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Stochastic Models for the Growth Rate of Animal Populations

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Stochastic Models for the Growth Rate of Animal Populations Variance Scaling and Decomposition
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Ferguson, Jake M
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[Gainesville, Fla.]
Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Zoology
Biology
Committee Chair:
PONCIANO CASTELLANOS,JOSE MIGUEL
Committee Co-Chair:
MCKINLEY,SCOTT
Committee Members:
ST MARY,COLETTE MARIE
HOLT,ROBERT D
LORENZEN,KAI
Graduation Date:
8/9/2014

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Autocorrelation ( jstor )
Demography ( jstor )
Ecological modeling ( jstor )
Ecology ( jstor )
Modeling ( jstor )
Peas ( jstor )
Population dynamics ( jstor )
Population growth ( jstor )
Statistical discrepancies ( jstor )
Time series ( jstor )
Biology -- Dissertations, Academic -- UF
community -- ecology -- population -- time-series
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born-digital ( sobekcm )
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Zoology thesis, Ph.D.

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All animal populations fluctuate through time. These fluctuations are due both to the intrinsic properties of populations such as the random nature of births and deaths, and to extrinsic environmental factors driving birth, survival, and migration dynamics. This dissertation is concerned with building models to describe these fluctuations using simple but general models describing how populations interact with their environment. The results are not just interesting for their own sake, but also have important implications for wildlife conservation and management efforts. Chapter 2 looks at how autocorrelated environments drive fluctuations in natural populations. Our results provide an interesting new perspective on when environmental autocorrelation will be important to account for, and how ecological processes can disrupt the autocorrelation present in environmental factors. The model developments in Chapter 3 include a derivation of demographic and environmental stochasticity and a single-species stochastic model that accounts for interspecific interactions. We tested this model with an experimental time series that tracked the extinction of populations in microcosms, finding that we could greatly improve predictions about the time to extinction when accounting for fluctuations induced by interspecific interactions. Finally, in Chapter 4 we looked at the influence of stochastic environments on the strength of density dependence. This is a perspective that has long been ignored but turns out to have interesting implications for population stability. We show that variation in the strength of density dependence will often yield more stable dynamics than the standard model of environmental variation but that there are cases where it can also destabilize dynamics, leading to periods of exponential growth followed by population crashes. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
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Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: PONCIANO CASTELLANOS,JOSE MIGUEL.
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Co-adviser: MCKINLEY,SCOTT.
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by Jake M Ferguson.

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LETTERPerformanceofseveralvariable-selectionmethods appliedtorealecologicaldataPaulA.Murtaugh*DepartmentofStatistics, OregonStateUniversity, Corvallis,OR97331,USA *Correspondence:E-mail: murtaugh@science.oregonstate. eduAbstractIevaluatedthepredictiveabilityofstatisticalmodelsobtainedbyapplyingseven methodsofvariableselectionto12ecologicalandenvironmentaldatasets.Crossvalidation,involvingrepeatedsplitsofeachdatasetintotrainingandvalidationsubsets, wasusedtoobtainhonestestimatesofpredictiveabilitythatcouldbefairlycompared amongmethods.TherewassurprisinglylittledifferenceinpredictiveabilityamongÞve methodsbasedonmultiplelinearregression.Stepwisemethodsperformedsimilarlyto exhaustivealgorithmsforsubsetselection,andthechoiceofcriterionforcomparing models(Akaike'sinformationcriterion,Schwarz'sBayesianinformationcriterionor F statistics)hadlittleeffectonpredictiveability.Formostofthedatasets,twomethods basedonregressiontreesyieldedmodelswithsubstantiallylowerpredictiveability. Iarguethatthereisno best methodofvariableselectionandthatanyoftheregressionbasedapproachesdiscussedhereiscapableofyieldingusefulpredictivemodels.KeywordsAIC,allsubsets,BIC,regressiontree,statisticalmodelbuilding,stepwise,subset selection,variableselection.EcologyLetters (2009)12:1061Ð1068INTRODUCTIONBuildingstatisticalmodelsofaresponseasafunctionof multipleexplanatoryvariablesisacommonexercisein ecology.Suchmodelsserveavarietyofpurposes,including predictionofresponsesfornewcases(Leigh etal. 2008); riskestimation(Gerritsen etal. 1996);understanding,orat leastforminghypothesesabout,cause-and-effectrelationships(Knick&Rotenberry1995);andconstructing parsimoniousmodelsofspatialortemporalcorrelation ofresponsevalues(Hoeting etal. 2006;Lee&Ghosh 2009). Manycriteriaareavailableforthestatisticalcomparison ofmultiple-variablemodels.Recently,information-theoretic criteriasuchasAkaike'sinformationcriterion(AIC)and Schwarz'sBayesianinformationcriterion(BIC)havegained favouramongecologists(Burnham&Anderson2002; Hobbs&Hilborn2006;Ward2008).Someoftheappealof theseapproachesseemstobebasedontheirindependence fromthehypothesis-testingframeworkoffrequentist statistics.Mazerolle(2006),forexample,writesthatthe AIC isremarkablysuperiorinmodelselection(i.e.variable selection)thanhypothesis-basedapproaches'',andLukacs etal. (2007)commentthat exploratorydataanalysisbased onnullhypothesistestingmethodssuchasstepwise selectionsimplyremovesthoughtfromdataanalysis''. Theoldestalgorithmsforselectingexplanatoryvariables arestepwiseprocedures,inwhichcandidatepredictorsare screenedforpossibleinclusionandvariablesalreadyinthe modelareconsideredforpossibleremovalinasequential fashion.Moremodernalgorithmsinvolveexhaustive searchesofmanymoresubsetsofpredictorsthanare usuallyevaluatedinstepwiseprocedures,leadingsome ecologiststodoubttheusefulnessofstepwisevariable selection.Whittingham etal. (2006),forexample,bemoan thewidespreaduseofstepwiseproceduresinecologicaland behaviouraljournals,giventhewell-established biasesand shortcomingsofstepwisemultipleregression ,andMundry &Nunn(2009) followothersinrecommendingthat biologistsrefrainfromapplyingthesemethods . Manyauthorshavediscussedandevaluateddifferent methodsofvariableselection(e.g.seeOlden&Jackson 2000;Sauerbrei etal. 2007;Ward2008;Lee&Ghosh2009). Raffalovich etal. (2008)provideanexcellentreviewofwork doneinthisarea.Simulationistheonlyapproachinwhich the true modelcanbeknown,buttheconclusionsfrom simulationstudiesareverydependentonthewaysthat dataaregeneratedinthesimulationsandthemeasuresthat EcologyLetters ,(2009) 12 :1061–1068doi:10.1111/j.1461-0248.2009.01361.x 2009BlackwellPublishingLtd/CNRS

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arechosenforcomparisonofdifferentmodel-building techniques.Murtaugh(1998),forexample,foundthat approachesbasedon F tests,Mallows Cp,andtheBIC hadsimilarfrequenciesofcorrectdecisionsaboutinclusion orexclusionofexplanatoryvariables.Regression-tree approachesweremarkedlyinferiortotheregression-based methods,butMurtaugh(1998)questionedthevalidityofthe comparison,giventhatthedataweresimulatedaccordingto alinearregressionmodel. HereIexaminetheperformanceofsevenmethodsof variableselectionappliedto12datasetsthatwere obtained,opportunistically,fromtheliteratureandfrom colleagues.Astherearenoknown true modelsof responsesasfunctionsofexplanatoryvariables,Iused cross-validationtoobtainhonestestimatesofpredictive abilitythatcanbefairlycomparedamongdifferent methodsofvariableselection.Thisapproachavoidsthe arbitrarinessofchoosinganalgorithmtogeneratedatain simulationstudies,butitproducesresultsofunknown generality,asthedatasetsusedareaselectsubsetofthe enormousvarietyofdatasetstowhichvariable-selection techniquesareappliedinpractice.ApproachestovariableselectionMultiplelinearregressionisafamiliarwayofmodellinga quantitativeresponsevariableasafunctionofmultiple explanatoryvariables.Approachestoselectingvariablesfor inclusioninthemodelfromapoolofcandidatepredictors include: (1)Stepwiseprocedures,inwhichsomequantitative criterionisusedtocompareregressionmodelswith andwithoutaparticularpredictor,andsequential additionand/ordeletionofexplanatoryvariables continuesuntilastoppingpointbasedonthevalue ofthecriterionisreached;and (2)Allsubsets,orexhaustive,variableselection,inwhich thesetofallpossiblegroupingsofexplanatory variablesissearchedandsubsetsofpredictorsgiving themostfavourablevaluesofthequantitativecriterion areidentiÞed. Thecriteriathatareusedtocompareregressionmodels include: 1. P -valuesfromextra-sum-of-squares F tests.Tocomparemodelswithandwithoutaparticularpredictor,we compute F¼ SSEwithout SSEwithd.f.without d.f.with SSEwithd.f.with; ð 1 Þ whereSSEistheerror,orresidual,sumofsquares,andd.f. isthenumberofresidualdegreesoffreedom.The P -valueis thenobtainedbycomparing F *toan F distributionwith thecorrespondingnumeratoranddenominatordegreesof freedom.Inthecaseofaquantitativepredictor, F *isthe squareofthe t statisticforthatpredictorintheregression output. (2)Akaike'sinformationcriterion.Foraregressionmodel withGaussianerrors,thisstatisticcanbewritten (Ramsey&Schafer2002,p.356): AIC ¼ n log ð MSE Þþ 2 p ; ð 2 Þ where n isthenumberofobservations, p isthenumberof regressioncoefficientsandMSEisthemeansquareerror, equaltotheresidualsumofsquaresdividedbyitsdegreesof freedom( n ) p ). (3)Schwarz'sBIC.ForregressionwithGaussianerrors, thiscanbewritten(Ramsey&Schafer2002,p.356): BIC ¼ n log ð MSE Þþ p log n : ð 3 Þ Noticehowallthreeofthesestatisticsbalanceexplained variationagainstthenumberofpredictorsinamodel:asthe numberofexplanatoryvariablesincreases,theresidualsum ofsquaresdecreases,butapenaltyformodelcomplexity increases(reßectedinthevaluesof p andtheresidual degreesoffreedom). ClassiÞcationandregressiontreesprovideamethodof modelbuildingthatisverydifferentfromtheregression approachesdiscussedabove(Breiman etal. 1984;De'ath &Fabricius2000).Startingwitha root correspondingto thewholedataset,themethodproducessuccessivesplits ofthedatasetbasedonvaluesoftheexplanatory variables.Forquantitativepredictors,ateachlevelofthe tree,theapproachconsidersallpossiblebinarysplitsof thepredictors,witheachsplitleadingtoapairof predictedresponsesequaltotheresponsemeansinthe twogroupscreatedbythesplit.Abifurcationiscreated forthepredictorandcutpointthatleadtothesmallest deviance,orsumofthesquareddifferences betweenobservedandpredictedresponses.Thisprocedureisrepeatedrecursivelytoproduceabranchingtree ofbinarysplitsbasedononeormoreoftheexplanatory variables. Unlikethelinearregressionapproaches,theregression treeisnotbasedonastatisticalmodelthatcanbeusedto quantifythetrade-offbetweenmodelcomplexityand explainedvariation.Instead,modelselectionisaccomplishedby pruning overlycomplicatedtreesbackto simplertreesthatwillpresumablyhavegreatergenerality. Acommonlyusedapproachinvolvescross-validation,in which,overmultipleiterations,atreebasedonallbuta smallsubsetofthedataisusedtopredictresponsesfor the validation subset(e.g.seeTherneau&Atkinson 1997). 1062 P.A.Murtaugh Letter 2009BlackwellPublishingLtd/CNRS

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METHODSThedatasetsTwelvedatasetswereobtainedbysearchingliteratureand on-linesources,andbymakingenquiriesamongcolleagues. ThegoalwastoÞndecologicalorenvironmentaldatasets thatincludedquantitativevariables,atleastoneofwhich couldbeconsideredasa response predictablebythe others.Table1presentssomefeaturesofthedatasets, whicharedescribedinmoredetailinAppendixS1andare availablefromtheauthoruponrequest.VariableselectionThevariablesineachdatasetwereexaminedindividually priortomodelÞtting.Responsevariablesandpredictors withextremeskewnesswerelog-transformedtoreducethe chancethatvariableselectionwouldbestronglyinßuenced byafewextremeobservations.Ithenappliedseven methodsofvariableselectionandtreebuildingthatI automatedinR(RDevelopmentCoreTeam2007).Inthe regressionmodelling,interactionsbetweenpredictorswere notincludedinthepoolofexplanatoryvariables. Itshouldbeemphasizedthatautomaticselectionof variablesisnotgoodstatisticalpractice;aniterativeand interactiveapproachismuchmorelikelytoyielduseful models(ChatÞeld2002).Amongotherthings,automating theprocessmakesitdifÞculttoconsidertheproper functionalformsofpredictors,possiblespatialandtemporal correlationofobservations,andinteractionsandcollinearity amongexplanatoryvariables. ThemethodsofmodelÞttingareasfollows. (1) StepwisevariableselectionwithFtests(Efroymson’salgorithm). P -valuesareobtainedbycomparingmodelswithand withoutaparticularexplanatoryvariablewithanextrasum-of-squares F test(eqn1).Inthispaper,Iuse P ¼ 0.05asthethresholdforinclusionorexclusionof predictors;differentvaluesof P wouldleadtodifferent levelsofmodelcomplexity.Startingwithamodel havingnopredictors,werepeatthefollowingsteps:(i) forthepredictorsnotinthemodel,addtheonehaving thesmallest,statisticallysignificant P toenter ,and(ii) ifanyofthepredictorsinthemodelnowhavea nonsignificant P tostay ,droptheonewiththelargest P -value.Theprocedureendswhenallofthevariables inthemodel,andnoneofthoseoutsidethemodel, have P <0.05. (2) StepwisevariableselectionusingAIC. Startingwithamodel havingnopredictors,werepeatthefollowingsteps:(i) forthepredictorsnotinthemodel,addtheoneleading tothelargestreductionintheAIC(eqn2),and(ii)if anyofthepredictorsinthemodelleadtoareductionin theAICwhendropped,removetheoneproducingthe largestreduction.Theprocedureendswhenthereare nofurthervariableadditionsordeletionsthatcanlower theAIC. Table1Descriptionofthedatasets. R2isthecoefÞcientofdeterminationforamodelcontainingallofthecandidatepredictors.The conditionindexisameasureofcollinearityofthepredictors(Belsley etal. 1980).AppendixS1hasmoredetailsaboutthedatasets LabelDescription No.of observations No.of predictors R2Cond. index AChlorophyll a innorth-easternU.S.lakes,predictedfromwaterchemistry348100.67102 BBirdspeciesrichnessaroundnorth-easternU.S.lakes,predictedfromlakeand watershedcharacteristics 18580.2232 CSpeciesrichnessofnativeÞshinnorth-easternU.S.lakes,predictedfromwatershed characteristicsandlakebiota 19480.4826 DSeedproductionbyweedyrice,predictedfromplantmorphology35670.9684 EWaveheightinthePaciÞcOcean,predictedfromweathervariables33570.94585 FFaecalcoliformbacteriainOregonrivers,predictedfromwaterchemistry7780.32121 GBiochemicaloxygendemandintheDeschutesRiver,predictedfromßow,water chemistryandtemperature 2680.58126 HSleepdurationformammalspecies,predictedfromlife-historycharacteristics,weight andexposure 5170.7029 JHumanpopulationdensityinstreamwatersheds,predictedfromwatershed characteristics 31070.3344 KSecchidepthinmid-Atlanticestuaries,predictedfromwatertemperatureand chemistry 87080.25180 LAbundanceofcaddisßylarvaeinanOregonstream,predictedfromlocalstreamand substratecharacteristics 31150.2723 MZooplanktonspeciesrichnessinNorthAmericanlakes,predictedfromarea,depth, elevationandproximitytootherlakes 6680.6429 Letter Variable-selectionmethods 1063 2009BlackwellPublishingLtd/CNRS

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(3) StepwisevariableselectionusingSchwarz’sBIC. Asin(2),but withtheBIC(eqn3). (4) AllsubsetsusingtheAIC. TheRfunction regsubsets identifiesthebestsubsetsofpredictorsusingabranchand-boundalgorithm(Miller2002).Irankedthe subsetsaccordingtotheirvaluesoftheAIC,and chosethesubsethavingtheminimumvalue. (5) AllsubsetsusingtheBIC .Asin(4),exceptthatrankings arebasedontheBIC. (6) Regressiontreesprunedbythe1-SErule (Breiman etal. 1984).IusedfunctionsinR's rpart package(Therneau &Atkinson1997)toÞtregressiontrees,whichwere thenprunedbacktoavoidoverÞtting.Thepruning algorithmusescross-validationtoidentifytreeswith smallvaluesofariskmeasurethatbalancesexplained variationagainsttreecomplexity.The1-SErule choosesthesimplesttreehavingriskthatiswithin1 standarderroroftheachievedminimum. (7) Regressiontreesprunedtotheminimumrisk. Asin(6),except thattheprunedtreewiththeminimumriskischosen. Thisintroducessomerandomnessintotreeselection thatthe1-SErule,above,seekstoavoid,butitalso providesalessaggressivepruningmethodforcomparisonwiththepreviousmethod.Cross-validationThepredictiveabilityofeachmethodappliedtoaparticular datasetwasestimatedusingcross-validation(e.g.seeHarrell 2001,p.93).Foreachdataset,thefollowingstepswere repeated2000times. (1)Randomlydividethedatasetintoatrainingsubset consistingofabout75%oftheobservationsanda validationsubsetconsistingoftheremaining25%of theobservations. (2)Foreachvariable-selectionmethod,usethemethodto buildapredictivemodelbasedonthetrainingdata. (3)Applythemodelobtainedinstep2(regression coefÞcientsorregressiontree)totheexplanatory variablesforobservationsinthevalidationsubsetto predictresponsesforthevalidationsubset.Compute thecross-validationmeansquaredpredictionerror: MSPE ¼ Pni ¼ 1ð Yi ^ YiÞ2n; where Yiistheobservedresponseforitem i ; ^ Yiisthe responsepredictedbythemodelbasedonthetrainingdata, andthesumisoverthe n *observationsinthevalidation subset.Incaseswherenovariableswereselectedfora particularsetoftrainingdata,themeanoftheresponsesin thetrainingdatawasusedasthepredictedresponseforall itemsinthevalidationsubset. Foreachdatasetandvariable-selectionmethod,the2000 valuesofthecross-validationMSPEwereaveragedtogive anoverallsummaryofpredictiveability.RESULTSAnexampleTable2showsmodellingresultsfordatasetC,inwhichthe responseisthespeciesrichnessofnativeÞshinnortheasternU.S.lakes.Ichosethisdatasetbecauseityieldsthe largestnumberofdifferentmodels,outofthe12datasets studiedhere.Theregression-basedmethodsresultedin threedifferentmodels,havingtwo,threeandfourexplanatoryvariables.Twomoredistinct models resultedfrom theregression-treeapproaches. Figure1showstheregressiontreeobtainedwithmethod 7.TheÞrstsplit,whichistheonlysplitproducedbymethod 6,indicatesthatthelakesshouldÞrstbediscriminatedon thebasisofarea:thosewithlog-transformedarealessthan 2.88haveameanlog-transformedspeciesrichnessof1.41, andtheremaininglakeshaveameanresponseof2.07. Thesesubgroupsoflakesarethenfurthersplitaccordingto theirvaluesofthezooplankton,elevation,depthandarea variables.NoticeinFig.1thattheassociationofdepthwith speciesrichnessisnegativeinthesplitontheleft-handside ofthetreeandpositiveintheright-handsplit.Thisisan exampleoftheveryßexible modelling ofthevarying associationofapredictorandresponseoversubgroupsthat isavailablewithregressiontrees.NumberofexplanatoryvariablesincludedTable3shows,foreachcombinationofdatasetand method,theaveragenumberofexplanatoryvariables includedinmodelsÞttothetrainingdatasets. Aconspicuouspatternisthegenerallysmallernumberof variablesincludedbytheregression-treemethods(6and7), comparedtotheothermethods.However,becauseasingle predictorcanoccuratmorethanonenodeofaregression tree(e.g.seelakeareainFig.1),thecomparisonofnumber ofpredictorsintreesvs.ordinaryregressionmodels providesanimperfectcontrastofmodelcomplexity. Figure2aisagraphicalsummaryoftheresultspresented inTable3.Onaverage,methodsusingtheAIC(2and4) yieldedthelargestmodels,whichisconsistentwiththe AIC'srelativelysmallpenaltyformodelcomplexity.Models ofintermediatesizewereproducedbythemethodsbased on F -tests(1)andtheBIC(3and5). Interestingly,modelsÞtusingtheall-subsetsalgorithm wereaboutthesamesizeasthoseÞtusingstepwise procedures,forboththeAIC(4vs.2inFig.2a)andBIC(5 vs.3). 1064 P.A.Murtaugh Letter 2009BlackwellPublishingLtd/CNRS

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PredictiveabilityTable4andFig.2bsummarizethepatternsofMSPEfound inthedifferentcombinationsofdatasetandvariableselectionmethod. Tree-basedvs.regression-basedapproaches Thetwotree-basedmethods(6and7)generallyhavehigher squaredpredictionerrorsthanthoseoftheregression-based methods(1Ð5),althoughthepatternisreversedfortwoof the12datasets(FandG;seeTable4). Takingacloserlookatthesetwodatasets,weseethat bothhaverelativelyhighconditionindicesÐameasureofthe collinearityoftheexplanatoryvariablesÐandsmallsample sizes.Highcollinearityusuallyinßatesthevariabilityof regressioncoefÞcientsandpredictedresponses,whichwould tendtoincreasethemagnitudeofsquaredpredictionerrors, especiallywhenthetrainingdatasetsaresmall.Insuchcases, themoreparsimoniousmodelsproducedbythetree-based methodsmayoutperformtheregression-basedapproaches, atleastwithrespecttothismeasureofpredictiveability. DatasetGisanextremeexample:in none ofthecrossvalidationsdidthetree-basedmethodsselect any explanatory variables,yettheassociatedMSPEs(equaltothepopulation variancesoftheresponsesinthetrainingdatasets)were usuallysmallerthanthosefortheregression-based approaches,which,onaverage,madeuseof1.3to3.3 predictors(Table3). Thisisnottosaythatregressiontreeswithoutbranches areusefulpredictivetools.Butthepruningalgorithmsthat resultedinthese root-only treesareinasenseprotection againsttheoverÞttingandvarianceinßationthatcanoccur whenoneÞtsregressionmodelstosmalldatasetshaving relativelylargenumbersofcollinearpredictors. Comparisonsamongtheregression-basedapproaches MaybethemostinterestingfeatureofFig.2bistheclose similarityoftheMSPEsassociatedwiththeregression-based approaches(1Ð5).Inspiteofthelargernumberof predictorsincludedbytheAIC-basedmethods(2and4), comparedtothe F -testandBIC-basedmethods(1,3and5; seeFig.2a),themeanpredictiveabilitiesofallÞvemethods werenearlyidentical.DISCUSSIONAppliedtothe12datasetsinthispaper,theÞveregressionbasedmethodsofvariableselectionproducedmodelswith verysimilarpredictiveability,whiletheperformancesofthe twotree-basedmethodswereusuallyinferior(Fig.2b). Therewassurprisinglylittledifferencebetweenstepwiseand all-subsetsapproaches(methods4vs.2,and5vs.3)with respecttoeithermodelsize(Fig.2a)orpredictiveability (Fig.2b).Theseresultscontrastwithsomeecologists assertionsabouttheshortcomingsofstepwiseprocedures (e.g.seeWhittingham etal. 2006;Mundry&Nunn2009). Asexpected,themodelswiththelargestnumberof predictorswereobtainedwiththetwoAIC-basedmethods Table2Resultsofapplyingthesevenvariable-selectiontechniquestodatasetC,predictingspeciesrichnessofnativeÞshinnorth-eastern U.S.lakes.Lakeareawaslogtransformed;chlorophyll a ,thenumberofnon-nativespecies,andtheresponsereceivedthelog( y +1) transformation.Entriesforthepredictorsareregressioncoefficientsforthevariablesincludedinthemodel,or,forregressiontrees(methods 6and7),thedirectionofthe effect ofthepredictorontheresponse(seeFig.1,anddiscussionintext).Thevalueofthecross-validation meansquaredpredictionerror(MSPE)reportedformodels2,4and5istheaverageoverthethreemethods MethodLake area(ha) Elev. (m) Mean depth(m) Chl. a ( l gL) 1) No.ofnonnativeÞshsp. Total zoopl.indiv. Cross-val. MSPE 10.180 ) 0.001250.08110.187 2,4,50.198 ) 0.001330.0884 ) 0.09960.185 30.168 ) 0.00131 0.185 6+ 0.226 7++/ ) +0.214 Figure1Regressiontreeforpredictingspeciesrichnessofnative Þshinnorth-easternU.S.lakes(datasetC),usingtheminimumriskcriterion(method7).Thenumbersintheovals( nodes )and rectangles( leaves )givethemeansofthelog(numberofspecies) forlakeswithdifferentcombinationsofpredictorvalues, determinedbysplitshigherinthetree.Thetreechosenbythe 1-SErule(method6)hasjustthesingle,initialsplitbasedonlog (area). Letter Variable-selectionmethods 1065 2009BlackwellPublishingLtd/CNRS

