. Unfortunately, this is impossible to do analytically and we need to adopt numerical techniques such as the Kalman filter in order to obtain classical estimates. Besbeas et al. (2002) provide a detailed description of the classical analysis.

Analysis and results We begin by specifying priors for our model parameters. In some cases we might have prior information that we want to include e.g., relating to productivity. Others who have analysed census or population index data alone from a Bayesian perspective have used informed priors (see e.g., Millar & Meyer, 2000; Thomas et al., 2004). In other cases, for instance with regard to regression coefficients, we may know very little about what to expect. In this paper we choose relatively vague priors to reflect this uncertainty. Hence we take N(0,100) priors for the regression parameters and an inverse gamma prior with parameters 0.001 for 2y. We also need to place priors on the initial population levels N1,2 and Na,2 (recall that our population index data begins in year 3, within our parameterisation). Again, we take vague Normal priors with mean 200 and 1,000 respectively and variances of 106 in order to avoid influencing the posterior with overly restrictive priors. An extensive sensitivity study in which each of these prior parameter values were increased by several orders of magnitude, gave essentially identical results, suggesting that the exact choice of prior had little influence on the results obtained. We ran our MCMC algorithm for one million iterations, discarding the first 100,000 as burn–in and thinning the remainder to one in every tenth observation to save storage space. In general, the selection of starting points should have no affect on the performance of the simulation nor on the final results. However, WinBUGS can occasionally get stuck or crash when certain starting point values are used. Posterior means often provide sensible starting values, though these would not generally be available in practice. MLE’s also provide suitable starting points for analyses in WinBUGS. The analysis of the combined data set took approximately 18 hours on a 850 MHz personal computer in WinBUGS, and we return to discuss the topic of computational overheads in "Discussion" section of the paper. Table 1 provides the posterior means and corresponding standard deviations for the model parameters from the analysis of the combined data, together with the same estimates under the analyses of the two data sets individually. The comparatively large posterior standard deviations for the majority

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Census model 1

m Recovery model

Fig. 1. Directed Acyclic Graph (DAG) corresponding to the combined model for the index and recovery data. Fig. 1. Gráfico Acíclico Dirigido (DAG) correspondiente al modelo combinado para los datos del índice y de recuperación.

of parameters under the population index data alone confirms our earlier assertion about the lack of information in the index data concerning survival. In particular, we can see that the posterior mean for

the slope parameter for the adult survival rate a, is barely negative, implying that the adult survival rate decreases only slightly with harsher winters, in terms of the number of frost days, but estimated

Table 1. Posterior means and standard deviations for the analyses of the population index data, the recovery data and the two combined. Tabla 1. Medias posteriores y desviaciones estándar para los análisis de los datos del índice poblacional, los datos de recuperación y ambos combinados.

Data Index only Mean SD

Recovery only Mean SD

Mean

7.167

7.014

0.536

0.069

0.543

0.069

–0.249

3.844

–0.208

0.062

–0.197

0.060

2.512

0.402

1.532

0.070

1.550

0.071

–0.004

0.219

–0.311

0.044

–0.243

0.039

–1.235

0.460

–

–0.668

0.095

–0.079

0.030

–

–0.027

0.005

–

–3.925

0.087

–3.910

0.087

–

–0.034

0.004

–0.034

0.004

24,170

7,427

–

28,599

8,615

Parameter

2 y

Combined SD

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with low precision. The posterior distribution for the first year survival rate also has a large variance, although the posterior mean for 1 is negative. We note also the similarity in the parameter estimates under the ring–recovery model and the combined analysis as in this application, it is the ring–recovery data that provide most of the information about survival. Finally, we note that by combining the two data sources, the posterior standard deviations for the productivity parameters decrease dramatically, because of the additional information about survival provided by the recovery data (table 2). Figure 2 provides plots of the corresponding posterior means and 95% highest posterior density intervals (HPDI’s) for the survival, recovery and productivity rates over time based upon the combined analysis. Clearly, the logistic and log regressions for and respectively on time provide very smooth estimates of recovery and productivity both of which decrease over time. We also note that the adult survival rate is always greater than the first year survival rate, for all times, as we would expect. The annual fluctuations in the survival rates relate to the number of frost days in the given year, with the noticeable drops in survival rates for both first years and adults corresponding to harsh winters. Figure 3 provides a plot demonstrating how survival decreases with the number of frost days for both first years and adults. This plot clearly illustrates the decline in survival associated with an increase in frost days corresponding to a generally harsher winter. Figure 4 provides plots of the posterior means and HPDI’s for the true underlying population levels for the birds aged 1, adults and entire population. Note the precision of the estimates of the underlying population levels for birds aged 1, despite the lack of any direct observations on the population size for these animals. Note also the wider credible intervals for the population levels in 1966. This is due to the reduced smoothing effect for this estimate which essentially has only one neighbour. In particular, we note the clear decline in the adult population from 1978 which is also apparent in the year 1 population levels but masked somewhat by the y–axis scale here. An attractive by–product of the Kalman filter approach, which essentially samples the underlying population levels and then averages over them, is that it can produce smoothed estimates of the numbers of animals in the different age–classes and also rt = (Nt+1/ Nt), where Nt is the total population size at time t. This is a quantity of particular interest to ecologists. However, within the classical paradigm, it is difficult to obtain error bands for this quantity, plotted against time, though a time–consuming bootstrap approach could be used. Replacing the Kalman filter by a Bayesian approach, it is relatively straightforward to obtain the desired error–bands. Once again, the WinBUGS code can be extended to draw samples from the posterior marginal distribution of the rt from which posterior

Table 2. Posterior means and associated standard deviations for the analyses of the census data from site 51, together with the recovery data: Po/Bin. Poisson/Binomial; P. Parameter. Tabla 2. Medias posteriores y desviaciones estándar asociadas para los análisis de los datos censales del emplazamiento 51, junto con los datos de recuperación: Po/Bin. Poisson/Binomial; P. Parametro.

Po/Bin P

Mean

Normal SD

Mean

0.521

0.068

0.526

0.068

–0.220

0.062

–0.214

0.062

1.477

0.067

1.496

0.068

–0.314

0.043

–0.316

0.043

0.049

0.732

1.025

1.013

–0.025

0.032

–0.084

0.053

–3.945

0.086

–3.939

0.086

–0.033

0.004

–0.033

0.004

6.320

3.531

6.140

3.037

2 y

means and variances can be derived. Figure 5 provides the corresponding estimates for the rt from our combined analysis together with error bounds corresponding to the 95% HPDI for the full data set. As we might expect from fig. 5, the values slowly decrease over time, but the width of the posterior HPDI’s is considerably smaller than the variation of the different values themselves over time, suggesting that a reasonably strong signal is coming through from the data. Non–normality State–space models have been used by many authors to describe ecological processes (Millar & Meyer, 2000; Newman, 1998; Sullivan, 1992; Jamieson & Brooks, 2004). Traditionally, (multivariate) normal approximations are made so that the Kalman filter can be used to form the likelihood. The need for a more flexible approach was appreciated by Carlin et al. (1992), who consider both non–linearity and non–normality, in the latter case by using mixtures of normals. Note also Durbin & Koopman (1997). However, the Bayesian approach provides an even more flexible framework in which assumptions of normality (Millar & Meyer, 2000) and linearity (Jamieson & Brooks, 2004; Millar & Meyer, 2000) can be relaxed.

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1.0

0.8

0.6

0.6 a

1.0

0.4

0.2

0.0

0.0 1965 1970 1975 1980 1985 1990 1995

1965 1970 1975 1980 1985 1990 1995

0.025

1.0

0.020

0.8

0.015

0.6

0.010

0.4

0.005

0.2

0.000

0.0 1965 1970 1975 1980 1985 1990 1995 Year

1965 1970 1975 1980 1985 1990 1995 Year

Fig. 2. Posterior means and corresponding HPDI’s for the population parameters over time under the analysis of the combined data. Note the reduced y–scale for the plot for clarity. Fig 2. Medias posteriores y consiguientes HPDI para los parámetros poblacionales a lo largo del tiempo, según el análisis de los datos combinados. La representación gráfica de y se reproduce a escala reducida para el eje y a efectos de claridad.

As an illustration of the ease with which the normal model described in this paper can be extended to the underlying Poisson/Binomial model, the WinBUGS code in the appendix describes the few lines which need to change in order to implement the Poisson/Binomial model. Running the WinBUGS code for the Poisson/Binomial model provided very similar parameter estimates to those obtained under the normal model, since the population levels are sufficiently large for the normality approximations to perform well. However, with smaller populations the approximation will begin to break down and the ability to fit the Poisson/Binomial model will be crucial. This is likely to be particularly important when we move from data at the national level to the local level. See Besbeas et al. (in press).

As an example, let us consider the raw census data from a single site. Single site data are considered by Besbeas et al. (in press); we shall focus on just one of the sites that they consider, site number 51, which accounts for approximately 1% of the total population. Table 2 provides the posterior means for the regression parameters under both the Poisson/Binomial and the Normal model. As with the full analysis, the recovery data dominate the estimation of the survival and recovery parameters, but the productivity rate is estimated essentially from the site census data and here we see a more dramatic difference between the Poisson/Binomial and Normal models. Figure 6 provides the corresponding plots for the productivity rate estimated under both models and

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3000

1.0

2500 2000 0.6 1500

Survival rate

0.8

0.4

1000

0.2

500

0.0

0 0

5 10 15 20 25 Number of frost days

1965 1970 1975 1980 1985 1990 1995 Year

Fig. 3. Plot of how survival changes with frost days for adults (top) and first years (bottom).

Fig. 4. Posterior means and corresponding HPDI’s for N1 (bottom), Na (middle) and total population size (top) over time under the combined data analysis.

Fig. 3. Representación gráfica de la variación de cómo la supervivencia varía con los días de helada en adultos (superior) y a los de primer año (abajo).

1.2 1.1

the total population size. The Normal model suggests a far sharper decrease in the productivity rate over time, with the curve finishing below the corresponding estimate under the Poisson/Binomial model at the end of the study. In terms of the underlying total population estimates, the two models provide very similar estimates except for the final few years, where the Normal model suggests a continued shallow decline, whilst the Poisson/Binomial model suggests a fairly rapid increase. This is a consequence of the Poisson/Binomial model estimate of productivity being flatter than that of the Normal model, as we see from Figure 6. This discrepancy clearly indicates an important divergence between the two approaches. Note that the bounds of the HPDI’s from one model barely cover the corresponding means under the other. Though the Normal model is supposed to provide an approximation to the Poisson/Binomial model, it is clearly beginning to break down for these data from a single site. For site 51 there were 26 consecutive annual observations, and counts covered the range, 1–16, with only 6 counts being $ 10. It is therefore not surprising that for these data the Bayesian analysis might differ from the analysis based on normal approximations. From the work of this section we have seen that the analysis based on the normal approximations is surprisingly robust.

Fig. 4. Medias posteriores y consiguientes HPDI para N1 (abajo), Na (centro) y tamaño poblacional total (arriba) a lo largo del tiempo, según el análisis de datos combinados.

1.0 0.9 0.8 1965 1970 1975 1980 1985 1990 1995 Year

Fig. 5. Posterior means and HPDI’s for the rt values over time, under the combined data analysis. Fig. 5. Medias posteriores y consiguientes HPDI para los valores de rt a lo largo del tiempo, según el análisis de datos combinados.

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3.0 2.5

2.0 N

15 1.5

10 1.0 5

0.5

0 1975

1980

1985 Year

1990

1995

1975

1980

1985 Year

1990

1995

Fig. 6. Plots of the posterior means and HPDI’s for the t and total population size over time, based upon both the Normal (continuous) and Poisson/Binomial (dashed) models using the census data from site 51. Fig. 6. Representaciones gráficas de las medias posteriores y los HPDI para t y el tamaño poblacional total a lo largo del tiempo, basadas en los modelos Normal (líneas continuas) y Poisson/Binomial (líneas discontinuas), utilizando los datos censales del emplazamiento 51.

In this paper we do not provide a detailed discussion of goodness–of–fit, but we note here that it is very simple to construct Bayesian p–values (Gelman et al., 1996) for the models of this paper; for example, for the Poisson/Binomial model applied to the national data, then using the likelihood as the discrepancy function we get a Bayesian p–value of 0.513, whereas using the Freeman–Tukey statistic we obtain a Bayesian p–value of 0.531. These values indicate very good agreement between the model and the data. Discussion We end with a brief discussion on the benefits and drawbacks of the WinBUGS package for ecological analyses. It is certainly our experience that for many simple and standard models, WinBUGS provides an invaluable tool for analysing small data sets (Brooks et al., 2003) and for initial exploratory analysis. WinBUGS is primarily designed for the Bayesian analysis of hierarchical models common within the medical literature, but many standard ecological models can be analysed with WinBUGS. Code is generally easy to write, especially when adapting existing code such as adding random effects for example. For moderately sized datasets, the computation times are generally fairly small and there are a variety of tools available within the

package for obtaining posterior summaries and for checking sampler performance. However, as models increase in complexity and data sets increase in size the advantages of having bespoke code in a lower–level language (the authors use Fortran) become very apparent. The data described here are a case in point. For example, the WinBUGS default choice of updating schemes appears to be rather inefficient for these data so that long run lengths are required. This not only means long run times, but can often cause memory problems on many desktop PC’s. In order to select and tune our own MCMC proposals, it was necessary for us to write our own code. For comparison, the MCMC run discussed in "Analysis and results" section that took 18 hours in WinBUGS took less than 1 hour in Fortran on machines of comparable power. Another big advantage of writing bespoke code to implement the MCMC algorithms is the ability to extend the code to incorporate reversible jump MCMC updates to allow for Bayesian model discrimination. See King & Brooks (2002a, 2002b, 2003), for example. Whilst it is possible to use the method of Carlin & Chib (1995) to calculate posterior model probabilities in WinBUGS and to use the DIC criterion developed by Spiegelhalter et al. (2002a) and implemented in WinBUGS version 1.4, these cannot be used to efficiently explore model spaces of even moderate dimensions.

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With any MCMC simulation, it is important to check convergence of the sampled values to their stationary distribution. WinBUGS incorporates the so–called Brooks–Gelman–Rubin Diagnostic (Gelman & Rubin, 1992; Brooks & Gelman, 1998) and also allows for sampler output to be written to a file for analysis outside of the WinBUGS package. The authors’ preferred method is to use standard diagnostic techniques (Brooks & Roberts, 1998) to determine the burn–in length and then discard ten times as many iterations as indicated. Several replications are run and compared and, if all agree, then convergence is assumed. However, the run– times within the WinBUGS package, coupled with the inability to improve mixing by controlling proposal schemes can mean that this over–cautious approach cannot easily be followed in WinBUGS. Indeed the built–in diagnostics in WinBUGS appeared to diagnose convergence after just 100,000 iterations, for the population index data alone though the answers obtained from the longer run–lengths adopted in this paper differ considerably from those obtained with such a short run (e.g., 1= –5.41; cf table 1). This problem is, of course, somewhat analogous to the problems encountered in classical optimisation routines used to find maximum likelihood estimates where it is difficult to check that the true maximum has indeed been found. Overall, more experienced MCMC users are likely to benefit from developing their own suite of lower–level codes to implement their own MCMC algorithms. However, as a tool for getting started, for performing exploratory analyses on moderately–sized datasets and for teaching, the WinBUGS package is invaluable (see the annex). Acknowledgements We thank the British Trust for Ornithology for permission to use and present the population index data for lapwings, and for providing the ringrecovery data. We also thank the BTO for providing access to the individual site census data, and we also gratefully acknowledge the work of the many BTO volunteers, whose labour results in the data analysed in this paper. We are grateful to Stephen Freeman at the BTO for his discussion of the issues involved. References Baillie, S. R. & Green, R. E., 1987. The Importance of Variation in Recovery Rates when Estimating Survival Rates from Ringing Recoveries. Acta Ornithologica, 23: 41–60. Besbeas, P., Freeman, S. N. & Morgan, B. J. T. (in press). The Potential of Integrated Population Modelling. Australian and New Zealand Journal of Statistics. Besbeas, P., Freeman, S. N. & Morgan, B. J. T. & Catchpole, E. A., 2001. Stochastic Models for

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Animal Abundance and Demographic Data. Technical report, University of Kent UKC/IMS/01/06. – 2002. Integrating Mark–Recapture–Recovery and census Data to Estimate Animal Abundance and Demographic Parameters. Biometrics, 58: 540–547. 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. & Harris, M., 2002. Bayesian Methods for Analysing Ringing Data. Journal of Applied Statistics, 29: 187–206. Brooks, S. P. & Gelman, A., 1998. Alternative Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics, 7: 434–455. Brooks, S. P., King, R. & Morgan, B. J. T., 2003. Bayesian Computation for Population Ecology. Technical Report 2003–11, Statistical Laboratory, Univ. of Cambridge. Brooks, S. P. & Roberts, G. O., 1998. Diagnosing Convergence of Markov Chain Monte Carlo Algorithms. Statistics and Computing, 8: 319–335. Carlin, B. P. & Chib, S., 1995. Bayesian Model choice via Markov Chain Monte Carlo. Journal of the Royal Statistical Society, Series B 5: 473–484. Carlin, B. P., Polson, N. G. & Stoffer, D. S., 1992. A Monte Carlo Approach to Nonnormal and Nonlinear State–Space Modelling. Journal of the American Statistical Association, 87: 493–500. Catchpole, E. A., Freeman, S. N., Morgan, B. J. T. & Harris, M. P., 1998. Integrated recovery/recapture data analysis. Biometrics, 54: 33–46. Dupuis, J., Badia, J., Maublanc, M.–L. & Bon, R., 2002. Survival and Spatial Fidelity of Mouflon (Ovis gmelini): a Bayesian Analysis of an Age– dependent Capture–Recapture Model. Journal of Agricultural, Biological and Environmental Statistics, 7: 277–298. Durbin, J. & Koopman, S. J., 1997. Monte Carlo Maximum Likelihood Estimation for non– Gaussian State Space Models. Biometrika, 84: 669–684. Gamerman, D., 1995. Monte Carlo Markov Chains for Dynamic Generalised Linear Models. Technical report. Univ. Federal do Rio de Janeiro, Brazil. Gelman, A., Meng, X. & Stern, H., 1996. Posterior Predictive Assessment of Model Fitness via Realized Discrepancies – with discussion. Statistica Sinica, 6. Gelman, A. & Rubin, D. B., 1992. Inference from Iterative Simulation using Multiple Sequences. Statistical Science, 7: 457–511. Gentleman, R., 1997. A Review of BUGS: Bayesian Inference Using Gibbs Sampling. Chance, 10: 48–51. Gilks, W. R., Richardson, S. & Spiegelhalter, D. J., 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall. He, C. Z., Sun, D. & Tra, Y. , 2001. Bayesian Modelling of Age–specific Survival in Nestling

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Studies under Dirichlet Priors. Biometrics, 57: 1059–1067. Jamieson, L. E. & Brooks, S. P., 2004. Density Dependence in North American Ducks. Animal Biodiversity and Conservation, 27.1: 113–128. King, R. & Brooks, S. P., 2002a. Bayesian Model Discrimination for Multiple Strata Capture–Recapture Data. Biometrika, 89: 785–806. – 2002b. Model Selection for Integrated Recovery/ Recapture Data. Biometrics, 58: 841–851. – 2003. Investigating the Effect of Location, Age and Sex on the Survival and Spatial Fidelity of Mouflon. Journal of Agricultural, Biological and Environmental Statistics, 8: 486–513. King, R., Brooks, S. P., Mazzetta, C. & Freeman, S. N., 2004. Identifying and Diagnosing Population Declines: A Bayesian Assessment of Lapwings in the U.K. Technical Report, University of St. Andrews. Link, W. A., Cam, E., Nichols, J. D. & Cooch, E. G., 2002. Of BUGS and Birds: Markov chain Monte Carlo for Hierarchical Modelling in Wildlife Research. Journal of Wildlife Management, 66: 277–291. 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 Science, 51: 2673–2687. Meyer, R. & Millar, R. B., 1999. BUGS in Bayesian Stock Assessments. Canadian Journal of Fisher-

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ies and Aquatic Science, 56: 1078–1086. Millar, R. B. & Meyer, R., 2000. Non–linear State Space Modelling of Fisheries Biomass Dynamics by Using Metropolis–Hastings within–Gibbs Sampling. Applied Statistics, 49: 327–342. Newman, K. B., 1998. State Space Modelling of Animal Movement and Mortality with Applications to Salmon. Biometrics, 54: 1290–1314. Spiegelhalter, D. J., Best, N. G., Carlin, B. P. & Van der Linde, A., 2002a. Bayesian Measures of Model Complexity and Fit (with discussion). Journal of the Royal Statistical Society, Series B, 64: 583–639. Spiegelhalter, D. J., Thomas, A. & Best, N. G.., 2002b. WinBUGS User Manual, Version 1.4. MRC Biostatistics Unit, Cambridge. Sullivan, P., 1992. A Kalman Filter Approach to Catch– at–Length Analysis. Biometrics, 48: 237–257. Ter Braak, C. J. F., Van Strein, A. J., Meyer, R. & Verstrael, T. J., 1994. Analysing of Monitoring Data with Many Missing Values: Which method? In: Bird Numbers 1992. Distribution, Monitoring and Ecological Aspects: 663–673 (W. Hagemeijer & T. Verstrael, Eds.). Beek–Ubbergeon, Sovon. Thomas, L., Buckland, S. T., Newman, K. B. & Harwood, J., in press. A Unified Framework for Modelling Wildlife Population Dynamics. Australian and New Zealand Journal of Statistics. Tucker, G. M., Davies, S. M. & Fuller, R. J., 1994. The Ecology and Conservation of Lapwings, Vanellus vanellus. Peterborough, Joint Nature Conservation Committee (UK Nature Conservation), 9.

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Annex. WinBUGS code, data and starting values which can be used to reproduce the results provided in this paper. The code was written and tested in WinBUGS version 1.4 and may not work on earlier versions. Anexo. Código WinBUGS, los datos y valores iniciales necesarios para reproducir los resultados dados en este trabajo. El código fue escrito y probado con la versión 1.4 de WinBUGS y puede no funcionar con versiones anteriores.

The code { # Define the priors for the logistic regression parameters alpha1 ~ dnorm(0,0.01) alphaa ~ dnorm(0,0.01) alphar ~ dnorm(0,0.01) alphal ~ dnorm(0,0.01) beta1 ~ dnorm(0,0.01) betaa ~ dnorm(0,0.01) betar ~ dnorm(0,0.01) betal ~ dnorm(0,0.01) # Define the observation error prior sigy <– 1/tauy tauy ~ dgamma(0.001,0.001) # Define the logistic regression equations for(t in 1:(T–1)){ logit(phia[t]) <– alphaa + betaa*f[t] log(rho[t]) <– alphar + betar*t # We assume here that t=1 logit(phi1[t]) <– alpha1 + beta1*f[t] # corresponds to the year 1963 logit(lambda[t]) <– alphal + betal*(t+1) } # Define r[t] for (t in 3:(T–1)){ r[t–2] <– (Na[t+1]+N1[t+1])/(Na[t]+N1[t]) } # Define the initial population priors for(t in 1:2){ N1[t] ~ dnorm(200,0.000001) Na[t] ~ dnorm(1000,0.000001) } # Define the system process for the census/index data using the Normal approximation for(t in 3:T){ mean1[t] <– rho[t–1]*phi1[t–1]*Na[t–1] meana[t] <– phia[t–1]*(N1[t–1]+Na[t–1]) tau1[t] <– 1/(Na[t–1]*rho[t–1]*phi1[t–1]) taua[t] <– 1/((N1[t–1]+Na[t–1])*phia[t–1]*(1–phia[t–1])) N1[t] ~ dnorm(mean1[t],tau1[t]) Na[t] ~ dnorm(meana[t],taua[t]) } # Define the system process for the census/index data using the Poisson/Binomial model # # NB. Need to change initial population priors as well to ensure N1 and Na take integer values

Animal Biodiversity and Conservation 27.1 (2004)

Annex. (Cont.)

