Spatial positioning in the selfish herd”, Lesley J. Morrell, Graeme D. Ruxton, Richard James by le-5015

OriginalArticle

Spatialpositioningintheselﬁshherd

LesleyJ.Morrell,a GraemeD.Ruxton,b andRichardJamesc aInstituteofIntegrativeandComparativeBiology,UniversityofLeeds,LCMiallBuilding,LeedsLS29JT, UK, bDivisionofEcologyandEvolutionaryBiology,FacultyofBiomedicalandLifeSciences,University ofGlasgow,GlasgowG128QQ,UK,and cDepartmentofPhysics&CentreforMathematicalBiology, UniversityofBath,BathBA27AY,UK

Theantipredatorbenefitsofgroupingarerelativelywellunderstood;however,predationriskoftendiffersforindividualsthat occupydifferentpositionswithinagroup.Theselfishherdhypothesisdescribeshowindividualscanreduceriskofpredationby movingtospecificpositionswithinthegroup.Inexistingtheory,thismovementoccursthroughtheadoptionofpossible ‘‘movementrules’’thatdifferintheircognitivecomplexity.Here,weinvestigatetheeffectivenessofdifferentpreviouslysuggested rulesinreducingriskforcentralandperipheralindividualswithinagroup.Wedemonstratethatinitialspatialpositionis importantindeterminingthesuccessofdifferentrisk-reducingmovementrules,asinitiallycentrallypositionedindividuals arelikelytobemoresuccessfulthanperipheralonesatreducingtheirriskrelativetoothergroupmembers,regardlessofthe movementrulesused.Simplerstrategiesareeffectiveinlow-densitypopulations;butathighdensity,morecomplexrulesare moreeffective.Wealsofindthatcomplexrulesthatconsiderthepositionofmultipleneighborsaretheonlyrulesthat successfullyallowindividualstomovefromperipheraltocentralpositionsormaintaincentralpositions,thusavoidingpredators thatattackfromoutsidethegroup.Ourresultssuggestthattheattackstrategyofapredatorshouldbecriticallyimportantin determiningpreyescapestrategiesinaselfishherdcontextandthatpreyshouldmodifytheirbehavioralresponsestoimpending attackinresponsetotheirpositionwithinagroup. Keywords: aggregation,antipredatorbehavior,grouping,groupcenter, individual-basedmodel,periphery,predationrisk. [BehavEcol22:16–22(2011)]

Predationriskhasbeenwidelydemonstratedtobeakeyfactordrivingthebehavioralecologyofmanyanimals(Caro 2005).Oneofthemost-studiedresponsestoheightenedpredationriskisgroupingbehavior(KrauseandRuxton2002).It iswidelyrecognizedthatunderheightenedpredationrisk, groupstendtobelarger(HagerandHelfman1991; Hoare etal.2004),morecompact(FosterandTreherne1981; Watt etal.1997; SpielerandLinsenmair1999),andindividuals withingroupstendtobemorecloselyassortedbyphenotype (AllanandPitcher1986; Theodorakis1989; Szulkinetal. 2006; Croftetal.2009).Theselﬁshherdtheory(Hamilton 1971)hasbeenparticularlyinﬂuentialinunderstandingsuch facultativeaggregationinresponsetoheightenedpredation risk.

Theselﬁshherdhypothesisassumesthattherelativepredationriskof2individualscanbedeterminedbycomparisonof theareasaroundeachthatisclosertothefocalindividual thantoanyotherindividuals:thedomainofdanger(DOD).

Predatoryattacksareassumedtobelaunchedfromrandom pointswithintheenvironment,withallpositionsbeing equallylikelytobelaunchpoints.Ifanattackislaunched fromwithinaparticularindividual’sDOD,thenthatindividualisassumedtobeattackedandkilled(Hamilton1971).

Thus,anindividualprey’sriskofpredationisproportional tothesizeoftheirDODandtoreducepredationrisk,each animalshouldendeavortoreduceitsDODsizerelativeto thoseofotherindividuals.Thisnaturallyleadstogreateraggregation.Morerecenttheoreticaldevelopmentshavefocusedonevaluatingcandidatebehavioral‘‘movementrules’’ thatcauseindividualstoreducetheirDOD(Mortonetal.

AddresscorrespondencetoL.J.Morrell.E-mail:l.j.morrell@leeds .ac.uk.

Received26April2010;revised31August2010;accepted1 September2010.