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(Fig.2a).Asthepredictiveabilityofthesemodelswasnot markedlybetterthanthatofmodelsproducedby approachesbasedon F -testsandtheBIC(Fig.2b),the choiceamongvariable-selectiontechniquesmustbeguided byotherconsiderations.Forexample,onecanweighthe relativecostsofthedifferentkindsofmistakesthatcanbe madeinmodelbuilding:includingapredictorthatistruly uninformative,orexcludingonethatisinfactinformative (Murtaugh1998).Dependingonwhichkindofmistakeis morecostlyorconsequential,onemightpreferamore conservative( F -testorBIC-based)ormoreliberal(AICbased)methodofvariableselection. Asmanystatisticianshavepointedout,itisdifÞcultor impossibletointerpretthe P -valuesfortheexplanatory variablesinaregressionmodelthatwasobtainedby winnowingamultitudeofotherpossiblemodels,asthose P -valuesdonotaccountfortheso-calledmodelselection uncertainty(e.g.seeMiller2002;Ramsey&Schafer2002). Someauthorshaveconsequentlydenigratedtheuseof F testsinvariableselection(e.g.seeBurnham&Anderson 2002;Mundry&Nunn2009).Butthe F statisticcan neverthelessbeausefulcurrencyforexpressingthetradeoffbetweenexplainedvariationandmodelcomplexity.In fact,the F -to-enter,theadjusted R2andtheAICcanallbe viewedasspecialcasesofageneralizedformofMallow's Cpstatistic(Miller2002,p.205). Asanexample,considerthevariousmodelsobtainedfor datasetC,predictingspeciesrichnessofnativeÞshin north-easternU.S.lakes(Table2).Applyingstepwise selection(method1)witha P -to-enterand P -to-stayof 0.05,weobtainthethree-variablemodelshowninthefirst Table3Meannumbersofvariablesincludedinmodelsproduced bythesevenmethodsappliedtothe12datasets(seeTable1for thecorrespondencebetweenlettersanddatasets).Eachentryis theaveragenumberofpredictorsinmodelsÞttothetrainingdata in2000randomsplits.Thisisanimperfectsummaryofthe complexityofregressiontrees,inwhichasingleexplanatory variablemaybeusedatmorethanonelevelofthetree.Methods: 1,stepwise F tests;2,stepwiseAkaike'sinformationcriterion (AIC);3,stepwiseBayesianinformationcriterion(BIC);4,all subsetsAIC;5,allsubsetsBIC;6,treewith1-SErule;7,treewith minimumrisk Dataset Method 1234567 A3.24.13.04.93.11.12.2 B2.83.42.53.42.60.41.6 C2.53.52.33.52.31.52.5 D2.12.91.63.11.61.01.0 E4.85.64.45.54.42.22.3 F1.62.11.52.01.50.81.5 G1.33.32.23.33.20.00.0 H2.43.42.53.32.61.01.4 J4.86.14.16.14.21.12.8 K3.03.12.83.22.81.84.0 L3.53.92.93.92.92.23.3 M1.92.21.92.21.90.91.7 MethodMean no. of variables 123456 7 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 MethodStandardized MSPE 1234567 1 0 1 2 (a) (b) Figure2Summarystatisticsforthesevenmethods,averagedover datasets:(a)meannumberofpredictorspermodel(seethe commentaboutregressiontreesinthelegendofTable3);and(b) meanvalueofthecross-validationmeansquaredpredictionerror (MSPE),standardizedasdescribedinthelegendofTable4.The verticallinesare95%conÞdenceintervalsbasedonthemean squareerror(MSE)fromlinearmodelsoftheresponsesasa functionofmethodanddataset: y t0 : 975 ; 66 MSE = 12 p . 1066 P.A.Murtaugh Letter 2009BlackwellPublishingLtd/CNRS

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lineofTable2.Ifweinsteaduseathresholdof0.01,we obtainthesametwo-variablemodelthatwasproducedby method3.Finally,usingathresholdof0.10,weobtainthe four-variablemodelproducedbymethods2,4and5.Rather thanaliabilityfromtherigidframeworkofhypothesis testing,the F -testsignificancelevelcanbethoughtofasa tuningparameterthatadjuststhepenaltyformodel complexityinawaythatisnotpossibleusingtheAICor BICalone(e.g.seeSauerbrei etal. 2007).Inthiscontext, itseemsunfairtodismissstepwisevariableselectionwith F -testsasaviabletoolformodelselection,justbecauseit usesthemachineryofhypothesistesting(Whittingham etal. 2006;Lukacs etal. 2007). Itisimportanttorememberthatthecomparisonsamong variable-selectionmethodsthataresummarizedhereare basedonpredictiveability;ifthegoalofmodelbuildingis explanationoridentiÞcationofpossiblecausalrelationships, thecriteriaforcomparingapproachescouldbedifferent fromthoseconsideredhere(Sauerbrei etal. 2007). Anothercaveatisthattheuseofreal,ratherthan simulated,datamakesitdifÞculttoidentifytherelevant scopeofinferenceforthiswork.The12datasetswere identiÞedinadecidedlynon-randomway,anditispossible thatspecialorunusualfeaturesofthesedatahadastrong inßuenceontheresults.Additionaleffortcouldbedirected toÞndingmoredata,whichcouldbehelpfulinidentifying featuresofdatasetsthatmakethemmoreorless amenabletothedifferentmodel-buildingtechniques.But itishardtoenvisionamethodofsamplingdatasetsthat wouldpermitgeneralizationofresultstoanidentiÞable largerpopulation. Simulation,inwhichthe true modelisknown,wouldseem theonlydeÞnitivewaytocomparemodel-buildingtechniques.Investigatorshavesimulateddatainsomanydifferent waysandusedsuchavarietyofmetricsforcomparing methodsthatitisdifÞculttosynthesizetheirresults,although Raffalovich etal. (2008)makeadeterminedattempt.Intheir ownresearch,Raffalovich etal. (2008)evaluatedtheabilityof severalprocedurestoincludeimportantvariablesandexclude irrelevantones.TheyfoundthatstepwiseregressionandBICbasedapproachesperformedbest,whileAIC-basedmethods areclearlyinferiorandshouldbeavoided .Murtaugh(1998), ontheotherhand,foundlittledifferenceinthediscriminating abilityofmethodsbasedon F -tests,theBICandMallows Cp(similartotheAIC),consistentwiththeempiricalresults reportedhere. Thevarietyofmodelsthatcanbeobtainedforindividual datasets(e.g.seeTable2)andthesimilarpredictiveability achievedbysomefairlydifferentmethodsofvariable selection(Fig.2b)suggestthatthereisno best methodof selectingstatisticalmodels.Thisconclusionisconsistent withthefrequentlyquotedassertionthat allmodelsare wrongbutsomeareuseful (Box1979).Ifthereisno correct model,therecanbenobestmethodofmodel building. Single-mindedpromotionofonemethodofvariable selectionoveranotherplacesundueemphasisonpurely statisticalconsiderations,apracticethatsomeauthorshave grownwearyof(Guthery etal. 2005;Murtaugh2007; Chamberlain2008).Thereisawidearrayofapproachesto variableselection,anyofwhichcangeneratemodelsworthy ofconsiderationinaparticularapplication.Whichmodels Table4Standardizedmeanvaluesofmeansquaredpredictionerror(MSPE)basedoncross-validationforthesevenmethodsappliedtothe 12datasets(seeTable1forthecorrespondencebetweenlettersanddatasets).Ineachrow,themeanMSPEswerestandardizedby subtractingthemean(column2)anddividingbythestandarddeviation(column3).Methods:1,stepwise F tests;2,stepwiseAkaike's informationcriterion(AIC);3,stepwiseBayesianinformationcriterion(BIC);4,allsubsetsAIC;5,allsubsetsBIC;6,treewith1-SErule;7, treewithminimumrisk DatasetMeanMSPESDofMSPE StandardizedMSPEbymethod 1234567 A2.440.186 ) 0.67 ) 0.44 ) 0.67 ) 0.44 ) 0.671.571.34 B0.4200.0428 ) 0.62 ) 0.73 ) 0.30 ) 0.73 ) 0.511.471.42 C0.3570.0597 ) 0.55 ) 0.61 ) 0.55 ) 0.61 ) 0.551.781.08 D0.2910.0764 ) 0.60 ) 0.58 ) 0.57 ) 0.57 ) 0.591.601.31 E12.32.67 ) 0.57 ) 0.59 ) 0.58 ) 0.60 ) 0.581.561.36 F1.180.1230.250.580.390.490.82 ) 0.46 ) 2.08 G0.3480.01460.210.680.490.790.71 ) 1.42 ) 1.45 H0.2400.139 ) 0.45 ) 0.60 ) 0.60 ) 0.60 ) 0.671.531.38 J0.07610.0243 ) 0.56 ) 0.80 ) 0.40 ) 0.80 ) 0.311.481.40 K0.2830.0143 ) 0.59 ) 0.52) 0.45 ) 0.31 ) 0.382.220.03 L372702105 ) 0.58 ) 0.79 ) 0.23 ) 0.86 ) 0.161.940.68 M0.1950.0171 ) 0.60 ) 0.55 ) 0.60 ) 0.53 ) 0.591.751.12 Letter Variable-selectionmethods 1067 2009BlackwellPublishingLtd/CNRS

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aremostusefulisdeterminednotbythemethodbywhich theywereobtained,butratherbytheirappropriatenessfor thetaskathand.ACKNOWLEDGEMENTSIthankJeannieSifneosforguidanceontheuseof regressiontreesandformanyconstructivecommentson anearlierversionofthemanuscript.Ialsothankthree anonymousrefereesfordetailedsuggestionsforclarifying andstrengtheningthepaper,andSteveStehmanforhelpful discussions.REFERENCES Belsley,D.,Kuh,E.&Welsch,R.(1980). RegressionDiagnostics: IdentifyingInuentialDataandSourcesofCollinearity .Wiley,New York. Box,G.E.P.(1979).RobustnessinthestrategyofscientiÞcmodel building.In: RobustnessinStatistics (edsLauner,R.L.&Wilkinson, G.N.).AcademicPress,NewYork,pp.201Ð236. Breiman,L.,Friedman,J.H.,Olshen,R.A.&Stone,C.J.(1984). ClassicationandRegressionTrees .Wadsworth&Brooks/Cole, Monterey. Burnham,K.P.&Anderson,D.R.(2002). ModelSelectionandMultimodelInference:APracticalInformation-TheoreticApproach ,2ndedn. Springer,NewYork. Chamberlain,M.J.(2008).ArewesacriÞcingbiologyforstatistics? J.Wildl.Manage. ,72,1057Ð1058. ChatÞeld,C.(2002).Confessionsofapragmaticstatistician. Statistician ,51,1Ð20. De'ath,G.&Fabricius,K.E.(2000).ClassiÞcationandregression trees:apowerfulyetsimpletechniqueforecologicaldataanalysis. Ecology ,81,3178Ð3192. Gerritsen,J.,Dietz,J.M.&Wilson,H.T.,(1996).EpisodicacidiÞcationofcoastalplainstreams:anestimationofrisktoÞsh. Ecol.Appl. ,6,438Ð448. Guthery,F.S.,Brennan,L.A.,Peterson,M.J.&Lusk,J.J.(2005). Informationtheoryinwildlifescience:critiqueandviewpoint. J.Wildl.Manage. ,69,457Ð465. Harrell,F.E.,(2001). RegressionModelingStrategieswithApplicationsto LinearModels,LogisticRegression,andSurvivalAnalysis .Springer, NewYork. Hobbs,N.T.&Hilborn,R.(2006).Alternativestostatistical hypothesistestinginecology:Aguidetoselfteaching. Ecol. Appl. ,16,5Ð19. Hoeting,J.A.,Davis,R.A.,Merton,A.A.&Thompson,S.E.(2006). Modelselectionforgeostatisticalmodels. Ecol.Appl. ,16,87Ð98. Knick,S.T.&Rotenberry,J.T.(1995).Landscapecharacteristicsof fragmentedshrubsteppehabitatsandbreedingpasserinebirds. Conserv.Biol. ,9,1059Ð1071. Lee,H.&Ghosh,S.(2009).Performanceofinformationcriteria forspatialmodels. J.Stat.Comput.Simul. ,79,93Ð106. Leigh,G.T.,Read,A.J.&Halpin,P.(2008).Fine-scalehabitat modelingofatopmarinepredator:dopreydataimprovepredictivecapacity. Ecol.Appl. ,18,1702Ð1717. Lukacs,P.M.,Thompson,W.L.,Kendall,W.L,Gould,W.R., Dougherty,P.F.,Jr,Burnham,K.P. etal. (2007).Concerns regardingacallforpluralismofinformationtheoryand hypothesistesting. J.Appl.Ecol. ,44,456Ð460. Mazerolle,M.J.(2006).Improvingdataanalysisinherpetology: usingAkaike'sinformationcriterion(AIC)toassessthestrength ofbiologicalhypotheses. Amphib-reptil. ,27,169Ð180. Miller,A.(2002). SubsetSelectioninRegression ,2ndedn.Chapman& Hall/CRC,BocaRaton,FL. Mundry,R.&Nunn,C.L.(2009).StepwisemodelÞttingandstatisticalinference:turningnoiseintosignalpollution. Am.Nat. , 173,119Ð123. Murtaugh,P.A.(1998).Methodsofvariableselectioninregression modeling. Commun.Stat.Simul.Comput. ,27,711Ð734. Murtaugh,P.A.(2007).Simplicityandcomplexityinecologicaldata analysis. Ecology ,88,56Ð62. Olden,J.D.&Jackson,D.A.(2000).Torturingdataforthesakeof generality:Howvalidareourregressionmodels? Ecoscience ,7, 501Ð510.RDevelopmentCoreTeam(2007). R:ALanguageandEnvironment forStatisticalComputing .Availableat:http://www.R-project.orgR FoundationforStatisticalComputing,Vienna. Raffalovich,L.E.,Deane,G.D.,ArmstrongD.&Tsao,H.S. (2008).Modelselectionproceduresinsocialresearch:MonteCarlosimulationresults. J.Appl.Stat. ,35,1093Ð1114. Ramsey,F.&Schafer,D.(2002). TheStatisticalSleuth:ACourse inMethodsofDataAnalysis ,2ndedn.DuxburyPress,Belmont,CA. Sauerbrei,W.,Royston,P.&Binder,H.(2007).Selectionof importantvariablesanddeterminationoffunctionalformfor continuouspredictorsinmultivariablemodeling. Stat.Med. ,26, 5512Ð5528. Therneau,T.M.&Atkinson,E.J.(1997).Anintroductionto recursivepartitioningusingtheRPARTroutines. TechnicalReport SeriesNo.61 .DepartmentofHealthScienceResearch,Mayo Clinic,Rochester,MN. Ward,E.J.(2008).Areviewandcomparisonoffourcommonly usedBayesianandmaximumlikelihoodmodelselectiontools. Ecol.Modell. ,211,1Ð10. Whittingham,M.J.,Stephens,P.A.,Bradbury,R.B.&Freckleton, R.P.(2006).Whydowestillusestepwisemodellinginecology andbehaviour? J.Anim.Ecol. ,75,1182Ð1189. SUPPORTINGINFORMATIONAdditionalSupportingInformationmaybefoundinthe onlineversionofthisarticle:AppendixS1Descriptionofthedatasets. Asaservicetoourauthorsandreaders,thisjournal providessupportinginformationsuppliedbytheauthors. Suchmaterialsarepeer-reviewedandmaybere-organized foronlinedelivery,butarenotcopy-editedortypeset. Technicalsupportissuesarisingfromsupportinginformation(otherthanmissingÞles)shouldbeaddressedtothe authors. Editor,JessicaGurevitch Manuscriptreceived13July2009 Manuscriptaccepted14July2009 1068 P.A.Murtaugh Letter 2009BlackwellPublishingLtd/CNRS



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STOCHASTICMODELSFORTHEGROWTHRATEOFANIMALPOPULATIONS: VARIANCESCALINGANDDECOMPOSITION By JAKEM.FERGUSON ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014

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c ! 2014JakeM.Ferguson 2

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Amiesposahermosaysolidaria,Rosana,andtomomanddad,fortheirlove,support, andinspirationthroughtheyears 3

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ACKNOWLEDGMENTS IwouldÞrstliketothankmythechairofmycommittee,Jos « ePonciano,for motivatingmetobemoreanalyticalinmyworkandforhissupportinbringingme andmywifetotheUniversityofFlorida.Iwouldalsoliketothankmyco-chair,Scott McKinleyforhisvaluablecritiquesofmymodelsandinspiringmewithhisintenseand rigorousapproachtosolvingproblems.Mycommitteemembershaveallprovided wonderfulfeedbackthroughoutmydissertation.IwouldespeciallyliketothankColette St.Maryforbringingmeintoherlabtolearnsome'realbiology'andforproviding valuablehelpwithimprovingmyscientiÞccommunication.RobertHoltalsoprovided extremelyvaluablefeedbackregardingtheecologicalassumptionsunderlying mymodels.Evenhisbriefestcommentssometimestookmeseveralyearstofully appreciate.Finally,committeememberandmymaster'sadvisorMarkTaperhasbeen extremelyinßuentialonmyacademicdevelopmentandhasinspiredmetoworkon theorythatisapplicabletorealworldproblems.Iwouldliketothankhimforinitiatingmy metamorphosisintoanecologist,aprocessthatisongoing,andforalltheopportunities heprovidedformealongthewayincludingtheinvaluablesummersspentintheÞeld withBradleyShepard. SupportformyresearchcamethroughtheDepartmentofBiology,whichinaddition totheÞnancialsupportprovidesawonderfulcollegialatmosphereforgraduatestudents. Thedepartmenthasgiven,esomegreatopportunitiesforcollaboration.Iwouldlike tothankTrevorCaughlinandJamesNifongforlettingmeassistthemwiththeirown research.IwasalsosupportedfortwoyearsbytheNationalScienceFoundationasan IGERTfellowunderGrantNo.0801544intheQuantitativeSpatialEcology,Evolution andEnvironmentProgram.TheIGERTprovidedmanyopportunitiesforprofessional developmentandachancetoworkwithagreatgroupofgraduatestudentsfromother departments.WorkingwiththeIGERTgroupwasoneofthehighlightsofmytimeatthe UniversityofFlorida. 4

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Icouldnothavecompletedthisworkwithoutthesupportofmyfamily.Iespecially wanttothankmywife,Rosana,whoalwaysencouragedmewithherenthusiasmwhen sharingideasandtookupmorethanherfairshareofresponsibilitiesaroundthehouse whenIwasburiedwithwork.Mybrother,Mick,Iwouldliketothankforinstillinginme ahealthycompetitivespirit,anecessityfortheacademiclifewhereperformanceis oftenmeasuredagainstothers.Finally,Ineedtothankmyparents,GregandKathleen Ferguson,foralltheirloveandencouragementthroughtheyears.Icreditmydad forinspiringmeatayoungagetolovenature,throughourbackpackingtripsinthe Cascades,andfordevelopinginmeaworkethicthathasallowedmetobesuccessful academicallywhileremainingrelativelyunstressed.Icreditmymomforinspiringmeto beanindependentandcreativethinkerandforteachingmenottotakeanyonescrap. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2POPULATIONGROWTH,DENSITYDEPENDENCE,ANDENVIRONMENTAL AUTOCORRELATIONINANIMALPOPULATIONS ................ 17 2.1Background ................................... 17 2.1.1AutocorrelationinAnimalPopulations ................ 19 2.1.2AnOverviewofAutocorrelationModels ................ 22 2.1.2.1Long-andshort-memorymodels .............. 22 2.1.3StochasticPopulationModels ..................... 25 2.1.4EnvironmentalTrackingModels .................... 27 2.2Methods ..................................... 28 2.2.1EnvironmentalTrackingModels .................... 28 2.2.2JointEstimationofDensityDependenceandAutocorrelationStructure inAbundanceTimeSeries ....................... 29 2.3Results ..................................... 32 2.3.1EnvironmentalTrackingModels .................... 32 2.3.2EstimatesofPEAintheGPDD .................... 33 2.3.3AlternativeErrorModels ........................ 34 2.4Discussion ................................... 35 3PREDICTINGTHEPROCESSOFEXTINCTIONINEXPERIMENTALMICROCOSMS ANDACCOUNTINGFORINTERSPECIFICINTERACTIONSINSINGLE-SPECIES TIMESERIES .................................... 46 3.1Background ................................... 46 3.2Models&Methods ............................... 47 3.2.1ExperimentalData ........................... 47 3.2.2PopulationModels ........................... 48 3.2.3ModelComparisons .......................... 51 3.2.4ParameterEstimation ......................... 54 3.2.5MeasurementError ........................... 55 3.2.6MovingAverageModels ........................ 56 3.3Results ..................................... 56 3.3.1ModelSelection ............................. 56 6

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3.3.2MeasurementError ........................... 59 3.3.3MovingAverageModelDynamics ................... 59 3.4Discussion ................................... 60 4THEEVIDENCEANDIMPLICATIONSOFHIGHERORDERSCALINGIN THEENVIRONMENTALVARIATIONOFANIMALPOPULATIONGROWTH .. 68 4.1Background ................................... 68 4.2Models&Methods ............................... 71 4.2.1ModelProperties ............................ 71 4.2.2DataandTimeSeriesAnalysis .................... 74 4.3Results&Discussion .............................. 78 5CONCLUSIONS ................................... 89 APPENDIX ACURATINGTHEGPDD ............................... 92 BPARAMETERESTIMATIONANDMODELSELECTION ............. 93 CGPDDCOVARIATEANALYSIS ........................... 99 DDERIVATIONOFTHEABUNDANCEMEANANDVARIANCE .......... 101 EARMAMODELDERIVATION ............................ 105 FLIKELIHOODS .................................... 109 GDERIVATIONOFTHEPOPULATIONABUNDANCEMEANANDVARIANCE WITHDEMOGRAPHICANDENVIRONMENTALVARIATION .......... 113 HUNGULATEBICTABLES .............................. 115 REFERENCELIST ..................................... 116 BIOGRAPHICALSKETCH ................................ 129 7

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LISTOFTABLES Table page 2-1Glossaryoftermsrelatedtoenvironmentalautocorrelationinanimalpopulations. 44 2-2SummarystatisticsandsamplesizefordistributionsofPEAestimatedunder differentmodelsandwithdifferentdatasets. ................... 45 3-1Fixedandvariedcomponentsinmodelcomparisons. ............... 63 3-2 ! AICvaluesforeachmodelandcommunitytype. ................ 64 3-3Rootmeansquareerrorforeachmodelandcommunitytype. .......... 64 4-1BICvaluesformodelselectionsofSoaysheepandAlpineibexdatasets.Bold valuesaretheminimumforeachdataset. ..................... 82 F-1 ! AICvaluesforeachmodelÞttothepopulationabundancesbycommunity type.Note-ThenumberofparametersusedintheAICcalculationisgiven by k andboldnumbersrepresentthebestmodelwithinasetofcomparisons. . 112 H-1ModelselectiontablefortheAlpineibex. ..................... 115 H-2ModelselectiontablefortheSoaysheep. ..................... 115 8

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LISTOFFIGURES Figure page 2-1Thespectraldensities, S ( f ) ,ofthreedifferentmodelsonalog-logplot. .... 39 2-2Anillustrationofhowenvironmentaltrackingresponsescantransformenvironmental autocorrelation. .................................... 40 2-3Theeffectoftwodifferentnonlinearinteractionsonanautocorrelatedtime series. ......................................... 41 2-4ApplicationofthestochasticLeibig'slawoftheminimumtoanumber, n ,of autocorrelatedtime-series. ............................. 42 2-5Histogramof389estimatedpopulationenvironmentalautocorrelation(PEA) valuesfromtheGPDD. ............................... 43 3-1Populationtimeseriesforreplicatesofdifferentexperimentalconditions.Simple communitiesincludedaconsumer Daphniapulicaria andplanktonicresource. 65 3-2ThemeanpopulationabundanceforaRickermodelofdensitydependence withdemographicandenvironmentalstochasticity,plottedforeachobservation withtheapproximate95%conÞdenceintervals. .................. 66 3-3Alog-logplotoftheaffectofachangingMAparametervalueonthemean timetoextinction. ................................... 67 4-1Theimpactofenvironmentalvariationintheper-capitagrowthrateoftheRicker model. ......................................... 83 4-2BICselectionsofthebestenvironmentalvariancemodelsintheGPDDtime seriesthatdisplayeddensitydependence. ..................... 84 4-3Thelog-ratioofthecarryingcapacitiesestimatedunderbothenvironmental variancemodels, K ! / K B ,forallGPDDtimeseriesdetectedtohavedensity dependence. ..................................... 85 4-4ThecoefÞcientofvariation(CV)undermultiplicativeandadditiveenvironmental variationforeachdatasetwithdensitydependence. ............... 86 4-5PredictedpopulationvariancefortheAlpineibexandSoaysheepunderalternative environmentalvariancemodels. .......................... 87 4-6TheeffectofdistributionalskewontheÞrstpassagetime. ............ 88 B-1Biasandvariancein ö ! fordatageneratedfromagivenlagwiththeGompertz modelwhenperformingselectionbetweendensityindependence,andGompertz andRickerlag1modelsandwithanARMA(1,1)errorstructure. ........ 97 9

