Stochastic population dynamics of a montane ground-dwelling squirrel

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Stochastic population dynamics of a montane ground-dwelling squirrel
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Oli, Madan
Hostetler, Jeffrey A.
Kneip, Eva
Van Vuren, Dirk H.
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Understanding the causes and consequences of population fluctuations is a central goal of ecology. We used demographic data from a long-term (1990–2008) study and matrix population models to investigate factors and processes influencing the dynamics and persistence of a golden-mantled ground squirrel (Callospermophilus lateralis) population, inhabiting a dynamic subalpine habitat in Colorado, USA. The overall deterministic population growth rate l was 0.946SE 0.05 but it varied widely over time, ranging from 0.4560.09 in 2006 to 1.5060.12 in 2003, and was below replacement (l,1) for 9 out of 18 years. The stochastic population growth rate ls was 0.92, suggesting a declining population; however, the 95% CI on ls included 1.0 (0.52–1.60). Stochastic elasticity analysis showed that survival of adult females, followed by survival of juvenile females and litter size, were potentially the most influential vital rates; analysis of life table response experiments revealed that the same three life history variables made the largest contributions to year-to year changes in l. Population viability analysis revealed that, when the influences of density dependence and immigration were not considered, the population had a high (close to 1.0 in 50 years) probability of extinction. However, probability of extinction declined to as low as zero when density dependence and immigration were considered. Destabilizing effects of stochastic forces were counteracted by regulating effects of density dependence and rescue effects of immigration, which allowed our study population to bounce back from low densities and prevented extinction. These results suggest that dynamics and persistence of our study population are determined synergistically by density-dependence, stochastic forces, and immigration.
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This work was supported by California Agricultural Experiment Station, University of California Davis; School of Natural Resources and the Environment, University of Florida; and the Department of Wildlife Ecology and Conservation, University of Florida. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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StochasticPopulationDynamicsofaMontaneGroundDwellingSquirrelJeffreyA.Hostetler1,2,EvaKneip1¤,DirkH.VanVuren3,MadanK.Oli1*1 DepartmentofWildlifeEcologyandConservation,UniversityofFlorida,Gainesville,Florida,UnitedStatesofAmerica, 2 MigratoryBirdCenter,SmithsonianConservation BiologyInstitute,NationalZoologicalPark,Washington,D.C.,UnitedStatesofAmerica, 3 DepartmentofWildlife,Fish,andConservationBiology,UniversityofCalifornia Davis,Davis,California,UnitedStatesofAmericaAbstractUnderstandingthecausesandconsequencesofpopulationfluctuationsisacentralgoalofecology.Weuseddemographic datafromalong-term(1990–2008)studyandmatrixpopulationmodelstoinvestigatefactorsandprocessesinfluencingthe dynamicsandpersistenceofagolden-mantledgroundsquirrel( Callospermophiluslateralis )population,inhabitinga dynamicsubalpinehabitatinColorado,USA.Theoveralldeterministicpopulationgrowthrate l was0.94 6 SE0.05butit variedwidelyovertime,rangingfrom0.45 6 0.09in2006to1.50 6 0.12in2003,andwasbelowreplacement( l 1)for9out of18years.Thestochasticpopulationgrowthrate lswas0.92,suggestingadecliningpopulation;however,the95%CIon lsincluded1.0(0.52–1.60).Stochasticelasticityanalysisshowedthatsurvivalofadultfemales,followedbysurvivalof juvenilefemalesandlittersize,werepotentiallythemostinfluentialvitalrates;analysisoflifetableresponseexperiments revealedthatthesamethreelifehistoryvariablesmadethelargestcontributionstoyear-toyearchangesin l .Population viabilityanalysisrevealedthat,whentheinfluencesofdensitydependenceandimmigrationwerenotconsidered,the populationhadahigh(closeto1.0in50years)probabilityofextinction.However,probabilityofextinctiondeclinedtoas lowaszerowhendensitydependenceandimmigrationwereconsidered.Destabilizingeffectsofstochasticforceswere counteractedbyregulatingeffectsofdensitydependenceandrescueeffectsofimmigration,whichallowedourstudy populationtobouncebackfromlowdensitiesandpreventedextinction.Theseresultssuggestthatdynamicsand persistenceofourstudypopulationaredeterminedsynergisticallybydensity-dependence,stochasticforces,and immigration.Citation: HostetlerJA,KneipE,VanVurenDH,OliMK(2012)StochasticPopulationDynamicsofaMontaneGround-DwellingSquirrel.PLoSONE7(3):e34379. doi:10.1371/journal.pone.0034379 Editor: MattHayward,AustralianWildlifeConservancy,Australia Received January14,2012; Accepted March1,2012; Published March27,2012 Copyright: 2012Hostetleretal.Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermits unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited. Funding: ThisworkwassupportedbyCaliforniaAgriculturalExperimentStation,UniversityofCaliforniaDavis;SchoolofNaturalResourcesandtheEnviron ment, UniversityofFlorida;andtheDepartmentofWildlifeEcologyandConservation,UniversityofFlorida.Thefundershadnoroleinstudydesign,datac ollectionand analysis,decisiontopublish,orpreparationofthemanuscript. CompetingInterests: Theauthorshavedeclaredthatnocompetinginterestsexist. *E-mail:olim@ufl.edu ¤Currentaddress:DepartmentofEcologyandEvolution,StonyBrookUniversity,StonyBrook,NewYork,UnitedStatesofAmericaIntroductionUnderstandingfactorsandprocessesthatdeterminedynamics andpersistenceofbiologicalpopulationsisanimportantgoalof ecology[1,2,3,4].Becausemanyenvironmentsfluctuatestochastically,populationdynamicsoforganismsinhabitingsuch environmentsarestronglyinfluencedbyunpredictableenvironmentalvariations.Ontheotherhand,densitydependenteffects arepresumedtobeubiquitousaswellasanimportantforcein regulatingbiologicalpopulations[4,5,6].Itisgenerallybelieved thatbothdensity-dependent(DD)anddensity-independent(DID) processesinfluencepopulationdynamics,buttherelativerolesof DDregulationandDIDdestabilizationarestilldebated [3,7,8,9,10].TheeffectsofDIDprocessesonpopulationdynamics arelikelytobecomestrongerduetoglobalclimatechange,andit iscriticaltounderstandhowstochasticvariationsanddensitydependentmechanismsinteracttoinfluencepopulationdynamics [7,9]. Globalclimatechangeispredictedtoimpactboththemeanand varianceofclimaticparametersandconsequently,themeanand varianceofdemographicrates(e.g.,survivalandreproductive rates)[11,12,13,14].Thiscanpotentiallyexacerbatetheeffectsof environmentalvariationonpopulationdemographyasorganisms areexposedtonovelenvironmentalconditions.Therefore, understandingthedemographiceffectsofenvironmentalvariabilityiscriticalsincetheseperturbationsarelikelytoinfluencethe long-termgrowthrate,persistence,andresilienceofpopulations inhabitingvariableenvironments[13,15,16,17]. Thegolden-mantledgroundsquirrel( Callospermophiluslateralis ; formerly, Spermophiluslateralis ;hereafter,GMGS)[18]isahibernatingspeciesthatiswidelydistributedinwesternNorthAmerica, includingsubalpinehabitatsintheRockyMountains.Westudieda populationofGMGSattheRockyMountainBiologicalLaboratory inColorado,USA,whereclimatechangehasbeenshownto influencelifehistoryandpopulationdynamicsofseveralspecies [15,19].Sincethestudybeganin1990,ourGMGSpopulationhas exhibitedsubstantialfluctuationsinsize(Figure1A)[20].The availabilityoftheselong-term(1990–2008)demographicdata allowedustoestimateannualvitaldemographicrates(survival, breedingprobability,andlittersize)andtotestfortheeffectsof populationdensityaswellasclimaticvariablesonvitaldemographic rates.Usingdeterministicandstochasticdemographicanalyses,our PLoSONE|www.plosone.org1March2012|Volume7|Issue3|e34379

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goalwastoinvestigatehowdensity-dependentprocessesinteracted withenvironmentalstochasticitytoinfluencedynamicsand persistenceoftheGMGSpopulation.First,wecalculatedoverall andyearlydeterministicpopulationgrowthrates,andestimated potentialandactualcontributionsofdemographicvitalratesto temporalchangesinpopulationgrowthrates.Second,weused stochasticdemographicmethodstocalculatethestochastic populationgrowthrate( ls),anditselasticitytochangesinthe meanandvarianceofvitalrates.Finally,weexaminedhowdensity dependenceandenvironmentalstochasticityinfluencedpopulation persistenceparameters(probabilityof(quasi-)extinctionand distributionofextinctiontimes)whentheinfluenceofdemographic stochasticityandimmigrationwasorwasnotconsidered.Methods StudyareaandspeciesWeconductedourresearchattheRockyMountainBiological Laboratory(RMBL)nearCrestedButte,Colorado(38 u 58 9 N, 106 u 59 9 W,elevation2890m),USA,ona13-hasubalpine meadow.Thestudyareawasinterspersedwithwillow( Salixsp. ) andaspen( Populustremuloides )stands.Themeadowwasbordered bytheEastRivertothewestandCopperCreektothesouth, whichformedbarrierstodispersal,andbyaspenwoodlandstothe northandeastthatwereuninhabitedbyGMGS. Thegolden-mantledgroundsquirrelisanasocialanddiurnal speciesthatoccursatabroadrangeofelevations( 1000–4000m abovesealevel).Itprefersopenhabitatssuchasmountainmeadows androckyslopesthatareadjacenttograsslands[21,22,23],and suchhabitatswerepatchilydistributedinourstudyarea.The nearestlocalitiessupportingotherGMGSpopulationswere250m totheeastand300mtothenorth.Dispersalinthisspeciestypically involvemovementsof 250mbutcanexceed1000m[24].The GMGSsurviveslongwinters,andthereforefoodshortage,via hibernation.Bothaltitudeandamountofsnowfallinfluencewhen squirrelscommenceandendtheirhibernationperiod[21,22].In ourstudyareaadultGMGSusuallyemergefromhibernation aroundthetimeofsnow-melt(mid-MaytoearlyJune).The breedingseasoncloselyfollowsemergence,andpupsemergefrom natalburrowslateJunetomid-July.Attheendofsummer(late AugusttoearlySeptember)thesquirrelsenterhibernation. AtRMBL,GMGSprimarilyforageonherbaceousvegetation (forbsandgrasses).Snow-meltgreatlyinfluencesthegrowthofthese green,leafyplantsandhenceimpactsfoodavailabilityforsquirrels. Soonafteremergingfromhibernation,thesquirrelsbegingaining mass,rapidlystoringfattoimprovetheirchancesofsurvivalthe nextwinterandtosustaingestationthenextspring[25].FieldmethodsGMGSwerelive-trappedandobservedfor19successiveyears (1990–2008)duringtheactiveseason(MaytolateAugust).The annualcensus(markingtheentireresidentpopulation)tookplace fromlateMaytoearlyJune.Pupsweretrappedandmarked betweenlateJuneandmid-Julyaslittersemergedfromtheirnatal burrows.SquirrelsweretrappedalsoinlateJulyandlateAugust, inordertorecordtheirbodymassesastheywerebuildingfat reservesforhibernation.Throughoutthesummer,animalswere observeddailyandtrappedopportunisticallytocaptureandmark