# for(t in 3:T){ # bin1[t] <– N1[t–1]+Na[t–1] # bin2[t] <– phia[t–1] # # po[t] <– Na[t–1]*rho[t–1]*phi1[t–1] # # N1[t] ~ dpois(po[t]) # Na[t] ~ dbin(bin2[t],bin1[t]) # } # Define the observation process for the census/index data for(t in 3:T){ y[t] ~ dnorm(Na[t],tauy) } # Define the recovery likelihood for(t in 1 : T1){ m[t, 1 : (T2 + 1)] ~ dmulti(p[t, ], rel[t]) } # Calculate the no. of birds released each year for(t in 1 : T1){ rel[t] <– sum(m[t, ]) } # Calculate the cell probabilities for the recovery table for(t1 in 1 : (T1–1)){ # Calculate the diagonal p[t1, t1] <– lambda[t1] * (1–phi1[t1]) # Calculate value one above the diagonal p[t1, t1+1] <– lambda[t1+1] * phi1[t1]*(1–phia[t1+1]) # Calculate remaining terms above diagonal for(t2 in (t1+2) : T2 ){ for(t in (t1+1):(t2–1)){ lphi[t1, t2, t] <– log(phia[t]) } # Probabilities in table p[t1,t2] <– lambda[t2]*phi1[t1] * (1–phia[t2])* exp(sum(lphi[t1,t2,(t1+1):(t2–1)])) } for(t2 in 1 : (t1 – 1)){ # Zero probabilities in lower triangle of table p[t1, t2] <– 0 } # Probability of an animal never being seen again p[t1, T2 + 1] <– 1 – sum(p[t1, 1 : T2]) } # Final row p[T1,T1] <– lambda[T1]*(1–phi1[T1]) for(t in 1:(T1–1)){ p[T1,t] <– 0 } p[T1,T1+1] <– 1 – p[T1,T1] }

527

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Brooks et al.

Annex. (Cont.)

Data The data here correspond to index records from 1965 to 1998, (normalised) frost days from 1963 to 1997 and recovery data from 1963 to 1997. Zeroes are added to make all data run from 1963 until 1998. list (T = 36, y = c (0,0, 1092.23,1 100.01, 1234.32, 1460.85, 1570.38, 1819.79,1391.27,1507.60, 1541.44,1631.21,1628.60,1609.33,1801.68,1809.08,1754.74,1779.48,1699.13, 1681.39,1610.46,1918.45,1717.07,1415.69,1229.02,1082.02,1096.61,1045.84, 11 3 7 . 0 3 , 9 8 1 . 1 , 6 4 7 . 6 7 , 9 9 2 . 6 5 , 9 6 8 . 6 2 , 9 2 6 . 8 3 , 9 5 2 . 9 6 , 8 6 5 . 6 4 ) , f = c(0.1922,0.3082,0.3082,–0.9676,0.5401,0.3082,1.1995,0.1921,–0.8526, –1.0835,–0.6196,–1.1995,–0.5037,–0.1557,0.0762,2.628,–0.3877,–0.968, 1.9318,–0.6196,–0.3877,1.700, 2.2797,0.6561,–0.8516,–1.0835,–1.0835, 0.1922,0.1922,–0.1557,–0.5037,–0.8516,0.8880,–0.0398,–1.1995,0), T1 = 35, T2 = 35, m = structure(. Data = c( 13., 4., 1., 2., 1., 0., 0., 1., 0., 0., 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 . , 11 2 4 . , 0., 16., 4., 3., 0., 1., 1., 0., 0., 0., 0., 0., 1., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 1259., 0 . , 0 . , 11 . , 1 . , 1 . , 1 . , 0 . , 2 . , 1 . , 1 . , 1 . , 1 . , 2 . , 0 . , 0 . , 1 . , 0 . , 0 . , 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 1082., 0., 0., 0., 10., 4., 2., 1., 1., 1.,0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 1595., 0 . , 0 . , 0 . , 0 . , 11 . , 1 . , 5 . , 0 . , 0 . , 0 . , 1 . , 1 . , 1 . , 1 . , 1 . , 0 . , 0 . , 0 . , 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1596., 0., 0., 0.,0., 0., 9., 5., 4., 0., 2., 2., 2., 1., 2., 0., 1., 0., 0., 1., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 2091., 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 11 . , 9 . , 4 . , 3 . , 1 . , 1 . , 1 . , 3 . , 2 . , 2 . , 1 . , 0 . , 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 1964., 0., 0., 0., 0., 0., 0., 0., 8., 4., 2., 0., 0., 0., 1.,2., 3., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 1942., 0., 0., 0., 0., 0., 0., 0., 0., 4., 1., 1.,2., 2., 1., 3., 3., 0., 2., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 2444., 0., 0., 0., 0., 0., 0., 0., 0.,0., 8., 2., 2., 2., 6., 1., 5., 2., 1., 3., 1., 1., 1., 2., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 3055., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 16., 1., 1., 1., 2., 3., 2., 0., 1., 1., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 3412., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 13., 4., 4., 7., 3., 1., 1., 1.,1., 0., 0., 2., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 3907., 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 11 . , 4 . , 0 . , 2 . , 1 . , 1 . , 2., 2., 0., 3., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 2538., 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 0 . , 11 . , 3 . , 5 . , 1 . , 3 . , 3., 2., 3., 0., 1., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 3270., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 12., 5., 0., 5., 4., 2., 1., 2., 3., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 3443., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 15., 5., 2., 2., 0., 5., 3., 0., 0., 0.,1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 3132., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 7., 4., 6., 1., 3., 3.,2., 0., 1., 0., 0., 1., 0., 1., 0., 0., 0., 0., 0., 3275., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 13., 8.,1., 2., 4., 5., 3., 0., 1., 2., 0., 0., 1., 0., 0., 0., 0., 0., 3447., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 23., 2., 2., 3., 3., 3., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 3902., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 10., 0., 6., 2., 0., 1., 1., 0., 0., 1., 0., 0.,0., 0., 0., 0., 2860., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 19., 7., 6., 4., 0., 0., 2., 0.,0., 0., 1., 2., 0., 0., 1., 4077.,

Animal Biodiversity and Conservation 27.1 (2004)

529

Annex. (Cont.)

0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 12.,3., 2., 0.,0., 0., 0., 1., 0., 1., 0., 0., 0., 0., 4017., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 25., 2., 5., 2., 0., 2., 2., 2., 0., 0., 0., 0., 0., 4827., 0., 0.,0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 14., 4., 3., 4., 4., 2., 2., 1., 0., 2., 0., 1., 4732., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,0., 0., 0., 0., 0., 0., 0., 14., 2., 1., 2., 2., 3., 0., 0., 3., 0.,0., 5000., 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., 18., 4., 4., 3., 0., 2.,1., 0., 2., 1., 4769., 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., 10., 4., 2., 4., 2., 2., 3., 1., 1., 3603., 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., 12., 3., 3., 2., 1., 0., 2., 0., 4147., 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., 9., 4., 6., 1., 0., 1., 0., 4293., 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., 18., 3., 1., 2., 0., 1., 3455., 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., 6., 5., 2., 2.,1., 3673., 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.,12., 4., 6., 0., 3900., 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., 7., 5., 1., 3578., 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., 7., 0., 4481., 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., 5., 4334. ),.Dim = c (35,36)) )

Starting values list (tauy = 1, Na = c(1000.,1000,1092.23,1100.01,1234.32,1460.85,1570.38,1819.79, 1391.27,1507.60,1541.44,1631.21, 1628.60,1609.33,1801.68,1809.08,1754.74, 1779.48,1699.13,1681.39,1610.46,1918.45,1717.07,1415.69, 1229.02,1082.02, 1096.61,1045.84,1137.03,981.1,647.67,992.65,968.62,926.83,952.96,865.64), N1 = c(400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400, 400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400,400), alpha1 = 1, alphaa = 2, alphap = –2, alphal = –4, beta1 =–2, betaa = 0.1, beta p =–0.7, betal = –0.3 )

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)

531

Dispersal and survival rates of adult and juvenile Red–tailed tropicbirds (Phaethon rubricauda) exposed to potential contaminants E. A. Schreiber, P. F. Doherty Jr. & G. A. Schenk

Schreiber, E. A., Doherty, P. F. Jr. & Schenk, G. A., 2004. Dispersal and survival rates of adult and juvenile Red– tailed tropicbirds (Phaethon rubricauda) exposed to potential contaminants. Animal Biodiversity and Conservation, 27.1: 531–540. Abstract Dispersal and survival rates of adult and juvenile Red–tailed tropicbirds (Phaethon rubricauda) exposed to potential contaminants.— Annual survival and dispersal rates of adult and juvenile red–tailed tropicbirds were examined in connection with exposure to heavy metals. From 1990–2000 the incineration of a U.S. stockpile of chemical weapons stored at Johnston Atoll exposed nesting tropicbirds to increased levels of human disturbance, smoke stack emissions and potential leaks. Using a multi–state mark–recapture modeling approach, birds nesting in this site (downwind of the plant) were compared to those nesting in a reference site (upwind of the plant) with less human disturbance, no exposure to smoke stack emissions or other potential incineration emissions. We did not find any difference in survival of adults or juveniles when comparing the two sites. Adult breeding dispersal rates did not differ between the sites but we did find differences in the age–specific natal dispersal rates. Birds fledged from downwind areas were less likely to return to their natal area to nest and more likely to immigrate to the upwind area than vice–versa. This asymmetry in emigration rates is believed to be due to differing vegetation densities and has implications for vegetation management in relation to tropicbird nest success and population size. Key words: Phaethon rubricauda, JACADS, Johnston Atoll, Age–specific breeding and dispersal, Survival, Chemical munitions. Resumen Tasas de dispersión y supervivencia de adultos y juveniles del rabijunco colirrojo (Phaethon rubricauda) expuestas a contaminantes potenciales.— Se examinaron las tasas de dispersión y de supervivencia anual de adultos y jóvenes de los rabijuncos colirrojos en relación a la exposición a metales pesados. Entre los años 1990 y 2000, la incineración de un arsenal de armas químicas del ejército de Estados Unidos almacenadas en Johnston Atoll expuso a las aves del trópico que anidaban en la zona a niveles más elevados de perturbaciones antrópicas/contaminación humana, emisiones procedentes de chimeneas y fugas potenciales. Se comparó las aves nidificantes en este lugar (a favor del viento de la planta) mediante modelos de captura–recaptura de multiestados, con las que anidaban en un emplazamiento de referencia (en cuyo caso la planta quedaba situada en contra del viento), caracterizado por menos perturbaciones antrópicas y ninguna exposición a emisiones procedentes de chimeneas ni a ningún otro tipo de emisiones potenciales de incineración. Al comparar ambos emplazamientos, no se halló ninguna diferencia en cuanto a la supervivencia de las aves adultas o jóvenes. Las tasas de dispersión reproductiva de los adultos no difirieron entre los emplazamientos; en cambio, sí que se observaron diferencias en las tasas de dispersión natal por edades. Las aves jóvenes que abandonaron las áreas situadas a favor del viento era menos probable que regresaran a su área natal para anidar y más probable que inmigraran al área situada en contra del viento. Se considera que esta asimetría en las tasas de emigración obedece a las diferentes densidades de vegetación, repercutiendo en la gestión de la vegetación con respecto al éxito de los nidos de las aves del trópico y el tamaño poblacional. Palabras clave: Phaethon rubricauda, JACADS, Johnston Atoll, Dispersión y reproducción por edades, Supervivencia, Munición química. E. A. Schreiber, National Museum of Natural History, Smithsonian Institution, E607 MRC 116, Washington D.C. 20560, U.S.A.– P. F. Doherty, Jr., Dept. of Fishery and Wildlife Biology, Colorado State Univ., Fort Collins, CO 80523–1474, U.S.A.– G. A. Schenk, 4109 Komes Court, Alexandria, VA 22306, U.S.A. Corresponding author: B. A. Schreiber. E–mail: SchreiberE@aol.com

ISSN: 1578–665X

532

Introduction The Johnston Atoll Chemical Agent Disposal system (JACADS) is located on Johnston Atoll and was the first operational facility in which parts of the U.S. stockpile of chemical weapons were destroyed (Schreiber et al., 2001; U.S. Army, 1994). Seven states in the mainland United States also have stockpiles of chemical weapons that are slated to be incinerated and the lessons learned at the Johnston Atoll site are important for judging the safety of incinerating these chemical weapons. Johnston Atoll is also a U.S. Fish and Wildlife National Wildlife Refuge and a breeding location for approximately a half–million seabirds of 13 species. Further details on the incineration process and the site can be found in Schreiber et al. (2001) and Schreiber (2002). Our previous work (Schreiber et al., 2001) examined survival and movement rates of adult Red–tailed tropicbirds (Phaethon rubricauda) during eight years of the burning of chemical munitions and found no effect on the tropicbirds. This study follows the adults through an additional four years, including three years post–burning, and additionally analyzes juvenile survival and natal dispersal rates. At the time of our previous analysis we did not have enough years of data to estimate juvenile survival due to the fact that tropicbirds have delayed breeding (young birds stay at sea approximately for up to six years before returning to breed; Schreiber & Schreiber, 1993). This delayed breeding and the fact that tropicbirds first return to breed at varying ages (age–specific first breeding probabilities) further complicates the estimation of juvenile survival (Clobert et al., 1994; Pradel & Lebreton, 1999; Schwarz & Arnason, 2000; Spendelow et al., 2002; Williams et al., 2002; Lebreton et al., 2003). Juvenile survival and age–specific breeding probabilities are of particular interest on Johnston Atoll because of the potential for negative effects on these metrics from the JACADS operation. For instance, inhalation of heavy metals emissions from the smoke–stack of the JACADS plant could affect juvenile survival (see table 7 in Schreiber et al., 2001, for list of these emissions). With the additional data collected in the past four years, we now have 12 years of data. The recent developments in model structures to estimate juvenile and adult survival, as well as breeding and age– specific natal dispersal probabilities while taking detection probabilities into account, allow us to test the predictions that these population dynamic parameters are negatively affected in the area adjoining the JACADS plant when compared to a reference area. Materials and methods Study area, study species, and data collection Details on the study area, study species and data collection were presented elsewhere (Schreiber, 2002; Schreiber & Schreiber, 1993; Schreiber et

Schreiber et al.

al., 2001). In brief, Johnston Atoll (fig. 1) is located in the central Pacific approximately 1150 km southeast of Hawaii and is home to the Johnston Atoll National Wildlife Refuge and JACADS. The JACADS plant incinerated chemical weapons from 1990 through 2000. Johnston Atoll is in the easterly trade wind belt and winds are from the east, except during El Nino–Southern Oscillation (ENSO) events and on a few still days during the year. The incinerator was situated on the downwind side of Johnston Island (fig. 1). Birds nesting in the smokestack outfall plume (fig. 1) were considered to be in the "downwind" area, where there was potential for contamination from the smokestack emissions (see table 7 in Schreiber et al., 2001) or from chemical spills and leaks (material to be incinerated was stored in the downwind area). The 3 smokestacks were each 30.48 meters high. Emissions fell on land around the stacks and out towards the wind direction. In days of particularly high winds, there may have been times when no emissions fell on the land, but high winds were rare. The great majority of the time winds on the atoll blow between 18.5–30 km/h, when outfall from the stacks was determined to fall on land where birds were nesting (Anonymous, 1996). Birds nesting outside this plume were considered to be in "upwind" areas with no potential for contamination from the incineration process. No incineration was carried out during days of westerly winds. Tropicbirds invariably lay one egg (Schreiber, 1999). Tropicbirds are strongly nest–site philopatric, nesting in the same spot each year once they have successfully nested (Schreiber, 1999; Schreiber et al., 2001). Tropicbirds exhibit deferred maturity, with most birds returning to their natal colony to breed between ages two and six years of age (Schreiber & Schreiber, 1993). A few birds are caught for the first time at 7–12 years of age, but since only 40–50% of the breeding adults are recaptured each year, these birds could have bred earlier and not been caught. In previous analyses (Doherty et al., 2004) age–specific breeding probability could be reliable estimated up to age five. After this time, so few birds had not returned that estimating a first–time breeding probability for age 6+ could not be done. Thus, we feel comfortable with assuming in our modeling that any bird that has not bred by age six will attempt to do so in later years with probability equal to one. If adult breeders do not breed with probability equal to one, then age–specific first–time breeding probabilities are in relation to what this probability is. In each year most adult birds were generally caught by hand during a February or March field visit. A second field visit was made sometime during the period from May–July to band chicks, as well as additional adults not caught on the first visit. Bands showing wear were replaced each year and band loss is assumed to be negligible. Movement of adult birds between breeding seasons (breeding dispersal), both between islands

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1300 km to Midway Island

Upwind Downwind

Incinerator

N 1500 km to Hawaii 4500 km to San Francisco

Fig. 1. Johnston Island with the location of the incinerator, downwind and upwind breeding areas designated. Fig. 1. Isla Johnston, con la situación del incinerador y las áreas de reproducción designadas a favor del viento y en contra del viento.

and within islands (Schreiber et al., 2002) is at very low rates and generally owing to loss of the nest site or mate. However, natal dispersal rates were not previously known. Our analysis used only data from birds nesting on Johnston Island since 1992. Whereas in our previous analysis we were only able to analyze demographic parameters associated with adult birds, in this current analysis (with four more years of data) we are also able to analyze pre–breeding survival and age–specific natal dispersal rates. Statistical analysis We used a multi–state capture–mark–recapture approach (Arnason, 1972; 1973; Nichols & Kendall, 1995; Nichols, 1996; Williams et al., 2002; Lebreton et al., 2003) as available in program MARK (White & Burnham, 1999). In Schreiber et al. (2002) we used this approach to estimate recapture probabilities (probability that a bird is caught and band read in any one year, given that it is available for capture), adult survival probabilities and breeding dispersal probabilities between upwind and downwind areas of Johnston Island. Modeling age–specific breeding probabilities (or the transition from pre–breeding to breeding status) in a seabird (Roseate tern; Sterna dougallii) has recently received attention (Spendelow et al., 2002; Lebreton et al., 2003) and the modeling of transition (movement) rates between spatially different areas or transition rates between status (i.e. non–breeding and breeding) can be done in an analytically similar way using a multi–state model. The combining of recruitment and dispersal over several sites can also be accomplished using a multi–state model as has recently been outlined by Lebreton et al. (2003). We used this latter approach. For our data, we considered a bird to be in one of four states; downwind pre–breeding, upwind pre–

breeding, downwind breeding, or upwind breeding. A limited number of possible transitions exist from one survey period (or breeding season) to the next. For instance, a pre–breeder can become a breeder or stay a pre–breeder. We assume that a breeder cannot become a pre–breeder, a fledged bird can first breed at the upwind or downwind site and that once a bird has attempted to breed, it will always attempt to breed in subsequent years (but may change sites). All birds that are going to breed are assumed to attempt to do so by age six. Other general assumptions required by this model are similar to those required by Cormack– Jolly–Seber (CJS) models (Cormack, 1964; Jolly, 1965; Seber, 1965); (1) recapture and transition probabilities are the same for all marked birds found in a particular state and sampling period; (2) birds behave independently with respect to survival, recapture and transition probabilities; (3) bands are not lost; (4) all samples are instantaneous; and (5) state transition probabilities reflect a first order Markov process in the sense that the state of an animal at time t +1 is stochastically determined as a function of it’s state at time t only. (Williams et al., 2002). We believe that our study and the study species are an ideal candidate for this type of modeling because these assumptions are well met. Red–tailed tropicbirds do not flock and are thought to behave independently except during the breeding season. We restricted our sampling to the breeding season, while not instantaneous, it is a period in which high mortality is unlikely to occur (i.e. a dead adult is rarely found). We performed a goodness of fit test on our most general model and, to help correct for any over dispersion in our data, we also estimated an over dispersion factor (‡) to adjust estimates and other statistics (Burnham & Anderson 2002). Our goodness of fit test and ‡ were based on a Pearson 2 goodness– of–fit test.

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Table 1. The relative likelihood, number of parameters, and AIC rankings of each model. We used the small sample approximation and adjusted AIC using ‡ as denoted by the QAICc term. QAICc is the relative difference of each model from the best fit model (lowest QAICc). The QAICc weight is the relative weight of evidence for each model. Adult survival ( A), juvenile survival (defined as survival from banding until age 1; J), recapture probability (p), age–specific natal dispersal probabilities (age = 1 to 5; age), and breeding dispersal rates ( breeder) were modeled as functions of time (t) and site (upwind or downwind of the JACADS plant). Tabla 1. Verosimilitud relativa, número de parámetros y clasificaciones de cada modelo según el criterio de Información de Akaike (AIC). Utilizamos la aproximación de muestras pequeñas y ajustamos el AIC utilizando ‡ según lo indicado por el término QAICc. QAICc es la diferencia relativa de cada modelo con respecto al modelo mejor ajustado (QAICc más bajo). El peso de QAICc es el peso relativo de la evidencia para cada modelo. La supervivencia adulta ( A), la supervivencia juvenil (definida como la supervivencia desde el anillamiento hasta alcanzar 1 año de edad; J), la probabilidad de recaptura (p), las probabilidades de dispersión natal por edades (edad = de 1 a 5; age), y las tasas de dispersión reproductora ( breeder) se modelaron como funciones del tiempo (t) y del emplazamiento (en contra del viento o a favor del viento con respecto a la planta JACADS).

Model

QAICc

QAICc weight

Number of parameters

43397.49

0.00

1.00

43417.13

19.64

0.00

43417.48

19.99

0.00

43433.45

35.96

0.00

107

43447.74

50.26

0.00

43574.46

176.97

0.00

43816.14

418.66

0.00

43848.34

450.85

0.00

Model set We constructed models including adult survival ( A), juvenile survival (defined as survival from fledging until age one; J), recapture probability (p), and adult movement (or breeding dispersal) probabilities ( breeder). Each of these parameters was modeled as a function of time (t) and site (upwind or downwind of the JACADS site). The age–specific–natal dispersal probabilities (age = 1 to 5; age) were modeled as a function of site only due to the data requirements associated with temporal estimates. Due to the tropicbirds propensity to stay out at sea for up to six years after fledging, with our 12 years of data we were only able to estimate juvenile survival rates meaningfully for the first five years of our data set. Since, in this case, some individuals do attempt to breed at age one, our juvenile survival rates (or pre– breeding rates) are for the time from banding to the next breeding season. All birds are assumed to survive at the same rate after this first year. By definition, there is no recapture rate for juveniles (when a bird is recaught for the first time it is a

breeding adult). Thus our most general model was designated as . Our predictions of interest focused on the effects of site. We thus constructed a set of reduced models in which the effect of site was removed for each parameter independently, as well as for both of the survival rates together. These models were designated as for adult survival being constant over site, for juvenile survival being constant over site, for both adult and juvenile survival being constant over site, for recapture probabilities to be constant over site, for age–specific natal dispersal probabilities being constant over site, and for breeding dispersal rates being constant over site. We also, a posteriori, constructed a model with an additive effect of site for all parameters ( ) and report the associate beta estimates from this model. We used Akaike’s Information Criteria (AIC; Burnham & Anderson, 2002) to compare the fit of these models. We specifically used the small sample

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1.0 0.9 Recapture rate

0.8 0.7 0.6 0.5 0.4 0.3 Upwind Downwind

0.2 0.1 0.0 1993

1994 1995 1996 1997 1998 1999 2000 2001 2002 Year

Fig. 2. Probability of recapture for birds at upwind (triangles and solid line) and downwind (squares and dotted line) sites of the JACADS plant from 1993 to 2002. Model–averaged means and 95% confidence intervals are shown. Fig. 2. Probabilidad de recaptura de las aves en áreas situadas en contra del viento (triángulos y línea continua) y a favor del viento (cuadrados y línea discontinua) con respecto a la planta JACADS, desde 1993 hasta 2002. Se indican los promedios de las medias de los modelos y los intervalos de confianza del 95%.

approximation and ‡ adjusted form of AIC denoted as QAICc (Burnham & Anderson, 2002) and considered models with QAICc values differing by less than 2 to be equally suitable models. We also calculated AIC weights for each model to help us interpret the relative strengths of each model (Burnham & Anderson, 2002). This also allowed us to calculate model–averaged estimates. A model–averaged estimate is essentially a weighted average over a set of models based on each individual model’s weight. This allows inference to the entire model set. Such a model–averaging approach is preferable to relying on parameter estimates from a single model (Burnham & Anderson, 2002). All results present in the figures are model– averaged estimates with their associated 95% confidence intervals unless otherwise noted. Results

sal as well as breeding rates (table 1). None of the other models described the data well with QAICc > 19. Below we examine the parameter estimates associated with recapture, adult and juvenile survival, as well as age–specific natal dispersal and breeding dispersal rates. Recapture probability The probability of recapture differed between the upwind and downwind sites through time and ranged from 0.32 to 0.59 in the upwind area and from 0.39 to 0.63 in the downwind area. Generally the downwind site (0 = 0.48, SÊ = 0.09) had a higher probability of recapture than the upwind site (0 = 0.38, SÊ = 0.08; fig. 2). The beta estimate from a model with site as an additive effect was 0.50 with a 95% confidence interval of [0.41, 0.60] also indicating an effect of site. Adult survival rate

We followed 31,527 birds banded as juveniles and adults over 12 years of data collection (1992– 2003). Our goodness of fit test failed (P < 0.01), which indicated that our data were over dispersed. To help correct for this over dispersion we incorporated a ‡ = 2.55. This was more conservative than the correction we used in our previous analysis (‡ = 1.6; Schreiber et al., 2002). Overall the model that ranked highest by QAICc coded for temporal variation in adult survival, juvenile survival, probability of recapture and breeding dispersal rates and variation across sites for detection probability and both age–specific natal disper-

The most supported model indicated only temporal variation in adult survival with little variation associated with site (downwind 0 = 0.86, SÊ = 0.04; upwind 0 = 0.86, SÊ = 0.04; fig 3). We found no consistent effect of site for adult survival with the beta estimate being 0.02 with a 95% confidence interval of [–0.06, 0.11]. Juvenile survival rate Since juvenile tropicbirds can take six years (and in a few cases longer) to return to the island to

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Adult survival rate

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Upwind

0.2 0.1

Downwind

0.0 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Year Fig. 3. Adult survival rates for birds upwind (triangle and solid line) and downwind (squares and dotted line) of the JACADS plant from 1992 to 2001 (Year t to t + 1). Results are estimates and 95% confidence intervals from model .