TheAuthor2010.PublishedbyOxfordUniversityPressonbehalfof theInternationalSocietyforBehavioralEcology.Allrightsreserved. Forpermissions,pleasee-mail:journals.permissions@oup.com

1994; Viscidoetal.2002; Jamesetal.2004; WoodandAckland 2007; MorrellandJames2008; Morrelletal.2010).

Thecostsandbenefitsofgroupingarenotalwaysexperiencedequallybyallmembersofagroup(KrauseandRuxton 2002).Thetheoryofmarginalpredation(Hamilton1971; Vine 1971)suggeststhatifpredatorsattacktheclosestprey,then thoseontheedgeofgroupsshouldexperiencegreaterrisk, eitherbecausetheyhavethelargestDODsorbecausepredatorsaresimplymorelikelytoattackfromoutsideagroupthan withinit.Simulationmodelshaveshownthatperipheralindividualsshouldbeatgreaterrisk(Bumannetal.1997; Morrell andRomey2008),andempiricalevidencesuggeststhatthisis indeedoftenbutnotalwaysthecase(Okamura1986; Rayor andUetz1990; Sa´lekandSmilauer2002; Stankowich2003; Romeyetal.2008,butsee Parrish1989; QuinnandCresswell 2006).Anindividual’spositionwithinagroup,therefore, mightoftenbeexpectedtoaffectitsresponsetoapredation event.Previoustheoryhasassumedthatallindividualsfollow thesamemovementrulesinresponsetocuesofanimminent predatoryattack.Here,weinvestigatehowpositionwithin agroupaffectsantipredatorbehaviorinthecontextofthe selfishherd,byinvestigatingthesuccessofa‘‘mutant’’movementruleinvadingagroupofindividualsusingadifferent movementruleinresponsetoapredationthreat(seeMETHODSfordescriptionsofdifferentmovementrules).Wewill considerbothsituations,wherepredationoccursfromapositionoutsidethegroup,aswellassituationswhereattackscan belaunchedfromanypointinspace.Weconsiderhowthe position(centralorperipheral)inwhichanindividualfinds itselfwhenitinitiallyrespondstothepredationthreataffects thesuccessofdifferentmovementrulesinreducingrelative DODareas.Wepredictthatstartingpositionwillinfluencethe relativesuccessofdifferentmovementrulesinreducingDOD areasandthereforepredationriskofindividuals.

Previoustheoreticaldevelopmentsexploringtheeffectivenessofdifferentmovementrulesevaluatedtheeffectiveness

BehavioralEcology doi:10.1093/beheco/arq157 AdvanceAccesspublication12October2010

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ofsuchrulespurelyintermsoftheirabilitytoreducethe mover’sDOD.Thisisreasonablebecause,asoriginallyformulated,theSelﬁshHerdtheoryassumedthatpredatoryattackswereequallyaslikelytobelaunchedfromanyposition withintheenvironment.Manysubsequentworkshaveretainedthisassumption(e.g., Mortonetal.1994; Viscido etal.2002; Jamesetal.2004; MorrellandJames2008, Morrell etal.2010),andweusethisapproachheretoallowfor comparisonwithpreviousstudies.Insomecircumstances (e.g.,avianpredatorsattackingpreyfromabove; Quinnand Cresswell2006),suchanapproachisappropriate.However,as Jamesetal.(2004) argue,inmanypredator-preyassociations, attacksaregenerallyunlikelytooccurfrompositionswithin thegroup.Forexample,asagroupmovesthroughitsenvironment,itislikelythatanambushingpredatorwaitinginthe pathofthegroupwouldbedetectedbeforethegroupmoves overitsposition.Hence,inmanyecologicalsituations,predatoryattacksongroupedpreywilloccurexclusivelyfromoutsidethegroup.Insuchcircumstances,thereisastrong premiumtoagroupmemberinbeingintheinteriorofthe groupbecauseperipheralindividualswillbeatmuchgreater risk.Accordingly,inasecondanalysis,weinvestigatehoweffectivedifferentmovementrulesareatplacingtheiruser withinthecenterofagroup.