allnewimmigrantsandrefreshmarksonresidents[20,26]. Squirrelswerecapturedinsingle-doorTomahawklive-traps (12.7 6 12.7 6 40.6cm)baitedwithamixtureofsunflowerseeds andpeanutbutter.Oncecaptured,squirrelswereidentifiedvia numberedmetaltagsineachearandweredistinctlydye-marked withfurdyeforvisualrecognition.Sex,massandfemale reproductiveconditionwererecorded,andnewindividuals receivedeartags.Emergingpupswerecaptured,dye-marked, andear-taggedatfirstemergencefromtheirnatalburrow.Their mothers’eartagswererecordedaswellaslittersize. Atotalof831squirrelswascapturedduringthestudyperiod.Age wasknownfor704squirrelsbecausetheywerecapturedasjuveniles whenemergingfromtheirnatalburrows.Weestimatedagebased onmassforimmigrants,whoseexactageswereunknown.Field methodsfollowedprotocolsapprovedbytheAnimalCareandUse CommitteeattheUniversityofCalifornia,Davis,andmet guidelinesoftheAmericanSocietyofMammalogists[27].MatrixpopulationmodelWeconstructedandanalyzedage-structuredmatrixpopulation models,focusingonthefemalesegmentofthepopulationbecause itwasnotpossibletoestimatereproductiveparametersformales. Ageoffirstreproductionwas1yearbecausemanyfemalesquirrels reproduceasyearlings.Ageoflastreproductionwas6years;of 326known-agefemalesquirrels,onlyonesurvived 6yrs.The populationprojectionmatrixwasofthefollowingform: A ( t ) ~ Fj( t ) Fa( t ) Fa( t ) Fa( t ) Fa( t ) Fa( t ) Pj( t )00000 0 Pa( t )0000 00 Pa( t )000 000 Pa( t )00 0000 Pa( t )0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 Figure1.Annualpopulationsizeandgrowthrates. (A)Total, juvenile,andadultpopulationsizesand(B)deterministicpopulation growthrate( l 6 SE)foragoldenmantledgroundsquirrelpopulation attheRockyMountainBiologicalLaboratory,CrestedButte,Colorado, foreachyearofthestudy. doi:10.1371/journal.pone.0034379.g001 StochasticPopulationDynamics PLoSONE|www.plosone.org2March2012|Volume7|Issue3|e34379

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where( t )indicatestime-specificity; Pj(t) and Pa(t) denotesurvivalof juvenilesandadults,respectively.Thefertilityratesforyearlings (i.e.,ageclass1)andadults( Fj(t) and Fa(t) ,respectively)were calculatedusingpost-breedingcensusmethodsas[28]: Fj( t ) ~ Pj( t ) LS ( t ) BPj( t ),and Fa( t ) ~ Pa( t ) LS ( t ) BPa( t ) where LS islittersizeand BPjand BPadenotebreedingprobability (i.e.,probabilityofsuccessfulreproduction)ofyearlingandadult squirrels,respectively.Primarysexratiowasassumedtobe1:1,as istypicalofmostgroundsquirrels[29].DeterministicdemographicanalysisWeconstructedandanalyzedoverallandyear-specificdeterministicmatrixmodels.Fortheoverallortime-invariantmodel,a projectionmatrix A wasconstructedusingage-specificestimatesof vitalratesbasedoncapture-mark-recaptureandreproductivedata collectedduringtheentirestudyperiod(1990–2008).Fortheyearspecificmodels,aseparatepopulationprojectionmatrix A ( t )was compiledforeachyearofthestudyusingage-andyear-specific vitalrateestimates;thus,wehad18year-specificprojection matrices.Wecalculatedtheoverallandyear-specificpopulation growthratesasthedominanteigenvaluesoftheoveralloryearspecificpopulationprojectionmatrices,respectively.Elasticityof deterministicpopulationgrowthratestomatrixentriesandlowerlevelvitalrates,netreproductiverate( R0),andgenerationtime( A ) werecalculatedusingmethodsdescribedbyCaswell[28].Finally, weusedlife-tableresponseexperiment(LTRE)analysisto decomposeyear-to-yearchangesinpopulationgrowthrateinto contributionsfromchangesinmatrixentriesorunderlyingvital rates[28,30]as: l ( t z 1) { l ( t ) & X p( t z 1) i{ p( t ) i L l L pi p ( t ) i z p ( t z 1) i 2where l ( t + 1)and l ( t )aregrowthratesinyear t + 1andyear t respectively; piisamatrixentryoralower-levelvitalrate[31]. Theterm L l L pi p ( t ) i z p ( t z 1) i 2indicatesthatsensitivitieswereevaluated atthemidpointbetweenvaluesof piinthe2yearsbeing compared. Overallestimatesofdemographicvariablesfortheentirestudy periodarepresentedinTable1.Estimatesofdemographic variablesandnumbersofimmigrantsforeachyearofstudyare giveninFigureS1.StochasticdemographicanalysisAsnotedpreviously,wecompiledapopulationprojection matrix A ( t )foreachyearofthestudyusingyear-specificestimates ofvitalrates.Weassumedindependentandidenticallydistributed ( iid )environmentsuchthatvitalratesobservedineachofthe18 yearsofstudywereequallylikelytooccur.Weusedasimulationbasedapproach(50,000timesteps)toestimatethestochastic populationgrowthrateandstochasticelasticities[28,32,33].The stochasticpopulationgrowthrate log lswascalculatedas: log ls~ 1 T XT { 1 t ~ 1rtwhere rt=log( n ( t + 1)/ n ( t ))isaone-step populationgrowthrate(alllogsarebase e )[28,32].Weestimated varianceoflog ls( s2)usingthelog-normalapproximation[28]. Theapproximate95%CIof lswasthencalculatedas: explog ls+ 1 : 96 s2p .Weestimatedelasticityof lstomatrix entriesas[33]: EC ij~ limT ? ? 