Juvenile survival rate

Fig. 3. Tasas de supervivencia de las aves adultas en áreas situadas en contra del viento (triángulos y línea continua) y a favor del viento (cuadrados y línea discontinua) con respecto a la planta JACADS, desde 1992 hasta 2001 (del año t al t + 1). Los resultados son estimaciones e intervalos de confianza . del 95% del modelo

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Upwind Downwind

1992

1993

1994 1995 1996 1997 Year

breed, we do not present estimates of juvenile survival for the last 5 years (many birds from those cohorts would not have returned to breed yet). Over the first five years we did not find any differences in juvenile survival rate for tropicbirds upwind (0 = 0.77, SÊ = 0.09) as compared to downwind (0 = 0.77, SÊ = 0.09) of the JACADS (fig 4). However, there was variation associated with time with juvenile survival rates being lower in 1994 and 1995 and higher in the other years. The 1994–1995 time period was at the end of an extended El Niño period. The beta estimate associated with site was 0.02 with a 95% confidence interval of [–0.06, 0.11]. Breeding dispersal rates

Fig. 4. Juvenile survival rate for birds upwind (triangle and solid line) and downwind (square and dotted line) of the JACADS plant from 1992 to 1997 (year t to t + 1). Results are estimates and 95% confidence intervals from model . Fig. 4. Tasas de supervivencia de las aves jóvenes en áreas situadas en contra del viento (triángulos y línea continua) y a favor del viento (cuadrados y línea discontinua) con respecto a la planta JACADS, desde 1992 hasta 1997 (del año t al t + 1). Los resultados son estimaciones e intervalos de confianza del 95% del modelo .

Although there was much variation associated with the breeding dispersal rate estimates, overall the breeding dispersal rates were low; 0 = 0.012, SÊ = 0.008 for birds dispersing upwind to downwind and 0 = 0.022; SÊ = 0.018 for birds dispersing downwind to upwind (fig. 5). Age–specific first–time breeding probabilities and natal dispersal We found differences in the philopatry and emigration rates of tropicbirds that fledged upwind and downwind of the JACADS plant. For all ages at which birds recruited to breeding status, birds that fledged from upwind sites showed greater

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0.10

Upwind to downwind

0.09

Downwind to upwind

Movement rate

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 1992

1993

1994

1995

1996 Year

1997

1998 1999

2000 2001

Fig. 5. Adult movement (breeding dispersal) rates of red–tailed tropicbirds nesting upwind and downwind of the JACADS. Model–averaged means and 95% confidence intervals are shown.

Age–specific natal dispersal rate

Fig. 5. Tasas de movimiento (dispersión reproductiva) de las aves del trópico de cola roja adultas que anidaban en áreas situadas en contra del viento y a favor del viento con respecto a la planta JACADS. Se indican los promedios de las medias de los modelos y los intervalos de confianza del 95%.

1.0

Upwind to upwind

0.9

Downwind to downwind

0.8

Downwind to upwind

0.7

Upwind to downwind

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

3 Age

Fig. 6. Age–specific breeding probabilities for tropicbirds of age 1 to 5 for sites upwind (clear bars) and downwind (hatched bars) of the JACADS. Model–averaged means and 95% confidence intervals are shown. Fig. 6. Probabilidades de reproducción por edades de las aves del trópico, de entre 1 y 5 años de edad, en emplazamientos situados en contra del viento (barras transparentes) y a favor del viento (barras sombreadas) con respecto a la planta JACADS. Se indican los promedios de las medias de los modelos y los intervalos de confianza del 95%.

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philopatry and were less likely to first breed elsewhere than were birds that fledged downwind (fig. 6). For both sites the rate at which birds recruited into the breeding population increased with age (fig. 6). Discussion As previously discussed (Schreiber et al., 2002), the main concerns for survival of birds nesting downwind of the JACADS plant were inhalation of heavy metals from the smoke stack and direct mortality from leaks of chemical agents. Ingestion of heavy metals from soil was not considered to be a problem since tropicbirds do not feed on land and do not tend to pick objects with their bill while on land (Schreiber & Schreiber, 1993). In our previous work (Schreiber et al., 2002), we did not find any evidence of negative effects of the JACADS plant on tropicbird fledging success or adult survival. Nor did we find any evidence that adult tropicbirds were moving out of the downwind area as if avoiding nesting there for some reason. We did find a higher recapture probability downwind as compared to upwind, and this was attributed to greater effort made to monitor birds in the downwind area. Our current analysis supports these past results concerning adult survival, recapture and dispersal rates (figs. 2, 3, 5). The lack of any difference between the two areas now, two years after burning of munitions has ceased, may indicate that there was no build–up of heavy metals in the birds. We could not estimate a juvenile survival rate or associated age–specific natal dispersal rates in our previous analysis (Schreiber et al., 2001). This was due to the fact that the time series of data was not long enough, as well as the fact that models that allowed these parameters to be estimated were still under development (Clobert et al., 1994; Pradel & Lebreton, 1999; Schwarz & Arnason, 2000; Spendelow et al., 2002; Williams et al., 2002; Lebreton et al., 2003). We did not find any effects associated with the JACADS plant on juvenile survival (fig. 4). We also did not find any negative effects associated with JACADS on age–specific breeding probabilities: birds in both areas returned to breed at similar ages. However we did find an effect that birds fledging downwind were more likely to first return to breed upwind than were birds fledged upwind to first breed downwind (lower natal philopatry rates downwind than upwind; fig. 6). This result was unexpected given that we have no differences in adult movement rates. Several possibilities must be considered as a reason for this difference. We believe the higher emigration rate from the downwind area is due to the fact that this area has no large patches of growing vegetation to attract new nesters. Most vegetation downwind consists of narrow strips bordering roads, or small patches next to buildings. All of the large courting/nesting areas (also associated with vegetation) occur in the upwind section of Johnston Island. This causes two events that

can affect natal philopatry. The larger groups of aerially courting birds that form over large vegetation patches may be particularly attractive to first time breeders, thus attracting more of them to breed upwind, rather than downwind. This idea could be tested through the planting and manipulation of nesting vegetation over a period of years. Secondly, the downwind area (with all the operations associated with weapons incineration) has a greater density of buildings and roads. Here, human activity and development restrict the amount of habitat available and also restrict growth of new habitat. Since birds return to their historic nesting sites, areas with no new, growing vegetation offer little opportunity for first time nesters to claim a site: most sites are taken. Areas of growing and spreading vegetation, that have not had previous nesters, are the best opportunities for new nesters. Experienced nesters are aggressive about defending their nest site from interlopers, and site owners historically are known to win in battles over nest sites (Schreiber & Schreiber, 1993). Thus, the downwind area presents less opportunity for a new nester, even if attracted to the area, causing a returning downwind first–time breeder to consider other parts of the island. We believe that an indirect effect of the JACADS plant (vegetation control) has caused a difference in nesting philopatry in the downwind as compared to the upwind area. We predict that once the JACADS plant is dismantled and the area deserted of heavy human activities, the differences in natal philopatry rates between the two areas will disappear as the vegetation grows normally. Possibilities exist to test these ideas in the years subsequent to the dismantling of the incinerator and the abandonment of the atoll by the military. Another reason for the differences in natal philopatry might be a negative effect associated with emissions from the JACADS plant. We think this is unlikely due to no effects being detected in the adult movement rates or in any other parameters measured (reproductive success, chicks growth rates, adult mass, adult survival, adult movement rates; see Schreiber, 2002; Schreiber et al., 2001). It is also possible that the degree of human activity downwind caused returning first– breeders to look elsewhere for a nest site. However, there are other areas on Johnston Island that have the same degree of human activity, and where vegetation growth is not restricted, yet the nesting population continues to increase. Management implications Over the 10 years of operation of the chemical agent disposal system on Johnston Atoll we could document no effect of the operation on breeding red– tailed tropicbirds in adult survival, juvenile survival (this paper), reproductive success, egg mass, adult mass or chick growth (Schreiber, 2002). Although there was a difference in natal dispersal rates that could be attributed to the plant, we think that this result is most likely tied to differences in vegetation

Animal Biodiversity and Conservation 27.1 (2004)

and activity differences downwind of the plant as compared to upwind. If this is so, then this suggests that stack emissions of trace amounts of heavy metals and nesting close to human activities was not harmful to nesting tropicbirds. Since incineration plants will be built in seven mainland states of the United States, our results may have important implications for those sites. Careful monitoring of those sites and the wildlife associated with those sites is warranted. A priori consideration of possible confounding factors and incorporating these into the data collection design may help tease apart direct and indirect effects associated with these plants. The results of this study and the interpretation of those results point out the need to understand the ecology of a species when (1) making management decisions to maintain healthy, viable populations, and (2) making permitting decisions on issues that will expose birds to unusual levels of contaminants. Additional experiments associated with management include altering human disturbance and planting of vegetation in a controlled way to further test what makes good courting and nesting habitat. This analysis also points out the importance of jointly modeling survival, dispersal and capture rates in a way that can account for heterogeneity in such rates. If site heterogeneity in the recapture rates was not modeled correctly, differences in adult survival associated with site may have been erroneously reported due to an overoptimistic assessment of the statistical power of our modeling exercise. If age–specific breeding and dispersal probabilities were not included in the modeling effort, then juvenile survival may have been thought to be lower, especially in the downwind area, owing to emigration to other nesting sites. Acknowledgements We thank the Defense Nuclear Agency, Project Manager for Chemical Demilitarization, G. McClosky, R. Moll, and P. O’Connell for many years of grant and logistical support for our work on Johnston Atoll. The U.S. Fish and Wildlife Service, Dept. of Interior, permitted access to the refuge. We thank B. D. Smith and C. M. Francis for comments on a previous draft of the manuscript. Many people have assisted us over the years in catching and banding the birds (and in all our others studies on the atoll), and we are extremely grateful for their help and the great company while in the field. We are deeply indebted to R. W. Schreiber for having the idea to return here and begin work again in 1984 after having spent part of the Pacific Ocean Biological Survey Project working here. References Anonymous, 1996. Johnston Atoll Chemical Agent Disposal System, Human Health and Ecological Risk Aassessment. Raytheon Engineers and Constructors, Inc., Philadelphia, PA.

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Arnason, N., 1972. Parameter estimation from mark–recapture experiments on two populations subject to migration and death. Res. in Pop. Ecol. 13: 97–113. – 1973. The estimation of population size, migration rates and survival in a stratified population. Res. in Pop. Ecol., 15: 1–8. Burnham, K. P. & Anderson, D. R., 2002. Model selection and multimodel inference: a practical information–theoretic approach. Springer, New York, USA. Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438. 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. Doherty, P. F., Jr, Schreiber, E. A., Nichols, J. D., Hines, J. E., Link, W. A., Schenk, G. A. & Schreiber, R. W. Testing life history predictions in a long–lived seabird: a population matrix approach with improved parameter estimation. Oikos, 105: 606–618. Jolly, G. M., 1965. Explicit estimates from capture– recapture data with both death and immigration – stochastic model. Biometrika, 52: 225–247. 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. Oikos, 101: 253–264. Nichols, J. D., 1996. Source of variation in migratory movement of animal populations: statistical inference and a selective review of empirical results for birds. In: Population dynamics in ecological space and time: 147–197 (O. E. Rhodes, Jr., R. K. Chesser & M. H. Smith, Eds.). Univ. of Chicago Press. Chicago, Illinois, U.S.A. Nichols, J. D. & Kendall, W. L., 1995. The use of multi–state capture–recapture models to address questions in evolutionary ecology. J. of Appl. Stat., 22: 835–846. Pradel, R. & Lebreton, J. D., 1999. Comparison of different approaches to the study of local recruitment of breeders. Bird Study, 46: 74–81. Schreiber, E. A., 2002. Breeding biology and ecology of the seabirds of Johnston Atoll, central Pacific Ocean, 1998: long–term monitoring for effect of Johnston Atoll Chemical Demilitarization Project 1984–1998. Report to the Department of Defense, Aberdeen Proving Ground, Maryland, U.S.A. Schreiber, E. A., Doherty, Jr., P. F. & Schenk, G. A., 2001. Effects of a chemical weapons incineration plant on red–tailed tropicbirds. J. Wildl. Manage., 65: 685–695. Schreiber, E. A. & Schreiber, R. W., 1993. Red– tailed tropicbird (Phaethon rubricauda). The birds of North America, number 43. The American Ornithologists’ Union, Washington, D.C., U.S.A., and The Academy of Natural Sciences, Philadelphia, Pennsylvania, U.S.A.

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Schreiber, R. W. & Schreiber, E. A., 1984. Central Pacific Seabirds and the El–Nino Southern Oscillation – 1982 to 1983 Perspectives. Science, 225: 713–716. Schwarz, C. J. & Arnason, A. N., 2000. Estimation of age–specific breeding probabilities from capture–recapture data. Biometrics, 56: 59–64. Seber, G. A. F., 1965. A note on the multiple recapture census. Biometrika, 52: 249–259. Spendelow, J. A., Nichols, J. D., Hines, J. E., Lebreton, J.–D. & Pradel, R., 2002. Modelling postfledging survival and age–specific breeding

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probabilities in species with delayed maturity: a case study of roseate terns at Falkner Island, Connecticut. J. Appl. Stat., 29: 385–405. U. S. Army, 1994. U.S. Army’s alternative demilitarization technology. Executive Summary Report for Congress, Washington, DC, U.S.A. White, G. C. & Burnham, K. P., 1999 Program MARK: survival estimation from populations of marked animals. Bird Study 46 suppl., 120:138. Williams, B. K., Nichols, J. D. & Conroy, M. J., 2002. Analysis and Management of Animal Populations. Academic Press, San Diego, U.S.A.

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Estimating true age–dependence in survival when only adults can be observed: an example with Black–legged Kittiwakes M. Frederiksen, S. Wanless & M. P. Harris

Frederiksen, M., Wanless, S. & Harris, M. P., 2004. Estimating true age–dependence in survival when only adults can be observed: an example with Black–legged Kittiwakes. Animal Biodiversity and Conservation, 27.1: 541–548. Abstract Estimating true age–dependence in survival when only adults can be observed: an example with Black– legged Kittiwakes.— In long–lived birds, pre–breeders are often difficult or impossible to observe, and even though a proportion of marked adults may be of known age, the estimation of age–specific survival is complicated by the absence of observations during the first years of life. New developments in MARK now allow use of an updated individual covariate. We used this powerful approach to model age–dependence in survival of Black–legged Kittiwakes (Rissa tridactyla) at a North Sea colony. Although only 69 marked breeders were of known age, there was strong evidence for a quadratic relationship between true age and survival. We believe that this simple but powerful approach could be implemented for many species and could provide improved estimates of how survival changes with age, a central theme in life history theory. Key words: Survival, Age–dependence, Senescence, Rissa tridactyla, Seabirds, Mark–recapture. Resumen Cómo estimar el efecto real de la edad verdadera en la supervivencia cuando sólo es posible observar individuos adultos: ejemplo de la gaviota tridáctila.— En las aves de larga vida, a menudo resulta difícil o incluso imposible observar individuos prerreproductores, y si bien un porcentaje de adultos marcados pueden ser de edad conocida, la estimación de la supervivencia a una edad específica se convierte en una tarea compleja, puesto que no se dispone de observaciones de los primeros años de vida. Las nuevas características del programa MARK nos permiten utilizar una covarianza individual actualizada. Empleamos este impactante enfoque para modelizar el efecto de la edad en la supervivencia de la gaviota tridáctila (Rissa tridactyla) en una colonia del mar del Norte. Aunque sólo 69 aves reproductoras marcadas eran de edad conocida, contábamos con numerosas pruebas que apuntaban a una relación cuadrática entre la edad real y la supervivencia. Creemos que este simple pero eficaz enfoque podría aplicarse en muchas especies, proporcionando estimaciones mejoradas acerca de cómo la supervivencia varía con la edad, un tema central de la teoría de las historias vitales. Palabras clave: Supervivencia, Dependencia de la edad, Senescencia, Rissa tridactyla, Aves marinas, Marcaje–recaptura. Morten Frederiksen, Sarah Wanless & Michael P. Harris, Centre for Ecology and Hydrology, Hill of Brathens, Banchory, AB31 4BW, U.K.

ISSN: 1578–665X

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Introduction Robust estimates of age–related patterns in survival of wild animals are of considerable importance in life history theory (Stearns, 1992), where they contribute to the basis on which theories about the evolution of senescence and delayed maturity are evaluated. Projection of population growth through matrix models (Caswell, 2001) also requires age–specific survival estimates. Mark–recapture methods typically provide estimates of mean age–specific survival in the population. As pointed out by e.g. Cam et al. (2002b) the relevant currency for life history theory is how the underlying individual probability of survival changes with age. If individual survival probabilities are heterogeneous, within–generation selection may lead to substantial differences between the individual and population patterns of age–specific survival (Nisbet, 2001; Cam et al., 2002b). Nevertheless, population level patterns provide a conservative estimate of senescence at the individual level, as well as appropriate input for age–specific population models. Estimating age–specific survival probabilities is fairly straightforward when all marked animals are of known age and can be observed at all stages of the life cycle (e.g. Newton & Rothery, 1997; Nichols et al., 1997; Frederiksen & Bregnballe, 2000a, 2000b). However, many long– lived birds delay recruitment to the breeding population for several years, and in the intervening period pre–breeders are often more or less unobservable. This is a particular problem for many species of seabird, where pre–breeders do not return to the breeding colony for several years and thus cannot be observed (Furness & Monaghan, 1987; Hamer et al., 2002). Low philopatry to the natal site may lead to known– age birds only constituting a small proportion of all marked adults, even in studies with high chick ringing effort (Dunnet & Ollason, 1978; Frederiksen & Petersen, 1999). Furthermore, modelling survival over the full life–span is complicated by the absence of information from the pre–breeding period and the fact that birds belong to many different birth cohorts with potentially different juvenile survival. Most researchers studying age–specific survival in birds with delayed recruitment have therefore analysed only the adult part of the life history, using one of two options. The first option is employing the time since first observation/marking as breeder as a surrogate for age (Bradley et al., 1989; Aebischer & Coulson, 1990). Because birds first observed breeding in the same year may differ in true age, this approach decreases the power to detect trends in age–specific survival, particularly in studies where birds may have bred for several years before being marked and included in the study. The second option involves splitting known–age birds into birth cohorts and modelling age–specific survival separately for each cohort, with constraints across cohorts to identify general pat-

terns (e.g. Tavecchia et al., 2001). This approach uses all the available information and should produce robust estimates of age–specific survival, but when the number of birth cohorts gets large in long–term studies, the modelling process becomes very complex. Here, we illustrate an alternative approach that uses new features in program MARK (White & Burnham, 1999) to model survival as a function of true age without splitting the data set. Our approach is equivalent to the second option, but it is analytically simpler. Methods We studied colour–ringed breeding adult Black– legged Kittiwakes (Rissa tridactyla) on the Isle of May, southeast Scotland (56° 11’ N, 2° 33’ W) from 1986 to 2002. Birds were captured and ringed with a combination of three coloured plastic rings within defined study plots. Some observations were also made of birds that had left the study plots and bred elsewhere on the island. Out of 470 colour– ringed breeders involved in the study, 69 had originally been ringed with metal rings as chicks between 1970–1989, mostly in the same colony. Birds were generally not recaptured and colour– ringed immediately following recruitment; the mean age at colour ringing was 8.3 years (range 2–23 years), whereas the mean age of recruitment in other kittiwake studies was about 4 years (Porter & Coulson, 1987; Cam et al., 2002a). We collated adult capture histories of the 69 known–age kittiwakes and tested the goodness of fit of the time–specific model &t, pt with program U–CARE (Choquet & Pradel, 2002). The age of each bird at recapture as breeder (and colour ringing) was included as an individual covariate "age". To model survival and resighting probabilities as functions of true age, we used the design matrix commands "add" and "product" implemented in MARK version 3.0. With the parameter index matrix set up as fully age–specific, the "add" command allows the user to construct an updated individual covariate, which is the sum of the individual covariate "age" and the number of years since colour ringing, i.e. equal to true age (fig. 1). In order to model a quadratic relationship between age and survival, we used the "product" command to square the covariate created in the previous step (fig. 1). Because the data showed evidence of pronounced trap–happiness (see Results), we analysed capture histories in MARK using a recent multi–state approach to modelling trap–dependence (L. Crespin, pers. comm.). In this approach, two states are defined: "seen" (coded e.g. as 1) and "not seen" (coded e.g. as 2, but this code will never appear in the capture histories). Actual resighting probabilities are then estimated as transition probabilities to the "seen" state, separately for birds seen in the previous year and those not seen, whereas the nominal resighting probabilities

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in MARK are fixed to 1 for the "seen" state and 0 for the “not seen” state. We then modelled survival and resighting probabilities as constant or as functions of either true age or the number of years since colour ringing (hereafter termed "ring–years"). If "ring–years" provided a better fit than true age, this might imply that any decline in survival detected is likely to be related to accelerating wear and loss of colour rings rather than to senescence of birds (but see Discussion). Because of the sparse nature of the data set, we did not attempt to model variation among years in survival or resighting probabilities. Elsewhere, we analyse the full data set of colour–ringed breeders and explore the nature of among–year variation in survival (Frederiksen et al., in press). We also used the cohort separation approach (Tavecchia et al., 2001) to confirm that our method indeed provides identical results, and to fit models with true age as a factor. We used AICc (Burnham & Anderson, 1998) to select the most appropriate model to describe the data. Results The observed correlation coefficient between true age and "ring–years" in our data set was 0.56, indicating that modelling survival as a function of either of the two might produce quite different results. The directional test for trap–happiness in U–CARE was highly significant: z = –4.65, P < 0.001, whereas the other test components showed no evidence of lack of fit ((2 = 4.71, df = 30, P = 1). Model selection indicated that the data were best described by a model with a quadratic relationship between true age and survival, and a linear relationship between "ring– years" and resighting probability (table 1). The weight of evidence in favour of this model was quite strong (ratio of Akaike weights (Burnham & Anderson, 1998) between first– and second– ranked models = 3.1, likelihood ratio test for quadratic term: (2 = 4.36, df = 1, P < 0.05). Evidence that either survival or resighting probability was related to true age was even stronger ( AICc for best model without effect of true age = 5.74; table 1). Estimates from the selected model showed an increase in survival in early adult life, followed by a plateau from approximately 3 to 11 years and a subsequent strong decline (fig. 2). Very few young breeders (2– and 3–year–olds) were included in the data set, so the evidence for the initial improvement in survival was not very strong. Resighting probabilities declined with "ring–years", particularly for birds not seen during the previous breeding season (fig. 3). When we fitted the same model by separating the birth cohorts and constraining across them, both deviance and parameter estimates were identical to at least the third decimal place, indicating that the two approaches are equivalent. Models fitted using this approach also provided unconstrained

Fig. 1. MARK design matrix, showing the use of the "add" and "product" commands to model survival as a quadratic function of true age (individual covariate "age" plus number of years since colour ringing). The corresponding parameter index matrix was set up as fully age–specific (i.e. 16 age classes, no time–dependence). Fig. 1. Matriz del diseño MARK, donde se indica la utilización de los comandos "añadir" y "producto" para modelizar la supervivencia como una función cuadrática de la edad verdadera (covarianza individual de la "edad" más el número de años desde el anillamiento). La matriz correspondiente de índice de parámetros se configuró como si dependiera totalmente de la edad (por ejemplo, 16 clases de edades, sin dependencia del tiempo).

estimates of age–specific survival to compare against the parametrical estimates obtained above (table 1, last two models, fig. 2). Analysis of deviance showed that a quadratic effect explained 39.2% of the total variation in survival associated with true age (fig. 2), and linear effects explained 23.6% of the total variation in resighting probability associated with "ring–years" (fig. 3).