METHODS

Modelframework

Weusethemodelingframeworkdescribedby Jamesetal. (2004) and MorrellandJames(2008) asthebasisforour simulationmodelofaggregationbehaviorandprovideasummaryhere. N point-likeagents(theprey)areplacedina 2-dimensionalcirculararenaofradius R usingarandomuniformdistribution.Thatis,initially,thereisnoaggregation,

ApproximateUDODs(a)and LDODs(b)foranillustrative groupof11animals.IndividualswithinfiniteUDODsand peripheralpositionsarecoloredgray,whereasthosewith finiteUDODsandcentralpositionsarecoloredblack.Panels (c)and(d)illustratethepositionofamutantindividualusingLCHinagroupof individualsusingNN,showing howamutantindividualwhich beginsontheperipheryofthe group(c)canacquireacentral positionafterseveralseconds ofmovement(d).Initialpositionsareasin(a)and(b).Individualsstartinginperipheral positionsaregray,andcentral individualsareblack.Dotted linesin(d)donotshowmovementtrajectories(whichare usuallycurved),butsimply linkfinalpositionstooriginal startpositions(opendashed circles).Individualcolorsin (d)representstartingposition (central/peripheral/mutant). Thesefiguresareforillustrativepurposesonlyanddonot representpositionsusedinthe analysis.

andindividualsareplacedwithoutconsiderationtothepositionsofothers.Populationdensity, d,isdescribedby N/pR2 Ineachsimulation, Nm agentsareallocateda‘‘mutant’’movementrule,whereas Np agentsareallocateda‘‘population’’ movementrule(Nm 1 Np ¼ N).Weusethe‘‘limiteddomain ofdanger’’(LDOD)frameworktodescriberelativepredation risks,thisisthoughttomorerealisticallyportrayindividual predationriskthanthetraditionaldefinition,inwhichsome individualshaveinfiniteDODs(Jamesetal.2004; Morrelland James2008).Theconventionalor‘‘unlimiteddomainofdanger’’(UDOD)ofanagentin2Dspaceisdefinedastheregion closertothatagentthananyotherandisthemeasureof individualpredationriskintroducedin Hamilton(1971) TheLDODofanagentistheregionthatisinsideboththe UDODandinsideacircleofradius r centeredontheagent (see Figure1 forillustrationofUDODsandLDODs).For eachagent,wecalculatethearea(A)oftheirLDOD.The maximumLDODarea,whichoccursonlywhentheagentis atleast2r fromanyothers,isgivenby Amax ¼ pr 2.LDODareas arereducedbyanybisectorgeneratedbyanindividualwithin adistanceof2r.WealsocalculateUDODareasusingVoronoi tessellations:Thisinformationislaterusedtoidentifycentral andperipheralagents.

Movementrules

Weconsiderasubsetofprevious-proposedmovementrules, encompassingvariationincomplexity:nearestneighbor (NN; Hamilton1971),multipleNNs(3NN; Mortonetal. 1994),andlocalcrowdedhorizon(LCH; Viscidoetal. 2002).UndertheNNrule,agentsmovedirectlytowardtheir closestneighbor,underthe3NNrule,individualsmovetowardtheaveragelocationoftheir3closestneighbors,and undertheLCHrule,thepositionofmultipleneighborsis

Figure1

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takenintoaccount,withcloseroneshavingthegreatestinfluenceonmovementdirection.Todescribethiseffect,we weighttheinfluenceofindividualsusingtheperceptionfunctionsuggestedby Viscidoetal.(2002) asbeingthemostbiologicallyplausible:f x ¼ 1 110 375x,where x isthedistance fromthefocalindividual.Anindividual’smovementdirection isdeterminedsolelybyitsmovementrule:agentsdonotreceive anydirectionalinformationonpredatorattackdirection,and thehabitatcontainsnorefugeareas.Thatis,atthestartof asimulation(t ¼ 0),thepreyareassumedtoinitiatetheir behavioralrulesinresponsetopickingupcuesofanimminent predatoryattack.Thesecuesinformthepreythatanattackis likelyintheimmediatefuture,buttheydonotprovideinformationastothepositionfromwhichtheattackwillbe launched.Allpreyareassumedtopickupthesecuesatthe sametime.Thewarningcuesmightbethefleeingoralarm callingofnearbyheterospecifics,forexample.Wethensimulate themovementoftheanimalsfromthispointuntiltheattack actuallyoccurs.Simulationsendatthepointofattack(t ¼ tmax), wheretherelativevulnerabilitiesofindividualsareevaluated.

Ineachtimestep t,untilamaximum tmax,eachagentidentifiesitstargetlocation,basedonthemovementruleitis following,andthenmovestowarditataspeedof0.15m/s (Jamesetal.2004; MorrellandJames2008).Eachagentupdatesitstargetlocationanddirectionineverytimestep,and atimestepinoursimulationslasts0.1s.Aftereverytimestep andforeveryindividual,wecalculatedboththeLDODarea andtheUDODarea.UDODareasareusedtodefinecentral andperipheralpositions(peripheralindividualshaveinfinite UDODs),andLDODareasmeasureindividualpredation risk. Figure1a,b illustratesagroupofanimalswithUDODs (Figure1a)andLDODs(Figure1b)marked.Weran1000 simulationsforeachpossiblepairwisecombinationofmutant versuspopulationmovementrules(6combinationsintotal). However,westandardizedstartingpositionsbetweenmovement rules,suchthatallmutantversuspopulationcombinationswere runwiththesamesetof1000initialpositions.Allsimulations wereprogrammedinC.Resultingdatawereanalyzedusing Matlab R2007b.