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B-2Biasandvariancein ö ! fordatageneratedfromagivenlagwiththeGompertz modelandwithanARMA(1,1)errorstructure. .................. 98 D-1Contourlikelihoodofthemeanandvarianceassumingalognormaltransition distribution. ...................................... 103 F-1Darkbarsaretheempiricallyrealizedvaluesfortheautocorrelationateach lag.Horizontalbluelineistheapproximate95%conÞdenceinterval. ...... 111 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulÞllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy STOCHASTICMODELSFORTHEGROWTHRATEOFANIMALPOPULATIONS: VARIANCESCALINGANDDECOMPOSITION By JakeM.Ferguson August2014 Chair:Jos « eM.Ponciano Cochair:ScottA.McKinley Major:Zoology Allanimalpopulationsßuctuatethroughtime.Theseßuctuationsaredueboth totheintrinsicpropertiesofpopulationssuchastherandomnatureofbirthsand deaths,andtoextrinsic,environmentalfactorsdrivingbirth,survival,andmigration dynamics.Thisdissertationisconcernedwithbuildingmodelstodescribethese ßuctuationsusingsimplebutgeneralmodelsdescribinghowpopulationsinteract withtheirenvironment.Theresultsarenotjustinterestingfortheirownsake,but alsohaveimportantimplicationsforwildlifeconservationandmanagementefforts. Chapter2looksathowautocorrelatedenvironmentsdriveßuctuationsinnatural populations.Ourresultsprovideaninterestingnewperspectiveonwhenenvironmental autocorrelationwillbeimportanttoaccountfor,andhowecologicalprocessescan disrupttheautocorrelationpresentinenvironmentalfactors.Themodelsdeveloped inChapter3includeaderivationofdemographicandenvironmentalstochasticity, andasingle-speciesstochasticmodelthataccountsforinterspeciÞcinteractions. Wetestedthismodelwithanexperimentaltimeseriesthattrackedtheextinctionof populationsinmicrocosms,Þndingthatwecouldgreatlyimprovepredictionsaboutthe timetoextinctionwhenaccountingforßuctuationsinducedbyinterspeciÞcinteractions. Finally,inChapter4welookedattheinßuenceofstochasticenvironmentsonthe strengthofdensitydependence.Thisisaperspectivethathaslongbeenignored 11

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butturnsouttohaveinterestingimplicationsforpopulationstability.Weshowthat variationinthestrengthofdensitydependencewilloftenyieldmorestabledynamics thanthestandardmodelofenvironmentalvariationbutthattherearecaseswhere itcanalsodestabilizedynamics,leadingtoperiodsofexponentialgrowthfollowed bypopulationcrashes.TakentogetherthisworkfurtherjustiÞestheapplicationof single-speciestimeseriesmethodstocomplexecologicalsystems.Thishasimportant implicationsformanagementandconservationapplicationswhereinformationislimited andsingle-speciesapproachesareused,oftenwithoutunderstandingwhetherthe assumptionsbehindthesingle-speciesapproacharevalid. 12

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CHAPTER1 INTRODUCTION Noiseismusictotheecologist. ÑDanielSimberloff Whenmakingpredictionsofecologicalprocesses,weareofteninterestedin forecastingdynamicsoutsidetherangeofobservedoutcomes.Thisisespeciallytrue forpopulationecologywherepredictingrareevents,suchasextinction,isimportantfor conservationandmanagementapplications.Dynamicaldescriptionsallowustoproject futuretrajectoriesofpopulations,however,ourabilitytoexplaintheobservedvariability inecologicaldataisoftenlimitedmakingthesepredictionshighlyuncertain.Both statisticalandecologicalfactorscandrivethisuncertainty.Thisdissertationfocuses onaccountingfortheecologicalfactorsgoverningthevariabilityinanimalpopulation dynamics. Populationstructure,ecologicalinteractions,andenvironmentalperturbationscan allmakeimportantcontributionstopopulationßuctuations.Accountingforthesefactors isimportantforaccuratelyassessinglong-runpopulationbehaviorandhasbeenthe focusofmuchworkintheoreticalpopulationecology.Thisdissertationhasattemptedto buildmodelsthatcanaccountforecologicalcomplexitybutinamannerconsistentwith thesmalldatasetsmostoftenavailablefortimeseriesanalysisofanimalpopulations. Thishasconstrainedthetypeofmodelsweexaminetothosethatcanbeestimated fromÞelddata,butalsomeansthatourmodelsmayhavepracticalapplicationsfor wildlifeconservationandmanagement. Withlimiteddata,wemustchoosewhichpopulationcomponentsarebesttreated asdeterministicprocessesandwhichcanbestbemodeledasstochasticprocesses. Epigraph .Simberloff,D.(1980).ASuccessionofParadigmsinEcology: EssentialismtoMaterialismandProbabilism. Synthese ,43,3Ð39. 13

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Becausestochasticanddeterministicforcesareoftenofequalimportanceinanimal populations,explainingstructureandpatterninthevariancecanbeasimportantas explainingmeantrendsinthedata.Thisdissertationhasfocusedonhowchangesin theenvironmentcandrivepopulationßuctuations,andhowtocapturethoseßuctuations withvariancemodels.Astochasticapproachgetsaroundspecifyingdetailsaboutthe environment,andinsteadspeciÞesthefrequencyandpatternofitseffectsthrough time.Thisallowsustomodeltheemergenteffectsoftheenvironmentwithoutthe necessityforlargequantitiesofdata.Theapproachyieldsmodelsthatcanbeused totesthypotheseswithdata,thoughthemodelsalsoendbeingsimplisticrelativeto detailedtheoreticaltreatmentsofsimilarprocesses. Thecruxofthestochasticmodelingapproachistounderstandthebiologicaldetails ofhowtheanimalsinteractwiththeirenvironmentandtheprobabilisticpropertiesof environmentalßuctuations.Thishasbeenafocusofmydissertation,bothcritiquing currentmodelassumptionsandextendingexistingmodelstoaccountforpreviously neglectedecologicalprocesses.Ihopethatthesedevelopmentswillhavesome inßuenceonecologicalapplications. InChapter2weareconcernedwithimpactofautocorrelatedenvironmentson thedynamicsofpopulations.Wesetouttodeterminewhetherautocorrelationinthe environmentdrivesßuctuationsofanimalpopulations.Autocorrelatedenvironments havebeenpredictedbytheorytohavesigniÞcantimpactsonpopulationdynamics andseveralexperimentshavesupportedthesepredictions.However,itisnotclear whethertheseeffectswillpersistundernaturalconditions.Basedonourempirical estimatesindicatingverylowenvironmentalautocorrelationinthegrowthrateofnatural populationsweaskedwhythecouplingbetweenpopulationsandtheenvironment mightbeweakerthanexpectedfrompasttheoreticalwork.Wefoundthatthatnonlinear ecologicalinteractionscandecouplepopulationsfromtheautocorrelationpresent inenvironmentalßuctuations.Thisworkprovidesanupdatedperspectiveonthe 14

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importanceofautocorrelatedenvironmentsandpredictsthatpopulationswilldisplay atightcouplingtotheenvironmentwhentheyareontheedgeoftheirenvironmental tolerances. Chapter3isconcernedwiththeabilityforpopulationdynamicsmodelstoaccount forinterspeciÞcinteractions.Weusedasimpleinteractionmodeltodeterminehow ßuctuationsinapreyspeciesmightpropagatetoapredatorspecies.Underthismodel thedynamicsofthepredatorareinßuencedthrougharandomshockpropagatedbythe preyspecies.Ourresultssuggestpredator-preydynamicscanbeaccountedforinsome circumstances,evenwhenoneofthepopulationsisunobserved.Weconfrontedthis modeltoexperimentaldatatodeterminehowaccountingforinteractionscanimprove dynamicalpredictionsofextinctionandfoundthatourmodelsigniÞcantlyimproved modelpredictions.Webelievethatthistheoreticalandempiricaldemonstrationwillhave animpactonthepracticeofpopulationviabilityanalysis. Chapter4examinedhowtheenvironmentcandriveßuctuationsinthestrength ofdensitydependence.Limitedpreviousworkhassuggestedthatthesekindsof environmentalßuctuationsmaybedestabilizingbyallowingforthepossibilitydensity independentgrowth,followedbyrapidlarge-scaledeclinesinabundances.Wefound evidencethatinmanypopulationsenvironmentalperturbationsinthestrengthofdensity dependencecanincreasethestrengthofregulation,stabilizinglong-termdynamicsby loweringthevarianceassociatedlowabundances.Wealsoprovideexamplesofhow thismodelcanbeusedtodiagnosethekindofinteractionbetweenapopulationand theirenvironmentbyapplyingthemodeltotwowellstudiedungulatepopulations. IwillclosebybrießypointingoutseverallessonsIhavelearnedfromthisworkthat Ithinkareofbroadinteresttoquantitativeecologists.TheÞrstisthatstartingwitha stochasticviewpointcanleadtonewmodelformulationsthatwouldn'tbeaddressedby takingadeterministicformulaandsimplyaddingrandomvariationtotheobservations. ThederivationofARMAinteractionmodelsinChapter3illustratesthispoint.Here,the 15

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interactionmodelwasderivedbyaccountingforthevariationineachpopulationand keepingtrackofhowinterspeciÞcinteractionspropagatetheseßuctuationsthrough time.Thisdiffersfromthemorecommonapproachoftakingadeterministic'skeleton' andtackingonarandomvariabletoaccountfornoise.Thisapproachmissesthe importantpointthatnoisecaninteractwiththedeterministicprocessestogenerate noveldynamics.Asecondimportantpointisthatsimplemodelscanbeusedtomake generalargumentsaboutnature,thoughwemustbeawarethatotherprocessesmay generatethesamepatternsasourhypothesis.Thisiswellillustratedthroughthe multipleroutestogeneratetheRickermodelofdensitydependence.Ithasbeenderived undermanydifferentecologicalassumptionssuchasthroughspatialcompetitionfor alimitedterritories,predator-preydynamicswithyearlyreproduction,andthrough cannibalisticinteractions.Thisphenomenonhasalsooccurredinmyownwork,theMA modeldiscussedinChapter3canrepresentinteractionsbetweenspeciesbuthasalso beenusedtoaccountformeasurementerrorintimeseriesdata.Despitethepower thatsimplemodelshave,cautionmustbeappliedinattributingmechanisticcauses usinginferentialapproaches.Forthisreasonbuildingmodelsupfrombasicecological processesiskeytohavinginterpretablemodels.Thispracticecanlimitthescopeofthe modelsetconsideredinempiricalstudiesandprovidestrongerbiologicalinterpretations thancurveÞttingexercises. 16

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CHAPTER2 POPULATIONGROWTH,DENSITYDEPENDENCE,ANDENVIRONMENTAL AUTOCORRELATIONINANIMALPOPULATIONS 2.1Background Biologistshavelongunderstoodthatstochasticfactorsinßuencepopulationgrowth throughbothintrinsicandextrinsicprocesses( Goodman 1987 , HilÞnger&Paulsson 2011 ).Howtheseprocessesarearticulatedmathematicallycanbeacriticaldeterminant whenassessingpopulationextinctionrisk.Therefore,itiscrucialthatmathematical representationsofhypothesizedextinctionforcesaccuratelyreßecttheprocessesdriving populationvariability. Extrinsicvariation,alsoknownasenvironmentalvariationinpopulationecology,isa practicalwaytoaccountfortheemergenteffectsofthecommunity-andecosystem-level interactionsinsinglespeciespopulationdynamicsmodelswhendetailedinformationon environmentalcovariatesisnotavailable( Dennis&Costantino 1988 ).ForsufÞciently largepopulations,environmentalvarianceisconsideredtobetheprimarycontributor tothepopulationvariance( Lande 1993 ).Thefactorsdrivingenvironmentalvariation areoftenarguedtoconvergetoanormaldistributionfollowingthecentrallimittheorem ( Lewontin&Cohen 1969 ),whichassumesthattheprocessesdrivingvariabilityare independentthroughtime.Formanyecologicalandenvironmentalprocesses,however, thisassumptionmaynothold.Infact,manyecologicalandenvironmentaltimeseries arecharacterizedbystrongtemporaldependencies( Cyr&Cyr 2003 , Miramontes& Rohani 1998 , Steele 1985 , Vasseur&Yodzis 2004 ).Inparticular,itisoftenthecasethat thestateofthesetimeseriesattime t + k canbeforecastasafunctionofthestateat theearliertime t . Anumberoftheoreticalstudieshaveshownthatautocorrelatedenvironmental statescanhaveimportantconsequencesonestimatesofpopulationpersistence ( Caswell&Cohen 1995 , Cohen 1995 , Cuddington&Yodzis 1999 , Heino&Sabadell 2003 , Holt etal. 2003 , Kaitala etal. 1997 , Kamenev etal. 2008 , L ¬ ogdberg&Wennergren 17

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2012 , Mode&Jacobson 1987 , Morales 1999 , Petchey etal. 1997 , Ripa&Lundberg 1996 , Rotenberg 1987 , Roughgarden 1975 , Roy etal. 2005 , Ruokolainen etal. 2007 , Schwager etal. 2006 , Tuljapurkar&Haridas 2006 , Wichmann etal. 2003 ; 2005 ). However,moststudiesthathaveempiricallyestimatedthepopulationabundance autocorrelationhavebeenunabletodirectlylinkthistoenvironmentalcovariates( Cyr 1997 , Halley&Inchausti 2004 , Inchausti&Halley 2001 ; 2002 , Petchey 2000 , Pimm &Redfearn 1988 , Swanson 1998 ),orhavefoundthatthelinkbetweenenvironmental covariatesandpopulationstobeweak( Garcia-Carreras&Reuman 2011 , Knape& deValpine 2010 ).Becauseofthepotentialimportancethatenvironmentalautocorrelation mayhaveforapplicationssuchaspopulationviabilityanalysis,itisimportantto empiricallydeterminehowanimalpopulationgrowthisaffectedbyautocorrelated environmentalcovariatesinordertobuildappropriatemodelsofenvironmental variability.Inparticular,thepopulationautocorrelationsignalmaycomefromboth, intra-speciÞc,ecologicalregulatoryprocesses(suchasdensity-dependence( Akcüakaya etal. 2003 )),andfromautocorrelatedenvironmentalprocesses( Halley 1996 ). Totesttherolethatenvironmentalautocorrelationmayhaveinpopulationrisk assessmentweÞrstlookedatmodelsdescribinghowpopulationsinteractwith autocorrelatedenvironmentalvariablesusingasimulationstudy.Thisprovideda baselineexpectationforthebehaviorofrealpopulationsembeddedinautocorrelated environments.Wethenestimatedtheenvironmentalautocorrelationpresentinalarge numberofanimalpopulationsfromabroadrangeoftaxafromtheGlobalPopulation DynamicsDatabase(GPDD)( NERC 2010 ).Finally,weconductedextensivetestsofour modelassumptions,inordertodeterminetherobustnessofourestimates. Webeginthemanuscriptwithabriefoverviewonmethodsthathavebeenapplied tostudyanimalpopulationautocorrelationinthepast.Weintroduceenvironmental trackingmodels,whichdescribehowchangesintheenvironmentdrivechangesin populationgrowth.Thisisfollowedbytheintroductionofstochasticpopulationmodels 18

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thatintegratepopulationgrowthandregulationwithautocorrelatedenvironmental variability.Finally,weapplythesemodelstoalargedatasetcuratedfromtheGPDDto jointlyestimatepopulationgrowth,densitydependence,andenvironmentalautocorrelation inalargenumberofabundancetimeseries. 2.1.1AutocorrelationinAnimalPopulations Manyclimaticandenvironmentalcovariatesdisplaytemporalautocorrelation( Cyr& Cyr 2003 , Steele 1985 , Vasseur&Yodzis 2004 ).Ifthesecovariatesdrivepopulation growththentheadditionalautocorrelationtheyintroducemaybeanimportant contributortotheoverallpopulationdynamics.Therearemanyexamplesofextrinsic driversofpopulationgrowththatmayleadtoadditionalpopulationautocorrelation. Forexampleclimaticfactorssuchasrainfall( Dennis&Otten 2000 , Taper&Gogan 2002 )andtemperature( Savage etal. 2004 ),andenvironmentalandecologicalfactors suchasdemographyandinterspeciÞcinteractions( Abbott etal. 2009 , Ferguson& Ponciano 2014 )areallcommonlyusedaspredictorsofpopulationgrowthandmaylead toautocorrelatedenvironmentalvariation.However,determiningwhetherthegeneral patternsofautocorrelationinpopulationabundancesareduetotheseextrinsicfactors ortotheintrinsicprocessesofpopulationgrowthandregulationisstillanopenquestion ( Akcüakaya etal. 2003 ). Understandingthepotentialimpactofautocorrelatedenvironmentsonpopulation dynamicsrequiresuntanglinginteractionsbetweenmultipleautocorrelatedrandom variables.WehavedeÞnedanumberoftermsinordertokeeptrackofthesedifferent formsofautocorrelation.ThesedeÞnitionsandabbreviationswillbeusedforthe remainderofthepaperandaredeÞnedinTable 2-1 .Thesetermsareintroducedinthe followingpassages.Whenpopulationgrowthdependsonenvironmentalcovariateswith autocorrelation(ECA),itmaybeincorporatedintothepopulationenvironmentalvariance (PEV).BecausethePEVisusuallyassumedtocomefromtemporallyindependent, identicallydistributedprobabilitydistributions,includingpopulationenvironmental 19

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autocorrelation(PEA)canhavesubstantialeffectsonabundancepredictionsand populationriskassessment. IncorporatingPEAintopopulationprojectionshasbeenshowntoleadtolarge differencesinextinctionriskpredictions.ThePEAmayinteractwiththeformofdensity dependencetodecreaseorincreaseextinctionrisk,dependingonthestrengthof densitydependence.PEAcoupledwithundercompensatorydensitydependence tendstoincreaseextinctionrisk,whilecouplingtoovercompensatorydynamicstends todecreaseextinctionrisk( Ripa&Heino 1999 , Schwager etal. 2006 ).Theseresults stronglysuggestthataccuratelyassessingspeciesextinctionriskusingpopulation dynamicsmodelsrequiresestimatingboththeformofdensitydependence,thedegree ofPEApresentinapopulationinordertounderstandhowthesetwoprocessesinteract. Previousstudiesthathaveattemptedtoempiricallyquantifylevelsofautocorrelation inanimalpopulationtimeserieshavetypicallylookedatthepopulationabundance autocorrelation(PAA),theoverallautocorrelationpresentinpopulationabundances. ThesestudieshavefoundpositivevaluesforthePAA( Cyr 1997 , Halley&Inchausti 2004 , Inchausti&Halley 2001 ; 2002 , Miramontes&Rohani 1998 , Pimm&Redfearn 1988 ).BecausepaststudieslookingfortheempiricalsignalofPEAinPAAhave notdecomposedenvironmentalvariabilityfromdensitydependentprocesses,itis notknownhowmuchoftheobservedPAAsignalisduetoenvironmentalvariability versusintrinsicpopulationautocorrelation(IPA)generatedfromgrowthandregulation processes.Furthermore,thesestudieshavefailedtolinkestimatedPAAtospeciÞc environmentalcovariatesdespitesuggestionsthatenvironmentalcovariatesaredriving theseobservations. Severalrecentempiricalstudiesthathaveattemptedtodeterminewhether populationabundancesdirectlytrackspeciÞcenvironmentalcovariates.Knape& deValpine( Knape&deValpine 2010 )incorporatedweatherandclimaticindex dataascovariatesintoalargenumberanimaltimeseriesmodelsandshowed 20

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thatenvironmentalcovariatescanimprovemodelpredictions,butonlymarginally whencorrectingformodeloverÞtting.AnotherstudybyGarc « õa-Carreras&Reuman ( Garcia-Carreras&Reuman 2011 )alsoexaminedanumberofclimaticvariablesand animaltimeseries,theirresultssuggestthatECAinmeansummertemperatureis weaklycorrelated( r =0.312 )withthePAA.Together,theresultsofthesestudies suggestthatthecouplingbetweenclimaticvariablesandanimalpopulationsislikelyto beweak,onaverage.Thecomplexityoftheenvironmentthatpopulationsareembedded in,andnonlinearresponsestoenvironmentalperturbationsmaycontributetoexplaining theseresults. Wearealsoawareoffourmicrocosmexperimentsthathaveexaminedtherole ofECAonpopulationdynamics.TheÞrst( Petchey 2000 )foundnosigniÞcanteffects ofECAontheresultingPAA,contrarytotheoreticalpredictions.However,thisstudy didÞndthatpopulationsinpositivelyautocorrelatedenvironmentstendedtobemore correlatedtotheenvironmentalstatethanpopulationsinuncorrelatedenvironments, aneffectthatwhilestatisticallysigniÞcant,wasweak.Asecondstudy( Gonzalez&Holt 2002 )foundthataveragepopulationabundancesinautocorrelatedenvironmentswere higherthaninuncorrelatedenvironments,aspredictedbytheory.Theotherprediction tested,thatpopulationsinautocorrelatedenvironmentsdisplaygreatervariability,was supportedinonlyoneofthetwoexperimentaltreatments.Becausethisexperiment wasconductedwithopenpopulationstheabilitytomigratelikelyincreasedtheability ofpopulationstotracktheirenvironment.Athirdexperiment( Laakso etal. 2003b ) foundthatpopulationscorrelatedmorestronglytoautocorrelatedenvironmentsand thatoverallthePAAwasinßuencedbytheECA,supportingtheoreticalpredictions. TheresultsoftheÞnalexperiment( Pike etal. 2004 )wassuggestiveoftheeffectsof ECAonextinctiontimesinoneoftwoexperiments,partiallysupportingtheoretical predictions.Takentogethertheseexperimentalresultssuggestthatalthoughitmaybe possibleforpopulationstobestronglycoupledtotheirenvironmentinsomecases,even 21

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insimpleexperimentalsystems,oftenthisrelationshipmaynothold.Takentogether, pastexperimentalandempiricalstudiessuggestthatthetheoreticalworkdoneupto date,whichhasassumedadirect,linearrelationshipbetweenECAandPEA,islikelyan oversimpliÞcation.Asaresult,webelievethatdevelopingamoregeneralunderstanding ofhowautocorrelatedenvironmentalvariationaffectsanimalpopulationsisanimportant goalifwearetounderstandtheimplicationsofECAfornaturalpopulations. 2.1.2AnOverviewofAutocorrelationModels Severalmodelingchoicesmustbemadewhenconsideringhowtomodelenvironmental autocorrelation.Weintroducetwoalternativemodelformulationsoftemporally autocorrelatedprocessesanddiscusssomecommonlyusedmethodsforestimating temporalautocorrelationfromdatathathavebeenusedbypaststudiesexaminingPAA. 2.1.2.1Long-andshort-memorymodels Apotentiallyimportantdistinctioninautocorrelatedtimeseriesmodelsisbetween long-andshort-memoryprocesses.Short-memorymodelsdependonlyonrecent realizationsoftheprocesswhilelong-memorymodelsexhibitautocorrelationover manypastrealizationsoftheprocess.Short-memoryprocesseshaveautocorrelation functionsthatdecayexponentiallyto0asthelag k "# ,whilelong-memoryprocesses haveautocorrelationfunctionsthatconvergeto0accordingtoslow-decayingpower-law functionsasthelag k "# ( Shumway&Stoffer 2006 ).Thus,thoughbothprocesses mayhavethesamedegreeofautocorrelationandgoto0,short-memoryprocesses goto0fasterthanlong-memorymodels.CuddingtonandYodzis( Cuddington&Yodzis 1999 )showedthatincorporatingPEAinlong-andshort-termmemorymodelswiththe samedegreeofautocorrelationledtodifferentpredictionsaboutspeciespersistence inotherwiseidenticalmodels.ThisÞndingstronglysuggeststhatdistinguishing betweenlong-andshort-memoryPEAmayhavepracticalimplicationsforpopulation modeling.Accordingly,anythoroughexaminationoftheeffectsofPEAshouldconsider 22