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Table 1. Model selection for age–specific survival and resighting probabilities of known–age Black– legged Kittiwakes. All models included trap–dependence in resighting probability, plus an interaction term with true age or "ring–years". We also fitted models with quadratic relationships between resighting probability and true age or "ring–years", but these in no case led to a decrease in AICc (results not shown): * These models were fitted using the cohort separation approach (see text). Tabla 1. Selección de modelos para determinar la supervivencia a una edad específica y probabilidades de reobservación de gaviota tridáctila de edad conocida. Para la probabilidad de reobservación, todos los modelos incluían la dependencia de trampas, así como un término de interacción con la edad real o "años–desde–el–anillamiento". También ajustamos modelos con relaciones cuadráticas entre la probabilidad de reavistaje y la edad verdadera o "años–desde–el–anillamiento", pero éstos en ningún caso se tradujeron en un descenso en AICc (los resultados no se indican): * Estos modelos fueron ajustados usando el enfoque de separación de cohortes (ver el texto).

Deviance

AICc

True age, quadratic

"Ring–years"

474.67

True age, linear

"Ring–years"

479.02

2.28

True age

479.29

2.54

True age, quadratic

True age

477.85

3.19

"Ring–years", quadratic

True age

478.71

4.04

True age, quadratic

Constant

482.89

4.09

Constant

True age

483.07

4.27

True age, linear

True age

481.20

4.46

True age, linear

Constant

486.09

5.23

"Ring–years"

484.54

5.74

"Ring–years", linear

Constant

488.54

7.68

"Ring–years", linear

"Ring–years"

484.54

7.80

"Ring–years", quadratic

Constant

488.04

9.23

"Ring–years", quadratic

"Ring–years"

484.22

9.56

Survival

"Ring–years", linear

Constant

Resighting

Constant

499.78

16.88

"Ring–years", factor

448.08

19.70

True age, factor*

"Ring–years"

459.33

26.31

True age, factor*

"Ring–years", factor

430.28

56.67

True age, quadratic*

Discussion We have shown clear evidence of declining survival probability with increasing age at the population level in our study colony (fig. 2). Our results support previous studies that have found senescence in survival of Black–legged Kittiwakes (Coulson & Wooller, 1976; Aebischer & Coulson, 1990; Cam & Monnat, 2000; Cam et al., 2002b). That three long– term studies at different colonies, which have been thoroughly analysed with this question in mind, have found senescence in survival indicates that it is the typical pattern in this species, as also found for birds in general in a recent review (Bennett & Owens, 2002). The most convincing example is that of Cam et al. (2002b), who showed that within–generation

selection partly masked the effect of senescence in their study population, and that population measures of senescence (equivalent to those presented here) thus underestimated the decline in individual survival probability. Population–level estimates of age–specific survival are, however, the appropriate input for population modelling, and elsewhere we use the estimates derived here in a model of the Isle of May kittiwake population (Frederiksen et al., in press). There are at least three possible interpretations of the observed decline in resighting probability with "ring–years" (fig. 3). First, this decline may reflect reproductive senescence in the form of a declining probability of breeding among old birds, as also found by Cam et al. (2002b). However, if this was

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1.0

Survival

0.8 0.6 0.4 0.2 0.0 5

10 Age

Fig. 2. Estimated adult survival probability of Black–legged Kittiwakes as a function of true age. Lines show estimates and 95% confidence limits from the preferred model with a quadratic relationship, while symbols show unconstrained estimates from a model with true age as a factor. Fig. 2. Probabilidad estimada de la supervivencia de gaviotas tridáctilas adultas como una función de la edad real. Las líneas indican estimaciones y límites de confianza del 95% a partir del modelo preferido con una relación cuadrática, mientras que los símbolos indican estimaciones sin restricciones a partir de un modelo que utiliza la edad real como factor.

Resighting probability

1.0 0.8 0.6 0.4 0.2 0.0 2

4 6 8 10 12 14 Years since colour ringing

Fig. 3. Estimated resighting probability of Black–legged Kittiwakes as a function of the number of years since colour ringing ("ring–years"). Lines show estimates and 95% confidence limits from the preferred model with linear relationships, while symbols show unconstrained estimates from a model with "ring–years" as a factor. Upper set of lines and solid symbols: birds seen the previous year, lower set of lines and open symbols: birds not seen the previous year. Fig. 3. Probabilidad estimada de reobservación de la gaviota tridáctila como una función del número de años desde el anillamiento. Las líneas indican estimaciones y límites de confianza del 95% a partir del modelo preferido con relaciones lineales, mientras que los símbolos indican estimaciones sin restricciones a partir de un modelo que utiliza los "años–desde–el–anillamiento" como factor. Conjunto superior de líneas y símbolos marcados: aves observadas el año anterior; conjunto inferior de líneas y símbolos abiertos: aves no observadas el año anterior.

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the case we would expect a closer relationship with true age than with "ring–years", i.e. opposite to what we found (table 1). Second, gradual loss of colour rings as they become more worn could lead to a similar pattern, as birds that had lost one of three colour rings might still be identifiable, although with a lower probability. Birds with only two colour rings remaining have only very rarely been recorded, however. Third, the probability that a bird had left the study plots to breed elsewhere on the island would be likely to increase with time since colour ringing, and because birds breeding outside the study plots were less likely to be detected, this could have led to the observed pattern of declining resighting probability with "ring– years". Simulations showed that if individuals can move to a state with a much lower resighting probability, and if this process is not explicitly modelled, strong trap–happiness is apparent and estimated resighting probability declines with years since first observation (M. Frederiksen, unpubl. data). Both a previous study (Danchin & Monnat, 1992) and our own data document that kittiwakes do show breeding dispersal, and on balance we believe that this is most likely explanation for the observed pattern. Our approach provides a simple way of modelling age–specificity in survival or other parameters in a mark–recapture context. In order to run equivalent models without the "add" and "product" commands, we had to split our data set into (in this case) 15 birth cohorts, some of which contained only one individual, and subsequently construct complex cross–cohort constraints in the design matrix. Our approach thus makes the modelling of age–specific parameters in even sparse data sets simpler and more versatile, and we believe that this powerful approach could lead to improved estimates of age–specific survival in many wild populations. Recently, it has been proposed to test the fit of age–specific changes in survival probability to biologically relevant models such as the Gompertz and Weibull functions rather than simple polynomials (Gaillard et al., 2004). At the moment, fitting these models to data such as ours can only be done by cohort separation, but if future MARK versions allow a range of mathematical functions to be used in the design matrix, these models could also be fitted using our approach. Acknowledgements We thank all the people involved in ringing and resighting kittiwakes at the Isle of May for their efforts over many years, Scottish Natural Heritage for access to the island and the Joint Nature Conservation Committee for funding under their Seabird Monitoring Programme. Charles Francis, Emmanuelle Cam and an anonymous referee provided helpful suggestions to an earlier version of this manuscript.

References Aebischer, N. J. & Coulson, J. C., 1990. Survival of the kittiwake in relation to sex, year, breeding experience and position in the colony. Journal of Animal Ecology, 59: 1063–1071. Bennett, P. M. & Owens, I. P. F., 2002. Evolutionary ecology of birds: life histories, mating systems and extinction. Oxford University Press, Oxford. Bradley, J. S., Wooller, R. D., Skira, I. J. & Serventy, D. L., 1989. Age–dependent survival of breeding short–tailed shearwaters Puffinus tenuirostris. Journal of Animal Ecology, 58: 175–188. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference. A practical information– theoretic approach. Springer, New York. Cam, E., Cadiou, B., Hines, J. E. & Monnat, J.–Y., 2002a. Influence of behavioural tactics on recruitment and reproductive trajectory in the kittiwake. Journal of Applied Statistics, 29: 163–185. Cam, E., Link, W. A., Cooch, E. G., Monnat, J.–Y. & Danchin, E., 2002b. Individual covariation in life–history traits: seeing the trees despite the forest. American Naturalist, 159: 96–105. Cam, E. & Monnat, J.–Y., 2000. Stratification based on reproductive state reveals contrasting patterns of age–related variation in demographic parameters in the kittiwake. Oikos, 90: 560–574. Caswell, H., 2001. Matrix population models. Construction, analysis, and interpretation. Sinauer, Sunderland, MA. Choquet, R. & Pradel, R., 2002. U–CARE (Utilities – Capture–Recapture) user’s guide. Version 1.4, August 2002, pp. 1–24. Montpellier: CEFE/CNRS. Coulson, J. C. & Wooller, R. D., 1976. Differential survival rates among breeding kittiwake gulls Rissa tridactyla (L.). Journal of Animal Ecology, 45: 205–213. Danchin, E. & Monnat, J.–Y., 1992. Population dynamics modelling of two neighbouring Kittiwake Rissa tridactyla colonies. Ardea, 80: 171– 180. Dunnet, G. M. & Ollason, J. C., 1978. The estimation of survival rate in the fulmar, Fulmarus glacialis. Journal of Animal Ecology, 47: 507–520. Frederiksen, M. & Bregnballe, T., 2000a. Diagnosing a decline in return rate of one–year–old cormorants: mortality, emigration or delayed return? Journal of Animal Ecology, 69: 753–761. – 2000b. Evidence for density–dependent survival of adult cormorants from a combined analysis of recoveries and resightings. Journal of Animal Ecology, 69: 737–752. Frederiksen, M. & Petersen, A., 1999. Adult survival of the Black Guillemot in Iceland. Condor, 101: 589–597. Frederiksen, M., Wanless, S. Harris, M. P., Rothery, P. & Wilson, L. J., in press. The role of industrial fishery and oceanographic change in the decline of North Sea black–legged kittiwakes. Journal of Applied Ecology. Furness, R. W. & Monaghan, P., 1987. Seabird ecology. Blackie, Glasgow.

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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: Hamer, K. C., Schreiber, E. A. & Burger, J., 2002. Breeding biology, life histories, and life history– environment interactions in seabirds. In: Biology of marine birds: 217–261 (E. A. Schreiber & J. Burger, Eds.) CRC Press, Boca Raton, Florida. Newton, I. & Rothery, P., 1997. Senescence and reproductive value in Sparrowhawks. Ecology, 78: 1000–1008. Nichols, J. D., Hines, J. E. & Blums, P., 1997. Test for senescent decline in annual survival probabilities of Common Pochards, Aythya ferina. Ecology, 78: 1009–1018. Nisbet, I. C. T., 2001. Detecting and measuring

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senescence in wild birds: experience with long– lived seabirds. Experimental Gerontology, 36: 833–843. Porter, J. M. & Coulson, J. C., 1987. Long–term changes in recruitment to the breeding group, and the quality of recruits at a kittiwake Rissa tridactyla colony. Journal of Animal Ecology, 56: 675–689. Stearns, S. C., 1992. The evolution of life histories. Oxford University Press, Oxford. Tavecchia, G., Pradel, R., Boy, V., Johnson, A. R. & Cézilly, F., 2001. Sex– and age–related variation in survival and cost of first reproduction in Greater Flamingos. Ecology, 82: 165–174. White, G. C. & Burnham, K. P., 1999. Program MARK: survival estimation from populations of marked animals. Bird Study, 46 (suppl.): 120–139.

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Appendix. The data set of 69 known–age Black–legged Kittiwakes colour–ringed at the Isle of May and included in the analysis. For each bird, its breeding adult capture history 1986–2002 is shown, followed by its age at colour ringing and the year in which it was ringed as a chick. Apéndice. Relación de datos de las 69 gaviotas tridáctilas, de edad conocida con el anillamiento, de la isla de May incluidas en el análisis. Para cada ave, se presenta la historia de captura de adulto reproductor 1986–2002, seguido de su edad en el momento del anillamiento y el año en el cual fue anillada como pollo.

Breeding capture history

Age

Year

Breeding capture history

Age

Year

10000000000000000

1981

00000111101111111

1986

11111111111110000

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00000111111100000

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11100000000000000

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10111111111100000

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00000001001010000

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11111000000011000

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11010000000000000

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11111110000000000

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00000000111001100

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11100000000000000

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11110000000000000

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00000000010000000

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11000100000000000

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00000000011000000

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00000000001100000

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01110000000000000

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01111110000000000

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1981

00101110000000000

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00000000001000000

00111111001000000

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00000000001111111

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00111111111110000

1981

00000000000110000

1986 1986

00111111111000000

1984

00000000000111111

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1984

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1986

00011000000000000

1986

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1985

00011110111100000

1981

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1986

00001100000000000

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Exploring mark–resighting–recovery models to study savannah tree demographics G. Lahoreau, J. Gignoux & R. Julliard

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. Abstract Exploring mark–resighting–recovery models to study savannah tree demographics.— Despite their sessile nature, juvenile trees in savannah ecosystems are not always easy to encounter. Here, we evaluate the applicability to plants of the remedy of choice in animal studies: capture–recapture modelling. The plant equivalents, tagging and resighting, were caried outnin 7 censuses, involving 4,145 juvenile trees of 8 dominant savannah species. Using models with joint analysis of live and dead encounters, the resighting probabilities averaged 0.88 ± 0.15 and 0.92 ± 0.10 for seedlings and resprouts respectively; while dead recovery probabilities averaged 0.71 ± 0.25 for all age–classes. An ad hoc method that did not take into account encounter probabilities yielded biased survival estimates compared with estimates obtained using the mark–resighting–recovery approaches. This bias was observed even at high encounter probabilities, and we recommend therefore capture–recapture models where plant encounter is less than one. Finally, survival probabilities estimated by models based only on live or on dead data might both differ and be less accurate than estimates based on combined data. This highlights the advantages of models with joint analysis of live and dead encounters even the value of site fidelity is one. Key words: Plant, Encounter, Survival, Capture–recapture model, Recovery data, Savannah. Resumen Exploración de los modelos de marcaje–reavistaje–recuperación para estudiar la demografía de los árboles de sabana.— Pese a su naturaleza sésil, los árboles jóvenes no siempre resultan fáciles de hallar en los ecosistemas de sabana. En el presente trabajo se evalua la modelización de captura–recaptura, tan utilizada en estudios de animales, para su aplicación en los estudios de plantas. El equivalente al marcaje y reavistamiento para las plantas, se llevó a cabo a lo largo de siete censos, con un total de 4.145 árboles juveniles de ocho especies de la sabana dominantes. Mediante el empleo de análisis conjuntos de hallazgos de individuos vivos y muertos, las probabilidades de reaviastamiento medias fueron de 0,88 ± 0,15 y 0,92 ± 0,10 para las plántulas y los rebrotes, respectivamente, mientras que el promedio correspondiente a las probabilidades de recuperación de individuos muertos fue de 0,71 ± 0,25 para todas las clases de edad. Un método especialmente disenyado que no tenía en cuenta las probabilidades de recaptura dio unas estimaciones de supervivencia sesgadas, en comparación con las estimas obtenidas utilizando las aproximaciones basadas en el marcaje–reavistamiento– recaptura. Este sesgo aparecía incluso con altas probabilidades de recaptura, por lo que recomendamos los modelos de captura–recaptura en los que la probabilidad de encontarr la planta es menor de uno. Por ultimo, las probabilidades de supervivencia estimadas mediante los modelos basados únicamente en datos de plantas vivas o muertas pueden tanto diferir como ser menos precisas que las estimas basadas en datos combinados. Esto realza las ventajas de los modelos que emplean análisis conjuntos de hallazgos de plantas vivas y muertas, aun cuando el valor de la fidelidad al emplazamiento sea uno. Palabras clave: Planta, Hallazgo, Supervivencia, Modelo de captura–recaptura, Datos de recuperación, Sabana. G. Lahoreau & J. Gignoux, Lab. "Fonctionnement et Evolution des Systèmes Ecologiques", Ecole Normale Supérieure, 46 rue d’Ulm, 75230 Paris Cedex 05, France.– R. Julliard, Lab. "Biologie de la Conservation", Muséum National d’Histoire Naturelle, CRBPO, 55 rue Buffon, 75005 Paris, France. ISSN: 1578–665X

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Introduction Capture–recapture (CR) models were initially developed to estimate abundance, and later demographic parameters of animal populations, by overcoming the problem that not all animals in a sampling unit can be followed from one census to another (e.g. Bailey, 1952; Brownie et al., 1985; Pollock et al., 1990; Lebreton et al., 1992; Williams et al., 2001). Although plants are sessile organisms, they might also suffer similar problems of detectability, and CR models have consequently been applied recently to plant populations. Small or non–flowering plants might be difficult to detect for example, living in grass or other ecosystems where they are surrounded by dense vegetation. Kery & Gregg (2003) have shown through a double–observer survey and a closed population CR model that detectability of the vegetative stages of the orchid Cleistes bifaria in meadow was just 0.82. Another problem is that plants that have periods of dormancy where they produce no aboveground structures might go undetected. The Mead’s milkweed, Aclepias meaddi, presents both these problems of detection. Its detectability has been estimated to be as low as 0.25 in a four– year study carried out in a U.S. prairie (Alexander et al., 1997). The use of a closed population CR model avoided underestimating the population size of this rare plant (Alexander et al., 1997). Slade et al. (2003) extended this study using a larger data set and an open population CR model, taht take into account the possibility that some individuals might have died during the study period. Where all non–dormant plants can be detected, CR models appear very promising to study dormancy because they yielded bias–free estimates of demographic parameters, including dormancy periods (Shefferson et al., 2001; Kery & Gregg, in press). In this case, the probability of dormancy (d) has been defined as the complement of the recapture probability (p): d = 1 – p. CR models have been used to demonstrate that dormancy probabilities are strongly correlated to weather covariates such as number of frost days (Shefferson et al., 2001) and precipitation during the previous spring (Kery & Gregg., in press). In the present study of a humid African savannah in Côte d’Ivoire, similar detection problems were experienced, with it being impossible to track every individual juvenile tree. Savannah are defined as ecosystems where trees and grass coexist in the same area. In humid savannahs, fire is considered to be the main factor that limit abundance of trees, mainly because it burns juvenile trees growing within the flammable grass layer (e.g. Brookman– Amissah et al., 1980; San Jose & Farinas, 1983; Scholes & Archer, 1997; Gignoux et al., 1997). Few studies, however, have estimated juvenile survival probabilities in the field and in particular those of seedlings (Hochberg et al., 1994; Hoffman & Solbrig, 2003; House et al., 2003).

To estimate survival probabilities of juvenilen savannah trees, censuses were conducted in permanent plots every six months from June 1991 to July 1994. Despite intensive searches, not all juvenile trees were detected on each census, as they were often hidden by tall dense grass which exceeded 2 m in height from August to December (fig. 1). Moreover, because savannah trees are deciduous, it was not always possible to determine if they were alive or dead during censuses in Decembe, which is at the end of the humid season. To avoid bias in survival estimates (Kery & Gregg, 2003), a capture–recapture approach is therefore appropriate. As the plant censuses provided information on both live and dead encounters, combined mark–resighting–recovery (MRR) models can be used. The theory for MRR models was initially developed by Burnham (1993) and then extended by Catchpole et al. (1998) to incorporate age effects. As MRR models allow estimation of survival probabilities with greater precision and little bias from emigration, they are increasingly used to study populations of birds (Szymczak & Rexstad, 1991; Catchpole et al., 1998; Frederiksen & Bregnballe, 2000; Blums et al., 2002; Francis & Saurola, 2002), mammals (Catchpole et al., 2000), turtles (Bjorndal et al., 2003; Seminoff et al., 2003) and also gastropods (Catchpole et al., 2001). In this paper, (i) resighting, recovery and survival probabilities for juvenile trees of eight savannah species are estimated using MRR models; (ii) survival probabilities estimated in this way are then compared with those obtained using an ad hoc method which does not take encounter probabilities into account; and (iii) survival probabilities estimated using MRR models, and their precision, are also compared with those estimated by methods using data only from live resightings or on dead recoveries. Materials and methods Tree censuses in a humid savannah The study was conducted in Guinean savannah at the Lamto research station in Côte d’Ivoire (06° 13’ N, 05° 02’ W). These savannahs are regulated by fires which occur annually during the January dry season (Abbadie et al., in press). High rainfall, 1200 mm annually on average, means that the standing crop of grass can reach 1000 g·m–2 at the end of the growing season. But grass production is strongly affected by the presence of trees, hewever, and be almost zero under clumps of trees (Abbadie et al., in press). From 1991 to 1994, seven plant censuses were conducted six months apart in four 50 × 50 m plots at the height of the wet season (June 1991, June 1992, July 1993 and July 1994) and at the beginning of the dry season (December 1991, November 1992 and December 1993). This allowed estimates of survival probabilities both during fire and grow-

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Animal Biodiversity and Conservation 27.1 (2004)

ing seasons. To simplify analyses, all censuses were considered to have been done at exactly 6– month intervals, in June and in December. During the first census, the position of more than 5,000 trees, including tiny seedlings, was marked with a fire–resistant iron flag attached to a 50 cm stake (fig. 1). Subsequent censuses counted not only plants that had been tagged already, but also new individuals (around 1,000 per year). Trampling of grass tufts was avoided by walking between them, as grass affects the growth of juvenile trees through competition for light, water and nutrient (Scholes & Archer, 1997). At each census, data were recorded on species, height and demographic state —seedlings (recently germinated), resprouts (more than one year old but with no perennial stem), and adult trees (with a perennial trunk)— age of stems (annual/perennial) was easily determined by the presence/absence of fire scars. Dead plants–flag —but no plant in the vicinity— were also tracked. If the viability of an individual could not be determined, because of the absence of leaves on the stem, it was recorded as being leafless state and for the purpose of analyses was considered as in an "un–encountered" state, regardless of its state in the following census. Four thousands one hundred forty five individuals were studied from eight dominant Svannah tree species: Annona senegalensis (Annonacea, 917 individuals), Bridelia ferruginea (Euphorbiaceae, 702), Crossopteryx febrifuga (Rubiaceae, 234), Cussonia arborea (Araliaecae, 747), Piliostigma thonningii (Cesalpiniaceae, 317), Psorospermum febrifugum (Clusiaceae, 303), Pterocarpus erinaceus (Fabaceae, 354), Terminalia shimperiana (Combretaceae, 571). From censuses to encounter histories Encounter histories were generated from individual census data (table 1). As resprout survival is known to be size–dependent (Gignoux et al., 1997; Hoffman & Solbrig, 2003), all resprouts were classified into one of 3 age–classes: one–year old (R1), two–year old (R2) and three–year or older (R3) resprouts. Resprouts found in years subsequent to 1991, which had not previously been recorded as seedlings (S), were assumed to be R1. Resprouts found during the first census were assumed to be R3, given that their exact age was not known and that R1 and R2 should be a minority of this population. The proportion of the different resprout age–classes in July 1994 gives an estimate error: between 3% and 39% of resprouts were younger than three– years old. If a resprout became an adult, it was nonetheless still considered as resprout as integrating an adult tree state would have required a multi–state CR model. Individuals not recorded in one census, but found dead in a subsequent one (i.e., flag detected but no plant in the vicinity) were labelled as delayed recoveries (Catchpole et al., 2001): the timing of death

Fig. 1. Resprout of Bridelia ferruginea marked with a tagged steel stake in the savannah at the Lamto research station, Côte d’Ivoire. Fig. 1. Rebrote de Bridelia ferruginea identificado mediante una estaca de acero marcada en la sabana en la estación de investigación de Lamto, Costa de Marfil.

could not be determined accurately. In this case, the dead recovery was omitted (see the last case in table 1) to prevent the bias that would be caused if such individuals were considered to have died during the last observation period —this concerned 7.7% of all plants. For each tree species, the seven censuses allowed estimation of survival probabilities for six time periods for seedlings and for R3, four for R1 and only two for R2. Mark–resighting–recovery (MRR) models The parameters of the MRR model are: &, the apparent survival probability, S, the true survival probability, p, the resighting probability, r, the recovery probability and F, the fidelity probability. True and apparent survival & are linked by the fidelity parameter as & = S × F. Originally, site fidelity was defined as that probability an individual would remain in the study area (Burnham, 1993), on

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Table 1. Examples of encounter history provided by the seven censuses made from June 1991 to July 1994 with: S. Seedling; R1. One–year old resprout; R2.Two–year old resprout; R3. Three–year or older resprout; T. Adult tree; D. Dead individual; 0. Un–encountered individual. Tabla 1. Ejemplos de la relación de hallazgos proporcionados por los siete censos realizados entre junio de 1991 y julio de 1994 con: S. Plántula; R1 . Rebrote de un año de edad; R2. Rebrote de dos años de edad; R3. Rebrote de tres años de edad o más; T. Árbol adulto; D. Individuo muerto; 0. Individuo no encontrado.