Ourmodelclearlydoesnotaccuratelydepictany1particularreal-worldsystem.Ouraimwastoexploretheprinciples thatmayunderliethe‘‘movementtowardconspecifics’’aspect ofantipredatorresponses(seenintherealworldintheaggregationofpreyanimalsunderthreatofpredation;seethe INTRODUCTIONforempiricalexamplesofthis).Asoneof ouraimswastoprovidecomparisonwith,anddevelopment of,previousselfishherdmodels,weexploretheprinciplesin thesamemodelingframeworkasusedbypreviousworks.Although,forsimplicity,weexplorea2-dimensionallandscape inourmodel,ourresultscouldcertainlybeextrapolatedfrom to3Dones(suchasfishshoalsorbirdflocks).

Researchquestions

Weusethedatafromthemodeltoanswer2questions:1)how doesstartingposition(peripheralorcentral)inﬂuencethe successofamutantinvadingapopulationintermsofitspredationriskrelativetoothergroupmembers(proportionof totalLDODareaoccupiedbythemutantandanaverage memberofthepopulation),therebyavoidingpredatorsthat attackrandomlyand2)howsuccessfularethedifferentmovementrulesinallowinganindividualtomovefromperipheral tocentralpositions,therebyavoidingpredatorsthatattack fromoutsidethegroup?Theﬁrstofthesecomplementary questionsfollowsthetraditionalapproachofcomparing DODareasbetweenindividuals,whereasthesecondexplores riskavoidancewhenpredatorsareexpectedtoattackfrom outsidethegroup.

Approach1:avoidingriskbyreducingLDODarearelativeto others:theeffectofspatialposition

Toassessrelativepredationriskandtheabilityofamutant strategytoinvadeapopulationstrategy,ateachtimestep,we firstcalculatedthetotalLDODarea, Atot (thesumof A forall individuals).Wethencalculatedtheproportionof Atot occupiedbythemutantandtheproportionoccupiedbyeach populationmember.Wethencalculatedthemeanvaluefor thepopulation.Finally,wecalculatedthedifferencebetween theproportionof Atot occupiedbythemutantandthemean proportionoccupiedbyanaveragepopulationmember.This wasrepeatedforeachtimestepineachreplicatesimulation.A positivevalueindicatesthatthemutantoccupiesalargerproportionofthetotalareathananaveragepopulationmember, andasmallervalueindicatedthatthemutantoccupiesasmallerareaandcansuccessfullyinvadethepopulation.ThisapproachassumesthatpredationriskisbasedonlyonLDOD area(i.e.,thatpredatorscanattackanywherewithinthegroup andfollowsthesameapproachusedinpreviousstudiesofthe selfishherd(Hamilton1971; Mortonetal.1994; Viscidoetal. 2002; Jamesetal.2004; MorrellandJames2008, Morrelletal. 2010).Previousworkhasshownthatbothpopulationsize(N) anddensity(d)canaffectrulesuccess(MorrellandJames 2008),andwevarythesehere,predictingthatcomplexrules willperformbetterinsmallerhigherdensitypopulations. Weareinterestedinhowspatialpositionaffectschangein LDODarea(andthereforereductioninrelativerisk).Wedefineaperipheralindividualasanymemberofagroupwithan infiniteDOD,usingtheUDODframework.Anyindividual withafiniteDODisconsideredcentral(thisisequivalentto usingtheminimumconvexpolygonapproachtodefining peripheralindividualswithinagroup; KrauseandTegeder 1994; KrauseandRuxton2002).Werecordthestartingposition(peripheralorcentral)forthemutantindividualineach simulationandusethisinformationtosplitthedatacollected onLDODareas.Wethencalculatethemeanandstandard erroracrossreplicatesformutantsthatwereperipheral(414/ 1000replicateswhen N ¼ 20,233/1000replicateswhen N ¼ 50)andcentral(586and767replicates,respectively)atthe startofthesimulation.Wealsocalculatethemeanandstandarderroracrossallindividuals,regardlessofwhetherthey startedinaperipheralorcentralposition.Forvisualsimplicity,wesubsampleourresultsandshowonlythedifferencesin LDODareasat2pointsintime,following MorrellandJames (2008):after2sofmovement(tmax ¼ 2, Figure2,lefthand column)andafter10sofmovement(tmax ¼ 10, Figure2, righthandcolumn).Thefirstoftheserepresentsa‘‘earlyattacking’’predatorthatcancompleteanattackwithin2s ofcuesofitspresencebeingdetected;thesecondrepresentinga‘‘laterattacking’’predatorthatrequireslonger.Biologically,thisvariationmightbeinterpretedasreflectinghow quicklyanambushingpredatorcanclosethedistancefrom thepointthatitbreakscovertothepointwhereitcancompleteanattack:anearly-attackingpredatorcancoverthisdistancequickly,soaggregatingpreyhaverelativelylittletimeto respondtoanycuesofimminentattack(whicharetriggered whenthepredatorbreakscover).