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investigatingtheeffectsofvariouscombinationsofthedegreeofautocorrelationandthe typeofmemorymodel. Whileitisoftenconvenienttothinkofpopulationabundancesasfunctionsoftime, previousdiscussionsoflong-memoryprocessesintheecologicalliteraturehaveoften usedthefrequencydomainrepresentationoftimeseries( Cuddington&Yodzis 1999 , Halley 1996 , Vasseur&Yodzis 2004 ).Thisapproachdecomposesatimedependent signalintoaninÞnitesumofsinewaveswithdifferentfrequenciesthroughtheFourier transform.Thetransformationcalculatestherelativecontributionofeachsinewave frequencytotheoverallsignal.Thepowerspectraldensityisdenotedas S ( f ) andthis quantitymeasurestherelativecontributionofthefrequency, f ,totheoriginalsignal. Thisfrequencyrepresentationcanbeusedasaconvenientdiagnostictoolwhen determiningtheappropriatenessofaparticulartimedependentmodel( Box etal. 2011 ). Theautocorrelationfunctioninatimeseriesmodeland S ( f ) areFouriertransformsof eachotherandthuscontainequivalentinformation. Themostcommonlyusedlong-memoryprocessistheinversepowerlawmodel ( 1 / f model)( Johnson 1925 ).Forthismodelthespectraldensityfunctionisgivenby S ( f )=constant / f ! where " controlsthedegreeofautocorrelation.The 1 / f model hasbeenproposedseveraltimesasageneralmodelofenvironmentalvariationfor populationdynamicsduetoitsapparentubiquityinnaturalphenomena( Bak etal. 1987 , Halley 1996 , Montroll&Shlesinger 1982 ). Generatingtemporaldatawithalong-rangedependencyconsistentwitha 1 / f processcanbeachievedbyspecifyingasuitableautocorrelationmodel.Thefractional whitenoise(FWN)modelisadiscretetimemodelofanautocorrelatedprocessthat hasaspectraldensityfunctionsimilartothe 1 / f process( Hosking 1981 ).IntheFWN, thecurrentstateoftheautocorrelatedprocessisgivenasthesumofallthepast contributionstotheprocessandanindependentrandomshock.Themathematical representationofthemodelis 23

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E ( t )= ! ! k =1 $ " ( k $ d ) " ( k +1)"( $ d ) E ( t $ k )+ W ( t ), (2Ð1) where E ( t ) isthestateofaprocessattime t , " ( x ) isagammafunction, d isthe fractionaldifferencingparameterwhere $ 0.5 < d < 0.5 ,and W ( t ) isanormal distributionwithmeanequaltozeroandvarianceof # 2 ,andisindependentfrom E ( t $ k ) .Inthismodel,thestateatthecurrenttimestepdependsonallprevious timelags, k ,throughaninÞniteseriesrepresentation,leadingtothelong-memory property.Thedifferencingdegree, d ,controlsthedegreeofautocorrelationand correspondsapproximatelyto " / 2 inthe 1 / f model.Thus,theFWNandthe 1 / f model areapproximatelyequivalent(Figure 2-1 ). Thesimplestexampleofashort-memorymodel,whichdependonlyonrecent realizationsoftheprocess,istheautoregressivelag1(AR(1))model.TheAR(1)has foundwideuseinstatisticaltimeseriesmodeling( Shumway&Stoffer 2006 ).Itisgiven by E ( t )= ! E ( t $ 1)+ W ( t ), (2Ð2) where W ( t ) isanormaldistributionwithmeanequaltozeroandvariancegivenby # 2 ,and ! controlsthedegreeofautocorrelation.As ! goesto0,theautocorrelation disappears.Thespectraldensityofthismodelisgivenby S ( f )= " 2 (1 " # ) 2 +2 # (1 " cos( f )) ( Erland&Greenwood 2007 ).Asopposedto 1 / f densities,thespectraldensityofAR(1) modelsisßatatlowfrequencies(becausewhen f 2 % 0 , 1 $ f 2 =1 $ cos( f )=0 , and f ( S ) becomesaconstant).Thus,atlowfrequenciestheAR(1)modelcanbehave similarlytoanuncorrelated(whitenoise)process,whichhasauniformspectraldensity (Figure 2-1 ). 24

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Anadditionalshort-memoryautocorrelationmodelofinterestisthemovingaverage lag1(MA(1))model,whichcanbeinterpretedasstochasticinßuencesfromeither measurementerrorterms( Dennis etal. 2006 )orinterspeciÞcinteractions( Abbott etal. 2009 , Ferguson&Ponciano 2014 , Royama 1981 ).TheMA(1)modelisgivenby E ( t )= W ( t )+ $ W ( t $ 1), (2Ð3) wherethe W ( t ) 'saredrawnfromanormaldistributionwithmeanequaltozeroand variancegivenby # 2 .AsopposedtoAR(1)processeswheretheautocorrelationis introducedintotheprocess,intheMA(1)modeltheautocorrelationisintroducedinthe randomshockterm.Therefore,thisautocorrelationstructuredistributestheeffectof varianceperturbationsfromprevioustimelagsontothecurrentobservation.Finally,the AR(1)andMA(1)modelscanbeappliedsimultaneouslythroughtheARMA(1,1)model (Appendix B ). 2.1.3StochasticPopulationModels Inordertomakeinferencesabouttheimpactofenvironmentalautocorrelation onanimalpopulations,werequireaframeworkthatwilldecomposePAAintotwo components:theIPAandthePEA.Hereweintroducebasicstochasticpopulation modelsthatincludebothterms.Weassumethatthisdecompositionisadditiveinthe scaleofthelog-growthratebutmultiplicativeinthescaleoftheabundances. Timeseriesofabundancesareoftenmodeledusingdiscrete-timedensity dependentrecursiveequations,duetothediscretenatureofreproductionandlife historystructureinmanyspecies.Twodifferentmodelsrepresentingdifferentqualitative speciÞcationsoftheformofdensity-dependencearetheGompertzandtheRicker modelsofpopulationdynamics.TheGompertzhascompensatorydynamicswhilethe Rickerisovercompensatory,similartothelogisticmodel( Kot 2001 ).TheGompertz modelofdensitydependencehasbeenpreviouslyusedwidelyinecologytodetect 25

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densitydependenceinanimalpopulations( Morris 1959 , Sibly etal. 2007 , Ziebarth etal. 2010 ).Thismodelhasalsobeenthebasisforthedevelopmentofmeasurementerror models( Dennis etal. 2006 )andforanalysisofdensitydependencedynamicsinlarge datasets( Abbott etal. 2009 , Knape&deValpine 2010 ; 2012 , Sibly etal. 2007 , Ziebarth etal. 2010 ). TheGompertzmodelisgivenby N ( t )= N ( t $ 1)exp[ a + b ln( N ( t $ 1))+ E ( t )]. (2Ð4) Here, N ( t ) istheabundancemetricattime t , a istheintrinsicpopulationgrowth rate, b isthestrengthofdensitydependence,andthe E ( t ) areallindependentand identicallydistributed( iid )drawsfromanormaldistributionwithmean 0 andvariance # 2 representingenvironmentalvariability.Additionalcomplexityindensitydependence canbeintroducedbyallowinglagsinthedensitydependence,whichcanbeinterpreted asaccountingforinterspeciÞcinteractionsandmorecomplexlifehistories( Ferguson& Ponciano 2014 , Royama 1981 , Turchin 1999 ).Themulti-lagGompertzmodelisgivenby N ( t )= N ( t $ 1)exp[ a + L ! i =1 b i ln( N ( t $ i ))+ E ( t )] ,wherethe b k 'smeasurethestrength ofdensitydependenceatlag k ,andatotalof L lag'sarepresent. AnothercommonlyusedformofdensitydependenceistheRickermodel. OriginallyderivedbyWilliamRicker( Ricker 1954 )asthedynamicsthatemergefroma cannibalisticinteractionbetweenadultsandjuvenilesinapopulation,therehavebeen anumberofothermechanisticderivations( Br ¬ annstr ¬ om&Sumpter 2005 , Geritz&Kisdi 2004 , Royama 1992 )linkingavarietyofinterpretationstothismodel.Themultilag Rickermodeliswrittenas N ( t )= N ( t $ 1)exp[ a + L ! i =1 b i N ( t $ i )+ E ( t )], (2Ð5) 26

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wherethe E ( t ) 'sareagain iid and a andthe b i areallconstants.Variationsofthis modelhavebeenusedinmanypaststudiesexaminingthetheoreticalpropertiesofPEA ( Cuddington&Yodzis 1999 , Heino&Sabadell 2003 , Morales 1999 , Petchey etal. 1997 , Ruokolainen etal. 2007 )andinstudiesexaminingpropertiesofdensitydependencein theGPDD( Sibly etal. 2005 ). Generalizingtheenvironmentalvariationterm, E ( t ) ,intheGompertzand Rickermodelsbeyondthe iid assumptionmayallowustoaccountforeffectsdueto environmentalprocesses,speciesinteractions,andmeasurementerrorthatotherwise notbecaptured.AR,MAandFWNautocorrelationmodelsfor E ( t ) allowustodothis. However,interpretingtheautocorrelationestimatesrequiresunderstandinghowthe biologicalprocessesinteractingwithECAaretranslatedintovariabilityintherateof growthofpopulations. 2.1.4EnvironmentalTrackingModels Environmentaltrackingmodelscallintoquestiontheimplicitassumptionthat theECApresentinenvironmentalfactorswilltranslatedirectlyintoPEA.Nonlinear physiologicalandecologicalresponsescanaffectthewaythatanimalstrackenvironmental conditions,whichinturncanleadtoPEAvaluesthataremuchdifferentthanECA values.NonlineargrowthresponsestoenvironmentalconditionsdeÞnetheecological niche,acentralconceptinecology( Holt 2009 ). Nonlinearresponsestoenvironmentalstatesarerelativelycommonplace( Clark etal. 2003 , Coulson etal. 2001 , Jenouvrier etal. 2009 , Kausrud etal. 2008 )and havebeenshowntoaltertheECAwhentransformingintoPEA( Laakso etal. 2001 ; 2003a ).AnillustrationofthiseffectisgiveninFigure 2-2 .Asecondmechanismthat mayreduceECAwhentransformedintoPEAisastochasticversionofLiebig'slawof theminimum(SLLM)( Hooker 1917 )(Table 2-1 ).ThismodiÞcationofLeibig'slaw,which statesthatpopulationgrowthislimitedbytheminimumresourcenecessaryforgrowth, accountsformultiple,stochasticresources.ThismechanismcouldreducethePEA 27

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forpopulationsthatarelimitedbymultiplelimitingresourcesbychangingthelimiting resourceeachyear.Thelimitingenvironmentalfactorforeachyearisthentheminimum ofoneofseverallimitingresources,eachofwhichmaybeautocorrelatedwithitselfbut isuncorrelatedtotheotherresources.TheSLLMmayservetodisrupttheECApresent inanyoneoftheenvironmentalcovariatesandreducetheoverallPEA.Weillustrate theimpactsofthesetrackingmodelsontheobservedPEAinanimalpopulationswitha simulationstudy. 2.2Methods 2.2.1EnvironmentalTrackingModels Inordertoassesstheeffectsthatnonlineartrackingmodelshaveintransforming ECAtoPEA,wesimulatedtheimpactofseveralformsofpopulationresponsesto environmentalvariationontimeseriesgeneratedfromanAR(1)model.Various functionsthathavebeenusedwhenmodelingspeciesresponsestoenvironmental covariates.Twoofthemostcommonarethepowerandthelogistic(orS-shape) functions.ThepowerfunctionisoftenjustiÞedasanichemodelofenvironmental suitabilitybecauseatevenpowersthefunctionresultsinapeakedoptimalresponse alongtheenvironmentalaxis.Thelogisticfunctioncanbeseenasathresholddynamics responsewhereinbelowcertaincriticalleveloftheenvironmentalvariable,growthis closetozeroandbeyondthislevelthegrowthresponseofthepopulationdramatically increases. Westartedwiththesimulationofanenvironmentalcovariatetimeseries, X ( t ) as anAR(1)process(Equation 2Ð2 ).Wethentransformed X ( t ) intothePEV, E ( t ) ,using thepowertransform, E ( t )= X ( t ) $ andthelogistictransform E ( t )= e 10 X ( t ) ! 1+ e 10 X ( t ) ! ,where % wasvariedbetweentheintegervalues1to4.TheECA,givenby ! inEquation 2Ð2 , wasvariedbetween30equallyspacevaluesintheinterval [ $ 0.95,0.95] .Wedrew anobservationoflength 10 6 fromeachlevelofautocorrelation,withameanof 0 and varianceof 1 .Wethentransformedthetimesseries X ( t ) into E ( t ) usingthepowerand 28

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logistictransformsandestimatedthePEA,byregressingtheseriesonitspreviousstate E ( t )= ! # E ( t $ 1)+ W ( t ) ,wherethe W ( t ) were iid normallydistributedwithmean zeroandunknownvariance.Inthisequation ! # representsthePEA.Analyseswere performedintheRsoftwarepackage( RDevelopmentCoreTeam 2012 ). TheeffectoftheSLLMontheoverallPEAwasdeterminedbyusingasimulation procedure.Assumingthatanumber n factorsarecriticaldeterminantsofpopulation growthrate,weÞrstsimulated n independenttimeseriesoflength 10 6 fromanAR(1) processwithmean0andvariance1.Wethentooktheminimumvaluefromthe n ECArealizationsateachtimestep,sothat E ( t )=min { X 1 ( t ), X 2 ( t ),..., X n ( t ) } , where E ( t ) isthePEV.For n $ 1 ofthesetimeseries,weÞxedtheECAat 0.5 ,alevel frequentlyhypothesizedtobeareasonabledefaultforenvironmentaltimeseries ( Halley&Inchausti 2004 ).TheremainingtimeserieshadanECAlevelthatwas incrementallyvariedfrom-0.9to0.9,bystepsof0.1foratotalof20ECAvalues. WethenestimatedthePEA, ! # ,byregressingtheseriesonitspreviousstate E ( t )= ! # E ( t $ 1)+ W ( t ) ,wherethe W ( t ) areagain iid normallydistributedwithmeanzeroand unknownvariance.Werepeatedthisprocedurefor n =2,3,4,5 limitingenvironmental factors. 2.2.2JointEstimationofDensityDependenceandAutocorrelationStructurein AbundanceTimeSeries Tostudytherobustnessofthejointestimationoftheautocorrelationinducedby thedensitydependence(theIPA)andtheenvironmentalautocorrelationstructure (thePEA)weusedatwo-prongedapproach.First,weperformedasimulationstudy ofpotentialestimationproceduresusingtimeseriesgleanedfromtheGPDD( NERC 2010 )andappliedthoseprocedurestoadatasetfromtheGPDD.Second,wecompared ourshort-memoryestimatorstothelong-memorymodelsusedinpreviousstudiesof theGPDD.Thesemethodsallowedustotestboththeconsistencyofourestimation procedurewiththetruevalueofthePEA,andtheconsistencyofourmethodswith 29

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previouslypublishedstudies.OurcuratedGPDDdatasetconsistedofasetof389high qualitytimeseriesfromtheapproximately5000availabledatasetsintheGPDD.Details onthecriterionusedtodeterminethissubsetofhighqualityseriesarepresentedin Appendix A . Inoursimulationstudy,theabilitytoaccuratelyestimatethePEAwasdetermined usingamodeladequacyanalysis( Taper etal. 2008 ).SpeciÞcally,we1)simulatedmany timeseriesofabundancesassumingdifferentformsofdensitydependence,modellags, andPEAlevelswithparametersestimatedfromtheGPDDdataset.2)Foreachset ofsimulatedtimeseriesweÞttedamodelsetthatcontaineddensityindependentand bothGompertzandRickermodelsoflag1densitydependence,whereallmodelsalso containedanARMA(1,1)errorstructure.3)Wethenperformedmodelselectiononthis setusingAICcandevaluatedthestatisticalproperties( i.e. biasandvariance)ofthe estimatorofthePEA, ö ! ,underthebestAICcmodelsasafunctionoftheunderlying PEAandthegeneratingmodellag.Wealsoperformedanothersimulationanalysis wherethemodelsetcontaineddensityindependentandlag1tolag3Gompertzand RickerdensitydependencewithbothAR(1)andARMA(1,1)errorstructures.Adetailed accountofthismethodologyisgiveninAppendix B .ThisallowedustoprovidePEA estimatesunderdifferentmodelassumptions,anddeterminearealisticrangeofPEA estimates(conditionalonourestimationprocedure),aswellascarryingadetailed assessmentofthereliabilityoftheseestimates.Finally,inordertodeterminewhether therewereunderlyingpatternsinthedistributionofPEAestimateswetestedanumber ofcovariatesassociatedwiththeGPDDtodetermineiftheycouldexplainestimated PEAvalues.ThesemethodsandresultsarefullydescribedinAppendix C . Oursecondapproachaimedatstudyingtherobustnessoftheestimatesofthe AR(1)autocorrelationcoefÞcientundervariousalternativeerrormodels.Weestimated thePAAusingthespectralexponent,ameasureusedinpreviousanalysesofPAA ( Halley&Inchausti 2004 , Inchausti&Halley 2001 , Pimm&Redfearn 1988 ).Todothat, 30

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weÞrstestimatedthespectralexponentbyÞttingalinearregressiontotheFourier transformofthelog-transformedabundancesinourGPDDdataset,consistentwith previousmethods.ThetransformwascalculatedwiththefftfunctionintheRsoftware environment( RDevelopmentCoreTeam 2012 ).Thefrequencyamplitudeofthe transformedtimeseries, S ( f ) ,wasthenÞttoamodeloftheform, log( S ( f ))= a $ " log( f )+ W ,where f istheobservedsamplingfrequencyofthetimeseriessignal, " isthespectralexponent, a isaninterceptterm,and W is iid normallydistributed. WethencomparedthesePAAestimatesderivedfromthespectralexponenttothose foundusingtheFWNmodelandtheAR(1)model.ByestimatingthePAAusingthese modelswewereabletoassesstheaccuracyofthePAAestimateswhenthePEAis actuallyalong-memorymemoryprocess.Next,theconsistencyoftheFWNandAR(1) autocorrelationestimateswith ö " wasevaluatedbyregressingtheirautocorrelation estimates( d inEquation 2Ð1 and ! inEquation 2Ð2 )against ö " usingasimplelinear regressionmodel.FortheFWNmodelweÞt, ö " / 2= a d + b d ö d + W , (2Ð6) andfortheAR(1)modelweÞt ö " / 2= a # + b # ö ! + W , (2Ð7) where W wasarandomvariableassumedtobe iid normallydistributed.Consistency withthe 1 / f ! modelestimateswastestedbydeterminingwhethertheintercept parameters( a )weresigniÞcantlydifferentfrom0andtheslopeparameters( b )were signiÞcantlydifferentfrom1.Thiscomparativemethodwaspreferredoveradirectmodel selectionprocedureactingonAR(1)andFWNmodelsofPEAbecausetheinformation theoreticmethodswetesteddidn'treliablydistinguishbetweenlong-andshort-memory modelswithshorttimeseries( Wagenmakers etal. 2004 ).Wealsodeterminedifthe propertiesofthedatasetsusedherewereconsistentwithpreviousstudiesonthe 31

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GPDD( Inchausti&Halley 2001 )bycomparingourestimatesofthePAAusingthe 1 / f modeltopreviousestimatesofPAA( Halley&Inchausti 2004 , Inchausti&Halley 2001 ). 2.3Results Together,ourresultsillustratehowtheautocorrelationpresentinenvironmental covariatescanbereducedthroughnonlinearinteractionsorbyinteractionswithmultiple limitingresources.Weestimatedthedegreeofenvironmentalautocorrelationpresentin populationtimeseriesfromtheGlobalPopulationDynamicsDatabase,andfoundthat animalpopulationsatlargeareaffectedbylowlevelsofenvironmentalautocorrelation inmosttimeseries.Thisresultisconsistentwithpredictionsfromourinteractionmodels butiscontrarytopreviousanalysesthathavenotdecomposedtheoverallPAAinto density-dependenteffectsandthePEA.InwhatfollowsweÞrstpresenttheresults fromouranalysisofenvironmentaltrackingmodels,thensummarizetheresultsofPEA estimationintheGPDD,andÞnallydiscusstheresultspertainingtotherobustnessof alternativeautocorrelationmodels. 2.3.1EnvironmentalTrackingModels Wefoundthatthemagnitudeofautocorrelationwasreducedinallthenonlinear environmentaltrackingmodelsexplored(Figure 2-3 ).Consistentwithpreviouswork,we foundthatwhentrackingmodelswereevenfunctionsnegativeautocorrelationbecame positive( Laakso etal. 2001 ).However,thissignchangealsohadacorresponding decreaseinmagnitude,aneffectthathasnotbeendescribedintheliteraturetothebest ofourknowledge.ThedegreeofthedampeningofautocorrelationintranslatingECA toPEAdependedontheformoftherelationshipbetweenthepopulationgrowthand theenvironmentalcovariate(seeFigures 2-3 , 2-4 ).However,thenicheandthreshold functionsdisplayedsimilarresponsestoenvironmentalvariationwithrespecttoboth thesymmetryandthestrengthoftheresponse.Timeserieswithhigherlevelsof autocorrelationweredampenedrelativelyless,astheseseriescorrespondedto populationsthatspentmoreconsecutivetimeontheedgesoftheirenvironmental 32

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tolerances.Attheselimitsthecurvatureofthetrackingmodelsbecomeslessandthe transformationofECAtoPEAisclosertoalineartransformation.Theseresultssuggest thatthedampeningofECAduetononlinearresponsesofthepopulationgrowthto changesintheenvironmentisaubiquitousphenomenon.Wefoundthatthisdampening andisnotverysensitivetotheparticularformofthepopulationsinteractionwiththe environmentbutrather,dependsonwhetherthepopulationisnearthelimitsofitsniche toleranceornot. ApplyingtheSLLMtoasetofECAtimeseriesledtoareductioninthePEAby breakingthedependencypresentinlongrunsfromanyoneofthetimeseries.This dampeningeffectwasparticularlystrongforvaluesoftheECAbetween $ 0.5 and 0.5 , wherethetransformedPEAwasnearlyßat(Figure 2-4 ).Forexample,wefoundthat takingtheminimumof n AR(1)processeswith ECA=0.5 ledtoreductionsofECAto between0.2-0.4,dependingon n ,thenumberoflimitingenvironmentalcovariates.As n increasedthestrengthofthedampeningincreasedandtheoverallautocorrelationinthe environmentalsignalthatregulatespopulationgrowthisreduced. 2.3.2EstimatesofPEAintheGPDD OursimulationstudyindicatedthattheestimationofPEAdependedonboththe densitydependenceandtheerrorstructureofthemodel(Appendix B ).Thereforewe reportseveraldifferentestimatorsofthePEAinourGPDDdataset(Table 2-2 ).We foundthatthebestestimationprocedurewastoperformmodelselectionbetween densityindependentmodelsandlag1RickerandGompertzdensitydependence modelsthatincludedanARMA(1,1)errorstructure( Dennis etal. 2006 ).Thisprocedure ledtothePEAestimate ö ! =0.08 .Thisestimateismuchlowerthanpreviously hypothesizedlevelsofPEAofaround0.5.Whennotapplyingthetransformationof adding1toallobservations,thisPEAestimatewas ö ! =0.07 (Table 2-2 ),suggesting thatthistransformationhadlittleeffectonourestimates.Whenweincludeddensity independenceandGompertzdensitydependenceatlags1to3,themedianPEA 33

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estimateincreasedto ö ! =0.18 .WithRickerdensitydependencethePEAestimate decreasedto ö ! = $ 0.08 (Table 2-2 ).Theestimatorsthataccountedforhigher orderlagstendedtobelessreliableinoursimulationstudybecausetheytendedto displayhigherbiasandvariance(Figure B-1 ).However,theseestimatesof ! didnot stronglydifferfromouroptimalprocedure,whichsuggeststhatourgeneralÞndingoflow environmentalautocorrelationisindependentoftheparticularmodelused. WhenselectingbetweendensityindependenceandtheGompertzandRickerlag 1densitydependencefunctions,wefoundthatmostserieswereidentiÞedasbeing densityindependent(47%).TheGompertzmodelbestexplained35%ofthesewhilethe Rickermodelonlywasselectedin18%ofthecases.Thedegreeofdensitydependence foundwasconsistentwithpreviouswork( Knape&deValpine 2010 ),whichfoundthat 45%ofalargenumberoftimeseriesfromtheGPDDdisplayeddensitydependence. Wetestedanumberofpotentialcovariatestodetermineiftheycouldexplain variabilityinourestimatesofPEA,howeverwefoundnoimportantexplanatoryvariables (methodsandresultsinAppendix C ).Ourfurtherdiagnostictestsindicatethatestimates ofECAtendedtodecreasewithsamplesizeandthatspecialistspeciesmaybesubject todifferentlevelsofPEAthangeneralists(usingasubsetoftheGPDDdeÞnedby Murdoch etal. ( 2002 )),consistentwithpredictionsfromoursimulationsoftheSLLM. However,theseeffectswerenotstatisticallysigniÞcant(Appendix C ). 2.3.3AlternativeErrorModels Theestimatedlevelsofthespectralexponentwithourdataset, ö " =1.06 (Table 2-2 ), weresimilartopreviousestimatesintheGPDD, ö " =1.02 ( Inchausti&Halley 2002 ). Thisindicatesthatthedatasetusedhereiscomparabletodatasetsusedinprevious work.WealsocomparedPAAestimatesfromboththeAR(1)andFWNerrorprocesses to ö " inordertodeterminewhetherbiasmightariseduetomodelmisspeciÞcation.For theAR(1)model(seeeq. 2Ð7 ),wefoundaninterceptvalueof ö a # =0.002 ± 0.028 ,and ö b # =1.156 ± 0.057 .FortheFWNmodel(seeeq,. 2Ð6 ),wefoundaninterceptvalue 34