Encounter history

Observation

SSD0000

Seedling found in June 1991 and in December 1991, found dead in June 1992

000SR100 Seedling found in December 1992, hence not detected at that stage in June 1992. Found again in June 1993 but never encountered after. 00R1R1R20R3

Resprout found the first time in June 1992, expected to having been missed on seedling state in 1991 and so to be 1–year old. Seen every following year excepted in December 1993

R3R3R3R3R3R3R3

Resprout found in June 1991, assumed to be 3–year old or older resprout. Found alive during all subsequent censuses.

R3R3R3R3TTT Resprout which became an adult tree in June 1993, considered to stay in t R3R3R3R3R3R3R3 resprout state S0D0000 t S000000

Seedling found in June 1991, not found in December 1991 but found dead in June 1993. In such case, we omitted the delayed recovery.

the assumption that resightings take place locally, while recoveries can take place anywhere. On this basis, F would equal one for plants. Recently, Frederiksen (unpublished work) proposed that F might be more appropriately considered as a correction factor which allows unbiased estimation of survival when p or r are highly heterogeneous among individuals. To test this, we allowed F to vary with time; F was invariably estimated at one, suggesting little parameter heterogeneity. As a result, we fixed the value of F as one. In animal studies, dead individuals are usually found by the general public, often hunters and members of the recovery parameter r has therefore been defined as incorporating both the probabilities of finding a dead individual and of reporting its band (White & Burnham, 1999; Otis & White, 2002). In plant censuses, the demographers record all individuals found dead so that the probability of tag reporting equalles one, and r represents the probability of finding a dead plant. As we were interested in ascertaining survival probabilities for each species, we developed a general model for each incorporating all possible effects: (1) time and age–dependence for survival probabilities; (2) time and age–dependence for resighting probabilities (which might be dependent on individual size, itself related to age); and (3) time–dependence for recovery probabilities (age dependence was considered irrelevant, as the probability of finding only a flag should be identical for all dead individuals).

The general model was therefore: S(a4–t/t/t/t)p(a4–t/t/t/t)r(t)F

(1)

where a4 means that 4 age–classes were considered —namely S, R1, R2, and R3— t means time– dependence and therefore a4–t/t/t/t means time– dependence for each of the 4 age–classes (Cooch & White, 1999). We created the general models with the "Burnham model" using the MARK software programme (White & Burnham, 1999), which expands upon the MRR theory of Burnham (1993) to allow age–dependent parameters. Age–classes were coded into two groups: the first composed of individuals marked in a seedling, and the second of those marked as resprouts with the assumption that resprouts marked in 1991 started as R3 and those in 1992 and 1993 as R1. For each species, we investigated the fit of the global model to the data using the MARK bootstrap procedure, with simulation of 100 data sets and calculation their deviances (White & Burnham, 1999). If the deviance of the global model did not fall within that of the 100 simulated deviances, the global model was not a good fit (White & Burnham, 1999). The global model did not fit the data (P < 0.01) in any tree species except Terminalia shimperiana, and the significance of the fit on the latter was itself bordeline (P = 0.06). This indicated either overdispersion of the data or failure of the model to account for the data

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structure (Lebreton et al., 1992; Burnham & Anderson, 1998). In animal count data, overdispersion is very common and occurs when some assumptions of CR models, such as independence or homogeneity of parameters are violated in some groups of individuals (Lebreton et al., 1992; Burnham & Anderson, 1998). In plants, overdispersion of the data is likely given that their frequently clumped distribution could result in non–independent capture probabilities (Kery & Gregg, 2003). Assumpting the lack of fit was due to overdispersion, we therefore calculated a variance inflation factor ( ) as the ratio of our model deviance to the mean of the 100 simulated deviances with the MARK bootstrap procedure (White & Burnham, 1999). The variance inflation factors for the eight species range from 1.34 to 2.98 (table 2). We then used the corrected Quasi–Akaike Information Criterion, QAICc, which incorporates corrections for small sample and overdispersion to select among models (Burnham & Anderson, 1998). The best approximating model was that with the lowest QAICc value, but models with QAICc differences < 2 should also be considered as possible candidate models. In such cases, we chose the model with the highest number of parameters, to incorporate all potential biological effects. Several reduced–parameter models were considered. First, the age–dependence of the resighting probabilities was reduced by clumping age–classes. Models were constructed with only three age– classes, by clumping two resprout age–classes together (R1 and R2 or R2 and R3), with only two age–classes (seedlings and resprouts, with all resprout age–classes grouped), and finally with no age–effect. Second, models were developed allowing p and r to be dependent on season without year variation, or constant through years and seasons. It was assumed that p might be higher in June than in December, when leaf fall occurs. But as previous studies have shown that the time and duration of leaf fall vary considerably from year to year (Hopkins, 1970; Menaut & Cesar, 1979), models were also tested where p was constant in June but year– dependent in December.

Table 2. Variance inflation factor ( ) of the global model calculated with the bootstrap procedure of the program MARK for each of the eight savannah tree species. Tabla 2. Factor de inflación de la varianza ( ) del modelo global calculado mediante el procedimiento bootstrap del programa MARK para cada una de las ocho especies de árboles de sabana.

Tree species Annona senegalensis

2.02

Bridelia ferruginea

1.90

Crossopteryx febrifuga

2.22

Cussonia arborea

2.98

Piliostigma thonningii

1.87

Psorospermum febrifugum

1.42

Pterocarpus erinaceus

1.63

Terminalia shimperiana

1.34

The survival rate was then simply calculated as the fraction of survivors as is usually done in plant studies (e.g. Garnier & Dajoz, 2001; Hoffman & Solbrig, 2003). The underlying assumption of the DS estimation was therefore that the proportion of alive/dead individuals was the same for encountered and un– encountered plants and that resighting and recovery probabilities were threrefore identical. This assumption was hypothesized to be false, as live plants ware easier to find than dead ones (field observation). Bias was estimated as the difference between the survival rates estimated by the DS (SDS) and MRR (SMRR) methods: Bias = SDS – SMRR

Direct survival (DS) estimation An ad hoc analysis was used, named direct survival (DS) estimation, which assumed that the level of encounter equalled one (i.e. all live and dead individuals were found). Only data composed of individuals alive at time i–l and found dead or alive at time i was used to calculate survival rate at time i. This therefore required no assumptions about internal or external zeros. In such analyse, the encounter history "S0R1R1000" was not used to estimate the seedling survival rate between time 1 and 2 or between time 4 and 5, but was used to estimate the R1 survival rate between time 3 and 4. As in the MRR models, information from delayed recoveries was not used.

Bias was tested to determine if it was negatively correlated to the percentage of individuals encountered, defined as the proportion of individuals alive at time i–l and found dead or alive at time i. All statistics were done using SAS software (SAS Institute, 1990). Comparison with alive or dead models Following comments of Burnham (1993) and studies of Catchpole et al. (1998) and Francis & Saurola (2002), results from MRR were compared to those from models based only on live resightings (e.g. Lebreton et al., 1992) or on

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Table 3. Selected dependence for resighting (p) and recovery (r) probabilities for the eight savannah tree species with initially four age–classes considered: S. Seedlings; R1. One–year old resprouts; R2. Two–year old resprouts; R3. and three years or older resprouts; a3(S,R1R2,R3. The selected parameter were 3 age–classes dependent with R1 and R2 clumped; a3(S,R1,R2R3). The selected parameter were 3 age–classes dependent with R2 and R3 clumped; a2(S,R1R2R3). The selected parameter were 2 age–classes (seedlings vs resprouts); t. Probabilities were time–dependent; c. Constant over time; season. Probabilities were different in June and in December; and Dec(t)June(c). Parameters were time–dependent in December but constant in June. Tabla 3. Dependencia seleccionada con respecto a las probabilidades de reavistaje (p) y recuperación (r) para las ocho especies de árboles de sabana, inicialmente con cuatro clases de edad consideradas; concretamente: S. Plántulas; R1. Rebrotes de un año de edad; R2. Rrebrotes de dos años de edad ; R3. Rebrotes de tres años de edad o más; a3(S,R1R2,R)3. El parámetro seleccionado comprendía tres clases de edad dependientes, con R1 y R2 agrupados; a3(S,R1,R2R3). El parámetro seleccionado comprendía tres clases de edad dependientes, con R2 y R3 agrupados; a2(S,R1R2R3). El parámetro seleccionado comprendía dos clases de edad (plántulas frente a rebrotes): t. Las probabilidades dependían del tiempo: c. Constante a lo largo del tiempo; season. Las probabilidades diferían en junio y diciembre; y Dec(t)June(c). Los parámetros dependían del tiempo en diciembre, pero se mantenían constantes en junio.

Tree species

Annona senegalensis

a3(S, R1,R2R3)–t/season/t

Bridelia ferruginea

Crossopteryx febrifuga

season

Cussonia arborea

a2(S,R1R2R3)–t/Dec(t)June(c)

Piliostigma thonningii

a3(S,R1,R2R3)–season/season/season

Psorospermum febrifugum

a3(S,R1,R2R3)–season/t/t

Pterocarpus erinaceus

a3(S,R1R2,R3)–t/t/t

Terminalia shimperiana

a2(S,R1R2R3)–Dec(t)June(c)/t

dead recoveries (e.g. Brownie et al., 1985). Unlike animal studies (Francis & Cooke, 1993), the resighting models allowed estimation of survival probabilities free from emigration biass, as plants do not move (F = 1). For each model and making the same assumptions as in the combined analysis the following general models were considered: Live encounters model: S(a4–t/t/t/t)p(a4–t/t/t/t) Dead recoveries model: S(a4–t/t/t/t)r(t) Using the bootstrap procedure (White & Burnham, 1999), the live sightings model appeared to fit the data for four tree species (0.04 < P < 0.17), and the dead recoveries models appeared to fit for six (0.03 < P < 0.64). Nevertheless, the variance inflation factors were quite similar to those calculated using MRR models (live models: 1.19 < < 2.58; dead models: 0.88 < < 2.50). As in the MRR approach, submodels were also defined and model selection was based on QAICc values.

season t

Results Resighting and recovery probabilities The most parsimonious MRR models indicated that resighting probabilities, p, were (table 3): (1) similar for all age–classes in Bridelia ferruginea and Crossopteryx febrifuga; (2) different between seedlings and resprouts but not between resprout age–classes in Cussonia arborea and Terminalia shimperiana; and (3) different for 3 age–classes with R 1 and R2 clumped (Pterocarpus erinaceus) or with R2 and R3 clumped (Annona senegalensis, Piliostigma thonningii and Psorospermum febrifugum). Depending on species and age–classes, p was either constant, dependent on season (with probabilities constant in December or not), or dependent on time (table 3). Average resighting probabilities across species were 0.88 ± 0.15 for seedlings, 0.92 ± 0.08 for R 1, 0.89 ± 0.11 for R2 and 0.94 ± 0.11 for R3. Recovery probabilities were time–dependent for all species, with the exception of Piliostigma

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Annona senegalensis

Bridelia ferruginea

Crossopteryx febrifuga

Cussonia arborea

Piliostigma thonningii

Psorospermum febrifugum

Pterocarpus erinaceus

Terminalia febrifuga

1 0.8 0.6 0.4 0.2 0

1 Resighting and recovery probabilities

0.8 0.6 0.4 0.2 0

1 0.8 0.6 0.4 0.2 0

1 0.8 0.6 0.4 0.2 0 J–91 D–91 J–92 D–92 J–93 D–93 J–94

J–91 Time

D–91 J–92 D–92

J–93 D–93

J–94

Fig. 2. Resighting (p) and recovery (r) probabilities for the seedlings estimated by the mark– resighting–recovery models for the eight savannah tree species (with standard error), as estimated by models listed in table 3: J. June; D. December. Fig. 2. Probabilidades de reavistaje (p) y recuperación (r) con respecto a las plántulas estimadas mediante los modelos de marcaje–reavistaje–recuperación para las ocho especies de árboles de sabana (con error estándar), según lo estimado por los modelos especificados en la tabla 3: J. Junio; D. Diciembre.

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SMRR

Seedlings

1–year old resprouts

0.8 0.6

0.8

0.4 0.2

0.4

0.6 0.2 0

0.2

0.4

0.6

0.8

0 0.4

2–year old resprouts

0.8

0.9

SMRR

0.6

0.8

0.4

0.7 0.8

0.8

3–year old resprouts

0.2 0.7

0.6

0.9 SDS

0.6 0.85

0.9

0.95 SDS

Fig. 3. Comparison of the survival probabilities obtained by direct survival estimation (SDS) (confidence limits not shown) and by the mark–resighting–recovery model (SMRR) (with 95% confidence intervals). All survival probabilities estimated per each age–classes for the eight savannah tree species are represented, i.e. 48 for seedlings and for 3–year or older resprouts, 32 for 1–year old resprouts and 16 for 2–year old resprouts. The dashed lines represent the line x = y. Fig. 3. Comparación de las probabilidades de supervivencia obtenidas mediante la estimación directa de supervivencia (SDS) (límites de confianza no indicados) y mediante el modelo de marcaje– reavistaje–recuperación (SMRR) (con intervalos de confianza del 95%). Se representan todas las probabilidades de supervivencia estimadas para cada clase de edad de las ocho especies de árboles de sabana; es decir, 48 para las plántulas y para los rebrotes de tres años de edad o más; 32 para los rebrotes de un año de edad y 16 para los rebrotes de dos años de edad. Las líneas discontinuas representan la línea x = y.

thonningii and Pterocarpus erinaceus (table 3). Values varied widely among species, but the average r was 0.71 ± 0.25. Temporal variation was generally greater than variation in resighting probabilities, with recovery probabilities ranging from 0.25 to 1 (fig. 2). In more than 77% of estimates, recovery probabilities were lower than resighting probabilities. Bias in survival probabilities Across species, based on MRR, the survival probabilities averaged 0.60 ± 0.23 for seedlings, 0.80 ± 0.16 for R1, 0.93 ± 0.07 for R2 and 0.98 ± 0.23 for R3 (fig. 3). To compare survival probabilities between the MRR model (SMRR) and the DS estimation (SDS), all the individual period–specific survival probabilities

were estimated for all tree species resulting in a total of 48 survival probabilities (6 time periods x 8 species) for seedlings, 32 for R1, 16 for R2 and 48 for R3 (fig. 3). Estimates from DS models were biased almost as much positively as negatively compared with the MRR approach; but higher bias estimates were mainly positive. The DS approach overestimated the survival probabilities up to 0.57 for seedlings, up to 0.39 for R1, up to 0.07 for R2 and up to 0.04 for R3 (fig. 4). The magnitude of bias in absolute values was negatively correlated with the percentage of individuals encountered in seedlings (P < 0.001; R2 = 42%) and in R1 (P = 0.008; R2 = 32%), but in R2 (P = 0.84) and in R3 (P = 0.97). The apparent bias even where the percentage of encounters was 100%, resulted from the way the percentage of encounters and SDS were calculated.

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Seedlings

Relative biass

1.0

0.4

0.5

0.2 0.0

0.0

–0.5

–0.2 –0.4

–1.0 0

100

2–year old resprouts

0.1 Relative biass

1–year old resprouts

0.6

100

3–year old or older resprouts 0.04

0.05 0.02 0 0.00

–0.05

–0.02

–0.1 0

20 40 60 80 100 Percentage of individual encountered

Fig. 4. Relative bias in survival probabilities (Bias = SDS – SMRR) as a function of the percentage of individual encounters with SDS, the survival probabilities obtained by the direct survival estimation and SMRR by the mark–resighting–recovery model. Dashed lines represent the line x = 0. Fig. 4. Sesgo relativo en las probabilidades de supervivencia (Sesgo = SDS – SMRR), expresado como una función del porcentaje de hallazgos individuales con SDS, las probabilidades de supervivencia obtenidas mediante la estimación directa de supervivencia y SMRR mediante el modelo de marcaje– reavistaje–recuperación. Las líneas discontinuas representan la línea x = 0.

Indeed, encounter histories such as "R300R3R3R3R3" were not used to calculate the percentage of individual encounters and SDS between time 2 and 3 but were used to estimate SMRR. Alive, dead and combined models The resighting models did not always permit estimation of survival probabilities for the last period due to the selected model structure. The value and the precision of the survival probabilities using only resighting data were very similar to those obtained using the MRR models (fig. 5). For recovery–only models, all survival probabilities could be estimated, as time–dependence of r was never selected (results not shown). Many survival probabilities based only on recovery data differed from those obtained using MRR models. The precision of the survival estimates from recovery models was alson frequently much lower (i.e. higher standard error, fig. 5).

Discussion MRR models in the study of plant populations MRR models were applied here to plant populations and found to work well. Their use might thus increase in the future, as plant censuses often record both live and dead individuals without always detecting every individual on every visit. Variance inflation factors in the general model of the eight savannah tree species (1.34 < < 2.98) were quite similar to those of animal populations. Overdispersion is likely to occur in plants (Kery & Gregg, 2003), and might be even more frequent and substantial in savannahs due to the spatial heterogeneity of the ecosystem ranging from open grass areas to dense tree clumps. The high value of = 2.98 for Cussonia arborea reflect the interdependence of survival probabilities for its seedlings, which unlike those of the other species observed, aggregate under the mother tree crown.

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SMRR

A 1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

0.2

0.4

SAlive

0.6 SDead

0.8

D 0.4

0.6 0.5

0.3 " MRR

0 0

0.4

0.2

0.3 0.2

0.1 0

0.1

0.2 " Alive

0.3

0.4

0.1

0.2

0.3

0.4

0.5

0.6

"Dead

Fig. 5. 5A, 5B. Comparison between the survival probabilities estimated by the resighting–only models (SAlive) and the recovery–only models (SDead) to theses by the mark–resighting–recovery models (SMRR). 5C, 5D. Comparison between the standard errors estimated by the resighting–only models ("Alive) and the recovery–only models ("Dead) to theses by the mark–resighting–recovery models ("MRR). All survival estimates and their standard errors per time for all age–classes of the eight savannah tree species are represented. The dashed lines represent the line x = y. Fig. 5. 5A, 5B. Comparación entre las probabilidades de supervivencia estimadas mediante los modelos de sólo reavistaje (SAlive) y los modelos de sólo recuperación (SDead) y las tesis obtenidas mediante los modelos de marcaje–reavistaje–recuperación (SMRR). 5C, 5D. Comparación entre los errores estándar estimados mediante los modelos de sólo reavistaje ("Alive) y los modelos de sólo recuperación ("Dead) y las tesis obtenidas mediante los modelos de marcaje–reavistaje–recuperación ("MRR). Se representan todas las estimaciones de supervivencia y sus errores estándar por tiempo para todas las clases de edad de las ocho especies de árboles de sabana. Las líneas discontinuas representan la línea x = y.

For the eight savannah tree species, the average r = 0.71 ± 0.25 was lower than resighting probabilities in seedlings (p = 0.88 ± 0.15) and in resprouts (p = 0.92 ± 0.10) confirming the field impression that finding only a flag was more difficult than finding a flag and an associated plant. Bias in survival estimates Encounter probabilities less than 1.0 introduced bias into the survival probabilities as assumed by Kery & Gregg (2003), and could lead to misleading conclusions and predictions. Percentage of encounters was generally not a good predictor of the bias magnitude, suggesting that

CR models should be used even when encounter rates are high. Contrary to expectations, however bias was not consistently in the same direction. For three–year or older resprouts, survival probabilities were close to 0.9 and the biases did not exceed 0.04. Nevertheless, such small differences might have a major impact on estimates of long–term survival probabilities and estimated population dynamics, as resprouts might remain in that state for decades before becoming adults or simply dying. For example, the probability for a resprout to survive 10 years (20 seasons) equals 0.82 if S = 0.99 (0.99 20) but only 0.35 if S = 0.95 (0.91 20).