Approach2:avoidingriskbymovingtothegroupcenter Foramovementruletobeconsideredsuccessfulintermsof reducingtheriskofperipheralpredationforindividualsusing thatrule,theproportionofmutantindividualsoccupyingperipheralpositionsshouldbelowerthanexpectedbychance oncemovementhasbegun(seebelow).Usingthedeﬁnition ofperipheralabove,foreachindividual(mutantandpopulationmember)withineachgroup,werecordwhethertheir

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positionisperipheralorcentralatthestartofthesimulation, after2sofmovement,andafter10sofmovement.

Foreachreplicatesimulation,wecalculatetheproportionof individualsoccupyingcentralpositionsandtheproportion occupyingperipheralones.Wealsocalculatetheproportion ofthe1000simulations,wherethemutantindividualoccupies

Figure2

Mean(62standarderror)differencebetweentheproportionoftotalLDODarea occupiedbythemutantand themeanproportionoftotal LDODareaoccupiedbypopulationmembers,after2s(left handcolumn)and10s(right handcolumn)ofmovement. Dataarepresentedasthe meanformutantsstartingin allpositions(ﬁlledcircles) plusthosestartingincentral (opencircles)andperipheral (opendiamonds)positions.

Panelsshowasinglemutant (Nm ¼ 1)inagroupof20individuals(N ¼ 20),at3populationdensities:(aandb) d ¼ 2;(candd) d ¼ 4;(eandf) d ¼ 10,andasinglemutant (Nm =1)inagroupof50individuals(N =50)at d =4(gand h).Whereerrorbarsarenot shown,theyaresmallerthan thesizeofthesymbol.

acentralpositionandtheproportionwhereitoccupiesaperipheralone.Thisallowsustoassesswhethermutantsare moreorlesslikelytooccupycentralpositionsthanwould beexpectedbychance.Iftheprobabilityofamutantoccupyingaperipheralpositionislowerthanthemeanproportion ofthegroupthatisontheperiphery,thenthemutantismore

a)b) c) d) e)f) g)h)

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likelytooccupyacentralpositionthanwouldbeexpectedby chance.Forexample,ifonaverage,40%ofthegroupmembersarepositionedontheperipheryatanygivenpointinthe movementsequence,wewouldexpectasinglemutantwithin thatgrouptobeontheperipheryin40%ofthesimulations. Ifthemutantisperipheralinlessthan40%ofthesimulations,thisislowerthanonewouldexpectbychance,andthe ruleissuccessfulatmovingtheindividualtothecenterofthe group.Ifitisperipheralinmorethan40%ofthesimulations, thentheruleisunsuccessful. Figure1c,d illustratesthepositionsoftheindividualsinagroupofanimalsbefore(Figure1c) andafter(Figure1d)movement,demonstratinghowaperipheralmutantcanmovetoacentralposition.

RESULTS

Approach1:avoidingriskbyreducingLDODarearelativeto others:theeffectofspatialposition

Wedeﬁneasuccessfulinvasionasonewherethedifferencein theproportionofthetotalLDODareabetweenthemutant andthepopulation(seeMETHODS)isnegative.In Figure2, equalsuccessoccursat0,indicatedbyahorizontaldashed line.Ifthedatapointrepresentingthemeandifferencelies belowthisline(i.e.,isnegative),themutantcaninvade.Ifthe pointisabovethisline(i.e.,ispositive),thepopulationis stableagainstinvasion.