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of ö a d = $ 0.273 ± 0.022 and ö b d =1.13 ± 0.069 .SincetheinterceptissigniÞcantly differentfrom0(t-test,p-valueÁ 20 " 16 )theautocorrelationestimateforthelong-memory process ö d isn'tdirectlyequivalentto ö beta .Thisdiscrepancyislikelyduetoproblemswith thelong-memoryprocessautocorrelationestimator(Equation 2Ð1 ),whichisknownto beunreliablewhenworkingwithshorttimeseries( Lieberman 2001 ).Together,these resultsindicatethattheAR(1)modelwasmoreconsistentwiththe 1 / f modelthanthe FWNmodel.OurresultssuggestthatPEAestimationisaccuratefortherangeofvalues presentintheGPDDindependentofwhethertheunderlyingPEAprocessisalong-or short-memoryprocess. 2.4Discussion DecouplingpopulationdynamicsfromthePEVisnecessarytoprovideinterpretable PEAestimates.PreviousestimatesofPAA( Cyr 1997 , Halley&Inchausti 2004 , Inchausti &Halley 2001 ; 2002 , Pimm&Redfearn 1988 , Swanson 1998 )havenotdecomposed thissignalintoitsbasiccomponentsandthereforearelikelytoprovideambiguous estimatesofthePEApresentinanimalpopulations.Whenprocess-basedstatistical methodsareusedtoteaseapartexternalandinternalsourcesofPAA,lowlevelsof PEAemerge.Previoustheoreticalstudiesassessingtheeffectsofautocorrelated environmentsonextinctionriskstypicallyshowsigniÞcanteffectsonpopulation persistenceatmoderatelevelsofPEA,typicallywhen ! & 0.5 ( Cuddington&Yodzis 1999 , Petchey etal. 1997 , Schwager etal. 2006 ).Hereweshowthatmedianlevelsof PEAarefarlowerthan0.5,withourbestestimatearound0.08(seeTable 2-2 ).Inwhat followsweinterpret,andexaminetheconsequencesandimplicationsoftheseÞndings. Whenmultiplelimitingfactorsshapedensity-dependentprocesses,thegrowth dynamicsofapopulationlosesitsabilitytotrackenvironmentalcovariatesand theeffectsofECAaremarkedlydampened(see 2-4 ).Hereweshowedthatsuch dampeningcanbemodeledbythestochasticversionofLLM(theSLLM).Thiseffect isfurtherstrengthenedbynonlinearinteractions,asweshowinFig. 2-3 .Itisnot 35

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unreasonabletothinkthatoneorbothofthesemechanismsareoperatinginmost animalpopulations.Asaresult,animperfecttranslationofECAintoPEAshouldntbe consideredasasurprisingphenomenonandrecentempiricalresultsarebeginningto supportthisviewpoint( Garcia-Carreras&Reuman 2011 , Knape&deValpine 2010 , vandePol etal. 2011 ). Theseresultsarenotevidencethatenvironmentalchangesdonotstrongly affectpopulationgrowth.AtÞrstglancethepotentiallyweakrelationshipbetween environmentalcovariatesandpopulationabundancesuggestedbyourresultsisat oddswiththeviewthatpopulationsareoftenberegulatedbydensityindependent factors( Andrewartha&Birch 1954 ).Intheirclassicstudyonthepestspecies Thrips imaginis ,DavidsonandAndrewartha( Davidson&Andrewartha 1948 )foundthat environmentalvariablescouldbeusedtoexplainedover80%oftheannualvariation inpeaklog-densities.However,theirinvestigationintodailyßuctuationsofvariability showedthatenvironmentaleffectscouldonlyexplain10%ofthevariationnotdueto populationgrowthprocesses.Theuseofyearlypeaklogabundancesislikelytobe moresensitivetoßuctuationsundernonlinearenvironmentaltrackingthanpopulation averagesbecause,asourresultssuggest,populationswilloftenhavedisproportionately strongerresponsestoenvironmentalcovariatesasconditionsdeviatefurtherfrom optimality.Empiricalstudiesdemonstratingnonlinearresponsesofdemographicrates toenvironmentalfactorscanbefoundthroughouttheliterature( Coulson etal. 2001 , Jenouvrier etal. 2009 , Kausrud etal. 2008 ). DespiteourÞndingsexistingexperimentalworkshowsconvincinglythatECA canaffectpopulationdynamicsundersomescenarios( Gonzalez&Holt 2002 , Laakso etal. 2003b ).Atleasttwoargumentscouldreconciletheseexperimental Þndingsandourresults.First,carefulexperimentalsettingsareidealtoÞnetune thelimitsofapopulation'snicheuntilthisoneexhibitsapositivetrackingofECA ( e.g., ( Petchey 2000 )).TheGPDDmaynotprovideasampleofpopulationsstrongly 36

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coupledtoanenvironmentbecausethesedatasetstypicallybelongtolong-studied, stablepopulationsandthereforearenotlikelyneartheedgeoftheirenvironmental tolerances.ThissuggeststhatfutureworkwouldbeneÞtbyexaminingtheimpactof autocorrelatedenvironmentsonspeciesrangelimitsandonthedynamicsofpopulations inenvironmentalgradients.Second,oursimulationsdonotencompassallthebiological mechanismsbywhichanECAcanaffectthegrowthrateofapopulation.Inparticular, oursimulationsallfollowanadditivegrowthratemodelinwhichthecovariateeffect isaddedtothemaximumgrowthrateandthedensity-dependentterm(eqs. 2Ð4 and 2Ð5 ).AnalternativeformulationoftheeffectofECAistoassumethattheenvironment directlyaffectsthedensity-dependentcoefÞcient(the b parameterineqs. 2Ð4 and 2Ð5 ). Whichautocorrelationmodelbetterexplainsthedynamicsofanimalpopulationswhen subjectedtostrongenvironmentalßuctuationslikelyvariesfrompopulationtopopulation accordingtoitslife-historytraits(see( Berryman&Lima 2006 )). Wedidnotconsidermorecomplexlifehistoriesinouranalysis,althoughtheory hassuggestedthatPEAmayhavesigniÞcanteffectsonthepopulationgrowthrateof age-andstage-structuredpopulations( Tuljapurkar 1982 , Tuljapurkar&Haridas 2006 ). However,recentworkonEurasianoystercatchers( Haematopusostralegus )foundthat thepresenceofECAinatemperaturedependent,stage-structuredmodelonlyweakly affectedpopulationextinctionrisk.Thiswasprimarilyduetononlinearinteractions betweenthedemographicratesthatledtoadampeningoftheenvironmentaltracking ( vandePol etal. 2011 ).Theseresultsprovideanempiricalexamplethatcouplingreal life-historycomplexitytononlinearenvironmentalresponsescanleadtoreductionsin theinßuenceofECA.Ourresultssuggestthatthemechanismfoundin( vandePol etal. 2011 )ismoregeneralthantheirparticularsystem. Themechanismsthatlinkanimalpopulationstotheirenvironmentareundoubtedly morecomplexthanthesimpletrackingmodelsusedhere,yetourworkshowsthat simple,tractablemodelscanbeusedtorevealthestochasticpropertiesofthe 37

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population-environmentinteraction.Inthefaceofclimaticchangeswithlongterm effects,itisimportanttobeabletomakeinformedpredictionsregardingchangesin populationabundance.Afundamentalgoalofthescienceofconservationbiologyisto applytheoreticalinsightsfromecologytoelicitsoundmanagementstrategies.Todate, theproblemofmanagingpopulationgrowthbyunderstandingfeedbacksandregulation hasbeenexaminedeitherbydescribingtheeffectofnon-independentenvironments orbymodelingregulationthroughpuredensitydependentprocessesinrandom andindependentenvironments.However,littledirectevidencehasbeenprovided foreitherapproach.Ourworkprovidesanimportantsteptowardsunderstanding whenautocorrelatedenvironmentsmatter,andjustiÞcationforthecommonlymade assumptionofindependentenvironmentalstates.Finally,ourresultsshowthata constantexaminationofpubliclyavailabledatabasesisessentialfortheadvancementof ourconceptualandpracticalunderstandingofpopulationregulation. 38

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Figure2-1.Thespectraldensities, S ( f ) ,ofthreedifferentmodelsonalog-logplot.The AR(1)modelhaslesslowfrequencynoisethanthelongmemoryprocesses givenbythe 1 / f andFWNmodelswhenallmodelshaveapproximatelythe samedegreeofautocorrelation( " =1.8 , ! =0.9 , d =0.45 ).Theinsetgivea timeseriesrealizationoftheAR(1)andFWNmodelswiththese autocorrelationvalues. 39

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Figure2-2.Anillustrationofhowenvironmentaltrackingresponsescantransform environmentalautocorrelation. A, theenvironmentalcovariate autocorrelation(ECA).Webeginwithatimeseriesofanenvironmental variablethathassomelevelofautocorrelation. B, theÞlter.Atrackingmodel thatdescribesthechangeinthepopulationgrowthrate( ! pgr)todifferent valuesoftheenvironmentalvariable.Nonlineartrackingmodelsarejustone ofthepotentialfactorsthatmaymodifytheeffectofenvironmentalcovariates onapopulation. C, thepopulationenvironmentalautocorrelation(PEA).The resultingobserved ! pgrduetoenvironmentA.Thishassometransformed autocorrelationthatistheresultoftimeseriesApassingthroughtheÞlterB. 40

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Figure2-3.Theeffectoftwodifferentnonlinearinteractionsonanautocorrelatedtime series, X ( t ) .Thedashedblacklineistheenvironmentalcovariate autocorrelation(ECA)intheuntransformedtimeseries,whilethecolored linesarethepopulationenvironmentalautocorrelation(PEA)valuesfor differenttransformations.Panel A correspondstopowertransformsofthe form X ( t ) $ .Panel B correspondstologistictransformsoftheform e 10 X ( t ) ! 1+ e 10 X ( t ) ! . 41

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Figure2-4.ApplicationoftheSLLMtoanumber, n ,ofautocorrelatedtime-series.The lawoftheminimumtendstoreducetheECAoverabroadrangeofvalues. Theoveralleffectdependsonthenumberofenvironmentalcovariates, n , thatlimitthepopulation. 42

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Figure2-5.Histogramof389estimatedpopulationenvironmentalautocorrelation(PEA) valuesfromtheGPDD. 43

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Table2-1.Glossaryoftermsrelatedtoenvironmentalautocorrelationinanimal populations. Term(abbr.) DeÞnition Populationenvironmentalvariance( PEV ) Thevarianceofpopulationabundancesdue toenvironmentalßuctuations.Thisquantityscalesasaconstantwiththepopulation growthrate. Populationenvironmentalautocorrelation ( PEA ) ThetemporalautocorrelationinthePEV.This mayarisewhenpopulationgrowthisaffected byautocorrelatedenvironmentalcovariates. Populationabundanceautocorrelation( PAA )Thisisthetotalautocorrelationpresentinthe populationabundances.Itcanbeamixture ofPEAaswellastheautocorrelationinduced byintrinsicpopulationgrowthandregulation processes. Intrinsicpopulationautocorrelation( IPA ) Thisistheportionofautocorrelationpresentin thepopulationabundancesduetogrowthand regulationprocesses. Environmentalcovariateautocorrelation( ECA )Autocorrelationinenvironmentalcovariates thatareimportanttoanimalpopulationsmay becommon.Thesecovariatescontributeto PEVandthereforemaydrivetheobserved PAA. Environmentaltracking Environmentaltrackingisthedegreetowhich changesinthepopulationgrowthratereßect changesinenvironmentalcovariates. StochasticLeibig'sLawoftheMinimum ( SLLM ) Astochasticmodelthatdescribeshowadifferentenvironmentalvariablecouldbelimiting inanyoneyear.Wemodelthisasasampling processwherethelimitingfactorforeachyear istheminimumobservedenvironmentalcovariatevalueforthatyearfromthesetofall environmentalcovariates. 44

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Table2-2.SummarystatisticsandsamplesizefordistributionsofPEAestimatedunder differentmodelsandwithdifferentdatasets. Densitydependence AutocorrelationmodelLagsSamplesizeMedian ö ! SD ö ! Gompertz&Ricker ARMA(1,1) 1389 0 . 08 0.44 Gompertz&Ricker(no+1transform)ARMA(1,1) 1161 0.07 0.46 Gompertz ARMA(1,1) ' 3389 0.18 0.51 Ricker ARMA(1,1) ' 3389 -0.08 0.43 None Spectralexponent NA389 1.02 0.96 Note-TheÞrst,boldrowhighlightstheestimationprocedurewefoundtobemost reliablefromoursimulationanalysis.Thedensitydependencecolumnreportsthe formsofdensitydependenceconsideredintheestimationprocedure.Thelagcolumn describesthelagsconsideredinthemodelselectionprocess.Lag1indicatesthatonly densityindependentandlag1densitydependentmodelswereÞttothedataset.Lag ' 3indicatesdensityindependentanddensitydependentmodelswithlags1-3wereÞtto thedataset. 45

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CHAPTER3 PREDICTINGTHEPROCESSOFEXTINCTIONINEXPERIMENTALMICROCOSMS ANDACCOUNTINGFORINTERSPECIFICINTERACTIONSINSINGLE-SPECIES TIMESERIES 3.1Background Oneofthemostimportantmodelingapplicationsinconservationecologyis predictingfuturepopulationabundances.ThepracticeofPopulationViabilityAnalysis (PVA)connectsstochasticpopulationmodelstodataandisusedtounderstandthe factorslimitingpopulationgrowthandtoassessriskofextinctionorfallingbeneath acertainabundance(e.g., Shaffer 1981 , Staples etal. 2005 ).Despiteabodyof well-developedtheoryonpopulationregulation,therehasbeenrelativelylittlework validatingquantitativepredictionsofPVAmodelsandmethods(butseeattempts by Brook etal. 2000 , Lindenmayer etal. 2003 ).Thisisofparticularimportancefor speciesofconservationconcernwherelittledatamaybeinhandregardinglife-history detailsandknowledgeoftheunderlyingbiologicalprocessesislimited.Inthesecases, ecologistsoftenrelyonverybasicpopulationsmodelsofgrowth,densitydependence, andvariabilitytomakepredictions. Characterizingtheextinctionprocesshasbeenanimportantgoalfortheoretical ecology( Dennis&Munholland 1991 , Foley 1994 , Lande&Orzack 1988 ).Classic resultshaveprovidedusefulscalinglawsforthetimetoextinction(e.g., Leigh 1981 , Ludwig 1976 ),rulesofthumbfordealingwiththetypesofvariationthatcandrive populationstoextinction(e.g., Boyce 1992 , Lande 1993 ),aswellastheeffectsofage ( Lande&Orzack 1988 ),andspatialstructure(e.g., Hanski&Gilpin 1997 , Holt 1985 ) onextinction.Despitethebroadscopeofcurrentextinctiontheory,determiningthe impactsofcommunityinteractionsonextinctionriskremainsanelusivetask.Though asmallhandfulofPVAstudieshavetakenamultispeciesapproachwhenassessing populationviability(asreviewedin Sabo 2008 ),weknowofnoworkthathasattempted toexplicitlyincorporatetheeffectsofinterspeciÞcinteractionsonextinctionwhenonly 46

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single-speciestimeseriesdataisavailable.Thus,developingmodelingmethodsthat includetheseeffectsisanimportantstepinextendingtheapplicationofextinction theorytorealecologicalsystemswhereavailabledataisoftenlimited. Microcosmexperimentscanfacilitatetheessentialtaskofanchoringtheoretical predictionswithexperimentalobservations.Althoughmicrocosmexperimentshave beenarguedtogreatlyoversimplifyecologicalsystems( Carpenter 1996 ),theyalsooffer atestbedfortheorythatisunambiguous.Pastexperimentaltestsofextinctionhave primarilyfocusedontestingqualitativedifferencesbetweenpredictions,ratherthan quantitativepredictions,thuspotentiallylimitingtheirapplicability( Griffen&Drake 2008 ). Itisnotclearwhetherstandardmodelsofpopulationvariabilitywilladequately accountforecologicalinteractions,suchaspredationandcompetition,asthese interactionsarelikelytoinducestructuredvariation( Abbott etal. 2009 , Boyce 1992 , Royama 1981 , Stenseth etal. 1998 ).Here,weexaminedtheabilityofasuiteof unstructuredandstructuredvariancemodelstopredictextinctiontimesobservedin anexperimentalmicrocosmsof Daphniapulicaria ,whileatthesametimeassessing theeffectsofanumberofcommonmodelassumptions,includingtheparticularformof densitydependence,themodelforthevarianceofthegrowthrate,andthetransition probabilitydistributionofthediscretegrowthprocess.Importantly,thenatureofthe experimentaldatasetsthatweusedallowedustotesttheabilityofthesemodel variationstopredicttheextinctionprocessfortwodifferentclassesofdynamicsoften dealtwithinPVAanalyses.IntheÞrstsetofexperimentsthepopulationsßuctuate aroundasteady-state,whileinthesecondexperimentpopulationtendstodeclineover time.Thus,thescenariostestedallowsourresultstobeextendedbeyondthisstudy. 3.2Models&Methods 3.2.1ExperimentalData Thedatasetsre-analyzedinthisstudycomefrompreviousworkby Grover etal. ( 2000 )whofollowedexperimentalpopulationsof D.pulicaria fornearly2years(Figure 47

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3-1 ).TheoriginalexperimentdiscardedtheÞrst140daysofdatainordertoexclude theinitialtransientphaseofpopulationdynamicsandwefollowedthesamepractice. Weconsideredtwooftheirexperimentaltreatmentswhereeachtreatmenthadthree replicates.Inonetreatment,populationabundancesßuctuatedaroundthesteady statedisplayingquasi-stationarydynamics,whileintheothertreatmentabundances displayedadeclinetowardsextinction.Thesedifferentdynamicscorrespondtothe small-populationparadigmandthedeclining-populationparadigmasdeÞnedby Caughley ( 1994 ).Bothscenariosareofinterestformanagementandconservation purposes. Theexperimentalconditionsconsistedofthreereplicatesofsimplecommunity microcosmsandthreereplicatesofcomplexcommunitymicrocosms.Simplecommunities werecomposedofconsumer-resourceinteractionsbetween D.pulicaria feedingon greenalgae( Scenedesmusacutus , Scenedesmusquadricauda ,and Chlorellasp. ) andotherindigenousmicroorganisms,whilecomplexcommunitieswerecomposed oftheconsumer-resourcesystemplusadditionalgrazers( Simocephalusvetulus and Cypridopsisobesa ).While D.pulicaria andthe S.vetulus areÞlterfeedersandhave beenshowntocompeteinmicrocosmexperiments( Frank 1952 ), C.obesa isascraper ( Roca etal. 1993 ).Samplingoccurredevery4days.Ineachsampleapproximately58% ofthemicrocosmwasrecordedbyvideo-cameraandanimalswerecounted. D.pulicaria displayagenerationtimeof10-50daysatthetemperatureused(15 $ Celsius).More experimentaldetailsareprovidedin Grover etal. ( 2000 ). 3.2.2PopulationModels Ouranalysiswasbasedonaderivationforthemeanandvarianceofadiscrete-time abundancemodelwithdemographicandenvironmentalvariance.Afullderivationofthe meanandvarianceforabundancesisprovidedinAppendix D ,thoughsomedetailsare presentedhere.Throughoutthetextweusetheconventionthatrandomvariablesare denotedusingcapitalletters,suchas N t ,whilerealizationsoftherandomvariableare 48

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lowercase(e.g., n t ).Inwhatfollows, N t denotestherandompopulationabundancesat time t . Westartbyassumingthateveryindividualgivesbirthonaverageto & offspring pergenerationwithavariancegivenby ! 2 ,thedemographicvariationinreproduction (seeAppendix D formoredetailsregardingtheassumptionsofthisvariancemodel). Nextweassumethat & canvaryrandomlyovertimewithvariancegivenby ' 2 ,the environmentalvariationinreproduction.Afterreproductionoffspringandparentssurvive withprobability p t " 1 = p 0 p ( n t " 1 ) ,where p 0 isthedensityindependentsurvivaland p ( n t " 1 ) isthedensitydependentsurvival.Wenotethattheindividualparametersof survivalandreproductionintheproduct & p 0 arenon-identiÞable(asshowninAppendix D ),thereforewedeÞne r = & p 0 .Followingtheseassumptions,theexpectedvalueand varianceforcurrentabundancesasafunctionofthepreviouslyobservedabundances are: E[ N t | N t " 1 = n t " 1 ]= rn t " 1 p ( n t " 1 ) (3Ð1) Var[ N t | N t " 1 = n t " 1 ]= " rp ( n t " 1 ) ( 1 $ p ( n t " 1 ) ) + ! 2 p ( n t " 1 ) 2 # n t " 1 +[ n t " 1 p ( n t " 1 )] 2 ' 2 . (3Ð2) TheÞrstterminEquation 3Ð2 isafactorof n t " 1 andisreferredtoasthedemographic stochasticity.Thisexpressioncomesfromcomputingtheaveragevariabilityofthe reproductionandsurvivalprocess(seeAppendix D ).Thus,thisvariancecomponent correspondstotheoverallcontributionofdemographicvariancetothetotalpopulation variance,alwaysscalinglike n t " 1 regardlessoftheparticularformofthereproduction andsurvival.Notethatonlyinthecaseofdensity-independence,when p ( n t " 1 )=1 , isthedemographicvarianceofthepopulationprocessisexactlyproportionalto n t " 1 . Whensurvivalisdensitydependentthedemographicstochasticitywillscaleasamore complexfunctionof n t " 1 ( Drake 2005 , Saether etal. 1998 ). 49

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ThesecondhalfofEquation 3Ð2 scalesby n 2 t " 1 andisreferredtoastheenvironmental stochasticity.Thistermcomesfromcomputingthevariabilityovertimeoftheaverage numberofoffspringproducedthatsurvivetothenextgeneration(Appendix D ).This explicitlyaccountsfortemporalßuctuationsinthereproductionprocess,thustranslating theconceptoftheenvironmentalvariabilityintoananalyticalvariancecomponentofthe populationgrowthprocess Previousworkhasshownthatautocorrelationcanbeinducedthroughpopulation interactionsthataresubjectedtovariability( Abbott etal. 2009 , Royama 1981 ).Asimple modeloftwointeractingspeciesonthelog-scale( X ( t ) ( ln N t )canbewrittenas: X 1 ( t )= c 1 + b 11 X 1 ( t $ 1)+ b 12 X 2 ( t $ 1)+ W 1 ( t ) (3Ð3) X 2 ( t )= c 2 + b 21 X 1 ( t $ 1)+ b 22 X 2 ( t $ 1)+ W 2 ( t ). (3Ð4) The b 11 and b 22 termscorrespondtodensitydependenceregulationterms,the b 12 and b 21 termscorrespondtointeractionsbetweenlife-historystagesorspecies,andthe W ( t ) termsrepresentsdemographicandenvironmentalstochasticity.Thissystemcan beviewedasalinearizedapproximationtomorecomplexfunctionalresponses( Ives etal. 2003 ).Thesystem(Equations 3Ð3 and 3Ð4 )canbereexpressedasaunivariate autoregressiveÐmoving-average(ARMA)modelforthespeciesofinterest,either X 1 ( t ) or X 2 ( t ) ,obtaininganARMA(2,1)model(seeAppendix E forderivation).Inthisworkwe refertotheAR(1)componentasthedensitydependencemodel,whilewerefertothe AR(2)componentassimplyanARautocorrelationmodelconsistentwithanextensive literatureontheeffectsofenvironmentalautocorrelation(e.g., Royama 1981 ).This AR(2)termalsohasbeencalledthelaggeddensitydependenceby Turchin ( 1990 ). Thisapproachcanbefurthergeneralizedsuchthattheunivariatestochastic dynamicsforaspeciesinacommunityof n specieswillfollowanARMA( n , n $ 1 ) model( Abbott etal. 2009 ).Inthetwospeciescase,themagnitudeoftheautoregressive 50