1 T X vit Cijt ujt { 1 l t S v t u t T ~ E vit Cijt ujt { 1 l t S v t u t T where u (t)and v (t)vectorsrefertostochasticpopulationstructure andreproductivevalueattime t l (t) onetimesteppopulation growthrate,andtheterm S v ( t ), u ( t ) T isthescalarproductof vectors v (t)and u (t).Wecalculatedthreetypesofstochastic elasticities[33].First,theoverallstochasticelasticities ES ijwere calculatedbysetting Cij( t ) ~ Aij( t ) foreveryyear t ;elasticitiesof lstothemeanofmatrixelements ES m ijandvarianceofthematrix entries ES s ijwereobtainedbysetting Cij( t ) ~ mij,and Cij( t ) ~ Aij( t ) { mij,Cijt ~ Aijt { mij,respectively[33].Elasticitiesof lstolower-levelvitalrateswerecalculatedusingmethods describedbyCaswell[34]. Influenceofenvironmentalanddemographicstochasticity, densitydependence,andimmigrationonpopulationpersistence. Weusedasimulation-basedapproachtopopulationviability analysis(PVA)usingmethodssimilartothosedescribedinMorris andDoak[35].Weestimatedpopulationpersistenceparameters (probabilityofextinction/quasi-extinctionandtimetoextinction) underavarietyofscenarios,dependingonwhetherandhowthe effectsofenvironmentalanddemographicstochasticities,density dependenceandimmigrationweremodeled. Weusedtwoapproachesforincorporatingenvironmental stochasticityinoursimulations.Inthefirstapproach,weused yearlyestimatesofvitalratesasdescribedpreviously;simulations wereconductedundertheassumptionthatvitalratesobservedin eachyearofthestudywereequallylikelytooccur(hereafter,ES: year).Thesecondapproachwasbasedonourearlierfindingsthat averagerainfallinJuneandJulyaffectedage-specificsurvival directly,andage-specificbreedingprobabilitywitha1-yrtimelag (hereafter,ES:rainfall)[20].Thefunctionalrelationshipsbetween rainfallandage-specificsurvivalrateswere: Pj~ 1 1 z e{ b j z b Rj rain t Pa~ 1 1 z e{ b 0 z b R rain t where b0and bjaretheintercepttermsforadultandjuvenile survival,respectively; bRand bRjareslopeparametersrelating rainfalltoadultandjuvenilesurvival,respectively,and raintisthe meanJune–Julyrainfallforyear t Thefunctionalrelationshipsbetweenrainfallandage-specific breedingprobabilitywere: Table1. Meanandstandarderror(SE)ofvitalrates,aswellas sensitivityandelasticityofoveralldeterministicpopulation growthrate(i.e.,basedonvitalratesestimatedfortheentire studyperiod)tochangesinvitalratesforagolden-mantled groundsquirrelpopulationinGothic,Colorado.ParameterMeanSESensitivityElasticity Pj0.3100.0241.1250.404 Pa0.5190.0291.0810.596 LS 4.7930.1410.0790.404 BPj0.3130.0470.3000.100 BPa0.8160.0370.3500.304 Vitalratesare:juvenilesurvival( Pj),adultsurvival( Pa),littersize( LS ),and breedingprobabilityforyearlings( BPj)andolderfemales( BPa). doi:10.1371/journal.pone.0034379.t001 StochasticPopulationDynamics PLoSONE|www.plosone.org3March2012|Volume7|Issue3|e34379

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BPj~ 1 1 z e{ y j z y Rj rain t { 1 BPa~ 1 1 z e{ y 0 z y R rain t { 1 where y0and yjareintercepttermsforbreedingprobabilityof adultandjuvenilefemales; yRand yRjareslopeparameters relatingaverageJune–Julyrainfalltobreedingprobabilityofadult andjuvenilefemalesquirrels,respectively. Wemodeleddemographicstochasticity(DS)usingasampling approach[28].Ateachtimestep t ,thenumberofsurvivorsforall ageclasses i ( i 1)wassampledfromabinomialdistributionwith parameter p =age-specificsurvivalprobability,and n =numberof femalesat t 2 1inageclass i 2 1.Likewise,thenumberoffemalesthat reproducedinyear t wassampledfromabinomialdistributionwith parameter p =age-specificbreedingprobability,and n =numberof femalesinclass i 2 1survivingfromyear t 2 1 toyear t .Wesampled thenumberofoffspringproducedbyeachfemalethatreproduced fromazero-truncatedPoissondistribution,withthePoisson parameter m =meanlittersize(adjustedtoaccountforthezerotruncation).Thetotalnumberofoffspringproducedbyfemalesofa givenageclasswasthencalculatedasthesumofoffspringproduced byallfemalesinthatageclass.Thenumberoffemaleoffspringwas sampledfromabinomialdistributionwithparameter n =total numberofoffspring,and p =primarysexratio(0.5).Thenumberof individualsinageclass1(juveniles)wastheprojectedtotalnumberof femaleoffspringproducedbyfemalesofallageclasses. Densitydependencehasbeensuggestedtobeanimportant factorinfluencingdynamicsandpersistenceofbiologicalpopulations[4,5],sowealsoevaluatedpopulationleveleffectsofdensity dependence.Inanearlierstudy,wefoundstrongevidencefora delayed,negativeeffectofpopulationdensityonage-specific survival[20].ThefunctionalDDrelationshipforsurvival( Pjand Pa)isdescribedbythefollowinglogisticregressionequations: Pj~ 1 1 z e{ b 0 j z b N n t { 1 Pa~ 1 1 z e{ b 0 0 z b N n t { 1 where b ’srepresentregressioncoefficientsrelatingpopulationsizeto age-specificsurvival( b ’0:intercepttermforadultsurvival, b ’j: intercepttermforjuvenilesurvival,and bN:slopeparameterrelating thepreviousyear’spopulationdensitytosurvival).Thisdensitydependentrelationshipwasestimatedusingtotalpopulationsize (bothsexes)andourpopulationmodelwasfemale-only,butthe observedsexratiodidnotvarymuchbyyear.Therefore,wedivided correspondingfemalepopulationsizebytheobservedoverallsex ratio(i.e.,proportionoffemales)inthepopulation(0.515)to extrapolatetheapproximatetotalpopulationsizefromthenumberof females. SomePVAscenariosincludedbothESandDD(seebelow).For ES:year,thiswassimulatedbymodelingsurvivalasDDandall otherparametersasvaryingbyyear.ForES:rainfall,wemodeled BPasdescribedpreviouslyandusedthetop-rankedmodel(based onAkaikeInformationCriterioncorrectedforsmallsamplesize [AICc][36])incorporatingESandDDonvitalratestomodel survival[20]: Pj~ 1 1 z e{ b # j z b # Rj rain t z b # N n t { 1 Pa~ 1 1 z e{ b # 0 z b # R rain t z b # N n t { 1 where b#indicatesvaluesofcoefficientsrelatingrainfalland populationsizetoage-specificsurvival.Valuesofcoefficients relatingtheeffectofpopulationdensityandaverageJune–July rainfallonvitalratesaregiveninTableS1. Wealsomodeledimmigrationasastochasticprocess,usingdata onthenumberofimmigrantfemales( $ 1yr)observedduring