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Comparison between capture–recapture and recovery models As observed by Catchpole et al. (1998) and Francis & Saurola (2002), survival probabilities estimated by models based only on live or only on dead encounters sometimes different from those estimated using combined models, and were generally less precise. Estimates from models with live sightings or dead recoveries alone were potentially more biased than MRR models. This was particularly true in recovery models. In plant censuses, dead individuals are, however, rarely recorded without there being information about live individuals. Conclusion Mark–resighting–recovery models are well–suited to retrospective analyses of existing and valuable historical datasets on plant ecosystems. In future experiments in savannah ecology, improvements in methodology such as electronic tagging of plants may ameliorate resighting and recovery, although field studies will, for the foreseeable future, often by necessity be conducted under onerous logistical conditions. A combination of improved techniques, combined with imaginative modelling, will provide a means to advance the study of tree dynamics in savannah ecosystems and enhance understanding of the factors limiting their distribution and abundance. Acknowledgements We thank T. Boulinier, J–F. Le Gaillard, M. Massot and T. Tully for advice and G. White for help with MARK. This work would not have been possible without the field facilitation of the former and present directors of the Lamto research Station: R. Vuattoux and S. Konaté and the field assistance of N’Dri Konan, Prosper Savadogo and François Kouamé N’Guessan. We thank Charles Francis and two anonymous reviewers for critical comments on an earlier draft. References Abbadie, L., Gignoux, J., Le Roux, X., & Lepage, M. (in press). Lamto: Structure, functioning and dynamics of a savannah ecosystem. Springer– Verlag, New–York. Alexander, H. M., Slade, N. A. & Kettle W. D., 1997. Application of mark–recapture models to estimation of the population size of plants. Ecology, 78 (4): 1230–1237. Bailey, N. T. J., 1952. Improvements in the interpretation of recapture data. Journal of Animal Ecology, 21: 120–127. Bjorndal, K., Bolten, A. & Chaloupka, M., 2003. Survival probability estimates for immature

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green turtles Chelonia mydas in the Bahamas. Marine Ecology Progress Series, 252: 273–281. Blums, P., Nichols, J. D., Hines, J. E. and Mednis, A., 2002. Sources of variation in survival and breeding site fidelity in three species of European ducks. Journal of Animal Ecology, 71: 438–450. Brookman–Amissah, J., Hall, J. B., Swaine, M. D. and Attakorah, J. Y., 1980. A re–assessment of a fire protection experiment in north–eastern Ghana savanna. Journal of Applied Ecology, 17: 85–99. Brownie, C., Anderson, D. R., Burnham, K. P., and Robson, D. S., 1985. Statistical inference from band recovery data: a handbook. Second Edition. United States Fish and Wildlife Service Resource Publication, 156. Burnham, K. P, 1993. A theory for combinated 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.). Verlag. Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference. Model selection and inference: a practical information–theoretic approach. Springer–Verlag, New York. Catchpole, E. A., Morgan, B. J. T., Coulson, T. N., Freeman, S. N. & Albon, S. D., 2000. Factors influencing Soay sheep survival. Applied Statistics, 49(4): 453–472. Catchpole, E. A., Freeman, S. N., Morgan, B. J. T. & Nash, W. J., 2001. Abalone I: Analyzing mark– recapture–recovery data incorporating growth and delayed recovery. Biometrics, 57: 469–477. Catchpole, E. A., Freeman, S. N., Morgan, B. J. T., & Harris, M. P., 1998. Integrated recovery/recapture data analysis. Biometrics, 54: 33–46. Cooch, E. & White, G., 2002. Program MARK: A gentle introduction, 2nd edition. Francis, C. M. & Cooke, F. 1993. A comparison of survival rate estimates from live recaptures and dead recoveries of lesser snow geese. In: Marked individuals in the study of bird population: 169– 183 (J. D. Lebreton & P. M. North, Eds.). Verlag. Francis, C. M. & Saurola, P., 2002. Estimating age– specific survival rates of tawny owls –recapture versus recoveries. Journal of Applied Statistics, 29(1–4): 637–647. Frederiksen, M. & Bregnballe, T., 2000. Evidence for density–dependent survival in adult cormorants from a combined analysis of recoveries and resightings. Journal of Animal Ecology, 69: 737–752. Garnier, L. K. M. & Dajoz, I., 2001. The influence of fire on the demography of a dominant grass species of West African savannas, Hyparrhenia diplandra. Journal of Ecology, 89: 200–208. Gignoux, J., Clobert, J. & Menaut, J.–C., 1997. Alternative fire resistance strategies in savannah trees. Oecologia, 110: 576–583. Hochberg, M. E., Menaut, J.–C. & Gignoux, J., 1994. Influences of tree biology and fire in the spatial structure of the west African savannah.

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Journal of Ecology, 82: 217–226. Hoffman, W. A. & Solbrig, O. T., 2003. The role of topkill in the differential response of savannah woody species to fire. Forest Ecology and Management, 180: 273–286. Hopkins, B., 1970. Vegetation of the Olokemeji forest reserve, Nigeria: IV. The litter and soil with special reference to their seasonal changes. Journal of Ecology, 58(3): 687–703. House, J. I., Archer, S., Breshears, D. D., Scholes, R. J. & NCEAS Tree–Grass Interactions Participants, 2003. Conundrums in mixed woody–herbaceous plant systems. Journal of biogeography, 30: 1763–1777. Kery, M. & Gregg, K. B., 2003. Effects of life–state on detectability in a demographic study of the terrestrial orchid Cleistes bifaria. Journal of Ecology, 91: 265–273. – (in press). Demographic analysis of the terrestrial orchid Cypripredium reginae. Journal of Ecology. 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(1): 67–118. Menaut, J.–C. & Cesar, J., 1979. Structure and primary productivity of Lamto savannas, Ivory Coast. Ecology, 60(6): 1197–1210. Pollock, K. H., Nichols, J. D., Brownie, C. & Hines, J. E., 1990. Statistical inference of capture– recapture experiments. Wildlife Monographs, 107: 1–97. Otis, D. L. & White, G. C., 2002. Re–analysis of a band study to test the effects of an experimental increase in bag limits of mourning doves. Jour-

Lahoreau et al.

nal of Applied Statistics, 29 (1–4): 479–495. San Jose, J. J. & Farinas, M. R., 1983. Changes in tree density and species composition in a protected Trachypogon savanna, Venezuela. Ecology, 63(4): 447–453. SAS Institute, 1990. SAS/STAT user’s guide. SAS Institute, Cary. Scholes, R. J. & Archer, S. R., 1997. Tree–grass interactions in savannahs. Annual Review Ecological System, 28: 517–544. Seminoff, J. A., Jones, T. T., Resendiz, A., Nichols, W. J. & Chaloupka, M. Y., 2003. Monitoring green turtles (Chelonia mydas) at a coastal foraging area in Baja California, Mexico: multiple indices to describe population status. Journal of the Marine Biological Association of the United Kingdom, 83(6): 1355–1362. Shefferson, R. P., Sandercock, B. K., Proper, J. & Beissinger, S. R., 2001. Estimating dormancy and survival of a rare herbaceous perennial using mark–recapture models. Ecology, 82 (1): 145–156. Slade, N. A., Alexander, H. M. & Kettle, W. D., 2003. Estimation of population size and probabilities of survival and detection in Mead’s milkweed. Ecology, 84(3): 791–797. Szymczak, M. R. & Rexstad, E., 1991. Harvest distribution and survival of a gadwall population. Journal of Wildlife Management, 55(4): 592–600. 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. 1st edition. Academic Press.

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Methods for investigating parameter redundancy O. Gimenez, A. Viallefont, E. A. Catchpole, R. Choquet & B. J. T. Morgan

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. Abstract Methods for investigating parameter redundancy.— The quantitative study of marked individuals relies mainly on the use of meaningful biological models. Classical inference is then conducted based on the model likelihood, parameterized by parameters such as survival, recovery, transition and recapture probabilities. In classical statistics, we seek parameter estimates by maximising the likelihood. However, models are often overparameterized and, as a consequence, some parameters cannot be estimated separately. Identifying how many and which (functions of) parameters are estimable is thus crucial not only for proper model selection based upon likelihood ratio tests or information criteria but also for the interpretation of the estimates obtained. In this paper, we provide the reader with a description of the tools available to check for parameter redundancy. We aim to assist people in choosing the most appropriate method to solve their own specific problems. Key words: Mark–recapture data, Mark–recovery data, Profile–likelihood, Analytical–numerical method, Symbolic algebra software. Resumen Métodos para investigar la redundancia de parámetros.— El estudio cuantitativo de individuos marcados se basa fundamentalmente en el uso de modelos biológicamente significativos. Posteriormente, la inferencia clásica se lleva a cabo a partir de la probabilidad del modelo, parametrizada mediante parámetros tales como las probabilidades de supervivencia, de recuperación, de transición y de recaptura. En la estadística clásica, intentamos obtener estimaciones de parámetros maximizando la probabilidad. Sin embargo, los modelos a menudo se parametrizan en exceso, por lo que algunos parámetros no pueden estimarse por separado. Por consiguiente, identificar qué parámetros, cuántos y qué funciones de los mismos son estimables resulta crucial, no sólo para poder efectuar una adecuada selección de modelos basada en pruebas de razón de verosimilitud o criterios de información, sino también para la interpretación de las estimaciones obtenidas. En este trabajo presentamos una descripción de las herramientas disponibles para verificar la redundancia de parámetros. Nuestro objetivo es ayudar a elegir el método más apropiado para la resolución de sus problemas específicos. Palabras clave: Datos sobre recaptura de marcas, Datos sobre recuperación de marcas, Probabilidad del perfil, Método numérico analítico, Software de álgebra simbólica. Olivier Gimenez, CEFE/CNRS, Equipe Biométrie et Biologie des Populations, 1919 route de Mende, 34293 Montpellier Cedex 5, France; Inst. de l'Ingénierie de l'Information de Santé, Lab. TIMC UMR CNRS 5525, Équipe TIMB, Fac. Médecine, 38706 La Tronche Cedex, France.– Anne Viallefont, Equipe de Recherche en Ingénierie des Connaissances, Univ. de Lyon 2, 69676 Bron Cedex, France.– Edward. A. Catchpole, Univ. of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia.– Rémi Choquet, CEFE/CNRS, Equipe Biométrie et Biologie des Populations, 1919 route de Mende, 38706 La Tronche Cedex 5, France.– Byron J. T. Morgan, Inst. of Mathematics, Statistics and Actuarial Science, Univ. of Kent, Canterbury CT2 7NF, England. Corresponding author: O. Gimenez. E–mail: gimenez@cefe.cnrs-mop.fr ISSN: 1578–665X

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Introduction Capture–mark–recapture (CMR) models in the broad sense include all the models developed to estimate demographic parameters based on data from marked animals. The initial papers are those of Cormack (1964), Jolly (1965) and Seber (1965). Over the past two decades, many improvements of the methods have been provided, that have led to both a diversification and a generalization of the tools available. We can now describe briefly the situation in this field as follows: when the marked animals are recaptured (or resighted) alive, the models used are "mark– recapture" models stricto sensu. The parameters estimated in this case are survival and capture probabilities. For a thorough review of the developments of the method since the sixties, including constancy over time, group effects, constraints on the parameters, etc. see Lebreton et al. (1992). In other cases, for example game species, the animals are not seen again during their lifetime, but their time of death is known. Brownie et al. (1990) provide a clear review of many of the models that can be used, called "mark–recovery" models. The parameters estimated are survival probabilities, and "return probabilities", i.e. the probability that the mark of a dead animal is found. See also the development in Freeman & Morgan (1992). Integrated modelling of mark– recovery and mark–recapture data is considered by several authors, see e.g. Catchpole et al. (1998). Models have also been developed to estimate transition rates between sites or states jointly with the survival probabilities, in either of the two main situations above (Arnason, 1973; Schwarz et al., 1993; Brownie et al., 1993). All these models have a common structure and can be combined in the framework of multi–state models, as has been shown by Lebreton & Pradel (2002). We shall here outline the common structure of many CMR models. For details concerning the one– site capture–recapture models, see Lebreton et al. (1992). Let us call the set of all the different "encounter histories" that have been observed in a data set. Let q denote the number of encounter histories, i (i = 1,…, q) the i–th capture history, and ni the number of animals with this encounter history. The i constitute the q cells of a multinomial model with individual probabilities i, where the i is the probability of observing the encounter history i, conditionally on the time of marking and first release of the corresponding individuals. The i can be expressed as functions of the parameters to estimate survival and/or transition probabilities, capture and/or return probabilities, etc. The likelihood of a model L is proportional to the product of the i namely . Estimating the parameters of the model by the maximum likelihood method will thus consist in finding the values of the parameters that maximize L. A model is defined to be identifiable if no two values of the parameters give the same distribution function. A particular case of non–identifiability usually occurs due to overparameterization. The likeli-

hood of a redundant model can be expressed as a function of fewer than the original number of parameters (Catchpole & Morgan, 1997). When CMR protocols are considered, one is often faced with this form of non–identifiability and that is probably the reason why "non–identifiable" has been widely used in place of "parameter redundant" in the literature. To fix ideas, let us consider the standard mark– recapture Cormack–Jolly–Seber (CJS) model with K capture occasions. The raw encounter histories can be fruitfully summarized in the so–called reduced m–array (Burnham et al., 1987, p.36) which summarises the data in the form of the number of individuals released per occasion i denoted Ri (1 [ i [ K–1) and the number of first recaptures given release at occasion i at the succeeding occasions j (2 [ j [ K) denoted mij (table 1). For instance, throughout this paper, we will consider the well known Dipper example (Lebreton et al., 1992). During the breeding season, over a period of 7 years (1981–87), a total of 294 birds were marked and resighted. The data are summarized in table 2. Conditioning on the releases and assuming independence among cohorts, the CJS model likelihood can therefore be easily written down as a product of multinomial probability distributions with the m–array cell probabilities given in table 3 (e.g. Lebreton et al., 1992). For the CJS model, it is well known that the last survival probability and the last recapture probability cannot be estimated separately, only their product being estimable (e.g. Lebreton et al., 1992). Because 2 and p3 only appear together in the cell probabilities, the likelihood can be rewritten in terms of 1, p2 and = 2 p3 as shown in table 4.

Table 1. The observed m–array for the CJS model with 3 capture occasion: Ocr. Occasions of release; Nr. Number released; FRc. First recapture occasion; Ri. Number released at occasion i; mij. Number of first recaptures at occasion j, given release at occasion i. Tabla 1. Matriz m observada para el modelo CJS con 3 capturas: Ocr. Lliberaciónes; Nr. Cantidad de liberados; FRc. Primera ocasion de recaptura; Ri. Número liberado en la ocasión i; mij. Número de primeras recapturas en la ocasión j, cuando la liberación se ha producido en la ocasión i.

FRc Ocr

m 12

m 13

m 23

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Table 2. The m–array of the Dipper data set. In 1981, 22 birds were released among which, 11 were first recaptured in 1982, 2 in 1983, and 9 (= 22 – 11 – 2) were never observed again: Yr. Year of release, Nr. Number released; YFRr. Year of first recapture.

Table 3. The m–array cell probabilities for the CJS model with 3 capture occasions: Ocr. Occasion of release; Nr. Number released; FRc. First recapture occasion; i. Survival probability between i and i + 1; pj. Detection probability at occasion j.

Tabla 2. Matriz m para los datos del Mirlo acuático. En 1981, se libararon 22 aves de las cuales 11 fueron recapturadas por primera vez en 1982, 2 en 1983 y 9 no sa han vuelto a observar: Yr. Año de liberación; Nr. Cantidad liberada; YFRc. Año de la primera recaptura.

Tabla 3. Probabilidades de las celdas de la matriz m para el modelo CJS con 3 ocasiones de captura: Ocr. Lliberación; Nr. cantidad de liberados; FRc. Primera ocasión de recaptura; i. Probabilidad de supervivencia entre i e i + 1; pj. Probabilidad detectada en la ocasión j.

YFRc Yr

1981

1982

1983

1984

1985

1986

FRc

1982 1983 1984 1985 1986 1987 11

Ocr

(1 – p2) 2

0 52

As a consequence, it has to be borne in mind that both estimates of 2 and p3 cannot be separately discussed, except under the form of their product, otherwise one will certainly get misleading conclusions. E.g. for the Dipper data set, maximum likelihood estimates (MLEs) for 6 and p7, both equal to 0.728, were obtained both with software MARK (White & Burnham, 1999) and M– SURGE (Choquet et al., 2003), resulting in a MLE for the product equal to 0.53. In fact, infinitely many other combinations also work e.g. 6 and p7 equal respectively to 0.59 and 0.9. In this particular case, one should not look for a complex explanation of a decrease in the survival probability at the end of the study, but rather, consider it as an artefact due to the redundancy. Moreover, model selection is often achieved in capture–recapture studies via the Akaïke information criterion (AIC), as recommended by Burnham & Anderson (1998). This criterion is:

Table 4. The re–parameterization of the m– array cell probabilities for the CJS model with 3 capture occasions: Ocr. Occasion of release; Nr. Number released; FRc. First recapture occasion; i. Survival probability between i and i + 1; pj. Detection probability at occasion j. Tabla 4. Reparametrización de las probabilidades de las celdas de la matriz m para el modelo CJS con 3 ocasiones de captura. Ocr. Liberación; Nr. Cantidad de liberados: FRc. primera oportunidad de recaptura; i. Probabilidad de supervivencia entre i e i + 1; pj. Probabilidad detectada en la ocasión j.

FRc Ocr

2 1

3 1

(1 – p2)

AIC = –2 log (Lmax) + 2 np where Lmax is the maximum likelihood, and np is the number of estimable (functions of) parameters. To calculate this criterion in the CJS case, one has to subtract one from the total number of parameters in order to obtain the number of actually estimable parameters. Consequently, a naive computation of the number of parameters may lead to a wrong AIC–ranking of the models.

Determining how many and which functions of the original parameters are estimable is thus crucial in model selection and in the interpretation of estimates. Two questions naturally arise, and we will focus on them in the next section: Question 1 (Q1) – How many parameters are estimable? This number is called the rank of the model.

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Question 2 (Q2) – Which parameters are estimable? In the preceding didactic example, the conclusions could have been reached by visual inspection, trying to find the parameters which appear only together. However this approach becomes intractable for complex models. Additionally, parameter redundancy is not always intuitive. Indeed the common belief that parameter redundancy is the result of "too many" parameters can be completely misleading. A good example occurs in modelling data from mark– recovery studies of animals banded at birth. The so–called Seber model (Seber, 1971) has a fully age–dependent survival and a constant recovery probability, with no time dependence in survival or recovery probabilities. This model is parameter redundant, and yet when extra parameters are added by allowing first–year survival to be time dependent, the model becomes non redundant (for details see Morgan & Freeman, 1989; Catchpole et al., 1996). Another issue arises demonstrating that there exists no simple "rule of thumb" that allows us to compute the number of non redundant parameters in a model. When constraints are used on the parameters, the number of non redundant parameters may depend on the type of link function that is used. Viallefont (1995) illustrated such a situation with group dependence (see also Catchpole et al., 2002). What precedes is an inherent model property, called intrinsic redundancy. Such a situation can be detected a priori, by methods allowing us to detect redundancy problems in the structure of a model, independently for any specific data set. It could also be detected a posteriori, i.e. after fitting the model to the data set of interest.

However, there exists a second sort of redundancy, called extrinsic redundancy, due to a particular structure of the data, usually missing or sparse data. E.g. it may happen that no individuals are detected at time i, inducing the redundancy of and i, with only the survival probability bei–1 tween i – 1 and i + 1 being estimable. Such redundancy can only be detected "a posteriori", i.e. when the model has been fitted to the specific data set for which the problem appears. The purpose of this paper is to provide the reader with the tools available to check for intrinsic and extrinsic parameter redundancy and to choose the most relevant method to solve their own problems. In the next section, we review the procedures available for checking for parameter redundancy, giving explanations and illustrations. Four approaches are considered and illustrated with the CJS model in conjunction with the Dipper data. We emphasize relative drawbacks and advantages and provide recommendations concerning parameter redundancy for the user of models for marked individuals. Methods to redundancy

check

for

parameter–

To our knowledge, there exist four different methods that can be used to detect parameter–redundancy. Table 5 presents these methods according to their ability to detect a priori intrinsic redundancy, or a posteriori both types of redundancy, and to answer Q2. The first two methods are more "intuitive", a posteriori methods, whereas the next two methods are more appropriate to a priori detect intrinsic redundancy problems.

Table 5. Summary of the conditions of use and relative advantages of the four methods proposed. For details see the text. Tabla 5. Resumen de las condiciones de uso y ventajas relativas de los cuatro métodos propuestos. Para detalles al respecto ver el texto.

Name of the method

Detection of intrinsic redundancy

Detection of extrinsic redundancy

Necessary software

Answer to Q2

Profile likelihood

Possible on simulated data

Yes

Any CMR software for "by hand" plots; routinely implemented in M–SURGE

The Hessian

Possible on simulated data

Yes

Implemented in MARK, M–SURGE

Simulated data with large numbers released

RELEASE (see also MARK) for computation of expected numbers and any CMR software for the optimization step

Partial

Yes

MAPLE or MATHEMATICA

Yes

Simulation

The formal derivative matrix

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Profile likelihood

Deviance values

680 This a posteriori method is based on the fact that redundancy results in a flat ridge in the likelihood, hence inducing an infinity of solutions. The redundancy can thus be shown by plotting the profile likelihood i.e. the likelihood as a function of a parameter of interest and simultaneously maximized over the other parameters (e.g. Freeman et al., 1992; Lebreton & Pradel, 2002). This method is normally used a posteriori, i.e. on a given data set with a given model, and more specifically it is used to detect problems concerning a specific subset of parameters. One needs first to have an idea of which parameters are redundant. For example, if in two quite close models, very different estimates are found for one parameter, this may mean that it is redundant. This method can also be used a priori on simulated data to detect intrinsic redundancy, or to distinguish between extrinsic and intrinsic redundancy problems: indeed, an intrinsic problem is model–dependent and remains so whatever the data set used, whereas an extrinsic problem is due to the structure of a specific data set, and disappears if a simulated data set with a different structure is used. For the Dipper example, graphs of the profile deviance are shown in figure 1. Several model– fitting steps are necessary, with different fixed values at each time for the parameter under investigation. Then the different deviance values are plotted against the corresponding fixed value of the parameter. If the parameter is estimable uniquely by maximum likelihood, for example 1, one will get a higher deviance for any value of this parameter other than the MLE. On the contrary, if the parameter is not estimable, e.g. p7, one will get a flat ridge according to the direction of this parameter. Actually, this flat ridge does not extend completely from 0 to 1. In this example, because we have to deal with probability, once the value of p7 has been fixed between 0 and 1, the value of 6 has to be between = 6 p7 and 1, otherwise the deviance is no longer constant, but as long as the ridge exists in the neighbourhood of the MLE, the concerned parameter must be considered redundant. Further detail on the extent of a ridge in the context of Seber’s mark–recovery models is given in Catchpole et al. (1993) and Catchpole & Morgan (1994). This method can be used "by hand" with any CMR software, while the program M–SURGE can automatically give the plot of profile deviance. Applications can be found in Viallefont (1992), Freeman & Morgan (1992), Catchpole et al. (1993), Catchpole & Morgan (1994), Lebreton & Pradel (2002), Pradel et al. (in prep.) and Gimenez et al. (submitted). This method is a graphical diagnostic only of the model parameter redundancy. It should not be used systematically, which would necessitate drawing as many graphs as there are parameters in the model, which would be very time–consuming.

675 1

670 665 660 655 0.3

0.4

0.5 0.6 0.7 0.8 0.9 Fixed parameter value

Fig. 1. Profile deviance for two parameters of the CJS model: application to the Dipper data set. Fig. 1. Desviación del perfil para dos parámetros del modelo CJS: aplicación al conjunto de datos del Mirlo acuático.

The Hessian This a posteriori method is based on detecting zero eigenvalues of the matrix of the second derivatives of the log–likelihood with respect to the parameters —namely the Hessian matrix— evaluated at the MLE. The model rank is computed as the number of non–zero eigenvalues (Viallefont et al., 1998) i.e. the numerical rank of the Hessian (the number of linearly independent rows). It is also possible to determine the parameters that are separately estimable by computing the eigenvectors associated with the zero eigenvalues and identifying their null co–ordinates (Reboulet et al., 1999). For the Dipper example, the 12 x 12 Hessian matrix and the associated eigenvalues are given in table 6. The eigenvalue in bold is close to zero meaning that the model rank equals 11. Out of the 12 original parameters, only 11 are estimable, confirming what is known about the CJS model. In addition, by considering the entries in the eigenvector corresponding to the smallest eigenvalue, it is confirmed that only the last survival and capture probabilities are redundant parameters (values in bold). This method should be used cautiously because: (1) as with any a posteriori method, it does not distinguish between intrinsic and extrinsic redundancy, thus it does not allow generalisation of the results found for one specific data set to other data sets with the same structure mode; (2) it requires the Hessian matrix, which is often done numerically through an approximation via a finite difference scheme; and (3) a perfect tuning of a zero threshold value probably does not exist (Viallefont et al., 1998).