First,weconsiderthesuccessofmutantindividualsinobtainingsmallerLDODsthanpopulationmembers,regardless oftheirstartingposition(ﬁlledcirclesin Figure2),andﬁnd thattherearecleardifferencesintheabilityofmutantsto successfullyinvadeapopulation.Forexample,inasmall lowdensitypopulation(N ¼ 20, d ¼ 2),andifpredatorsattack after2sofmovement(Figure2a),theonlymutantsableto invadeareNNmutantsinLCHpopulationsand3NNmutants inLCHpopulations.Inallothermutantpopulationcombinations,thepopulationisstableagainstinvasion.Incontrast, after10sofmovement(Figure2b),NNpopulationsareunstableandcanbeinvadedbymutantsusingboth3NNand LCH.

However,ifwedividethedataandconsiderseparatelythose mutantsthatstartinthecenterofthegroup(opencircles), andthosethatstartontheperiphery(opendiamonds),somewhatdifferentpatternsemerge.Ingeneral,centrallypositionedmutantsaremorecapableofinvadingpopulations thanaretheirperipherallypositionedcounterparts.Thatis (regardlessofbehavioralrulesadoptedbyindividuals),those individualsthat—forwhateverreason—ﬁndthemselvesaway fromtheperipherywhenmovementsbeginareatanadvantage.Indeed,ifamutantisontheedgeofthegroupatthe startofsimulations,itisunabletoinvadethegroupafter2sof movement(i.e.,ifpredatorsattackearly; Figure2a,opendiamonds).Ifpredatorsattacklate(after10sofmovement, Figure2b),thenNNpopulationsbecomeunstableagainst evenperipheralmutantsusing3NNandLCHandarethemselvesunabletoinvadepopulationsusingthesestrategies,regardlessoftheirstartingposition.

Wealsoconsideredtheimpactofalteringthestartingdensity andsizeofthepopulationontherelativeinvasionsuccessof peripheralandcentralmutants(Figure2).After2sofmovement(Figure2,lefthandcolumn),increasingthestarting densityofthegroup(Figure2a–e)resultsisadecreasein performanceforNNmutants;theybecomeincreasinglyless abletoinvadepopulationsusingotherstrategiesandincreasinglysusceptibletoinvasionby3NNandLCHmutants.Centrallypositionedmutantscontinuetobemoresuccessfulin theirinvasionthanperipheralones;inparticular,central mutantsusingsimplerrulesareabletoinvadeapopulation

usingmorecomplexrules.Ifpredatorsattacklater(after10s, Figure2,righthandcolumn),thenincreasingpopulation densityresultsinincreasedstabilityofcomplexrulesand increasedinvasionsuccessofcomplexrules(Figure2b,d,f). Even,peripheralmutantsusingmorecomplexrulesareable toinvadepopulationsusingsimplerrules.

Increasingpopulationsizehassimilareffectstodecreasing populationdensity(Figure2g,h).Increasinggroupsizefrom 20to50individuals(whilecontrollingfordensity;comparing Figure2c,g)suggeststhatinlargerpopulations,simplerrules aremoreeasilyabletoinvade.Forexample,consideringall mutants,NNcaninvadeLCHafter2sofmovementinagroup of50(Figure2g)butnotinagroupof20(Figure2c).After 10sofmovement(Figure2,righthandside),weseedecreasedstabilityofpopulationsusingcomplexrulesaspopulationsizeincreases(Figure2d,h).Forexample,centrally positioned3NNmutantscaninvadeLCHpopulationsat N ¼ 50(Figure2h)butnotat N ¼ 20(Figure2d),and 3NNpopulationsarestableagainstinvasionbycentralLCH mutantsat N ¼ 50,butnotat N ¼ 20.

Approach2:avoidingriskbymovingtothegroupcenter Whenweconsidertherelativeproportionsofmutantsand populationmembersoccupyingperipheralpositions,weﬁnd thatforallpopulationdensitiesstudied,theresultsaresimilar (Figure3).NNmutantsaremorelikelytooccupyperipheral positionsafter2and10sofmovementthanexpectedby chance,whenattemptingtoinvadeboth3NNandLCHpopulations.Conversely,both3NNandLCHmutantsareless likelytooccupyperipheralpositionsinprimarilyNNgroups (Figure3).LCHisalsolesslikelytooccupyperipheralpositionsinNN3groups,althoughNN3mutantsaremorelikely tooccupyperipheralpositionsinLCHgroups.Thesepatterns holdwhenthegroupissampledat2or10sandaremore pronouncedasthestartingdensityofthegroupincreases (Figure3b,c),suggestingthatLCHisthemostsuccessful methodofoccupyingcentralpositions,followedby3NN. Thereisalsoverylittleeffectofincreasingpopulationsize (to N ¼ 50):patternsremainthesame,althoughoverallproportionsofperipheralindividualsarereducedincomparison tosmallergroupsaswouldbeexpected(datanotshown).