annualcensusesthattookplacesoonaftertheemergencefrom hibernation(FigureS1).Dispersalinthisspeciespredominantly occurslateinthesummerofbirth[24],andmostimmigrantsinto ourpopulationwerefirstrecordedearlythefollowingyearatage 1.Mostimmigrantswereofunknownage;knownagedsquirrels wereyearlingswhentheywerefirstrecordedintheannualcensus. Toincorporatetheinfluenceofimmigrationonpopulation dynamicsandpersistence,werandomlyselectedthenumberof yearlingfemaleimmigrantseachyearfromthe18yearsofdata (FigureS1).FortheES:yearandimmigrationsimulations,the numberofimmigrantswasselectedfromthesameyearasthevital rates.Becausetheimmigrantswerecountedbeforethebirthpulse (i.e.,pre-breedingcensus)butourpopulationmodelwasbasedon apost-breedingcensusformulation,itwasnecessarytoinclude immigrantsintotheanalysisinsuchawaythatmortalityofthe immigrantswasnotincludedbothexplicitly(inthepopulation model)andimplicitly(mortalitybetweenthebreedingseasonand whentheimmigrantswerefirstcounted).Todoso,forsimulations thatincludedimmigrationanddemographicstochasticity,we addedtheimmigrantsbetweenthemortalityandreproduction steps(assumingthatimmigrantshadsimilarreproductiveparametersasresidentsForsimulationsthatincludedimmigrationbut notdemographicstochasticity,weprojectedthepopulationas: n ( t z 1) ~ An ( t ) z I ( t ) Pj( t ) 00000 T where I(t) isthenumberofimmigrantyearlingfemalesattime t Thisapproachassumesthatimmigrantfemaleshadsimilar survivalandreproductiveratesasresidents. Weprojectedpopulationsizefor50yearsusingtheappropriate populationprojectionmatrix(oranequivalentalgorithmwhen demographicstochasticitywasconsidered)andaninitialpopulationvector n (0).Theaveragenumberoffemalesobservedduring ourstudy(i.e.,30females)wasmultipliedbythestableage distributionvectortoobtaintheinitialpopulationvector n (0).We projectedthepopulationsizeandcalculatedprobabilitiesof(quasi)extinctionanddistributionofextinctiontimesunder24scenarios. Thesescenariosincludedallcombinationsofenvironmental stochasticity(none,ES:year,andES:rainfall);demographic stochasticity(noneorallvitalrates);density-dependence(noneor densitydependentsurvival);andimmigration(noneorrandom immigrationofjuvenilefemales). WeusedMATLAB[37]forallcalculations.Results DeterministicdemographicanalysisTheoveralldeterministicpopulationgrowthrate( l ),calculated usingvitalratesestimatedfortheentirestudyperiod,suggesteda populationdeclineof6%peryear( l =0.94 6 SE0.05)inthe absenceofimmigration.However,95%confidenceinterval included1.0(0.84–1.04),offeringnostatisticalevidencefora populationdecline.Matrixentryelasticityanalysisrevealedthat l wasproportionatelymostsensitivetochangesinsurvivalof juveniles( Pj),followedbythatinsurvivalof2-yroldfemales. Resultsoflower-levelelasticityanalysisshowedthat l wasStochasticPopulationDynamics PLoSONE|www.plosone.org4March2012|Volume7|Issue3|e34379

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proportionatelymostsensitivetochangesinsurvivalofadults( Pa) (elasticity=0.596),followedbythatin Pjandlittersize(elasticity forboth=0.404),breedingprobabilityforadults(elasticity=0.304),andbreedingprobabilityforjuveniles(elasticity=0.10; Table1).Thenetreproductiveratewas0.804daughtersper femalepergeneration;generationtimeandlifeexpectancyat emergencefromthenatalburrowwas2.74and1.62years, respectively. Allvitalratesvariedsubstantiallyovertime(FigureS1); coefficientofvariationwas29.61%,21.76%,19.05%,97.61%, and18.28%for Pj, Pa, LS BPyand BPa,respectively.Consequently,populationgrowthratealsowashighlyvariableovertime, rangingfrom0.45 6 0.09in2006to1.50 6 0.12in2003(Figure1B); itwas 1in9years,and 1in9years.Thepatternofelasticity wasidenticaltothatdescribedabovefortheoverallpopulationin mostyears,exceptthatin2000whenelasticityof l tosurvivalof juvenilesandlittersizeexceededthattosurvivalofadults,and elasticityof l tobreedingprobabilityofjuvenilesexceededthatto breedingprobabilityofadults.Thiswaslikelyaconsequenceofthe factthatallfemalesoneyearofageoroldersuccessfully reproducedin2000. Contributionofvitalratestoyear-to-yearchangesinpopulation growthratealsovariedovertime;thiswasasexpectedgiventhat bothvitalratevaluesaswellassensitivityof l tovitalratesvaried overtime(Figure2a–2c).TheabsolutevalueofLTRE contributionwasinthefollowingorder(largesttosmallest): Pa, Pj, LS BPyand BPa.AKruskal-Wallistestrevealedthatabsolute valuesofLTREcontributionsdifferedamongvitalrates ( x2=14.24, P =0.007).However,thecontributionofvitalrates toyear-yearchangesin l variedovertime(Figure2c).Changesin Pa,followedbythatinLSmadethelargestcontribution(absolute values)toyear-to-yearchangesin l ,in6and5years,respectively. Twoofthelargestcontributionsof BPyoccurredduring1999– 2000and2000–2001transitions,mostlikelybecauseofthefact thatallfemales1yearofageoroldersuccessfullyreproducedin thatyear;thus,changesinbreedingprobabilitiesfrom1999to 2000,andfrom2000to2001wererathersubstantial.The contributionof Pjranked4thintermsoffrequencyoflargest contribution,althoughmean(absolutevalue)LTREcontribution ofthisvariablewassecondonlytothatof Pa(Figure2c).StochasticdemographicanalysisThestochasticpopulationgrowthrate lswas0.92(95%CI: 0.52–1.60);thisvaluewaslessthan,butstatisticallyindistinguishablefrom,theoveralldeterministicpopulationgrowthrate calculatedfrompooledestimatesofvitalrates( loverall=0.94;95% CI:0.84–1.04)orthatbasedonthemeanmatrix( lmean=0.95),as istypical.Stochasticvitalrateelasticitiesrevealedapatternsimilar