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(AR)componentisthedifferencebetweeninterspeciÞcinteractionsandintraspeciÞc interactions, b 12 b 21 $ b 11 b 22 .ThesignandmagnitudeoftheARparametergivesa measureofthebalancebetweeninterspeciÞcinteractions( b 12 b 21 )andgrowthand intraspeciÞcinteractions( b 11 b 22 )inthesystem.Incontrast,themoving-average(MA) componentthatisgeneratedisafunctionofhowwelltheinterspeciÞcinteraction propagatesperturbations.TheMAparameterismaximizedwhenthespeciesinteracting withthefocalspeciesfollowsarandomwalk,suchthatperturbationsfromequilibrium arepropagatedbytheinteractingspecies,ratherthanbeingdampenedbyadensity dependentresponse(Appendix E ).Inourmodelformulationtheautocorrelation componentseffectboththedemographicandenvironmentalvariances,andwelimit ourselvestoÞttingoneARcomponent(inadditiontothedensitydependence)andone MAcomponentbasedontime-seriesdiagnosticspresentedinAppendix F . 3.2.3ModelComparisons Weconsideredanumberofpossibleassumptionsthatmightbemadewhen conductingaPVAinordertoconstructarealisticbuttractablesetofmodels.The effectsofeachmodelassumptionidentiÞedwereassessedindependently.These assumptionsare:1)theformofthedensitydependence,2)thedistributionalformofthe transitionpdf,3)theformoftheautocorrelationand4)thevariancemodeloftheprocess error.Foreachoftheassumptions,weconsideredahandfulofexplicitalternative speciÞcationsandevaluatedtheinferentialconsequencesofconsideringeachofthese components,oneatatime.Thus,foursetsofindependentcomparisonsweredone.The comparisonsaresummarizedinTable 3-1 . Incomparison1(seeTable 3-1 ),wevariedthemodelofdensitydependencewhile othercomponentswereÞxedusingfunctionalformsfor p ( n t " 1 ) inEquations 3Ð1 and 3Ð2 thatrepresentdifferenthypothesesabouttheunderlyingpopulationdynamics. Theseformsmaybestrictlyinterpretedasdifferenthypothesesaboutpopulation regulatorymechanisms(suchastheformofinterspeciÞccompetition),orasstatistical 51

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modelsthatapproximatecomplexunderlyingpopulationregulatorymechanisms.In anycase,theformofdensitydependencecontrolstherateatwhichnewindividuals areaddedtothepopulationasafunctionofabundance.SpeciÞcally,weconsideredthe overcompensatoryRicker( p ( n t " 1 )= e " bn t ! 1 )andlogistic( p ( n t " 1 )=1 $ bn t " 1 )models, aswellastheundercompensatoryBeverton-Holt( p ( n t " 1 )= 1 1+ bn t ! 1 )andGompertz ( p ( n t " 1 )= e " b ln n t ! 1 )models.Wealsoincludedthedensityindependent( p ( n t " 1 )=1 ) modelofpopulationgrowthasa"nullhypothesis".Althoughprimarilyassociatedwith themeanresponseofthepopulation,thistermalsoaffectsthevariance(Equation 3Ð2 ).Allformsofdensitydependencewererequiredtobeboundedbetween0and1, thereforeweimplementedboundaryconstraintsonparameterswhennecessary(e.g., forthelogisticmodel).Whencomparingeachofthesedensity-dependencemodels,the transitiondistributionwasassumedtobegammaandthevariancetohavedemographic andenvironmentaltermswithnoautocorrelation. Forcomparison2(seeTable 3-1 ),weexaminedthepropertiesofthetransitionpdf whileothercomponentswereÞxed.Wetestedtheimpactofthehighermomentsofa distributionandlattice(rounding)errorsonpredictions.AlthoughtheimpactoftheÞrst twomoments(themeanandvariance)onextinctionpredictionshasbeenbeenstudied usingdiffusionapproximations( Lande 1993 , Ludwig 1976 )andsimulations( Drake 2005 , Melbourne&Hastings 2008 ),theimpactofhighermomentsonpredictionshasnot beenconsidered.Thus,wevariedthehighermomentsbyÞttinggamma,log-normal, andnegativebinomialdistributionstothedata.Also,wetestedforthepresenceof latticeeffectsbycomparingpredictionsfromthediscretestatenegative-binomial distributiontogammadistribution,asimilarcontinuousstatedistribution.Latticeeffects, orroundingerrorsintroducedintheapproximationofanintegervaluevariablebya continuousvariablecanappearindiscretestatemodels,andcanhavepotentialimpacts onpredictions( Henson etal. 2001 ).Moredetailsonhowwematchedthemeanand varianceofEquations 3Ð1 and 3Ð2 totheparametersofthesedistributionsisprovided 52

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inAppendix F .Forthedifferenttransitionpdf'sincomparison2,weassumedaRicker modelofdensitydependenceandademographicandenvironmentalvariancemodel withnoautocorrelation. Incomparison3(seeTable 3-1 )wetestedtheimpactoftheautocorrelation structureinthevariancemodelwhiletheothermodelcomponentswereÞxed.Although Equation 3Ð2 isshownasconditionalon n t " 1 ,ecologicalinteractionsmayinduceother dependenciesnotproperlycapturedbythedensitydependencemodel.Bothmoving average(MA)andautoregressive(AR)autocorrelationmodelshavebeenshown tocaptureinformationaboutinterspeciÞcandintraspeciÞcinteractionsinecological systems( Abbott etal. 2009 ).IthasalsobeenshownthatMAmodelsalsoarisedue tomeasurementerror( Dennis etal. 2006 ).ARandMAmodelscanbecombinedinto theautoregressive-moving-average(ARMA)correlationmodeltoincludeeffectsofboth modelssimultaneously( Shumway&Stoffer 2006 ).WeincorporatedAR,MA,andARMA correlationstructuresintothevariancemodelandtestedtheirimpactonpredictions oftheextinctionprocess.Forallmodelsincomparison3,weassumedagamma transitiondistribution,aRickermodelofdensitydependence,andademographicand environmentalvariancemodel. Forcomparison4(seeTable 3-1 )weexaminedseveralassumptionsaboutthe demographicandenvironmentaltermsinthevariancemodelwhileÞxingothermodel components.Wetestedwhetherpurelydemographicorenvironmentalvarianceterms improvedpredictionsoverthefulldemographicandenvironmentalmodel.Theform inEquation 3Ð2 assumesthatpopulationsaresubjecttobothdemographicand environmentalprocesses,anassumptionthatisnotalwaysmade(e.g., Ellner& Holmes 2008 ).Wealsotestedtheimpactofremovingthedensitydependencein thedemographicvarianceofthepopulationgrowthrate R t =ln ( N t / N t " 1 ) similartoa testperformedby Drake ( 2005 ).Thisledtothevariancemodel V [ N t | N t " 1 = n t " 1 ]= rn t " 1 p ( n t " 1 ) ! 2 + n 2 t " 1 p ( n t " 1 ) 2 ' 2 .Underthecomparison4model-set,weÞxedthe 53

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transitiondistributiontobegamma,theformofdensitydependencetobeRicker,and hadnoautocorrelationsintheerrorstructure. 3.2.4ParameterEstimation ToÞtthedataweusedthelikelihoodfunctionsdeÞnedinAppendix F ,wherethe jointprobabilityoftheobservationsisgivenbytheproductoftheone-steptransition distributionofthepopulationprocess.Thelikelihoodsofthetransitionswereconstructed bymatchingthemomentsofEquations 3Ð1 and 3Ð2 tothemeanandvariancesofthe transitiondistribution(detailsprovidedinAppendix F ).AllmodelswereÞttodatausing maximumlikelihoodestimationintheRstatisticalsoftwareenvironment( RDevelopment CoreTeam 2012 ). Afterparameterswereestimated,weusedsimulationstoexaminetheconsistency ofourmodelswiththeobservedextinctionprocess.Themetricweusedtocharacterize extinctionwastheÞrstpassagetime(fpt),deÞnedasthetimeittakesforapopulation toÞrstreachaquasi-extinctionabundance n fromsomeinitialabundance n 0 .The distributionoffptsisdenotedas T ( n ):=min { t & 0: N ( t ) ' n | N 0 = n 0 } ( Taylor&Karlin 1984 ).Thelikelihoodofthefptwascalculatedbydeterminingtheprobabilityofobtaining theobservedquantity ( ( n ) fromthedistributionoffptspredictedbymodel M ,writtenas T M ( n ) ,whereournotationemphasizesthefactthatpredictionsaremadeconditional onaparticularmodel.The ( ( n ) 'swereobtainedforallobservedabundances,and 10 5 simulationswereusedtoobtainthedistribution, T M ( n ) ,foreachmicrocosmpopulation. Becausethefptforthesemodelsisadiscreterandomvariable,theprobabilityofthe observing ( ( n ) , P ( T M ( n )= ( ( n ) ) ,wassetequaltotheproportionofsimulationswhich displayapredictedfptequaltotheobservedfpt.Forthe i thmicrocosmweusedthe likelihoodofall s i observedfptsforeachcommunitytype.Becausetherewerethree microcomspercommunitytypethislikelihoodisgivenby 3 $ i =1 s i $ j =1 P ( T M ( n i , j )= ( ( n i , j ) ) for allobserved n i , j . 54

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Thefptlikelihoodsforeachmodelandeachdatasetwereusedtocalculatevalues oftheAkaikeInformationCriterion(AIC),whichwerethenusedtoselectbetween modelsforeachmodel-setcomparison.Thenumberofparameters, k ,usedinthe AICcalculationwasthenumberofparametersÞtintheone-steptransitionlikelihood. ModelswithlowerAICvalueswereinterpretedasdoingabetterjobofexplainingthe extinctionprocess.UsingtheFPT,ratherthantherawabundances,canleadtodifferent predictionsastheFPTisadifferentsummaryofthedataandmayprovidedifferent information.WealsocalculatedtheBICcriterionforallmodelsandfoundthatallofour conclusionswererobusttotheformofcriterionused. Finally,inordertoobtainanabsolutemeasureofthepredictiveerrorwecalculated therootmeansquareerror(RMSE),deÞnedas RMSE= % ! 3 i =1 ! s i j =1 ( % i , j " E [ T M ( n i , j )] ) 2 ! 3 i =1 s i .This measuregivesanabsolutemeasureoftheerrorassociatedwiththemeanfptpredicted frommodel M , E [ T M ( n i , j )] ,ratherthanarelativemeasureofevidenceprovidedbyAIC values. 3.2.5MeasurementError Wetestedtheimpactofmeasurementerroronparameterestimatesinorderto determineiftheobservedMAparameterestimateswereduepurelytomeasurement error.Theeffectsofmeasurementerrorweretestedbysimulating100timeseriesusing parametersfromtheRickerdensitydependent,demographicandenvironmental variancemodelwithnoautocorrelationforeachofthethreesimplemicrocosm experiments.Werescaledtheobservationsby 1 / 0.58 inordertoaccountforthe unobservedvolumeineachmicrocosm( Grover etal. 2000 ).Abinomialsamplingmodel wasthenappliedtothesimulateddatawheretheprobabilityofdetectionwas 0.58 .We estimatedmodelparametersfortheresultingobservedtime-seriesandtheaverage andstandarderrorofthemovingaverageparameterwascalculatedforeachsetof microcosmsimulations.Wecomparedtheseestimatestovaluesobservedfromthe microcosmexperiments.Ifsamplingerrorexplainedthemovingaverageparameter 55

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estimatedinthesimplecommunitymicrocosmsthentheobservationsshouldfallwithin approximately2standarderrorsfromtheestimatedmean. 3.2.6MovingAverageModels Inordertotestthepotentialimpactsofnon-measurementerrorMAprocesseson populationpersistencewesimulatedabundancetime-serieswithparametersestimated fromthemodelwithRickerdensitydependence,environmentalanddemographic stochasticity,andanMAautocorrelationmodelinmicrocosm1.WethenvariedtheMA parameterwhilekeepingtheotherparametersÞxedover20equallyspacedvaluesfrom -0.9to0.0.Wecalculatedthemeantimetoextinction(MTE)bysimulatinguntilthetime seriesreached1individualorless.Werepeatedthisprocedure10,000timestoobtain stableestimatesoftheMTE. 3.3Results 3.3.1ModelSelection Inthesimplecommunityexperiment2outof3populationspersistedtheentire durationoftheexperimentsuggestingthatthesepopulationswerestablebutsubject tostochasticextinctioneventsoverlongertime-scales(Figure 3-1 ).Incontrast,all thecomplexcommunitypopulationswentextinctoverthecourseoftheexperiment asabundancestendedtodecreaseovertime,suggestingthatextinctionsweredue todeterministicprocesses.AmodelwithRickerdensitydependenceandagamma errorstructurecontainingbothdemographicandenvironmentalstochasticitywith noautocorrelationappearedstatisticallyconsistentwiththedataforallmicrocosm experiments(Figure 3-2 ),however,whencomparingmodelÞtswiththefptAICfurther modelimprovementswereapparent. Theformofdensitydependencehadamoderateeffectonpredictionsandremoving densitydependencegreatlyworsenedpredictions.WefoundthattheRickermodel ofdensitydependenceminimizedthe ! AICvalueinsimplecommunitiesthoughthe Beverton-Holtalsoperformedwell( ! AIC=0.91 ),whileinthecomplexcommunitiesthe 56

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Gompertzmodelperformedbestoverall(Table 3-2 ).Theundercompensatorydynamics displayedbytheGompertzmodelmaybecapturingtheweakconsumer-resource couplinginthecomplexcommunity,whilestrongercompensationoftheRickermodel bettercapturedthestrongcouplinginthesimplecommunities.BecausetheGompertz modelislesssensitivetodensitythantheothermodelsconsideredhere,itmaybea bettermodelforpopulationsembeddedincomplexsystemswhereextrinsicfactorsplay asigniÞcantroleinregulationprocesses.Consistentwiththisinterpretation,previous workhasshownthatconcaveformsofdensitydependencesuchastheGompertz modelaregenerallyfoundintimeseriesofnaturalpopulations( Sibly etal. 2005 ).Inall microcosmsdensitydependentmodelsgreatlyoutperformedthedensityindependent exponentialgrowthmodel(Table 3-2 ). WhenusingtheRickermodelandassumingindependentlydistributedobservations withbothdemographicandenvironmentalvariability,therangeof ! AICvaluesamong transitiondistributionshadapproximatelythesamemagnitudeasdifferencesamong densitydependencemodels.Thissuggeststhatthechoiceoftransitiondistributionisa potentiallyimportantconsiderationwhenbuildingmodelsforextinctionriskassessment butonethatisrarelytested.Thegammadistributionpredicted ( ( n ) bestinthesimple communitieswhilethelog-normalwasbestinthecomplexcommunities(Table 3-2 ). Theweakperformanceofthenegativebinomialmodelrelativetothecontinuous modelsimpliesthatlatticeeffectsmaynotbeanimportantfactorwhenmodelingthese populations. WhenremovingtheassumptionofindependenterrorstructuresMAmodels outperformedothermodelsinthesimplecommunityexperiment,whiletheARMA modelperformedbestinthecomplexcommunity(Table 3-2 ).Asdiscussedinthe modelssections,ARMAmodeltermsmeasurethemagnitudeofinter-andintra-speciÞc interactionsandthetendencyforinterspeciÞcinteractionstodampenstochasticity.A negligibleARterminthesimplecommunityisindicativethatinter-andintra-speciÞc 57

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forcesarebalanced,whilethepresenceofallnegativeARtermsinthecomplex communitiesindicatesthatintraspeciÞcinteractionsareweakerthantheintraspeciÞc growthandinteractionterms.TheimportanceoftheMAterminbothsimpleand complexcommunitiesisindicativeofweakregulationinthepreypopulationanda correspondingtendencyforperturbationsinthepredatorpopulationtobepropagated bythepreypopulation.PureARmodelsperformedrelativelypoorlyinbothcommunity types,asurprisingresultduetotheemphasisofpreviousworkonthisandsimilar models(e.g., Cuddington&Yodzis 1999 , Halley 1996 , Morales 1999 ). Comparingmodelvariances,wefounddemographicandenvironmentalstochasticity werebothimportantforpredicting ( ( n ) .Removingthedensitydependentterminthe demographicvariancedidnotappeartoleadtoadifferenceinthesimplecommunities ( ! AIC=1.64 ),thoughitdidinthecomplexcommunities( ! AIC=3.41 ).Interestingly, whenlookingatAICvaluesfortheabundances(Appendix D ,Table F-1 ),ratherthan ( ( n ) ,ourresultsareconsistentwithpreviousworkthatsuggestsincludingthedensity dependenceindemographicstochasticityisimportant( Drake 2005 ).Thedifference between ( ( n ) 'sandabundancesmaybeduetothefactthatthedensitydependent terminthedemographicvariancebecomesapproximately1atlowabundances.For abundancetimeserieslowpopulationsizesarerelativelyrareandtheestimateswill tendtobedominatedbytheregionwheremostabundanceobservationsoccur,whilein the ( ( n ) 'slowabundanceswilltendtohavealargerrelativeeffectbecauseobservations aremoreevenlydispersedacrosstherangeofobservedabundances. Finally,theresultsfromtheRMSEcalculation(Table 3-3 )differedinsomeways fromtheAICvalues(Table 3-2 ).ThisislikelyduetothefactthattheRMSEonly considersthemeanresponseofthepredicted ( ( n ) 'sratherthanthewholedistribution asintheAIC.DespitethesedifferencestheoverallbestRMSEmodelswerethesame astheAICcomparison.ThesimplecommunityhadaRMSEin ( ( n ) of19.58days, predictedbytheRickerdensitydependencemodelwithdemographicandenvironmental 58

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variabilityandaMAautocorrelationmodel.TheminimumRMSEforthecomplex communitywasbetterthanthesimplecommunitiesat5.91days,predictedbythe Rickerdensitydependencemodelwithdemographicandenvironmentalvariabilityand anARMAautocorrelationmodel.Theinclusionofasuitableautocorrelationmodel improvedtheoverallRMSEbyabout20%inthesimplecommunityandbyabout30%in thecomplexcommunity. 3.3.2MeasurementError WetestedtheimpactofabinomialsamplingmodelonMAparameterestimates withdatasimulatedfromthesimplecommunityexperiment.Wefoundthatthe magnitudesoftheMAparameterestimatesduetomeasurementerrorwereless(mean, -0.04,-0.04,-0.06)thantheMLEestimatesoftheMAtermsinthesimplecommunity experimentmicrocosms(mean(se), $ 0.23(0.012) , $ 0.48(0.017) , $ 0.11(0.015) ).These resultssuggestthattheMAtermspresentinthedataarenotduetosamplingerrorand thatanothermechanismwaslikelyresponsible. 3.3.3MovingAverageModelDynamics WefoundthatincorporatingMAmodelsintopopulationgrowthprocessescan dramaticallyaffecttheMTEofaspecies(Figure 3-3 ).AsthemagnitudeoftheMA parameterincreased,persistenceincreasedoverthreeordersofmagnitudeinafaster thanpower-lawrelationship.Theseresultsaresimilartoprevioussimulationstudies exploringtheimpactofARmodelsonpopulationpersistence( Cuddington&Yodzis 1999 , Morales 1999 , Petchey 2000 )havingsimilarorderofmagnitudeimpactson populationpersistenceaspreviousstudiesonARmodels. GiventhepotentialimportanceofMAmodelsinpredictingabundancesitisworth furtherunderstandingtheimpactsofthesemodelsondynamics.Inourformulation themovingaverageaffectsthegrowthratethroughthelog-scaleabundancesandthe magnitudeofMAcomponentsareproportionaltomagnitudeofperturbationsfromthe expectedvalueoftheone-steptransitions.Thecontributiontothegrowthratebythe 59

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movingaveragecomponentcanthenbeinterpretedasarescalingofthelog-ratioofthe theobservedandexpectedpopulationabundances, ln ( n t " 1 / E [ N t " 1 | N t " 2 ] ) (Appendix F ).Whenthepopulationishigherthantheexpectedvalue,anegativeMAparameter (asfoundinallofthesemicrocosms)pullsthepopulationtowardstheexpectedvalue, increasingthestrengthofpopulationregulation(sensu Ziebarth etal. 2010 )througha restorativeforce. 3.4Discussion Inthisstudy,weassessedtheeffectsinthepredictedextinctionrisksoffour differentpopulationmodelingassumptions(thedensity-dependenceform,theformof thetransitionpdfandautocorrelationinthegrowthrate,andthenatureoftheprocess errorvariancemodel),butmuchremainstobedone.Despiteawelldevelopedbody ofliteraturedescribingthepropertiesofpopulationextinctions,verylittlehasbeen donetoquantitativelytestthesepredictions( Griffen&Drake 2008 ).Hereweused experimentaldatatoshowthatspeciesinteractionscanbeanimportantcontributorto extinctionriskwhilesuccessfullytestingtheabilityofautocorrelationmodelstoaccount fortheseinteractions.Additionally,wedemonstratehowandwhymanyofthecommon assumptionsassociatedwithÞttingpopulationmodelstodatacanimpactextinctionrisk predictionsinrealecologicalsystems.Ourapproachsuggeststhatstrongmodeling tests-usuallyreservedonlyforsimulationstudies-coupledwithdetailedexperimental data,canprovideusefulinsightsonmodelingchoiceseveninrelativelysimplesystems. Oureffortssuggestthatcombiningempiricalandanalyticalmethodscanleadto abetterunderstandingoftheprocessesgoverningpopulationdynamics.However,as suggestedby Dennis&Taper ( 1994 )theinferenceofecologicalmechanismsmustbe treatedwithcaution.Both Wolda ( 1991 )and Dennis&Taper ( 1994 )pointoutthatitis impossibletodistinguish"ßuctuatingequilibriumvalues"from"ßuctuatingdeviationsfrom thoseequilibriumvalues"usingsimpletimeseriesofabundance.Wewouldattributethis variationtotheenvironmentalvarianceforbothcases,thoughtheymayhavedifferent 60

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ecologicalandconservationimplications.Despitetheselimitationsthemodelswe exploredherehavethepotentialtoincorporateextrinsicfactorsthatcanbeusedto explainvariationinvitalratesandimprovepredictiveability.However,thisinformation needstobebothavailableandresolvable,whichisnotalwaysthecase.Asshown by Knape&deValpine ( 2010 )evenwhentimeseriesofenvironmentalvariablesare available,resolvingthesecomplexnonlinearinteractionsisnotatrivialtaskthough importantexceptionsdoexist(e.g., Ponciano&Capistr « an 2011 ).Inmorecomplex abundancemodelsevenmoredifÞcultiesexist.Theprocessofdistinguishingamong mechanismstranslatesintouniquelyidentifyingparametersfromtheavailabledata.Itis possible,andsurprisinglyeasytointroduceparameternon-identiÞabilityintostochastic models. Ponciano etal. ( 2012 )showhowDataCloningcanbeusedasadiagnostic tooltodetectparameternon-identiÞability.Thesecomplexitiessuggestthattestingthe inferentiallimitationsoftimeseriesabundancemodelsforPVAwillcontinuetobean importantlineofresearch. OurÞrstÞndingwasthattheformofdensitydependenceforthesamespecies differeddependingonthecommunitycompositioninwhichitwasgrowing,ina waythatisconsistentwithecologicaltheory.Thus,strongdensitydependence (over-compensatorydynamics),impliedbytheRickermodel,wasfoundtobebest inthesimplecommunitywhileinthecomplexcommunityaweakerformofdensity dependence,embodiedbytheGompertzmodel,resultedinbetterpredictions.The discrepancyinoptimaldensitydependentmodelsforotherwisesimilarpopulations reßectscommunity-leveldifferencesbetweenastronglycoupledconsumer-resource systemandamorecomplexcommunitywithanumberofbioticandabioticinteractions. Thisresultimpliesthatthebestmodelforonepopulationmaysimplynottranslate tootherpopulationsofthesamespecieswhenecologicalforcesdifferbetween communities( Murdoch&McCauley 1985 ),anassumptionthatisoftenmadewhen aparticularsystemlacksadequatedata(e.g., Ferguson etal. 2012 ). 61

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Althoughitiscommonpracticetoassumealognormaltransitiondistributionwhen Þttingmodelstodataofpopulationgrowth,thereareplausiblebiologicalreasonsunder whichthisdistributionmaynothold( Diserud&Engen 2000 , Henson etal. 2001 ).We foundthatthegammadistributiondidbestinthesimplecommunitywhilethelog-normal didbestinthecomplexcommunity.Additionally,bothdemographicandenvironmental stochasticitywereimportantforpredictionsinsimpleandcomplexmicrocosms. Predictionsofquasi-extinctionoftenonlyincludeanenvironmentalvarianceterm (e.g. Holmes etal. 2007 ),thedominantcontributionathigherabundances.However,for populationstrulyatriskofextinctionourresultssuggestthatthisisnotsufÞcientandthat demographicandenvironmentalmodelsofstochasticityshouldbeused. Animportantopenquestionforconservationandmanagementistoidentifywhen morecomplexmultispeciesprocessmodelscanbeusedtoimproveabundance predictions(e.g., Sabo 2008 ).OurresultsshowthataccountingforinterspeciÞc interactionsinPVA'smaybepossibleinsingle-speciestimeseriesthroughtheuseof appropriateautocorrelationmodels.ThesimpliÞedstatisticalrepresentationofspecies interactionsthatareprovidedbyautocorrelationstructurescanimprovepredictions withouttheneedformulti-speciestime-seriesdatamakingthisapowerfulandtractable approach. Whenfacingthepressingneedofaquantitativeassessmentofextinctionrisk, modelersandscientistsrelybynecessityonanumberofmodelassumptionsand simpliÞcations.ChoosingwhichsimpliÞcationsandassumptionsshouldberetainedor else,discarded,seemstobeakeycomponentofamodeler's savoirfaire ,thatalltoo oftenistakenforgranted.Asputby Taper etal. ( 2008 ),modelscarrythemeaningof science,andthisputsatremendousburdenontheprocessofmodelselection.Inthat sense,ourstudyrepresentsoneoftheÞrstexamplesweareawareofthatillustrates whythestructuraladequacy( sensu Taper etal. ( 2008 ))ofecologicalmodelsshould beroutinelyandextensivelyexplored.Wehopethatourstudyrepresentsastarting 62