DISCUSSION

Ourresultssuggestthatthepositionofanindividualwithin agroupiscriticallyimportantindeterminingthesuccessof movementrulesinallowingtheindividualtoavoidpredation. IfweconsiderpredationriskintermsofrelativeLDODarea (approach1)andassumethatpredatorsattackfromarandomlychosenposition(includingpositionswithinthegroup, withaprobabilitydependentonLDODarea; Hamilton1971), thenweseepatternsreflectingpreviousfindings,wherethe densityofthegroupandpredatorattacktimingstronglyaffect thesuccessofdifferentmovementrules(Morrelletal.2010). Inlowdensitypopulations,simplestrategiesareabletoinvade populationsusingmorecomplexstrategiesandarestable againstinvasionwhenpredatorsattackrapidly,butmorecomplexstrategiessucceedwhenpredatorsattackmoreslowly.In particular,weshowthatcentrallypositionedmutantsaremore capableofinvasionthanareperipherallypositionedmutants. Individualsusingsimplestrategiescanalsobenefitviathe encounter-dilutioneffect(TurnerandPitcher1986)inlowdensitypopulationswhenpredatorsattackrapidly(Morrell andJames2008).Intheencounter-dilutioneffect,allindividualsinagroupbenefitequallyfromaggregationreducingthe rateatwhichpredatoryattacksarelaunched.

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Mean(6standarddeviation)proportionofagroupofindividuals occupyingaperipheralposition(opencircles)andproportionof simulationrunswherethemutantoccupiedaperipheralposition (ﬁlledcircles)atthestartofthesimulations,after2sandafter10s, foreachmutant-populationcombination.(a) d ¼ 2,(b) d ¼ 4(c) d ¼ 10.Otherparametervalues: N ¼ 20, Nm ¼ 1.

However,ifpredatorsattackonlyfromoutsideagroup (approach2),primarilytargetingperipheralindividuals,we ﬁndthattheresultsdiffer.Simplerules(NN)nolongerperformwellagainstmorecomplexrulesastheydonotallow individualsusingthemtogaincentralpositionsandaremore likelytomeanthatanindividualthatbeginsinthecenter cannotmaintainthatposition.Populationdensityisnolonger importantindeterminingsuccessmeasuredinthisway.The morecomplexstrategiesperformsigniﬁcantlybetterinallowingindividualstogainandmaintaincentralpositions.

Itisperhapsunsurprisingthatthemorecomplexrulesresult inahigherproportionofindividualsendingupincentralpositions.Ifweimagineasimplegroupof3individuals,anindividualthatchoosestomovetotheaveragelocationofits2 companionswillbeaimingtoendupbetweenthem(i.e.,in thecenter,althoughinagroupof3ina2-Denvironment,all willhaveinﬁniteUDODs).Extendingtolargergroupsand

rulestakingintoaccountmoreneighbors,itbecomesclear thatcomplexruleswillusuallyleadtomovementtowardthe centerofthegroup.Thiswillresultinthekindofcompactionofgroupsnormallyseeninthewild.Inexperimental trials,sticklebacks(Gasterosteusaculeatus; KrauseandTegeder 1994),toadtadpoles(Bufobufo and Bufomaculatus; Wattetal. 1997, SpielerandLinsenmair1999),andbandedkilliﬁsh (Fundulusdiaphanus; Hoareetal.2004)allshowincreased densitiesfollowingapredatorstimulus,althoughtheprecise mechanismsandmovementrulesaremostlyunknown.

Ifgroupsareprimarilyattackedfromoutsidethegroup (Vine1971),ratherthanfromanywherewithintheenvironment(Hamilton1971),thenonecanimaginethatmovement towardthegroupcenterwillevolve,eitherasaruleinitselfor viaarulethatinvolvesanumberofNNs(perhapscognitively simplerthanmovementtowardthecenter,whichwouldrequireknowledgeofthepositionsofallindividualswithinthe groupratherthanafewclosestneighbors).Inmanycases, preyanimalswillreceivesomedirectionalinformationregardingthepositionofthepredator,andtheirmovementwill incorporatethisaspectinadditiontomovementtowardconspeciﬁcsaseachseekstheiroptimalpositionwithintheﬂeeinggroup(Viscidoetal.2001).