todeterministicelasticities,andshowedthat lswasproportionately mostsensitivetochangesinthemeanandvarianceof Pa,followed bythatof Pjand LS .Theelasticityof lstovitalratevarianceswas negative,indicatingthatanincreaseinvitalratevariancewould reducestochasticpopulationgrowthrate(Figure3).Theoverall stochasticelasticitiesdisplayedessentiallythesamepattern. Theinfluenceofenvironmentalanddemographicstochasticities,density-dependenceandimmigrationonpopulationdynamicsandpersistence. Theprobabilityofextinction(theprobabilitythattheprojected populationsizefallsbelow1female)aswellasmediantimeto extinctionvariedwidelyacross24scenariosdependingonwhether andhowdensitydependence,immigration,demographicstochasticity,andenvironmentalstochasticityweremodeled(Figure4). Whendensitydependenceandimmigrationwereignoredbut someformofstochasticitywasincluded,probabilityofextinction (PE)within50yearswasgenerallyhigh( $ 0.75,mostlyveryclose to1).Incontrast,theprobabilityofextinctionwasatornearzero whenbothdensitydependenceandimmigrationwereconsidered; theprobabilityofextinctionremainedclosetozeroevenwhen demographicand/orenvironmentalstochasticitywereconsidered. Probabilityofextinctionandmediantimetoextinctionwere intermediate(0.05 PE 0.25)wheneitherdensitydependenceor immigration(butnotboth)anddemographicstochasticitywere considered.Includingadditionalsourceofstochasticitygenerally increasedprobabilityofextinction.Asexpectedforsmall populations,demographicstochasticitygenerallyhadagreater impactonpopulationpersistencethanenvironmentalstochasticity. Forthescenariowithnostochasticity,densitydependenceor immigrationthepopulationsizeatyear50wasjustabove1,with PE=0(Figure5);whenenvironmentaland/ordemographic stochasticitywasincluded,however,PEincreasedsubstantially. Forscenarioswithnon-zeroextinctionprobabilities,mediantime toextinctionvariedbetween15and35years,withgenerally highervaluesofmediantimetoextinctionforscenarioswith densitydependenceandimmigration(Figure4). Figure2.Contributionsofvitalratestoannualchangesin populationgrowth. Resultsoflifetableresponseexperiment(LTRE) analysis:(a)differencesinvitalratesbetweenconsecutiveyears,(b) sensitivityofthedeterministicpopulationgrowthratetochangesin vitalrates,evaluatedatthemidpointbetweentwosuccessiveyears beingcompared,and(c)LTREcontributionofeachvitalratetoyear-toyearchangesinthedeterministicpopulationgrowthrate.Vitalrates are: Pj=juvenilesurvival, Pa=adultsurvival, LS =littersize, BPj=breedingprobability(i.e.,probabilityofsuccessfulreproduction)foryearlings, and BPa=breedingprobabilityforadults. doi:10.1371/journal.pone.0034379.g002 StochasticPopulationDynamics PLoSONE|www.plosone.org5March2012|Volume7|Issue3|e34379

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Meanprojectedpopulationsizesvariedacrossscenarios (Figure5).ScenarioswithDIDandnoimmigrationhadvery loworzeroaveragepopulationsizesbyyear50,inpartbecause extinctpopulations( N 1)wereincludedinthecalculationof averages.Scenarioswithdensitydependence,noimmigration,and noDShadmeanpopulationsizesof26–30femalesbyyear50; whendemographicstochasticitywasadded,thisdroppedto18–23 females.ScenarioswithDDandimmigrationhadmean populationsizesof30–36femalesbyyear50;demographicor environmentalstochasticitygenerallyhadlittleeffect.Projected meanpopulationsizesweregenerallyhigherforscenariosthat consideredimmigration(Figure5). Wealsocalculatedprobabilityofquasi-extinction(i.e.,probabilitiesthatthepopulationfallsbelowacriticalpopulationsize, Ncrit)for Ncrit=5and Ncrit=10females.Quasi-extinctionprobabilitiesweregenerallyhigherthanextinctionprobabilities,and increasedfurtherasthecriticalpopulationsizeincreased(Figure S2,FigureS3).Probabilitiesofquasi-extinctionwereparticularly highforscenarioswithDS,especiallyfor Ncrit=10.Mediantimeto quasi-extinctionwaslowerthanmediantimetoextinction,and variedbetween11and30yearsfor Ncrit=5and2and30for Ncrit=10.DiscussionVirtuallyallnaturalpopulationsexperiencestochasticenvironmentalvariationswhichcaninfluencedemographicvariables,and populationdynamicsandpersistence[12,33,38].Whereasenvironmentalstochasticitytendstodestabilizepopulationdynamics [39,40],density-dependentmechanismstendtohavestabilizing effectsandeventuallyleadtopopulationregulation[4,6,8,41,42]. Whenpopulationsizesaresmall,demographicstochasticitycan alsobeanimportantinfluenceonpopulationpersistence[43],and immigrationcanhelpreduceextinctionrisksinopenpopulations. Understandinghowthesefactorsinteracttoaffectpopulation dynamicsandpersistenceisespeciallyimportantforspeciesthat occupyhabitatssensitivetoclimatechange.Thisisbecauseglobal climatechangecanpotentiallyaccentuatethedestabilizingeffect ofenvironmentalstochasticity,andthuscanprofoundlyinfluence populationdynamicsandpersistence[16,17,44,45]. Ourgoalwastounderstandfactorsandprocessesinfluencing dynamicsandpersistenceofagolden-mantledgroundsquirrel (GMGS)populationinhabitingamontanehabitatwherethe changingclimateisaffectinglifehistoryandpopulationdynamicsof severalspecies[15,19].Thetotalsizeofourstudypopulation rangedfrom24squirrelsin1999and2000to140in2005,almosta 6-folddifference(Figure1A)[20].Likewise,thepopulationgrowth ratevariedovertime(Figure1B)reflectingsubstantialtemporal environmentalvariation,apatternalsoobservedinothersympatric hibernatingsquirrels[46,47].Deterministicprospectiveand retrospectiveperturbationanalysesrevealedthatchangesinsurvival ofjuvenileandadultfemaleswereprimarilyresponsiblefor