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Moreover, using this method without caution induces another problem when some parameters are estimated at a boundary value of the parameter interval. For example, if one of the survival probabilities is estimated at the value of one or zero, the numerical computation of the Hessian by finite differences may induce one eigenvalue of this matrix to be null, thus leading to counting the corresponding parameter as if it was not separately estimable, whereas it obviously is (with a value at a boundary). Despite all these problems, this numerically tractable method has been implemented extensively in the mark–recapture software SURGE (Viallefont et al., 1998; Reboulet et al., 1999), MARK (White & Burnham, 1999) and M–SURGE (Choquet et al., 2003). We advise the reader to be very careful when using MARK, where a wrong computation of the number of independent parameters in the mod-

els can lead to an unreliable ordering of the models via AIC. When using M–SURGE, the first derivatives are analytically computed which improves the precision. Simulation The first step of the simulation method (also called the analytical–numerical method) generates an artificial data set, by assuming a (realistic) set of parameter values and (large) ringing numbers, and then using the model to generate expected encounter histories. In a second step, this generated data set is analyzed with the model of interest using standard software such as MARK, M–SURGE, SURVIV (White, 1982) or MSSURVIV (Hines, 1994). For large numbers of released individuals per occasion, the MLEs and standard errors produced are approximately the expected values of the param-

Table 6. Numerical diagnostics for CJS model parameter redundancy with the Dipper data set. Tabla 6. Diagnósticos numéricos para la redundancia de parámetros en el modelo CJS con el conjunto de datos del Mirlo acuático.

Hessian matrix computed by a finite difference scheme 2.524 0.779 0.027 0.001 0.000 0.000 1.343

0.055

0.002

0.000

0.001 0.046

0.000 0.001

–0.542 –0.019

1.022 –0.468

0.031 1.166

0.002 0.075

0.000 0.004

0.000 0.001

0.716 17.067 1.100 0.046 1.100 20.076

0.028 0.503

–0.001 –0.000

–0.018 –0.662 1.790 –0.001 –0.043 –1.017

0.084 1.528

0.028 0.503

0.000 0.000 0.001 0.028 0.503 8.325 1.343 –0.542 –0.019 –0.001 –0.000 –0.000

–0.000 2.407

–0.000 –0.001 –0.025 –0.485 8.325 –0.038 –0.001 –0.000 –0.000 –0.000

0.055 0.002

1.022 –0.468 –0.018 –0.001 –0.000 0.031 1.166 –0.662 –0.043 –0.001

–0.038 –0.001

1.035 –0.029 –0.002 –0.000 –0.000 –0.029 1.965 –0.069 –0.003 –0.001

0.000 0.000

0.002 0.000

0.075 0.004

1.790 –1.017 –0.025 0.084 1.528 –0.485

–0.000 –0.000

–0.002 –0.069 3.008 –0.077 –0.025 –0.000 –0.003 –0.077 2.042 –0.485

0.000

0.001

0.028

–0.000

–0.000 –0.001 –0.025 –0.485

0.779 14.478 0.495 0.027 0.495 18.485 0.001 0.000

0.019 0.001

Hessian eigenvalues

0.019 0.716

0.503

8.325

Eigenvector associated with the null eigenvalue

0.00000364 0.94321127

–3.484365e–18 1.371273e–16

1.05686038 1.85022632

–8.138873e–17 2.486811e–17

1.87616909 2.70801343

6.028106e–17 –7.071068e–01

3.80631847 14.56896925

8.958153e–19 –1.775061e–15

p2 p3

16.52430693 16.87618746

2.851347e–16 2.764039e–16

p4 p5

18.83059874 20.69498495

5.154638e–17 7.071068e–01

p6 p7

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Table 7. Expected m–array using Ri = 10,000 for all i and the parameter values 1 = 0.4; 2 = 0.5; = 0.6; 4 = 0.6; 5 = 0.7; 6 = 0.7; p2 = 0.9; p3 = 0.9; p4 = 0.9; p5 = 0.9; p7 = 0.7. Values are 3 rounded to the nearest integer: YFRc. Year of first recapture; Yr. Year of release; Nr. Number released. Tabla 7. Matriz m esperada utilizando Ri = 10.000 para todos los i y los valores paramétricos 1 = 0,4; = 0,5; 3 = 0,6; 4 = 0,6; 5 = 0,7; 6 = 0,7; p2 = 0,9; p3 = 0,9; p4 = 0,9; p5 = 0,9; p7 = 0,7. Los valores 2 se redondean al número entero más próximo: YFRc. Año de la primera recaptura; Yr. Año de liberación; Nr. Cantidad liberada. YFRc Yr

1982

1983

1984

1985

1986

1981

10,000

4,900

756

1982

10,000

3,600

180

1983

10,000

1984

10,000

1985

10,000

1986

10,000

eter estimators and their standard errors. If a parameter estimator is unbiased to the 5th decimal place and has a coefficient of variation less than 100%, then it is declared estimable (Kendall & Nichols, 2002). For the CJS model example for the Dipper study, we used Ri = 10,000 for all i and parameter values shown in table 7. The resulting expected m–array values are given in table 7. Using SURVIV e.g., we analyze these data fitting the CJS model. The results in table 8 suggest that all parameters are estimable except for 6 (biased estimate) and p7 (biased estimate and large coefficient of variation). Recent applications can be found in Schaub et al. (2004) and Kendall & Nichols (2002). This method can be used as a completely a priori method, to investigate the intrinsic redundancy of a model, by generating the simulated data set with arbitrarily fixed values of the model parameters. Note however that the ringing numbers used in the simulation should be large enough to ensure that there are no zero cells in the m– array, to ensure that any redundancy found must be intrinsic. However, it is also often used a posteriori, using the point estimates obtained to generate the simulated data. The investigation of the extrinsic problems of redundancy (i.e. due to the data) cannot be assessed by this method, because it relies on the data set being very large, which requires simulated rather than actual data. Also, the simulation method is only valid for the particular values of the parameter that are chosen to compute the expected probabilities. We recommend using several set of different values to be sure that you do not have to deal with a very particular case (a model conditionally of full rank i.e. a model of full rank, but parameter redundant

4,500

1987

270

5,400

324

5,400

480 8,000

Table 8. MLEs obtained from the simulated data of table 6: P. parameters; MLEs. Maximum likelihood values; SE. Standard error; B. Bias; Cv. Coefficient of variation (%). Tabla 8. MLE obtenidos a partir de los datos simulados de la tabla 6: P. Parametros; MLEs. Valores de probabilidad máxima; SE. Error estandar; B. Sesgo; Cv. Coeficiente de variación (%). P

MLEs

0.700

0.078

0.11

0.400

0.070

–0.00001

0.17

0.500

0.072

–0.00000

0.14

0.600

0.071

–0.00000

0.12

0.600

0.070

–0.00000

0.12

0.894

0.067

0.19442

0.07

0.700

0.092

0.00000

0.13

0.900

0.080

0.00002

0.09

0.900

0.073

0.00002

0.08

0.900

0.070

0.00001

0.08

0.900

0.065

0.00001

0.07

0.894

225.184

0.19442 251.76

for one or several values of the parameters). For the Dipper example, we tried several sets of different values for the parameters and were led to the same conclusions.

Gimenez et al.

568

Given that the variances computed by SURVIV should not be trusted when one or several parameters are redundant in the model (Hines, pers. com.), we greatly encourage using MARK or M–SURGE which are more reliable in computing the Hessian matrix. Of course, generating all possible encounter histories for a large fixed number of capture occasions, and then writing down their expected values under a complex model, can quickly become time– consuming and often intractable. Note however that simulations of the CJS model can be conducted with program RELEASE which can be used as a standalone application (Burnham et al., 1987) or found as part of program MARK. Finally, it is important to notice that this method also allows us to study relevant statistical quantities, such as bias, precision of estimators, or power of likelihood–ratio and goodness–of–fit tests (Viallefont et al., 1995; Pollock et al., 1985).

(functions of) parameters are the formal solutions of a system of partial differential equations (PDEs). The successive steps required to perform this method are shown in table 9 and will now be detailed using the CJS model example in conjunction with the Dipper data set. Symbolic calculus software such as Maple or Mathematica can be used at each step to greatly ease the mathematical burden. Intrinsic parameter–redundancy The first step requires forming the vector q of original parameters and a vector m of m–array cell probabilities under the CJS model. In both vectors the order is arbitrary. We choose

= ( 1,..., 6, p2,...,p7)T and

The formal derivative matrix This a priori method is based on the analytical computation of the matrix D of derivatives of the vector of the multinomial distribution cell probabilities with respect to the vector of model parameters. This method gives the answer to both Q1: the number of estimable parameters is the symbolic row rank of D, and Q2: the estimable

= ( 1p2, 1(1 – p2) 2p3,..., 6p7)Τ

Then, step 2, we calculate the symbolic derivative matrix D of log( ) with respect to q. The result is: Then the answer to Q1 (step 3) is simply the symbolic rank of D which can be easily obtained, again with Maple or Mathematica. For the CJS model, we find a deficiency of 1 i.e. a rank equal to 11. The eigenvector corresponding to the zero eigenvalues is (step 4)

Table 9. Different steps required to perform the formal method. Tabla 9. Distintos pasos requeridos para ejecutar el método formal.

Steps

Mathematical objects and notation

1. Write down the vector of log–probabilities as a function of parameters

Vector of parameters: Vector of log–probabilities:

2. Differentiate formally log ((µ( )) wrt the components of

The derivative matrix:

3. Determine the number q of estimable parameters: if q < p the model is parameter redundant then go to step 4, otherwise the model is of full rank

Symbolic rank of the derivative matrix: r = rank (D)

4. Write down formal solutions of:

(θ)T D(θ) = 0,

i = 1,...,d

5. Determine position i1,...,is of 0 in common to all s

The s separately non–redundant parameters:

6. Write down the system of partial differential equations and solve it formally to obtain the estimable functions

The system of PDEs: j = 1,...,d

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Animal Biodiversity and Conservation 27.1 (2004)

where

= 1 – pj, j = 2,...,7

Since this (sole) eigenvector has zeroes in all entries except those corresponding to 6 and p 7 , all parameters except those two are estimable step 5). To determine the full set of estimable functions, step 6 requires the solution of the following partial differential equation (PDE):

The solutions of this PDE are 1, and p2, p3, p4, p5, p6 and the product

2 6

, 3, p7 .

Extrinsic parameter–redundancy What precedes deals only with inherent properties of the CJS model. If the behaviour of parameter estimates is needed, one has to take the data into account. By considering non–structural zeroes i.e. missing data in the m–array, it is easy to adjust the method for checking for extrinsic parameter redundancy. One has just to modify

step 1 by forming a vector m of m–array cell probabilities incorporating only the probabilities corresponding to non–zero mij (cell probabilities corresponding to zero mij do not play any role in building the likelihood, since they are raised to the power zero). With the Dipper data set, there is a substantial amount of missing data, so that only 11 cells do not contain missing data (see table 2); m12, m 13, m23, m24, m 34, m35, m 45, m46, m47, m 56 and m67. The expression of the derivative matrix D is simpler, consisting of just 11 of the columns of the previous D, but its rank still remains equal to 11 so that in this case the missing data do not render any extra parameters redundant. Recent applications can be found in Schaub et al. (2004). Other examples with more details about the theory and references can be found for single–state models in Catchpole et al. (2002) with associated Maple code freely available from http://www.ma.adfa.edu.au/~eac/Redundancy/Maple and for multistate capture–recapture models in Gimenez et al. (2003), for which Maple code is freely available from ftp://ftp.cefe.cnrs–mop.fr/bio/ PRM.

570

Discussion The four methods all give some information about the redundant parameters in a particular model or for the case when that model is fitted to a specific data set. They should be used in preference to any "rules of thumb". If, using one of these methods, the number of non–redundant parameters in a model is known with certainty, then information criteria can safely be computed. Otherwise, it is advisable to use the value automatically computed with the Hessian method by software such as MARK or M–SURGE. It has to be noted that the intrinsic number of non–redundant parameters is known for many models (Lebreton et al., 1992 for CJS–type models; Gimenez et al., 2003 for multistate models; see also Viallefont, 1995; Kendall & Nichols, 2002; Schaub et al., 2004). Hence, if the Hessian method yields different results, the structure of the data has to be checked to determine whether extrinsic reasons induce redundancy. Another easy way to count the number of non– redundant parameters is the simulation method. But, both the Hessian and simulation methods may be flawed by numerical issues, such as deciding what "close to zero" means. There is a slight advantage of the simulation method as it does give an immediate answer to the question: "Which parameters are not separately estimable?" (e.g. 6 and p7 in the CJS model with the Dipper example). Concerning the Hessian method, the variance of the redundant parameters is not always very different from that of the separately estimable parameters, and some insight in the eigenvectors is thus required. Knowing which parameters are not separately estimable only constitutes a partial answer to Q2 in the introduction of this paper. In the Dipper example, only the formal method yields the information that the product 6 p7 is estimable, rather than any other function of these two parameters. Hence, only the formal method gives the full answer to Q2. To make efficient use of the profile–likelihood method, one needs to have an idea about which parameters might be redundant, in other words, one should have an idea of the "partial answer" to Q2. The profile–likelihood is then an easy–to–use method, but the results only apply to the parameters for which the profile–likelihood plot has been drawn. To obtain a complete answer to Q1 with this method, one needs to plot the profile–likelihood for all the parameters, which will be very time–consuming in many cases. Even if a model is non–redundant, difficulties in estimation can occur because of local minima (Lebreton & Pradel, 2002) or duality phenomena (Pradel et al., in prep.) when two different estimates of the same parameters can give the same value of the smallest deviance (this is a case of non–identifiability). In such cases, a graph of the profile– deviance can add valuable information. While building the profile deviance for a parameter, it is also quite easy to derive at the same time profile–deviance–based confidence intervals which

Gimenez et al.

are known to be more robust than the classical Wald confidence interval with boundary estimates (Catchpole & Morgan, 1994 for mark–recovery models; Gimenez et al., submitted for multistate mark–recapture models). To know a priori the intrinsic number of estimable parameters in a model, all four methods proposed here may be used on simulated data for the profile– likelihood, the Hessian or the analytical–numerical methods: it is then advisable to generate two or more data sets to check that the result is not due to a special case in the simulated data. The formal method that does not require any simulated or expected data is not only more reliable, but also the only one to provide a clear answer to Q2. Moreover, having worked out a particular case (e.g. CJS with 3 capture occasions) it is often possible to extend the conclusions to larger examples with the same structure (i.e. more years of data) without having to repeat the analysis (e.g. Catchpole & Morgan, 1997, 2001; Gimenez et al., 2003). We are currently trying to extend these ideas in order to provide a taxonomy of intrinsic redundancy of many standard models. We thus advise anyone wanting to develop new models for the analysis of marked data to use the formal method to assess the properties of the new model they develop. Clearly, it requires the use of specific software and some knowledge of algebra. We hope to see an automatic implementation in standard CMR software in the near future. Pending this progress, we anticipate that in the near future, combining the formal method with other methods will be important. It could be the way to go towards a reliable routine computation of numbers of estimable parameters. Such an approach is adapted in program M–SURGE where the first derivatives are analytically computed. A second example is given by Choquet & Pradel (unpublished results). They developed practical rules in order to simplify the structure of derivative matrix D which makes calculation of the formal rank easier. From a Bayesian perspective, maximum likelihood theory is equivalent to finding the mode of the joint posterior distribution of the parameters, given uniform priors. Since Bayesians usually examine posterior means, rather than modes, issues of parameter redundancy are not apparent, and might well be thought to be of no importance. If there is parameter–redundancy, then the likelihood surface is flat, however if a Bayesian approach is adopted, then the posterior can result in remarkably precise estimators. This is investigated and explained in Brooks et al. (2000) for a particular example. In their case, Brooks et al. were aware that the ridge existed, but one could envisage examples arising when that was not the case. It is therefore important to be aware of parameter redundancy. Barry et al. (2000) found that the existence of parameter redundancy can have substantial impact on posterior means and standard deviations. Carlin & Louis (1996, p. 203) recommend against the use of Markov

Animal Biodiversity and Conservation 27.1 (2004)

chain Monte Carlo methods in the presence of parameter redundancy. Knowing that a model is full–rank may not be enough. This has been shown by Catchpole et al. (2001). It is shown there that if a full–rank model has a parameter–redundant sub–model, and is insufficiently different from that submodel, then it may perform badly in practice. Thus in practice one should think carefully about the models to be used, and make use of whatever knowledge and general results that are available (see, eg., Catchpole et al., 1996). Even though a model may be full–rank, it can still be useful to check the values of the eigenvalues. This then produces a kind of synthesis of intrinsic and extrinsic procedures. As a conclusion, the choice between methods clearly depends on the purpose of the study. As the tools presented here are enough to tackle all sorts of problems concerning parameter redundancy, we do hope that people will use them, in order to ensure valid biological conclusions. References Arnason, A. N., 1973. The estimation of population size, migration rates and survival in a stratified population. Researches on Population Ecology 15: 1–8. Barry, S. C., Brooks, S. P., Catchpole, E. A. & Morgan, B. J. T., 2003. The Analysis of Ring– Recovery Data Using Random Effects. Biometrics, 59(1): 54–65. Brooks, S. P., Catchpole, E. A., Morgan, B. J. T. & Barry, S. C., 2000. On the Bayesian analysis of ring–recovery data. Biometrics, 56: 951–956. Brooks, S. P. & Morgan, B. J. T., 1995. Optimization using simulated annealing. The Statistician, 44: 241–257. Brownie, C., Anderson, D. R., Burnham, K. P. & Robson, D. S., 1985. Statistical inference from band recovery data — a handbook, 2nd Ed. U.S. Fish and Wildlife. Service, Washington, DC. 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 Populatio: 199–213 (J.–D. Lebreton & P. M. North, Eds.). Birkhauser Verlag, Basel, Switzerland Burnham, K. P. & Anderson, D. R., 1998. Model selection and inference, a practical Information theoretic approach. Springer, New–York. 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. Bethesda, Maryland, American Fisheries Society. Carlin, B. P. & Louis, T. A., 1996. Bayes and empirical Bayes methods for data analysis. Chapman and Hall, London.

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Catchpole, E. A., Freeman, S. N. & Morgan, B. J. T., 1993. On boundary estimation in ring recovery models and the effect of adding recapture information. In: Marked individuals in the study of Bird Populations: 215–228 (J.–D. Lebreton & P. M. North, Eds.). Birkhauser–Verlag, Basel, Switzerland. – 1996. Steps to parameter redundancy in age– dependent recovery models. J. R. Statistic. Soc., B, 58: 763–774. Catchpole, E. A., Freeman, S. N., Morgan, B. J. T. & Harris, M. P., 1998. Integrated recovery/recapture data analysis. Biometrics, 54: 33–46. Catchpole, E. A., Kgosi, P. M. & Morgan, B. J. T., 2001. On the near–singularity of models for animal recovery data. Biometrics, 57: 720–726. Catchpole, E. A. & Morgan, B. J. T., 1994. Boundary estimation in ring recovery models. J. R. Statist. Soc. B, 56: 385–391. – 1997. Detecting parameter redundancy. Biometrika, 84: 187–196. – 2001. Deficiency of parameter–redundant models. Biometrika, 88: 593–598. 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. Montpellier, France, CEFE/ CNRS. Cormack, R. M., 1964. Estimates of survival from the sighting of marked animals. Biometrika, 51: 429–438. Currie, I. D., 1995. Maximum likelihood estimation and Mathematica. Applied Statistics, 44: 379–394. Freeman, S. N. & Morgan, B. J. T., 1992. A modelling strategy for recovery data from birds ringed as nestlings. Biometrics, 48: 217–236. Freeman, S. N., Morgan, B. J. T. & Catchpole, E. A., 1992. On the augmentation of ring–recovery data with field information. Journal of Animal Ecology, 61: 649–657. Gimenez, O., Choquet, R. & Lebreton, J. D., 2003. Parameter Redundancy in Multi–state Capture– Recapture Models. Biometrical Journal, 45(6): 704–722. Gimenez, O., Choquet, R., Lamor, L., Scofield, P., Fletcher, D., Lebreton, J. D. & Pradel, R. (submitted). Efficient profile likelihood confidence intervals for capture–recapture models. Journal of Agricultural, Biological, and Environmental Statistics. Hines, J. E., 1994. MSSURVIV User’s Manual. Laurel, MD, National Biological Survey: 21. 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., 2002. Estimating state–transition probabilities for unobservable states using capture–recapture/resighting data. Ecology, 83: 3276–3284. Lebreton, J.–D., Burnham, K. P., Clobert J. & Anderson, D. R., 1992. Modeling survival and

572

testing biological hypotheses using marked animals: A unified approach with case studies. Ecological Monographs, 62(1): 67–118. Lebreton, J. D. & Pradel, R., 2002. Multi–state recapture models: modelling incomplete individual histories. Journal of Applied Statistics, 29: 353–369. Reboulet, A. M., Viallefont, A., Pradel, R. & Lebreton, J. D., 1999. Selection of survival and recruitment models with SURGE 5.0. Bird Study, 46 (suppl.): 148–156. Schaub, M., Gimenz, O., Schmidt, B. R. & Pradel, R., 2004. Estimating survival and temporary emigration in he multistate cature–recapture framework. Ecology, 85: 2107–2113. 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. – 1971. Estimating age–specific survival rates from bird–band returns when the reporting rate is

Gimenez et al.

constant. Biometrika, 78: 917–919. Viallefont, A., 1992. Robustesse et flexibilité des modélisations en capture–recapture. Application à l’oie des neiges (Anser caerulescens caerulescens). Mémoire de DEA, Ecole Nationale Supérieure d’Agronomie de Montpellier, France. – 1995. Robustesse et flexibilité des analyses démographiques par capture–recapture. De l’estimation de la survie à la détection de compromis évolutifs. Unpublished Ph. D. Thesis. Univ. of Montpellier II. Viallefont, A., Lebreton, J.–D., Reboulet, A.–M., & Gory, G., 1998. Parameter identifiability and model selection in capture–recapture models: a numerical approach. Biometrical Journal, 40: 1–13. 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: survival estimation from populations of marked animals. Bird Study, 46 (suppl.): 120–139.

VII

Animal Biodiversity and Conservation 27.1 (2004)

Animal Biodiversity and Conservation

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Animal Biodiversity and Conservation (abans Miscel·lània Zoològica) és una revista interdisciplinària publicada, des de 1958, pel Museu de Zoologia de Bar celona. Inclou articles d'investigació empírica i teòrica en totes les àrees de la zoologia (sistemàtica, taxonomia, morfologia, biogeografia, ecologia, etologia, fisiologia i genètica) procedents de totes les regions del món amb especial énfasis als estudis que d'una manera o altre tinguin relevància en la biología de la conservació. La revista no publica catàlegs, llistes d'espècies o cites puntuals. Els estudis realitzats amb espècies rares o protegides poden no ser acceptats tret que els autors disposin dels permisos corresponents. Cada volum anual consta de dos fascicles. Animal Biodiversity and Conservation es troba registrada en la majoria de les bases de dades més importants i està disponible gratuitament a internet a http://www.bcn.es/ABC, de manera que permet una difusió mundial dels seus articles. Tots els manuscrits són revisats per l'editor execu tiu, un editor i dos revisors independents, triats d'una llista internacional, a fi de garantir–ne la qualitat. El procés de revisió és ràpid i constructiu. La publicació dels treballs acceptats es fa normalment dintre dels 12 mesos posteriors a la recepció. Una vegada hagin estat acceptats passaran a ser propietat de la revista. Aquesta es reserva els drets d’autor, i cap part dels treballs no podrà ser reproduïda sense citar–ne la procedència.

Els treballs seran presentats en format DIN A–4 (30 línies de 70 espais cada una) a doble espai i amb totes les pàgines numerades. Els manuscrits han de ser complets, amb taules i figures. No s'han d'enviar les figures originals fins que l'article no hagi estat acceptat. El text es podrà redactar en anglès, castellà o català. Se suggereix als autors que enviïn els seus treballs en anglès. La revista els ofereix, sense cap càrrec, un servei de correcció per part d'una persona especialitzada en revistes científiques. En tots els casos, els textos hauran de ser redactats correctament i amb un llenguatge clar i concís. La redacció del text serà impersonal, i s'evitarà sempre la primera persona. Els caràcters cursius s’empraran per als noms científics de gèneres i d’espècies i per als neologis mes intraduïbles; les cites textuals, independentment de la llengua, seran consignades en lletra rodona i entre cometes i els noms d’autor que segueixin un tàxon aniran en rodona. Quan se citi una espècie per primera vegada en el text, es ressenyarà, sempre que sigui possible, el seu nom comú. Els topònims s’escriuran o bé en la forma original o bé en la llengua en què estigui escrit el treball, seguint sempre el mateix criteri. Els nombres de l’u al nou, sempre que estiguin en el text, s’escriuran amb lletres, excepte quan precedeixin una unitat de mesura. Els nombres més grans s'escriuran amb xifres excepte quan comencin una frase. Les dates s’indicaran de la forma següent: 28 VI 99; 28, 30 VI 99 (dies 28 i 30); 28–30 VI 99 (dies 28 a 30). S’evitaran sempre les notes a peu de pàgina.