Theremaybesomelimitations,however,tothesuccessof complexrules.First,complexmovementruleshavebeencriticizedonthegroundsthattheymaybetoodifﬁcultforanimalstofollow(Mortonetal.1994; Viscidoetal.2002; Morrell andJames2008),forexample,individualsmaybelimitedin theirknowledgeofthepositionsofothergroupmembers. Second,ifanimalsareunawareofwhereapredatorattackis likelytocomefrom(i.e.,ifpredatorsattackrandomlyandthis sometimesrepresentsanattackfromoutside,andatother times,anattackfromwithintheareaoccupiedbythegroup), thentheremaybeatrade-offbetweenmovingtowardcentral locationsandreducingriskthrough,forexample,encounterdilutioneffects.Recentworkstudyingsticklebacksattacking swarmsof Daphnia hasshownthatdenserareasofgroupsare moreconspicuoustopredators(Ioannouetal.2009),yetindividualsbeneﬁtfrombeingindenserpartsofgroups throughaperceptualinabilityofpredatorstotargetindividualswithinthegroup(theconfusioneffect; Krakauer1995). Widelyspacedredshanks(Tringatetanus)arealsomorelikely tofallvictimtosparrowhawk(Accipiternisus)predators (QuinnandCresswell2006).

Insomespecies,theabilityofanindividualtooccupycentral positionsmaybelimitedbydominancehierarchieswithin agroup,asdominantindividualsmayforcesubordinate onestotheperiphery(HallandFedigan1997; Ruckstuhl andNeuhaus2005; Hirsch2007),wherepredationriskis higher.Thus,optimalmovementstrategiesmaydifferbetweendominantandsubordinateindividuals:Dominantindividualspreferringastrategythatensurestheymaintaincentral positions,subordinateonespreferringonethatallowsthemto reducetheirindividualriskrapidly.Theinteractionbetween dominanceandpredationavoidanceisaninterestingroute forfurtherinquiry.

Inotheranimalgroups,anindividual’spositionisnotlimitedbydominancehierarchiesbutmaydependinsteadon factorssuchashungerlevels(foragingneeds)andantipredatordefenses(Krause1994; Romey1995; MorrellandRomey 2008).Althoughcentralindividualsareoftensaferfrompredation,theyarealsooftensubjecttoreducedfeedingrates (Krause1994).Predatorattacksmaythereforearisewhen aparticularindividualisinthegroupcenteronsomeoccasionsandthegroupperipheryonothers.Individualbehavioralresponsestopredatorsmaythereforebeﬂexible, allowingindividualstorespondoptimallydependingonsome measureofstate.Thissuggeststhatthebestmovement

a) b) c)

Figure3

Morrelletal. • Spatialpositionsinselﬁshherds 21 by guest on December 3, 2016 http://beheco.oxfordjournals.org/ Downloaded from

strategydependsnotonlyonthepositionwithinagroupbut alsoonthebehavioraldecisionsofothers.Awell-protected centralindividualwithasmallDODwouldbeneﬁtthemostby maintainingthatposition:Individualsmovingtowarditfrom theperipherywouldservetoreduceitsDODfurtherthrough thecompactionofthegroup,potentiallyreducingitsrelative risk,buttheindividualmayalsoneedtoensurethatitisnot pushedtotheperipheryofthegroup.Agame-theoretical approachinvestigatingthepositionsofindividualswithin groupsbeforeandafterapredatorattackmaybeusefulhere, anditislikelythatantipredatormovementwithinrealanimal groupsismuchmorecomplexthanthemovementrulessofar proposed.

Ourresultsdemonstratetheimportanceofconsidering ‘‘how’’apredationeventmayoccurwhenconsideringthe successofdifferentantipredatoryaggregativebehaviors.If predatorspreferentiallytargetperipheralprey,thenmovementthattakesanindividualtoacentralpositionshould evolve.Ifpredatorstargetbasedonothercriteria,suchas spacing,preferentiallytargetingmoreisolatedprey,thenavarietyofescaperulesmayarise.Therolesofdominance,state (hungerlevelsorantipredatordefenses)andthewaythey interactwithpredatoravoidancehaveyettobestudiedin thecontextoftheselﬁshherd,andexperimentalworkinvestigatingtherulesusedbyrealanimalsisalmostcompletely lacking(butsee KrauseandTegeder1994).However,the predictionsprovidedbyourtheoryshouldprovidefurther stimulustosuchempiricalinvestigation.

FUNDING

NaturalEnvironmentResearchCouncilPostdoctoralFellowship(NE/D008921/1)toL.J.M.

Wewouldliketothank2anonymousrefereesforinsightfulcomments.PaulBennettwrotetheoriginalprogramonwhichthiswork isbased.

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