observedannualvariationinpopulationgrowthrate,although reproductiveparametersalsowereimportantespeciallywhenthey experiencedlargechanges.Thestochasticgrowthrate lswaslower thanthedeterministicgrowthrateofthemeanmatrix.Stochastic elasticitypatternsweresimilartothepatternofdeterministic elasticities,andrevealedthat lswasproportionatelymostsensitive tochangesinmeanandvarianceofadultandjuvenilesurvivalrates. Inadditiontothebroadpopulationfluctuations,wehave witnessedpopulationlowswithasfewasfiveadultfemalesquirrels in1999and2001[20].Duringthesummerof2001,theadult femalepopulationsizedippedtothreeindividualsbecausetwo femalesdisappearedfromthestudysiteaftertheannualcensus, mostlikelyduetopredation.Yet,thepopulationprovedresilient asitre-boundedandhasnotyetgoneextinct.Whatarethefactors andprocessesthatallowedtherelativelysmallpopulationof GMGStopersist?Toaddressthisquestion,weperformed populationviabilityanalysisunder24scenarios,dependingon whetherandhowdensitydependence,environmentalstochasticity,demographicstochasticityandimmigrationweremodeled. Whendensitydependenceandimmigrationwereignoredbut stochasticitywasconsidered,thepopulationhadaveryhigh ( 0.75;mostlycloseto1.0)probabilityofextinction,andthemost Figure3.Proportionalsensitivityofstochasticpopulation growthratetovitalrates. Resultsofstochasticelasticityanalysis: (a)elasticityofstochasticpopulationgrowthrate( ls)tochangesin meanvaluesofvitalrates,(b)elasticityof lstochangesinvarianceof vitalrates,and(c)overallstochasticelasticities.Vitalratesare: Pj=juvenilesurvival, Pa=adultsurvival, LS =littersize, BPj=breeding probability(i.e.,probabilityofsuccessfulreproduction)foryearlings, and BPa=breedingprobabilityforadults. doi:10.1371/journal.pone.0034379.g003 StochasticPopulationDynamics PLoSONE|www.plosone.org6March2012|Volume7|Issue3|e34379

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likelytimetoextinctionwasasearlyas15years.Theprobabilityof extinctiondeclinedsubstantiallyandsomewhatsimilarlywhen eitherimmigrationordensitydependencewasconsidered;when theeffectsofimmigrationanddensitydependencewere consideredsimultaneously,theprobabilityofextinctionpractically declinedtozero(Figure4).Finally,theinfluenceofdemographic stochasticitywasstrong,aspredictedbytheoryforsmall populations[28,48].Theseresultsconclusivelydemonstratethat stabilizingeffectsofdensitydependenceandrescueeffectsof immigrationcounteracteddestabilizingstochasticinfluenceson ourstudypopulations,andthatintheabsenceofdensitydependentregulationandimmigration,smallpopulationsare undersubstantialextinctionrisk. Boththeoveralldeterministicandstochasticpopulationgrowth rateswereproportionatelymostsensitivetochangesinsurvivalof adultandjuvenilesurvival–thetwovitalratesthathavealsobeen showntobedensity-dependent[20].Basedontheseresults,we concludethatdensity-dependentsurvivalandrescueeffectof immigrationhaveallowedourstudypopulationtopersistinthe faceofstochasticinfluences.Ourresultsaddtothebodyof evidencesuggestingthatmanybiologicalpopulationsarelikely regulatedbysynergisticeffectsofdeterministic(e.g.,density dependence)andstochastic(e.g.,environmentalanddemographic stochasticity)factors[3,7,9]. Inthelast18years,thetotalfemalepopulationsize(including juveniles)hasneverdroppedbelow10[20].Thisisconsistentwith ourPVAresultswheretheprobabilityofthepopulationdropping below10femaleswithin18yearswasonly21–24%forscenarios thatconsidereddemographicandenvironmentalstochasticity, immigrationanddensity-dependence(FiguresS2andS3).Over Figure4.Probabilityofextinctionbysimulationscenario. Thecumulativeprobabilityofextinctionduringa50-yrperiod(i.e.,probabilitythat thepopulationfallsbelowonefemale)across24simulationscenariosdependingonwhetherandhowdensitydependence,immigration, demographicstochasticity,andenvironmentalstochasticityweremodeled.Scenariosareasfollows:density-dependence-DID(density-independ ent vitalrates)andDD(density-dependentsurvival);immigration-immigrationignored(NoImmigration),andimmigrationincluded(Immigration); environmentalstochasticity-environmentalstochasticityignored(ES:None),environmentalstochasticitymodeledwithannualestimatesofvit al rates(ES:Year),andenvironmentalstochasticitymodeledwitheffectsofaverageJune–Julyrainfallonvitalrates(ES:Rainfall);anddemographi c stochasticity-demographicstochasticityignored(DS:None),anddemographicstochasticityconsideredinallvitalrates(DS:All).Cumulative probabilityofextinctionforeachscenariobasedon10,000simulationsisrepresentedbysolidline.Probabilityofextinctionwithin50yearsand medianextinctiontimeforeachscenarioarepresentedinlargetextwithineachfigurepanel.Allscenariosstartedwith30females,distributedtoa ge classesaccordingtostableagedistributionfortheoverallpopulation. doi:10.1371/journal.pone.0034379.g004 StochasticPopulationDynamics PLoSONE|www.plosone.org7March2012|Volume7|Issue3|e34379