Normes de publicació Els treballs s'enviaran preferentment de forma elec trònica (abc@mail.bcn.es). El format preferit és un document Rich Text Format (RTF) o DOC que inclogui les figures (TIF). Si s'opta per la versió impresa, s'han d'enviar quatre còpies del treball juntament amb una còpia en disquet a la Secretaria de Redacció. Cal incloure, juntament amb l'article, una carta on es faci constar que el treball està basat en investiga cions originals no publicades anteriorment i que està sotmès a Animal Biodiversity and Conservation en exclusiva. A la carta també ha de constar, per a aquells treballs en que calgui manipular animals, que els autors disposen dels permisos necessaris i que compleixen la normativa de protecció animal vigent. També es poden suggerir possibles assessors. Quan l'article sigui acceptat, els autors hauran d'enviar a la Redacció una còpia impresa de la versió final acompanyada d'un disquet indicant el progra ma utilitzat (preferiblement en Word). Les proves d'impremta enviades a l'autor per a la correcció, seran retornades al Consell Editor en el termini de 10 dies. Aniran a càrrec dels autors les despeses degudes a modificacions substancials introduïdes per ells en el text original acceptat. El primer autor rebrà 50 separates del treball sense càrrec a més d'una separata electrònica en format PDF. ISSN: 1578–665X

Format dels articles Títol. El títol serà concís, però suficientment indicador del contingut. Els títols amb designacions de sèries numèriques (I, II, III, etc.) seran acceptats previ acord amb l'editor. Nom de l’autor o els autors. Abstract en anglès que no ultrapassi les 12 línies mecanografiades (860 espais) i que mostri l’essència del manuscrit (introducció, material, mètodes, resultats i discussió). S'evitaran les especulacions i les cites bibliogràfiques. Estarà encapçalat pel títol del treball en cursiva. Key words en anglès (sis com a màxim), que orientin sobre el contingut del treball en ordre d’importància. Resumen en castellà, traducció de l'Abstract. De la traducció se'n farà càrrec la revista per a aquells autors que no siguin castellanoparlants. Palabras clave en castellà. Adreça postal de l’autor o autors. (Títol, Nom, Abstract, Key words, Resumen, Pala bras clave i Adreça postal, conformaran la primera pàgina.)

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Introducción. S'hi donarà una idea dels antecedents del tema tractat, així com dels objectius del treball. Material y métodos. Inclourà la informació pertinent de les espècies estudiades, aparells emprats, mèto des d’estudi i d’anàlisi de les dades i zona d’estudi. Resultados. En aquesta secció es presentaran úni cament les dades obtingudes que no hagin estat publicades prèviament. Discusión. Es discutiran els resultats i es compa raran amb treballs relacionats. Els suggeriments de recerques futures es podran incloure al final d’aquest apartat. Agradecimientos (optatiu). Referencias. Cada treball haurà d’anar acompanyat de les referències bibliogràfiques citades en el text. Les referències han de presentar–se segons els models següents (mètode Harvard): * Articles de revista: Conroy, M. J. & Noon, B. R., 1996. Mapping of spe cies richness for conservation of biological diversity: conceptual and methodological issues. Ecological Applications, 6: 763–773. * Llibres o altres publicacions no periòdiques: Seber, G. A. F., 1982. The estimation of animal abundance. C. Griffin & Company, London. * Treballs de contribució en llibres: Macdonald, D. W. & Johnson, D. P., 2001. Dispersal in theory and practice: consequences for conserva tion biology. In: Dispersal: 358–372 (T. J. Clober, E. Danchin, A. A. Dhondt & J. D. Nichols, Eds.). Oxford University Press, Oxford. * Tesis doctorals: Merilä, J., 1996. Genetic and quantitative trait vari ation in natural bird populations. Tesis doctoral, Uppsala University. * Els treballs en premsa només han d’ésser citats si han estat acceptats per a la publicació: Ripoll, M. (in press). The relevance of population studies to conservation biology: a review. Anim. Biodivers. Conserv. La relació de referències bibliogràfiques d’un treball

Animal Biodiversity and Conservation 27.1 (2004)

serà establerta i s’ordenarà alfabèticament per autors i cronològicament per a un mateix autor, afegint les lletres a, b, c,... als treballs del mateix any. En el text, s’indicaran en la forma usual: “...segons Wemmer (1998)... ”, “...ha estat definit per Robinson & Redford (1991)...”, “...les prospeccions realitzades (Begon et al., 1999)...” Taules. Les taules es numeraran 1, 2, 3, etc. i han de ser sempre ressenyades en el text. Les taules grans seran més estretes i llargues que amples i curtes ja que s'han d'encaixar en l'amplada de la caixa de la revista. Figures. Tota classe d’il·lustracions (gràfics, figures o fotografies) entraran amb el nom de figura i es numeraran 1, 2, 3, etc. i han de ser sempre ressen yades en el text. Es podran incloure fotografies si són imprescindibles. Si les fotografies són en color, el cost de la seva publicació anirà a càrrec dels au tors. La mida màxima de les figures és de 15,5 cm d'amplada per 24 cm d'alçada. S'evitaran les figures tridimensionals. Tant els mapes com els dibuixos han d'incloure l'escala. Els ombreigs preferibles són blanc, negre o trama. S'evitaran els punteigs ja que no es reprodueixen bé. Peus de figura i capçaleres de taula. Els peus de figura i les capçaleres de taula seran clars, concisos i bilingües en la llengua de l’article i en anglès. Els títols dels apartats generals de l’article (Intro ducción, Material y métodos, Resultados, Discusión, Conclusiones, Agradecimientos y Referencias) no aniran numerats. No es poden utilitzar més de tres nivells de títols. Els autors procuraran que els seus treballs originals no passin de 20 pàgines (incloent–hi figures i taules). Si a l'article es descriuen nous tàxons, caldrà que els tipus estiguin dipositats en una institució pública. Es recomana als autors la consulta de fascicles recents de la revista per tenir en compte les seves normes.

Animal Biodiversity and Conservation 27.1 (2004)

Animal Biodiversity and Conservation Animal Biodiversity and Conservation (antes Miscel·lània Zoològica) es una revista inter disciplinar, publicada desde 1958 por el Museo de Zoología de Barcelona. Incluye artículos de investigación empírica y teórica en todas las áreas de la zoología (sistemática, taxonomía, morfología, biogeografía, ecología, etología, fisiología y genéti ca) procedentes de todas las regiones del mundo, con especial énfasis en los estudios que de una manera u otra tengan relevancia en la biología de la conservación. La revista no publica catálogos, listas de especies sin más o citas puntuales. Los estudios realizados con especies raras o protegidas pueden no ser aceptados a no ser que los autores dispongan de los permisos correspondientes. Cada volumen anual consta de dos fascículos. Animal Biodiversity and Conservation está re gistrada en todas las bases de datos importantes y además está disponible gratuitamente en internet en http://www.bcn.es/ABC, lo que permite una difusión mundial de sus artículos. Todos los manuscritos son revisados por el editor ejecutivo, un editor y dos revisores independientes, elegidos de una lista internacional, a fin de garan tizar su calidad. El proceso de revisión es rápido y constructivo, y se realiza vía correo electrónico siem pre que es posible. La publicación de los trabajos aceptados se realiza con la mayor rapidez posible, normalmente dentro de los 12 meses siguientes a la recepción del trabajo. Una vez aceptado, el trabajo pasará a ser propie dad de la revista. Ésta se reserva los derechos de autor, y ninguna parte del trabajo podrá ser reprodu cida sin citar su procedencia.

Normas de publicación Los trabajos se enviarán preferentemente de forma electrónica (abc@mail.bcn.es). El formato preferido es un documento Rich Text Format (RTF) o DOC, que incluya las figuras (TIF). Si se opta por la versión im presa, deberán remitirse cuatro copias juntamente con una copia en disquete a la Secretaría de Redacción. Debe incluirse, con el artículo, una carta donde conste que el trabajo versa sobre investigaciones originales no publicadas anteriormente y que se somete en exclusiva a Animal Biodiversity and Conservation. En dicha carta también debe constar, para trabajos donde sea necesaria la manipulación de animales, que los autores disponen de los permisos necesa rios y que han cumplido la normativa de protección animal vigente. Los autores pueden enviar también sugerencias para asesores. Cuando el trabajo sea aceptado los autores de berán enviar a la Redacción una copia impresa de la versión final junto con un disquete del manuscrito preparado con un procesador de textos e indicando el programa utilizado (preferiblemente Word). Las pruebas de imprenta enviadas a los autores deberán ISSN: 1578–665X

remitirse corregidas al Consejo Editor en el plazo máximo de 10 días. Los gastos debidos a modifica ciones sustanciales en las pruebas de imprenta, intro ducidas por los autores, irán a cargo de los mismos. El primer autor recibirá 50 separatas del trabajo sin cargo alguno y una copia electrónica en formato PDF. Manuscritos Los trabajos se presentarán en formato DIN A–4 (30 líneas de 70 espacios cada una) a doble espacio y con las páginas numeradas. Los manuscritos deben estar completos, con tablas y figuras. No enviar las figuras originales hasta que el artículo haya sido aceptado. El texto podrá redactarse en inglés, castellano o catalán. Se sugiere a los autores que envíen sus trabajos en inglés. La revista ofrece, sin cargo ningu no, un servicio de corrección por parte de una persona especializada en revistas científicas. En cualquier caso debe presentarse siempre de forma correcta y con un lenguaje claro y conciso. La redacción del texto deberá ser impersonal, evitándose siempre la primera persona. Los caracteres en cursiva se utilizarán para los nombres científicos de géneros y especies y para los neologismos que no tengan traducción; las citas textuales, independientemente de la lengua en que estén, irán en letra redonda y entre comillas; el nombre del autor que sigue a un taxón se escribirá también en redonda. Al citar por primera vez una especie en el trabajo, deberá especificarse siempre que sea posible su nombre común. Los topónimos se escribirán bien en su forma original o bien en la lengua en que esté redactado el trabajo, siguiendo el mismo criterio a lo largo de todo el artículo. Los números del uno al nueve se escribirán con letras, a excepción de cuando precedan una unidad de medida. Los números mayores de nueve se escribirán con cifras excepto al empezar una frase. Las fechas se indicarán de la siguiente forma: 28 VI 99; 28, 30 VI 99 (días 28 y 30); 28–30 VI 99 (días 28 al 30). Se evitarán siempre las notas a pie de página. Formato de los artículos Título. El título será conciso pero suficientemente explicativo del contenido del trabajo. Los títulos con designaciones de series numéricas (I, II, III, etc.) serán aceptados excepcionalmente previo consen timiento del editor. Nombre del autor o autores. Abstract en inglés de 12 líneas mecanografiadas (860 espacios como máximo) y que exprese la esencia del manuscrito (introducción, material, métodos, resulta dos y discusión). Se evitarán las especulaciones y las citas bibliográficas. Irá encabezado por el título del trabajo en cursiva. Key words en inglés (un máximo de seis) que especifiquen el contenido del trabajo por orden de importancia. © 2004 Museu de Ciències Naturals

Resumen en castellano, traducción del abstract. Su traducción puede ser solicitada a la revista en el caso de autores que no sean castellano hablantes. Palabras clave en castellano. Dirección postal del autor o autores. (Título, Nombre, Abstract, Key words, Resumen, Palabras clave y Dirección postal conformarán la primera página.) Introducción. En ella se dará una idea de los ante cedentes del tema tratado, así como de los objetivos del trabajo. Material y métodos. Incluirá la información referente a las especies estudiadas, aparatos utilizados, me todología de estudio y análisis de los datos y zona de estudio. Resultados. En esta sección se presentarán úni camente los datos obtenidos que no hayan sido publicados previamente. Discusión. Se discutirán los resultados y se compara rán con otros trabajos relacionados. Las sugerencias sobre investigaciones futuras se podrán incluir al final de este apartado. Agradecimientos (optativo). Referencias. Cada trabajo irá acompañado de una bibliografía que incluirá únicamente las publicaciones citadas en el texto. Las referencias deben presentarse según los modelos siguientes (método Harvard): * Artículos de revista: Conroy, M. J. & Noon, B. R., 1996. Mapping of spe cies richness for conservation of biological diversity: conceptual and methodological issues. Ecological Applications, 6: 763–773 * Libros y otras publicaciones no periódicas: Seber, G. A. F., 1982. The estimation of animal abundance. C. Griffin & Company, London. * Trabajos de contribución en libros: Macdonald, D. W. & Johnson, D. P., 2001. Dispersal in theory and practice: consequences for conserva tion biology. In: Dispersal: 358–372 (T. J. Clober, E. Danchin, A. A. Dhondt & J. D. Nichols, Eds.). Oxford University Press, Oxford. * Tesis doctorales: Merilä, J., 1996. Genetic and quantitative trait vari ation in natural bird populations. Tesis doctoral, Uppsala University.

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* Los trabajos en prensa sólo se citarán si han sido aceptados para su publicación: Ripoll, M. (in press). The relevance of population studies to conservation biology: a review. Anim. Biodivers. Conserv. Las referencias se ordenarán alfabéticamente por autores, cronológicamente para un mismo autor y con las letras a, b, c,... para los trabajos de un mismo autor y año. En el texto las referencias bibliográficas se indicarán en la forma usual: "... según Wemmer (1998)...", "...ha sido definido por Robinson & Redford (1991)...", "...las prospecciones realizadas (Begon et al., 1999)..." Tablas. Las tablas se numerarán 1, 2, 3, etc. y se reseñarán todas en el texto. Las tablas grandes deben ser más estrechas y largas que anchas y cortas ya que deben ajustarse a la caja de la revista. Figuras. Toda clase de ilustraciones (gráficas, figuras o fotografías) se considerarán figuras, se numerarán 1, 2, 3, etc. y se citarán todas en el texto. Pueden incluirse fotografías si son imprescindibles. Si las fotografías son en color, el coste de su publicación irá a cargo de los autores. El tamaño máximo de las figuras es de 15,5 cm de ancho y 24 cm de alto. Deben evitarse las figuras tridimensionales. Tanto los mapas como los dibujos deben incluir la escala. Los sombreados preferibles son blanco, negro o trama. Deben evitarse los punteados ya que no se reproducen bien. Pies de figura y cabeceras de tabla. Los pies de figura y cabeceras de tabla serán claros, concisos y bilingües en castellano e inglés. Los títulos de los apartados generales del artículo (Introducción, Material y métodos, Resultados, Discusión, Agradecimientos y Referencias) no se numerarán. No utilizar más de tres niveles de títulos. Los autores procurarán que sus trabajos originales no excedan las 20 páginas incluidas figuras y tablas. Si en el artículo se describen nuevos taxones, es imprescindible que los tipos estén depositados en alguna institución pública. Se recomienda a los autores la consulta de fascículos recientes de la revista para seguir sus directrices.

Animal Biodiversity and Conservation 27.1 (2004)

Animal Biodiversity and Conservation

Manuscripts

Animal Biodiversity and Conservation (formerly Miscel·lània Zoològica) is an interdisciplinary journal which has been published by the Zoological Mu seum of Barcelona since 1958. It includes empirical and theoretical research in all aspects of Zoology (Systematics, Taxonomy, Morphology, Biogeography, Ecology, Ethology, Physiology and Genetics) from all over the world with special emphasis on studies that stress the relevance of the study of Conservation Biology. The journal does not publish catalogues, lists of species (with no other relevance) or punctual records. Studies about rare or protected species will not be accepted unless the authors have been granted all the relevant permits. Each annual volume consists of two issues. Animal Biodiversity and Conservation is registered in all principal data bases and is freely available online at http://www.bcn.es/ABC, thus assuring world–wide access to articles published therein. All manuscripts are screened by the Executive Edi tor, an Editor and two independent reviewers in order to guarantee the quality of the papers. The process of review is rapid and constructive. Once accepted, papers are published as soon as practicable, usually within 12 months of initial submission. Upon acceptance, manuscripts become the prop erty of the journal, which reserves copyright, and no published material may be reproduced without quoting its origin.

Manuscripts must be presented on A–4 format page (30 lines of 70 spaces each) with double spacing. Number all pages. Manuscripts should be complete with figures and tables. Do not send original figures until the paper has been accepted. The text may be written in English, Spanish or Catalan. Authors are encouraged to send their con tributions in English. The journal provides a FREE service of correction by a professional translator specialized in scientific publications. Care should be taken in using correct wording and the text should be written concisely and clearly. Wording should be impersonal, avoiding the use of the first person. Italics must be used for scientific names of genera and species as well as untranslatable neologisms. Quotations in whatever language used must be typed in ordinary print between quotation marks. The name of the author following a taxon should also be written in small print. The common name of the species should be writ ten in capital letters. When referring to a species for the first time in the text, both common and scientific names must be given when possible. Place names may appear either in their original form or in the language of the manuscript, but care should be taken to use the same criteria throughout the text. Numbers one to nine should be written in full in the text except when preceding a measure. Higher numbers should be written in numerals except at the beginning of a sentence. Dates must appear as follows: 28 VI 99, 28,30 VI 99 (days 28th and 30th), 28–30 VI 99 (days 28th to 30th). Footnotes should not be used.

Information for authors Electronic submission of papers is encouraged (abc@mail.bcn.es). The preferred format is a do cument Rich Text Format (RTF) or DOC, including figures (TIF). In the case of sending a printed version, four copies should be sent together with a copy on a computer disc to the Editorial Office. A cover letter stating that the article reports on original research not published elsewhere and that it has been submitted exclusively for consideration in Animal Biodiversity and Conservation is also necessary. When animal manipulation has been necessary, the cover letter should also especify that the authors follow current norms on the protection of animal species and that they have obtained all relevant permissions. Authors may suggest referees for their papers. Once an article has been accepted, authors should send a printed copy of the final version together with a disc. Please identify software (prefer ably Word). Proofs sent to the authors for correction should be returned to the Editorial Board within 10 days. Expenses due to any substantial altera tions of the proofs will be charged to the authors. The first author will receive 50 reprints free of charge and an electronic version of the article in PDF format.

ISSN: 1578–665X

Formatting of articles Title. The title must be concise but as informative as possible. Part numbers (I, II, III, etc.) should be avoided and will be subject to the Editor’s consent. Name of author or authors. Abstract in English, no longer than 12 typewritten lines (840 spaces), covering the contents of the article (introduction, material, methods, results and discussion). Speculation and literature citation must be avoided. Abstract should begin with the title in italics. Key words in English (no more than six) should express the precise contents of the manuscript in order of importance. Resumen in Spanish, translation of the Abstract. Summaries of articles by non–Spanish speaking authors will be translated by the journal on request. Palabras clave in Spanish. Address of the author or authors. (Title, Name, Abstract, Key words, Resumen, Palabras clave and Address should constitute the first page.) Introduction. The introduction should include the historical background of the subject as well as the aims of the paper.

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Material and methods. This section should provide relevant information on the species studied, materials, methods for collecting and analysing data and the study area. Results. Report only previously unpublished results from the present study. Discussion. The results and their comparison with related studies should be discussed. Suggestions for future research may be given at the end of this section. Acknowledgements (optional). References. All manuscripts must include a bibliogra phy of the publications cited in the text. References should be presented as in the following examples (Harvard method): * Journal articles: Conroy, M. J. & Noon, B. R., 1996. Mapping of spe cies richness for conservation of biological diversity: conceptual and methodological issues. Ecological Applications, 6: 763–773. * Books or other non–periodical publications: Seber, G. A. F., 1982. The estimation of animal abundance. C. Griffin & Company, London. * Contributions or chapters of books: Macdonald, D. W. & Johnson, D. P., 2001. Dispersal in theory and practice: consequences for conserva tion biology. In: Dispersal: 358–372 (T. J. Clober, E. Danchin, A. A. Dhondt & J. D. Nichols, Eds.). Oxford University Press, Oxford. * Ph. D. Thesis: Merilä, J., 1996. Genetic and quantitative trait vari ation in natural bird populations. Ph. D. Thesis, Uppsala University. * Works in press should only be cited if they have been accepted for publication: Ripoll, M. (in press). The relevance of population studies to conservation biology: a review. Anim. Biodivers. Conserv. References must be set out in alphabetical and

Animal Biodiversity and Conservation 27.1 (2004)

chronological order for each author, adding the letters a, b, c,... to papers of the same year. Bibliographic citations in the text must appear in the usual way: "... according to Wemmer (1998)...", "...has been defined by Robinson & Redford (1991)...", "...the prospections that have been carried out (Begon et al., 1999)..." Tables. Tables must be numbered in Arabic numerals with reference in the text. Large tables should be narrow (across the page) and long (down the page) rather than wide and short, so that they can be fitted into the column width of the journal. Figures. All illustrations (graphs, drawings or photographs) must be termed as figures, num bered consecutively in Arabic numerals (1, 2, 3, etc.) and with reference in the text. Glossy print photographs, if essential, may be included. Colour photographs may be published but its publication will be charged to authors. Maximum size of figures is 15.5 cm width and 24 cm height. Figures will not be tridimensional. Both maps and drawings must include scale. The preferred shadings are white, black and bold hatching. Avoid stippling, which does not reproduce well. Legends of tables and figures. Legends of tables and figures must be clear, concise, and written both in English and Spanish. Main headings (Introduction, Material and methods, Results, Discussion, Acknowledgements and Refe rences) should not be numbered. Do not use more than three levels of headings. Manuscripts should not exceed 20 pages including figures and tables. If the article describes new taxa, type material must be deposited in a public institution. Authors are advised to consult recent issues of the journal and follow its conventions.

Animal Biodiversity and Conservation 27.1 (2004)

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Animal Biodiversity and Conservation Subscription Form  Please enter our subscription to Animal Biodiversity and Conservation  66.11 e Spain  68.52 e Europe  69.12 e rest of world  Single use subscription:  21.04 e Spain  23.4 4 e Europe  24.04 e rest of world  Please despatch my issues by air mail (supplement of 6.01 e for outside Europe)  Please send me the Instructions to authors

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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

Animal Biodiversity and Conservation 27.1 (2004)

Welcome to the electronic version of Animal Biodiversity and Conservation

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http://www.bcn.cat/ABC

Animal Biodiversity and Conservation joins the recent worldwide Open Access Initiative of providing a permanent online version free of charge and access barriers. This is the result of the growing point of view that open access to research is essential for efficient and rapid scientific communication.

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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

Les cites o els abstracts dels articles d’Animal Biodiversity and Conservation es resenyen a / Las citas o los abstracts de los artículos de Animal Biodiversity and Conservation se mencionan en / Animal Biodiversity and Conservation is cited or abstracted in: Abstracts of Entomology, Agrindex, Animal Behaviour Abstracts, Anthropos, Aquatic Sciences and Fisheries Abstracts, Behavioural Biology Abstracts, Biological Abstracts, Biological and Agricultural Abstracts, Current Primate References, Ecological Abstracts, Ecology Abstracts, Entomology Abstracts, Environmental Abstracts, Environmental Periodical Bibliography, Genetic Abstracts, Geographical Abstracts, Índice Español de Ciencia y Tecnología, International Abstracts of Biological Sciences, International Bibliography of Periodical Literature, International Developmental Abstracts, Marine Sciences Contents Tables, Oceanic Abstracts, Recent Ornithological Literature, Referatirnyi Zhurnal, Science Abstracts, Serials Directory, Ulrich’s International Periodical Directory, Zoological Records.

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