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pointforfutureexplorationsofthevariancescalinganddecompositionofpopulation dynamicsmodelswiththegoalofimprovingquantitativepredictionsinconservationand management. Table3-1.Fixedandvariedcomponentsinmodelcomparisons. Comparison1Comparison2Comparison3Comparison4 Densitydependence Ricker Beverton-Holt Gompertz Exponential Ricker Ricker Ricker TransitiondistributionGamma LN NB Gamma Gamma Gamma Autocorrelation None None None AR MA ARMA None Variance D+E D+E D+E D+E None Eonly Donly Note-Foreachmodel-setcomparisonwevariedoneassumptionoutofthefourmodel componentswhileÞxingtheothers.Componentsthatwerevariedareinbold.For entriesthatwereÞxed,weusedtheRickermodelofdensitydependence,thelognormal (LN)transitiondistribution,andthedemographicandenvironmental(D+E)modelof stochasticity.Additionalabbreviationsusedarenegativebinomial(NB),Environmental variationonly(Eonly)andDemographicvariationonly(Donly). 63

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Table3-2. ! AICvaluesforeachmodelandcommunitytype. Model M k SimplecommunityComplexcommunity Ricker 4 26 . 72 400.86 Beverton-Holt 427.63 386.54 Logistic 479.92 271.38 Gompertz 468.93 245 . 04 Exponential 3128.90 569.12 Log-normal 445.81 355 . 11 Negativebinomial 4113.42 868.37 Gamma 4 26 . 72 400.86 Noautocorrelation 426.72 400.86 AR 513.44 370.04 MA 5 0 . 00 21.76 ARMA 633.84 0 . 00 DemographicandEnvironmental4 26 . 72 400.86 Nodensitydependenceinvariance428.36 397 . 45 Environmentalonly 3119.25 1231.23 Demographiconly 337.20 1385.27 Note-ThenumberofparametersusedpermicrocosmintheAICcalculationisgivenby k andboldnumbersrepresentthebestmodelwithinasetofcomparisons. Table3-3.Rootmeansquareerrorforeachmodelandcommunitytype. Model M SimplecommunityComplexcommunity Ricker 23.23 7.84 Beverton-Holt 22.51 7.18 Logistic 26.38 8.73 Gompertz 23.11 7.18 Exponential 36.91 13.05 Log-normal 22.09 8.44 Negativebinomial 23.66 12.07 Gamma 23.23 7.84 Noautocorrelation 23.23 7.18 AR 23.28 8.02 MA 19.58 6.15 ARMA 20.93 5.91 DemographicandEnvironmental 23.23 7.84 Nodensitydependenceinvariance 23.30 7.74 Environmentalonly 23.46 12.23 Demographiconly 27.71 8.18 Note-Boldnumbersrepresentthebestmodelwithinasetofcomparisons. 64

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Figure3-1.Populationtimeseriesforreplicatesofdifferentexperimentalconditions. Simplecommunitiesincludedaconsumer Daphniapulicaria andplanktonic resource.Complexcommunitiesincludedtheconsumerandresourcealong withcompetitors. 65

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Figure3-2.ThemeanpopulationabundanceforaRickermodelofdensitydependence (solidline)withdemographicandenvironmentalstochasticity,plottedfor eachobservation(points)withtheapproximate95%conÞdenceintervals (greyarea). 66

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Figure3-3.Alog-logplotoftheaffectofachangingMAparametervalueonthemean timetoextinction.AllparametersotherthanMAtermwerevaluesestimated frommicrocosm1. 67

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CHAPTER4 THEEVIDENCEANDIMPLICATIONSOFHIGHERORDERSCALINGINTHE ENVIRONMENTALVARIATIONOFANIMALPOPULATIONGROWTH 4.1Background Akeyquestionforecologistsisdetermininghowenvironmentalßuctuationsdrive populationvariability.Thestochasticmodelingframeworkinpopulationdynamics considersenvironmentalßuctuationsastemporalperturbationstothemeanofthebirth anddeathratesofindividualsinapopulation( Engen etal. 1998 , Levins 1969 ).Because speciÞcinformationonenvironmentalcovariatesisnotrequiredinthisapproach,ithas allowedustomakesigniÞcantprogressunderstandinghowenvironmentalperturbations drivepopulationvariability.Theapproachhasalsoproventobeusefulinempirical settingswhereastochasticprocessmodeliscombinedwithanobservationmodelto constructalikelihoodoftheobservations( Dennis etal. 2006 , deValpine&Hastings 2002 ).AnimportantaspectofthesemodelsisthespeciÞcationofhowenvironmental forcesdrivevariationinmodelparameters. Currentpracticeassumesthatenvironmentalvariationoccursasanadditive terminthelogoftheper-capitagrowthrate,deÞnedas R t =ln( N t / N t " 1 ) .This variationcanbederivedbyassumingthatthedensityindependentreproduction rateisarandomvariableleadingtoavariancemodelwiththewellknownquadratic scalingofthepopulationvarianceonabundances( Lande 1993 , Levins 1969 ). Environmentalfactorsliketemperaturedirectlyaffectthemaximumreproductive rateofindividualscanbecapturedbythismodelofenvironmentalvariation( Brown etal. 2004 ).However,thisadditivemodelmaynotaccuratelycaptureenvironmental variationinthestrengthofdensitydependence.Forinstancethecarryingcapacitymay ßuctuateintime.Thisvariabilityismultiplicativewithabundanceintheper-capitagrowth rateandcancauseßuctuationswithahigherorderscalinginthepopulationvariance ( Br ¬ annstr ¬ om&Sumpter 2006 , Feldman&Roughgarden 1975 ).Despitealonghistory 68

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ofapplyingmodelsofenvironmentalvariationtoanimalpopulations,thescalingofthe environmentalvariancehasrarelybeentested. Weareawareofonlytwoempiricalexamplesthathaveconsideredwhetherthe environmentaffectstheper-capitagrowthrateadditivelyormultiplicatively. Hanski &Woiwod ( 1993 )testedwhichmodelwasabetterdescriptionofmothandaphid populationvariation,concludingthatthestandardadditivemodelwasbetterforthese populations.Amorerecentstudyby Fowler&Pease ( 2010 )foundthattreatingthe carryingcapacityasarandomvariablebetterexplainedvariationintheperennialgrass, Boutelouarigidiseta ,thanthestandardmodelofenvironmentalvariation.Incontrast tothesefewempiricaltestsoftheenvironmentalvariancescaling,anumberofstudies havelinkedenvironmentalcovariatestointeractmultiplicativelywithabundances(e.g., Berryman&Lima 2006 , Coulson etal. 2001 , Jacobson&Provenzale 2004 , Lima &Berryman 2006 , Lima etal. 2006 , Mignatti etal. 2012 , Owen-Smith 2000 ).This empiricalworkissuggestivethattheadditivemodelofenvironmentalvariancescaling maynotalwayscaptureimportantstochasticpropertiesofanimalpopulations. Someoftheearlytheoreticalworkonstochasticpopulationdynamicsconsidered bothadditiveandmultiplicativemodelsofenvironmentalvariability. Levins ( 1969 ) showedthatwhenthecarryingcapacityoftheenvironmentßuctuatesthroughtimethe medianpopulationabundanceisaweightedharmonicmeanofthecarryingcapacities. Inaddition,heshowedthatthismultiplicativemodelledtopopulationswithhighgrowth ratesbeingdepressedbelowtheircarryingcapacitymorethanpopulationswithlow growthrates. Feldman&Roughgarden ( 1975 )followedupthisworkbyderivingthe populationvarianceunderarandomcarryingcapacityinthelogisticmodel.They foundthatthismultiplicativemodelleadstoapopulationvariancescalingthatis quartic,insteadofthestandardquadraticscalingthatemergeswhenenvironmental variationoccursadditively.Amorerecentderivationindiscrete-timemodelshasshown similarvariancescaling,butwithadependencyontheformofdensitydependence 69

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( Br ¬ annstr ¬ om&Sumpter 2006 ).Otherworkby Gyllenberg etal. ( 1994 )and Fagerholm &H ¬ ogn ¬ as ( 2002 )examinedtherecurrencepropertiesofdiscretetimemodelswithboth additiveandmultiplicativeenvironmentalvariation.Theyillustratedhowmultiplicative environmentalvariationcanleadtogrowth-catastrophebehavior,thatisexponential growthfollowedbyaswitchbacktodensitydependentdynamics,leadingtoextreme populationdeclines. Multiplicativeenvironmentalvariationhaslikelybeenoverlookedintheecological literatureduetoseveralhistoricalfactors.Importantearlyworkonparameterestimation andtesting( Dennis&Munholland 1991 )followedearliertheoreticaldevelopmentsin diffusionprocessesthatreliedonmodelingenvironmentalvariabilityasanadditive contributiontotheper-capitagrowthrate( Tuljapurkar&Orzack 1980 ).Inaddition, standardlinearregressiontoolscanbeusedtoaccountforadditiveenvironmental variation( Dennis&Taper 1994 )makingthismodelavailabletomostecologists. However,therecentandongoingadoptionofhierarchicalmodelingtoolsbyecologists meansthattheestimationandtestingofmodelswithmultiplicativeenvironmental variationshouldnowbeaccessibletomanypopulationecologists. Hereweshowthatpopulationswilloftenbestabilizedbymultiplicativeenvironmental variation,butthattheformoftheprobabilitydistributionusedtomodelenvironmental variabilitycanaffectthisstabilizingbehavior.Inordertodothis,weÞrstderiveamodel ofdemographicandenvironmentalvariationfordiscrete-timeunstructuredpopulations toexaminethepotentialpopulation-levelconsequencesofmultiplicativeenvironmental variation.Thisisfollowedbyanexaminationofalargenumberofdatasetsinthe GlobalPopulationDynamicsDatabase(GPDD),testingthedifferentenvironmental variancemodels.Wethenperformamoredetailedanalysisassessingtheformof thepopulationvarianceinlong-termtimeseriesofAlpineibex( Capraibex )andSoay sheep( Ovisaries ).Becausepreviousstudies( Berryman&Lima 2006 , Mignatti etal. 2012 )haveidentiÞedclimatecovariatesthatdrivevariationinthecarryingcapacity 70

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ofthesepopulationswehavehypothesizedthatthesepopulationswilldisplayhigher ordervariancescalingconsistentwiththisclimate-abundanceinteraction.Finally,we examinedthestochasticdynamicsofpopulationswithmultiplicativeenvironmental variationthroughsimulation.Ourresultssuggestthatmultiplicativeenvironmental variationmaybeacommonphenomenonthatcanstrengthenorweakenthepopulation regulationrelativetothestandardmodelofenvironmentalvariation. 4.2Models&Methods 4.2.1ModelProperties Weusedasimpleone-dimensionalmodelofstochasticpopulationgrowthsimilarto previouswork( Br ¬ annstr ¬ om&Sumpter 2006 , Engen etal. 1998 ).Weassumethatthe populationisunstructuredandcontainsdensitydependentsurvivalfollowing Ferguson &Ponciano ( 2014 ).Therandomvariables # and B describerandomßuctuationsin modelparametersthroughtime,wherethedeterministicmaximumrateofreproduction isgivenby & ,while b isthedeterministicparameterrelatedtothedeterministiccarrying capacity.Weareusingtheconventionthatrandomlettersrepresentrandomvariables. Weassumethatthestochasticanalogsoftheseparametersfollownormaldistributions, # ) N (ln( & ), # 2 ! ) (4Ð1a) B ) N ( b , # 2 B ). (4Ð1b) OurmodelfortheabundancesandgrowthratesarethendeÞnedas N t | ( N t " 1 = n t " 1 , #, B )= n t " 1 exp[#+ Bf ( n t " 1 )] (4Ð2a) R t | ( N t " 1 = n t " 1 , #, B ) % # $ Bf ( n t " 1 ). (4Ð2b) Theformofdensitydependenceisdeterminedby f ( n t " 1 ) ,weexaminedRicker ( f ( n t " 1 )= n t " 1 ) ,theta-Ricker ( f ( n t " 1 )= n & t " 1 ) ,andGompertzmodelsofdensity dependence ( f ( n t " 1 )=ln( n t " 1 )) .ItcanbeseenfromEquation 4Ð2b thatrandom ßuctuationsin # areadditiveto f ( n t " 1 ) whileßuctuationsin B aremultiplicativewith 71

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f ( n t " 1 ) .Forexample,whentheformofdensitydependenceisRicker,additivevariation in # isequivalenttoarandominterceptmodelintheper-capitagrowthrate,whilethe multiplicativevariationin B isequivalenttoarandomslopetermintheper-capitagrowth rate(Figure 4-1 ). Theexpectedvalueoftherandomvariabledescribingthecurrentpopulation abundance, N t ,canbeshowntobe(Appendix G ): E[ N t | N t " 1 = n t " 1 ]= & n t " 1 e " bf ( n t ! 1 )+ " 2 ! / 2+ " 2 B f ( n t ! 1 ) 2 / 2 (4Ð3) % & n t " 1 exp( $ bf ( n t " 1 )). (4Ð4) Thefullpopulationvariance, Var[ N t | N t " 1 = n t " 1 ] isthedemographicvarianceplusthe contributionsoftheenvironmentalcomponentsandisderivedinAppendix G .Weuse thefollowingnotationforconvenience, Var[ N t | N t " 1 = n t " 1 ]=Var dem ( n t " 1 )+Var ! ( n t " 1 )+Var B ( n t " 1 ), (4Ð5) wheretheÞrsttermarisesduetodemographicstochasticityandscalesby n t " 1 .This quantityisgivenby, Var dem ( n t " 1 )= & n t " 1 exp( $ bf ( n t " 1 ))(1 $ exp( $ bf ( n t " 1 )))+ # 2 dem n t " 1 exp( $ 2 bf ( n t " 1 )). Thedemographicvariancehastwopieces,theÞrstcorrespondingtothebinomial variationassociatedwithsurvivalandthesecondcorrespondingtovariationintheto fecundity.Acommonassumptionisthat # 2 dem = & ,correspondingtoaPoissonmodelfor thedemographicvariance.ThesecondterminEquation 4Ð5 isthestandardmodelof environmentalvariationthatarisesduetotemporalßuctuationsin # .Thiscontributionis 72

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additivetotheper-capitagrowthrateandgivesthefollowingpopulationvariance Var ! ( n t " 1 ) % # 2 ! & 2 n 2 t " 1 exp( $ 2 bf ( n t " 1 )). (4Ð6) Finally,thethirdterminEquation 4Ð5 arisesduetoenvironmentalßuctuationsin B , whichyieldaninteractionbetweentheenvironmentalvarianceandthepopulation abundance.Thismultiplicativeenvironmentalvariationintheper-capitagrowthrate givesthefollowingpopulationvariance Var B ( n t " 1 ) % # 2 B & 2 n 2 t " 1 f ( n t " 1 ) 2 exp( $ 2 bf ( n t " 1 )), (4Ð7) whichhastheadditionalscalingof f ( n t " 1 ) 2 comparedtoEquation 4Ð6 .Forthe Rickermodelthiscorrespondstoavariancetermthatscalesby n 4 t " 1 insteadofthe standard n 2 t " 1 scalinggiveninEquation 4Ð6 .Theenvironmentalvariancetermsgiven inEquations 4Ð6 and 4Ð7 areapproximationsandthefullexpressionsfortheseterms aregiveninAppendix G .Theapproximationsprovideabetterintuitionforthescalingof theenvironmentalvariancethanthefullexpressionsanddidnotaffectinferencesforthe datasetstestedinthismanuscript. Inordertocomparethepropertiesoftheenvironmentalvariancemodelswe examinedapopulationwithnodemographicstochasticityandassumedthatatcarrying capacity, K (theÞxedpointofthedeterministicmodel),theenvironmentalvariance inbothmodelsareequivalent,suchthat Var ! ( K )=Var B ( K ) ,leadingtotheequality # 2 B = # 2 ! / f ( K ) 2 .Wecomparethesedifferentenvironmentalvariancemodelsbylooking attheratioofthevariances: Var ! ( n t " 1 ) Var B ( n t " 1 ) = # 2 ! n 2 t " 1 exp( $ 2 bf ( n t " 1 )) # 2 ! f ( n t " 1 ) 2 n 2 t " 1 exp( $ 2 bf ( n t " 1 )) / K 2 (4Ð8) = f ( K ) 2 f ( n t " 1 ) 2 (4Ð9) 73

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Thevarianceofapopulationwithmultiplicativeenvironmentalvariancewillthereforebe lessthanapopulationwithadditiveenvironmentalvarianceforall n t " 1 < K because thisratiowillbegreaterthan 1 .Thus,populationsthatspendmuchoftheirtimebelow K willbestabilizedby Var B ( n t " 1 ) relativetothestandardmodelofenvironmentalvariation, whilepopulationsthatspendmuchtimeabove K ,oroccasionallyreachextremely highpopulationsizescanbedestabilizedbythisformofvariation.Thispotentialfor destabilizationhasbeendescribedinpreviousworkthathassuggestedthatpopulations withmultiplicativeenvironmentalvariationcangoextinctfromlargeßuctuationsin K coupledwithovercompensatorydynamics( Gyllenberg etal. 1994 , H ¬ ogn ¬ as 2000 ).The potentialstabilizingeffectduetomultiplicativeenvironmentalvariationhasnotbeen discussedintheliteratureasfarasweknow. 4.2.2DataandTimeSeriesAnalysis Empiricalanalyseswereconductedusingaone-steptimeseriesanalysis.WeÞt modelstodatausingalog-normaltransitiondistribution.Themeanandvarianceforthe transitiondistributionweregivenbyEquations 4Ð4 and 4Ð5 withthedensitydependence formdependingontheparticulartimeseriesanalysis,describedbelow.Inaddition, ourpreliminaryanalysisfoundthepresenceofamovingaveragecomponentinmany timeseries,thereforewealsoincludedthistermintheanalysesfollowing Ferguson& Ponciano ( 2014 ).WeperformedalloptimizationsintheRstatisticalenvironment( R CoreTeam 2014 )usingtheRgenoudpackage( Mebane&Sekhon 2011 ). Wecurated165highqualitytimeseriesfromtheGPDDtoexaminethegeneral propertiesofenvironmentalvariancescalinginanimalpopulations.Ourcriterionfor theinclusionofatimeseriesintoouranalysiswasthatthedatasethadaminimum of15observations,nozeroabundances,aqualitativereliabilityratingof4or5outof 5,andwedidnotincludedatasetsintheanalysisthatindicatedsamplingwasdueto harvestingasthesemaynotaccuratelyreßectunderlyingpopulationtrends.Foreach timeseriesselectedfromtheGPDDweaÞtGompertzdensitydependencemodels 74

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withdemographicstochasticityandeachoftheenvironmentalvariancemodels,one with Var ! ( n t " 1 ) andtheotherwith Var B ( n t " 1 ) ,aswellasadensityindependentgrowth modelforatotalofthreemodelsperdataset.Allmodelsalsoincludedamovingaverage termtoaccountforobservationerror( Dennis etal. 2006 ).Previousworkhasshown thatdensitydependenceintheGPDDismostlyconcave( Sibly etal. 2005 ),consistent withthetheGompertzform.OutofthethreemodelsÞttoeachtimeseries,thebest wasselectedusingBIC.ThisanalysisoftheGPDDallowedustodeterminewhich environmentalmodel,ifany,isasuitabledefaultwhenbuildingstochasticpopulation models. ForeachofourGPDDtimeserieswecalculatedtheestimatedcarryingcapacity underthedifferentenvironmentalvariancemodels,wherethecarryingcapacity, K ,was givenbytheÞxedpointofthemodelasthevariancegoestozero.Wedenote K ! as theestimatedcarryingcapacityunderamodelwithenvironmentalvariationin # ,and K B astheestimatedcarryingcapacityunderamodelwithenvironmentalvariationin B .Wethenusedtheratioofthesecarryingcapacities, K ! / K B ,totestthetheoretical predictionthatpopulationswithenvironmentalvariationin B shouldspendmoretime belowcarryingcapacitythanpopulationswithenvironmentalvariationin # ( Levins 1969 ).Ifthispredictionistruethenestimatesof K B shouldbehigherthan K ! forthe sametimeseriesandtheratio, K ! / K B willbelessthan1.Ournullhypothesisforthis one-sidedhypothesistestwasthatthenumberofserieswith K B lowerthan K ! was binomiallydistributedwithprobabilityof0.5. Inordertotestthestrengthofregulation(sensu Murdoch ( 1994 ))wesimulated abundancesforallpopulationsthatwereselectedtohavedensitydependence.We initiatedeachsimulationattheÞrstobservedabundanceforeachtimeseriesand projectedfuturepopulationswithgrowth,densitydependence,andenvironmental variationusingestimatedparametersforeachmodel.Simulatedtimeserieswereof length10000andforeachsimulationwecalculatedthecoefÞcientofvariation(CV) 75

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asameasureofthelong-termpopulationstabilitywherehighvaluescorrespond tolessstablepopulations.Wetestedwhetherseriesgeneratedwithmultiplicative environmentalvariationhadlowerCV'sthanseriesgeneratedwiththestandardadditive modelofenvironmentalvariation.Ournullhypothesisforthisone-sidedhypothesistest wasthatthenumberofserieswithalowerCVwasbinomiallydistributedwithprobability of0.5. WeusedtwoadditionalhighqualityabundancetimeseriesoftheAlpineibexand Soaysheepinordertofurtherexploretheseenvironmentalvariancemodels.Previous workontheseungulatepopulationshaslinkedclimatecovariatestotheircarrying capacity;giventhisinteractionwiththeenvironmentwepredictedthatthesepopulations willdisplayenvironmentalvariationin B .IntheAlpineibex,twopreviousstudies ( Jacobson&Provenzale 2004 , Mignatti etal. 2012 )foundthatmeanwintersnowdepth interactedwithabundanceintheper-capitagrowthrate.Thiswasfoundtooccurbothin theoverallpopulationabundancesandinmostofthestageandsexclasses.Weused thesamedatasetas Mignatti etal. ( 2012 )whichwasprovidedbytheauthors.TheSoay sheepdatasetfor1995-2010wasobtainedbydigitizingthetimeseriesplotavailable athttp://soaysheep.biology.ed.ac.uk/population-ecologyusingtheEngaugesoftware ( Mitchell 2010 ).Previousworkby Berryman&Lima ( 2006 )onthisdatahaslinkedthe populationcarryingcapacitytotheseverityofwinterweather,thoughmanyotherstudies havealsoexaminedtheinteractionbetweenthispopulationandtheenvironment(e.g., Coulson etal. 2001 , Grenfell etal. 1992 , Stenseth etal. 2004 ). Intheanalysisofthesehighqualitytimeserieswefollowedsimilarmethodsas previouswork,thoughwedidnotincorporateclimaticcovariatesintoouranalysis.For theAlpineibexweusedtheanalysisof Mignatti etal. ( 2012 )asareference,whilefor theSoaysheepweused Berryman&Lima ( 2006 ).TheAlpineibexfollowedaRicker modelofdensitydependenceforthetotalpopulationandforallstageandsexclasses. Weanalyzedthetotalpopulationabundancesbetween1956-1985,ignoringlaterdates 76

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duetothenonstationaryperturbationoccurringafter1985.Wealsopresentananalysis oftotalabundancesofthefulldataset,1956-2008aswellaseachstageclasstreated independentlyinAppendix H .FortheSoaysheepanalysiswedidnotincludethedata from1985-1994inthemainanalysisaspreviousworkhasfoundthatabreakpoint in1994ledtodifferentpopulationdynamicsbeforeandafterthisyear( Berryman& Lima 2006 ).InAppendix H wepresentananalysisoftheSoaysheepdatasetfrom 1985-2010.Following Berryman&Lima ( 2006 )weusedatheta-Rickermodelofdensity dependence.WeÞtmodelswithallcombinationsoftheenvironmentalanddemographic variancecomponentstoeachofungulatetimeseries,inadditionwetestedwhether includingamovingaveragetermimprovedmodelÞts.WeusedBICtoselectbetween models. Finally,weexaminedhowtheskewnessofmultiplicativeenvironmentalvariation increasedordecreasedthestrengthofregulationthroughsimulation.Usingestimated parametersfromouranalysisoftheAlpineibexpopulationwecalculatedtheexpected Þrstpassagetime,theexpectedtimeittakesforthepopulationtoÞrstreachagiven abundance,forenvironmentalvariationin # andin B .Formodelswithmultiplicative environmentalvariationwealsotestedtheimpactofskewnessonpredictionswhile holdingthemeanandvarianceconstant.Weusedaskew-normaldistributionfor B withthemeanandvariancesetequaltotheestimatedvaluesofthesequantitieswhile wevariedtheshapeparameter, ) ,whichcontrolstheskewnessofthedistribution. Theshapeparameter, ) ,tookonthevaluesof-10,5,0,5,and10.Wecalculatedthe expectedÞrstpassagetimebyinitiatingpopulationsatthecarryingcapacityof3309 individuals,thensimulating10000timeseries,eachwith10000timesteps,undereach variancemodelandforallvaluesofskewness.ThedistributionofÞrstpassagetimes wasthencalculatedforeachofthetimeseriesatallabundancesbelowtheinitialvalue. 77