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thelongertimespanof50years,however,thisprobability increasesto $ 50%. Severalauthorshavepointedoutthatenvironmentalstochasticitybasedonannualestimatesofvitalratesmaybebiasedhigh duetoconfoundingofsamplingerrorandprocessvariance [49,50,51].Conversely,estimatesofenvironmentalstochasticity basedonenvironmentalfactorsmaybebiasedlowduetoeffectsof unmeasuredenvironmentalcovariates.Wetestedfortheeffectsof environmentalstochasticityestimatedbasedonannualestimatesof vitalrates(ES:Year)andthosebasedontheeffectsofsummer rainfallonvitalrates(ES:Rainfall),andevaluatedhowthese alternativeapproachestoquantifyingenvironmentalstochasticity affectedextinctionparameters.Theprobabilityofextinctionand mediantimetoextinctionobtainedfromthetwoapproacheswere oftensimilar.Whenprobabilityofextinctionestimatedbasedon thetwoapproachestoESdiffered,theestimatefromES:Rainfall wasgenerallyclosertothatobtainedfromanalysesthatignored environmentalstochasticitythanthatbasedonES:Year;theonly exceptionwasthescenariosthatignoredDDandimmigration; Figure4).Itseemslikelythattheactualeffectofenvironmental stochasticityonthedynamicsandpersistenceofourstudy populationliesbetweenthetwoapproachesconsideredhere. Causesandpopulationdynamicconsequencesofimmigration (andemigration)havebeenanactiveareaofresearchinecology [52,53].Althoughimmigrationisthoughttobenecessaryfor metapopulationpersistence[54],itsroleinlocalpopulation dynamicsisstilldebated[55,56].Insomespeciesofsmall mammals,theroleofimmigrationinlocalpopulationdynamics isconsideredtobeminor(e.g.[57,58,59]).Ourresultssuggestthat immigrationisanimportantfactorcontributingtodynamicsand persistenceofourstudypopulation.Withoutimmigration,our studypopulationwouldhavefacedahighlikelihoodofextinction duringapopulationbottleneckthatoccurredfrom1999–2002;the populationsizeduringthatperiodwasreducedto # 14adults (Figure1A).Aninfluxofimmigrantsin2002and2004mostlikely preventedpopulationextinctionandlossofgeneticvariationand inbreeding[26].Despiteafairlylowrateofimmigrationtoour population(mean=1.17females/year),oursimulationresults suggestthatimmigrationdramaticallyreducestheprobabilityof extinction(from 20%within50yearstol 1%,whenDD,DS, Figure5.Populationsizesbysimulationscenario. Projectedmeanpopulationsize(solidlines)and90thpercentile(dashedlines)ofprojected populationsizesbasedon10,000simulationsacross24scenariosdependingonwhetherandhowdensitydependence,immigration,demographic stochasticity,andenvironmentalstochasticityweremodeled.Meanprojectedpopulationsizeand90thpercentilesinyear50(roundedtotheneares t integer)alsoaregivenforeachscenariowithinthefigurepanels.SeeFigure4forthedescriptionofscenariosandothersimulationdetails. doi:10.1371/journal.pone.0034379.g005 StochasticPopulationDynamics PLoSONE|www.plosone.org8March2012|Volume7|Issue3|e34379

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andESareconsideredsimultaneously).Althoughourstudy focusedonasinglelocalpopulation,itisclearthatthispopulation existsaspartoflargermetapopulationwithdemographicand geneticconnectionsamonglocalpopulations[26].Exchangeof individualsamonglocalpopulationswasclearlyimportantin populationpersistenceaswellasmaintenanceofgeneticdiversity. Theinfluenceofenvironmentalstochasticityislikelytobe exacerbatedbytheeffectofthepredictedglobalclimatechange. Indeed,theclimateischanginginourstudysite,andthechanging climatehasbeenshowntoinfluencethelifehistoryofseveral species[15,19].Onepossiblemechanismbywhichclimatechange couldinfluenceourstudypopulationisviachangesinsummer rainfallpatterns.Theaveragerainfallduringsummermonths (June–July)hasbeenshowntoinfluencebothsurvivaland probabilityofsuccessfulreproductioninourstudypopulation [20],andthiscaninfluencebothprobabilityofextinctionand mediantimetoextinction(Figs.4–5).Similarpopulation-level effectsofclimatechangehavebeenpredictedforseveralspecies [11,16,17]. Despitesubstantialpopulationfluctuations,ourstudypopulationhasbouncedbackfromlownumbersandpersistedtodate. Theregulatoryeffectofdensitydependenceandtherescueeffect ofimmigrationwilllikelyallowthispopulationtopersistforyears tocome.Nonetheless,theGMGSpopulationislikelytoface substantialextinctionrisk,especiallyifregulatoryinfluencesare weakenedorifhabitatorclimatechangereducestherateof immigrationintothestudypopulationsuchasthatobservedinthe endangeredIdahogroundsquirrel( Urocitellusbrunneus )[60]. Stochasticprocessessuchasenvironmentalanddemographic stochasticityaswellasincreasesinthemeanandvariabilityof summerprecipitationwouldundoubtedlyincreasevulnerabilityof ourstudypopulationtoextinction.Theearth’sclimateis changing,andthechangingclimatewillundoubtedlyaffectthe distribution,abundance,andpersistenceofpopulations[14,61].A dauntingfuturechallengeforecologyistobeabletounderstand andpredicthowthesechangeswouldinfluencebiological populationsandcommunities[17,42,45,62].SupportingInformationFigureS1Annualmean( 6 SE)valuesofvitalratesandannual numberofimmigrants. (DOC)TableS1Regressioncoefficientsrelatingsummerrainfalland populationdensitytoage-specificsurvivalandbreedingprobabilities. (DOC)FigureS2Cumulativeprobabilitiesofquasi-extinction(i.e.,the probabilitythatthesimulatedpopulationfallsbelow5females) acrosssimulationscenarios. (DOC)FigureS3Cumulativeprobabilitiesofquasi-extinction((i.e.,the probabilitythatthesimulatedpopulationfallsbelow10females) acrosssimulationscenarios. (DOC)AcknowledgmentsWethankC.Floyd,K.Jenderseck,andC.Muellerfortheircontributions todatacollection.M.E.Sunquist,V.Rolland,M.Haywardandtwo anonymousreviewersprovidedmanythoughtfulcomments,forwhichwe aregrateful.AuthorContributionsConceivedanddesignedtheexperiments:MKODHVVJAHEK. Performedtheexperiments:DHVVMKOJAHEK.Analyzedthedata: JAHMKOEK.Contributedreagents/materials/analysistools:DHVV MKOJAHEK.Wrotethepaper:JAHMKOEKDHVV.References1.AndrewarthaHG,BirchLC(1954)Thedistributionandabundanceofanimals. 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