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4.3Results&Discussion Wefoundthat28%oftheGPDDtimeserieshadminimumBICvaluesforthe densityindependencemodel.Althoughweexpectthatallpopulationsdisplaydensity dependenceoversomerangeofabundances,testsfordensitydependenceovera limitedrangeofabundancesmaynotdetectit.Thiscanbeduetotheinßuencesof delayeddensitydependence,convexpopulationgrowthrates,andsomeformsof measurementerror.Outofthetimeseriesthatwereselectedasdensitydependent, 52%werebetterexplainedwithenvironmentalvariationin B ratherthanin # (Figure 4-2 ).Althoughthemajorityofthedensitydependenttimeserieswereselectedtohave environmentalvariationin B ,manyoftheselectionswerenotstronglydistinguishable with61%ofthe ! BICvalues < 2 .Theseresultsarenottoosurprisinggiventhe difÞcultypastworkhashadinstatisticallydetectingdensitydependenceintheGPDD ( Knape&deValpine 2012 )andindistinguishingbetweenthesevariancemodels ( Mutshinda&O'Hara 2010 ).DespitethesestatisticaldifÞcultiestherewasevidence supportingthepresenceofhigherordervariancescalinginmanyofthesetimeseries, thoughneithervariancemodelappearstobeanappropriatedefaultmodelforall scenarios. Inourtestofthetimespentbelowcarryingcapacitywefoundthatthat63%ofthe ratioswerebelow1(p-value= 0.004 ,n=117)(Figure 4-3 ).Thehigher K B valuesindicate thatpopulationswithmultiplicativeenvironmentalvariationtendtospendmoretime belowcarryingcapacitythanpopulationssubjectedtoadditiveenvironmentalvariation. Thus,populationsarestabilizedbythemultiplicativeenvironmentalvarianceastheyare spendingmoretimeinstateswithlessvariationthanpredictedbythestandard,additive modelofenvironmentalvariation.Thiseffectisoftennotlarge,withmostdifferences within5%ofeachother(Figure 4-3 ). InourtestofstabilitymeasuredbytheCV,wefoundthat72%ofthepopulations generatedwithenvironmentalvariationin B hadthelowerCV(Figure 4-4 ).Therefore, 78

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populationswithvariationin B seemtobemorestronglyregulatedthanpopulations withvariationin # (p-value= 1.1 á 10 " 6 ,n=117).However,wealsofoundthatseveralof thepopulationsgeneratedwithenvironmentalvariationin B hadsigniÞcantlyhigher CV'sthanpopulationsgeneratedwithenvironmentalvariationin # .Furtherexamination ofthesetimeseriesshowedthattheydisplayeddynamicsqualitativelysimilartothe growth-catastrophedynamicsdescribedinpreviouswork.Thissuggeststhatitis possibletoachievethepredictedgrowth-catastrophedynamics,thoughtheymaybea rarephenomenon. OuranalysisoftheAlpineibexandSoaysheeptimeseriesindicatedthatboth populationswerebettermodeledwithenvironmentalvariationin B ratherthanthe standardmodelofenvironmentalvariationin # (Table 4-1 ).Thissupportsourpredictions aboutthescalingoftheenvironmentalvarianceinthesepopulations,basedonprevious worklinkingßuctuationsinthesepopulationstoclimaticvariables.Furthermore, itindicatesthatthesebasicvariancemodelscanbeusedasadiagnostictoolto determinethekindsofinteractionswiththeenvironmentthatresearchersshould lookforwhentryingtolinkpopulationgrowthtoenvironmentalfactors. TheestimatedtotalvarianceintheibexandsheeppopulationsisgiveninFigure 4-5 .Thevariancebehavesasexpectedfromthelinearizationanalysis,withthetotal variationdueto B beinglessthanthetotalvariationdueto # ,when n t " 1 isbelow K . Theeffectleadstomuchlowervarianceinthemultiplicativemodelthanwouldbe predictedbytheadditiveenvironmentalvariancemodelforasigniÞcantproportionofthe observations. InthesimulationofÞrstpassagetimes,wefoundthataccountingforvariationin B ,ratherthanin # ,ledtoslightlymorestronglyregulateddynamicsconsistentwithour prediction(Figure 4-6 ).Inaddition,theÞrstpassagetimesimulationsshowthatthe effectsofskewin B canbestrengthenorweakenregulationdependingonthetailof thedistribution.Whenthedistributionof $ B wasskew-right,correspondingtoasmall 79

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numberenvironmentsthatwereveryhighlyfavorablebutwithalowerqualitymedian environment,themeanÞrstpassagetimeafterabout10yearswaslowerrelativetothe symmetricmodel.Overshorttimescales,howeverthesedynamicsaremorestable duethelowprobabilityofexponential-likegrowthoccurring(Figure 4-6 ).Theinstability atlongertimescalesisduetothepotentialforraregoodyearstopushpopulations intohighvariancestates,whilethestabilityovershorttimescalesarisesduetothelow variationassociatedwiththemedianstate.Whenthedistributionof $ B wasskew-left, correspondingtoasmallnumberofyearswithverypoorconditionsbutahighermedian qualityenvironment,themeanÞrstpassagetimewasincreasedatlongertimescales butslightlydecreasedovershortertimescalesrelativetothesymmetricmodel(Figure 4-6 ).Thisinstabilityatshorttimescalesisduetothemedianpopulationbeinghigher thaninthesymmetricenvironment,leadingtomoretimespentinitiallyinahigher variationenvironment.However,atlongertimescalesthelikelihoodofbeinginaneven highervariationenvironmentislowleadingtolongtermstability. Theseresultshighlightstheimportanceofrareeventsonlong-termpopulation regulationandtheimportanceofusingthecorrectdistributionwhenassessingriskin populationviabilityanalyses.OverlongtimescalesthebeneÞcial/detrimentaleffects ofextremeenvironmentstendedtobeinßuencedbythetailsofthedistributions.Over shortertimescales,however,thesetailshadlessinßuenceandthedynamicswere morestronglycontrolledbythemedianpopulationdynamics.Ourresultsalsoshow thatthepredictionsofthemeanÞrstpassagetimesbegandivergingataround10 yearssuggestingthatrelativelysmalldifferencesinmodelformulationleadtodiverging predictionsoveronlymoderatelylongtimescales.Thesesimulationsalsohighlight thedifÞcultiesinparameterizingthesemodelsadequatelyaslongtermtimeseries willbenecessarytoadequatelyassesslong-termdynamics.Onepotentialstrategy fordetermingtheshapeofthedistributionofenvironmentaleffectsistoexamine 80

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thedistributionoftheenvironmentalcovariatesthoughttodriveßuctuationsinthe populationsandusedistributionsconsistentwiththesehypothesizedeffects. Wehavetreatedthesevariancemodelsasarisingfromparticularecological mechanisms.Theadditionofcomplexitynotaccountedforinourmodelscanlead todifferentpatternsoftheresidualsthatmayalsobecapturedbythesevariance models.Forinstance,theinclusionofhigherorderlagsinthedensitydependence throughagestructureorinterspeciÞcinteractionsmayfavorthetraditionalmodel ofenvironmentalvarianceasadditivetermswillwillscalebytheabundanceinthe per-capitagrowthrateandwillthereforefollowquadraticscaling.Thesameislikely truewhentheunderlyingformofdensitydependenceismisidentiÞedasthiswillleadto residualswithouthigherorderscalingsintheabundance.However,inmostpopulations bothformsofenvironmentalvariationarelikelytooccurandtheroleofstatistical inferencewillbetodeterminewhichprocessesdrivemostofthevariation. Despitethesimplicityofthesepopulationmodels,theprinciplesunderlyingthem arequitegeneral.Fluctuatingenvironmentalresourceswilloftenleadtoßuctuating carryingcapacities,therefore,higherordervariancescalingshouldbeconsideredwhen performingpopulationviabilityassessments.Ouranalysishassupportedthisintuitive claimandshownthathigherorderscalingcanhaveconsequencesonpopulation predictions.Multiplicativeenvironmentalvariationcanpromotestabilityinpopulationsby allowingthemtospendmoretimeinstatesassociatedwithlowerpopulationvariance thanpredictedbythestandardmodelofenvironmentalvariability,thoughdynamicscan alsobedestabilizedwiththismodelthroughpreviouslydescribedgrowthcatastrophe dynamics( Fagerholm&H ¬ ogn ¬ as 2002 , Gyllenberg etal. 1994 , H ¬ ogn ¬ as 2000 ).However, ourresultssuggestthatmultiplicativeenvironmentalvariancewillmostoftenstabilize dynamicsandthatgrowth-catastrophedynamicsarerare. Finally,ourempiricalresultsindicatedthatthesesimplevariancemodelscan serveasreliableindicatorsofhowpopulationsinteractwiththeirenvironment.This 81

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Table4-1.BICvaluesformodelselectionsofSoaysheepandAlpineibexdatasets.Bold valuesaretheminimumforeachdataset. PopulationYears Autocorrelation Var ! ( n t " 1 )Var B ( n t " 1 )Var ! ( n t " 1 )+Var B ( n t " 1 ) Alpineibex1956-1985MA(0) 429.14 425.14 428.54 Alpineibex1956-1985MA(1) 431.82437.54 436.64 Soaysheep1995-2010MA(0) 234.57 230.56 234.14 Soaysheep1995-2010MA(1) 235.90233.88 235.51 usefulresultmeansthathypothesesaboutthenatureofthepopulation-environment interactioncanbetestedwithoutspeciÞcknowledgeaboutthestateoftheenvironment. Understandingtheseinteractionsmaybeespeciallyimportantindetermingthe consequencesthatenvironmentalshiftshaveonpopulationdynamics.Forexample, previousworkinAlpineibexshowedthattheinteractionbetweensnowdepthand populationabundancesarecrucialfordetermingthefuturepopulationabundances Lima &Berryman ( 2006 ).Thisworkwasabletolinkanonstationaryshiftinibexpopulations inthelate1980'stochangesintheaveragesnowdepth,aphenomenoncouldnotbe accountedforbyanadditiveenvironmentalvariancemodel.Distinguishingbetween additiveandmultiplicativevariancemodelsmayleadtodifferentmanagementand conservationstrategiesinthefaceofclimaticchange,wherenonstationarychanges inclimatemayleadtodifferentoutcomesdependingontheimpactofclimateonthe populationecology. 82

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Figure4-1.Theimpactofenvironmentalvariationintheper-capitagrowthrateofthe Rickermodel.Additivevariationin # (panelA)isarandominterceptinthe per-capitagrowthratewhiletheimpactofmultiplicativevariationin B (panel B)isarandomslopeterm. 83

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Figure4-2.BICselectionsofthebestenvironmentalvariancemodelsintheGPDDtime seriesthatdisplayeddensitydependence.Additiveenvironmentalvariance correspondstoenvironmentalvariationin # ,whilemultiplicative environmentalvariancecorrespondstoenvironmentalvariationin B .The blackregioncorrespondstothenumberofselectionsmadewith ! BIC < 2 , whiletheblueregioncorrespondstothenumberofselectionsmadewith ! BIC & 2 . 84

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Figure4-3.Thelog-ratioofthecarryingcapacitiesestimatedunderbothenvironmental variancemodels, K ! / K B ,forallGPDDtimeseriesdetectedtohavedensity dependence.Log-ratio'sbelow0indicateahigherestimated K B and thereforemoretimespentbelowcarryingcapacityunderthemodelwith environmentalvariationin B . 85

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Figure4-4.ThecoefÞcientofvariation(CV)undermultiplicativeandadditive environmentalvariationforeachdatasetwithdensitydependence. RelativelyhighCV'sforthemultiplicativemodelscorrespondto growth-catastrophedynamics. 86

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Figure4-5.PredictedpopulationvariancefortheAlpineibexandSoaysheepunder alternativeenvironmentalvariancemodels.Theblacklinecorrespondsto additiveenvironmentalvariationthatoccursin, # ,whiletheredline correspondstomultiplicativeenvironmentalvariationthatoccursin B . Horizontaldashesonthex-axisdenotetheobservedabundanceswhilethe horizontaldashedlinesrepresenttheestimatedcarryingcapacitiesunder thedifferentvariancemodels.Thevariancebelow K ismuchlessforthe multiplicativeenvironmentalvariationmodel,thoughmuchhigherathigh abundances. 87

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Figure4-6.TheeffectofdistributionalskewontheÞrstpassagetime.Leftpanel,the meanÞrstpassagetimeastheskewnessof B ischanged.Thedashedline correspondstothestandardmodelofenvironmentalvariationin & ,whilethe solidlinescorrespondtoskew-normaldistributionfor B (withshape parameter ) ).Rightpanel,thecorrespondingprobabilitydistribution functionsfor B ,notethatthex-axis, $ b ,isproportionaltothecarrying capacity. 88

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CHAPTER5 CONCLUSIONS Theroleofsingle-speciesmethodsformanagementandconservationiswell-established. However,alackoftheorymotivatingthesemethodsincomplexsystemshasledtosome confusionregardingwhensingle-speciesmodelpredictionsmightfail.Callsforholistic approaches,suchasecosystem-basedmanagement,oftenimplicitlyassumethat populationswillhavestrongnonlinearresponsestoßuctuatingenvironmentalconditions andthataccountingfortheseresponseswillimprovepredictivepower.Inreality,most managedpopulationsappeartobeadequatelydescribedusingsingle-speciesmethods, anempiricalrealitysupportedbythetheoreticalworkinthisdissertation.Thereare certainquestionsthatsingle-speciesmethodsmaynotbeabletoadequatelyaddress suchastryingtopredicttheimpactoftheremovalorintroductionofonespecieson another.However,giventhecomplexityofpopulationresponsestoenvironmental changes,itislikelythatmodel-basedapproacheswilloftenhavedifÞcultiesinpredicting theseresponses.Thisworkhasbeguntodescribewhysingle-speciesmethodshave beensorobustincertainapplications. Chapter2examinedthereasonsweexpecttheautocorrelationpresentin environmentalcovariatesnottoaffectextinctionpredictions,asmanytheoretical modelshaveproposed.Giventhepotentialimportancethattheoreticalstudieshave placedonautocorrelatedenvironments,empiricalworkvalidatingthesepredictionshas laggedfarbehind.Ourworkpaintsamorenuancespictureoftheroleofenvironmental autocorrelationonpopulationpredictions,onethatsuggestsuncorrelatedenvironmental processesmaybesufÞcienttodescribemanypopulations.Onepotentiallyimportant resultisthatpopulationsontheedgeoftheirecologicalnichecanoftenbeexpected tohavestrongerresponsestoenvironmentalautocorrelationthanthoseintheoptimal niche,thisisduetothecurvatureoftheresponseofthepopulationgrowthtochangesin theenvironment. 89

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Chapter3builtonChapter2byconsideringhowinterspeciÞcinteractionscandrive variationinpopulations.Inthecaseofbioticinteractionsweshowedthatasimplelinear approximationtothepredator-preydynamicscanyieldausefulsingle-speciesmodel. Thismeansthatitispossibletoaccountfortheimpactsofthepredator-preydynamics, evenwhenweonlyobserveoneofthespecies.Wecombinedthismodelderivationwith atestusingexperimentaldata,showingthatwecanincorporatesomeoftheeffects ofinterspeciÞcinteractionsintosingle-speciesmodels.Combined,Chapters2and3 describetheimpactofcomplexabioticandbioticeffectsonpopulationdynamicsandthe ecologicalscopeofsingle-speciesmodels. Chapter4revisitedaclassicalproblemonthescalingofenvironmentalvariationin animalpopulations.Wewereabletoshowhowthevariationinteractswithpopulation parametersaffectsthisscaling.Besidesoverturningthecurrentdogma,thishasseveral practicalimplications.TheÞrstisthathigherorderscalingintheenvironmentalvariance canactuallyleadtomorestabledynamicsthanotherwisepredicted.Thesecondis thatthescalingoftheenvironmentalvariationcanprovideimportantcluesabouthow thepopulationisaffectedbyenvironmentalßuctuations,withoutdirectobservationson thisinteraction.Webelievethatdetermininghowtheenvironmentdrivespopulation ßuctuationstobeanimportantstepinunderstandinghowchangesintheenvironment affectsspeciesecology. DeÞningthescopeofanecologicalsystemhasneverbeensimpleandalthough disciplineshaveemergedstudyingdifferentlevelsofecologicalorganizationpotentially importantinßuencesoccuracrosstheselevels.Thisdissertationhasdemonstratedthat simplemodelscanbeusedtoinfertheecologicalmechanismsdrivingßuctuationsin animalpopulations,evenwithoutobservingthespeciÞcdetailsoftheseenvironmental interactions.Thishasimportantpracticalimplications,suggestingthatcomplex ecosystemmodelsareoftennotnecessarytoaccuratelypredictpopulationßuctuations. 90

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Thisbodyofworkcanserveasausefulroadmapforecologistsseekingtounderstand howtounderstandhowdealwiththefuzzyboundariesofecologicalsystems. 91

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APPENDIXA CURATINGTHEGPDD Wechoseahighqualitysubsetoftheover5000GPDDdatasetsusingthefollowing criteria:theremusthavebeenatleast15continuousobservationsinthetimeseries, thequalitativeGPDDreliabilityratingmusthavebeen4or5(withabestpossible ratingof5),andthedatamustnothavebeenconstantovertheÞrstÞveyears.In addition,weonlyalloweddatasetswheresamplingunitsindicatednonharvestindices, asharvestsmaynotreßecttruepopulationabundances.Alldatawastransformedby adding1toallobservations,inordertoremoveany 0 's.Wetestedtheeffectsofthis datatransformationbyanalyzingasubsetofthe389datatsetswherethistransformation wasnotnecessary,butfewerdatasets(161)wereavailableforthisanalysis(Table 2-2 ). 92

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APPENDIXB PARAMETERESTIMATIONANDMODELSELECTION Parameterestimation Weusedone-steppredictionstoÞtpopulationdynamicsmodelswithPEAusing abundancedatafromtheGPDDsimilartotheprocedurereportedin Ferguson& Ponciano ( 2014 ).Parameterestimationwasperformedusingbothmaximumlikelihood (ML)andrestrictedmaximumlikelihood(ReML)methods.AICcwasusedtoperform modelselectionamongthecandidatemodelsusingtheML.However,ourreported valuesof ö ! usedReMLparameterestimatesfromthebestAICcmodels.ReML estimationhasbeenshowntoperformbetterthanMLwhenestimatingvariance components,butReMLisnotvalidformodelselectionusinginformationcriterion Staples etal. ( 2004 ).MLandReMLestimationswereperformedusingthegeneralized leastsquares(gls)functionfromtheNLMEpackageintheRstatisticalenvironment Pinheiro etal. ( 2011 ), RDevelopmentCoreTeam ( 2012 ). Weassumedamultivariatenormaldistributionfortheobservedpopulationgrowth rate, ln & N ( t ) N ( t " 1) ' ,wherethemeanvectorwasgivenbythepredictedpopulationgrowth rateforthecorrespondingformofdensitydependencegiveninEquation 2Ð4 or 2Ð5 . Thecovariancematrixwasgivenby # 2 R ,wherethecorrelationmatrix, R ,wasgivenby thecorrelationstructurefortheARMA(1,1)model.For k observationsofthepopulation growthrate,theARMA(1,1)correlationstructurewithAR(1)parameter ! andMA(1) parameter $ isgivenby, R = (1+ $! )( $ + ! ) 1+2 $! + $ 2 ( ) ) ) ) ) ) ) * ! " 1 ! 0 ... ! k " 3 ! 0 ! " 1 ... ! k " 4 . . . . . . . . . . . . ! k " 3 ! k " 4 ... ! " 1 + , , , , , , , . (BÐ1) TheAR(1)modelcorrespondstoaspecialcasewhere $ =0 .Forexample,the log-likelihoodfortheGompertzlag1densitydependencemodelwith k observationsis 93

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thengivenby, ln( L )= $ k ln(2 * ) 2 $ k ln | # 2 R | 2 $ 1 2 # 2 . ln . N ( t ) N ( t $ 1) / $ a + b ln( N ( t $ 1)) / T R " 1 . ln . N ( t ) N ( t $ 1) / $ a + b ln ( N ( t $ 1) ) / . (BÐ2) Thevector, N ( t ) correspondstothe k $ 1 abundancesrepresentingthecurrentyears observationsandnotincludingtheinitialvalue,and N ( t $ 1) correspondstothe k $ 1 vectorofpreviousyearsabundances. Modelselection ForthisstudywewerefocusedonestimatingthePEAparameter, ! ,intheAR(1) modelofenvironmentalvariationgiveninEquation BÐ1 .Wefollowedamodelstructural adequacyprocedure(MSA) Taper etal. ( 2008 )todeterminetheestimabilityofthe PEAwiththeGPDDdataset.TheMSAprocedureisaformalnameforanoftenused approachinstatisticalanalysisthatusessimulationtodeterminetheabilityofan estimatorormethodtoprovidemeaningfulanswerstoscientiÞcquestions.Itconsists ofsimulatingdatafromcomplexprocessmodelsrepresentingreality,thenÞttingaset ofsuitablecandidatemodelstothegenerateddataandassessinghowwelltheÞtted modelsrecapitulatethecomplexprocessmodel.Thisprocedureisoftenusedincases wherecomplexitymakesamoreformalanalysisofmethodsdifÞcult.Weconductedtwo MSA'sinordertodeteriminethemostreliablemethodofestimating ! . Biasanduncertaintyin ö ! mayarisethroughlowsamplesize,modelmisspeciÞcation, andthroughthemodelselectionprocedure.IntheÞrstMSAwetestedtheimpactof selectingtheproperformofdensitydependencefromthemodelsetandtheimpact ofignoringhighermodellags.Thismodelsetcontaineddensity-independentaswell aslag-1RickerandGompertzdensitydependentmodels,alongwithanARMA(1,1) autocorrelationstructure.InthesecondMSAwetestedtheimpactofperformingmodel selectiononthenumberoftimelagsinthedensitydependenceandtheimpactof 94

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ignoringmeasurementerror,weassumedthattheformofdensitydependencewas knownforthisprocedure. WebegantheMSA'sbyÞttingmodelstoeachoftheGPDDtimeseriesincludedin ourstudyinordertoobtainparametersforgeneratingnew,realisticdatasets.ThisÞtting procedure(describedintheprevioussection)wasperformedfordensityindependent modelsanddensitydependentmodelwithlags1through3,withbothGompertzand Rickermodelsofdensitydependence.AnARMA(1,1)errorstructurewasusedforall models.WethengeneratedanewdatasetusingeachoftheÞttedmodel'sparameters excepttheAR(1)component, ! ,wassettooneofthevalues ( $ 0.3,0.0,0.3,0.9) . Thesegenerateddatahadthesameparameters,samplesize,andwereinitializedatthe sameinitialpopulationabundanceastheoriginaldataset. FortheÞrstMSAweÞtasetofcandidatemodelsthatincludeddensityindependence andlag1RickerandGompertzdensitydependence,withanARMA(1,1)errorstructure usingbothReMLandMLestimationoneachofthegenerateddatasets.Weusedthe MLestimationproceduretodeterminetheAICcminimummodel,thencompiledReML estimatesof ! foreachgenerateddatasetasafunctionofboththegeneratingmodellag andthetrue ! value.Wethenexaminedestimatorbiasandvariancebycomparingthe knownvalueof ! to ö ! . InoursecondMSAweÞtasetofcandidatemodelswithdensityindependenceand lag1to3RickerandGompertzdensitydependencemodels.Weperfomedestimation usingbothanAR(1)errortermandanARMA(1,1)errorterm.ForthisMSA,allÞtted modelswereassumedtohavethesameformofdensitydependenceasthegenerating model,eitherGompertzorRickerfunctions.Ineachmodelsetweperformedmodel selectionusingtheAICcthenthecompiledReMLestimatesof ! forthebestÞtmodel asafunctionofthegeneratingmodellagandtrue ! value.Weexaminedestimatorbias andvariancebycomparingtheknownvalueof ! to ö ! . 95

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Results TheÞrstMSAtestedtheabilitytoestimate ! whenselectingbetweendensity indepedenceandtwoformsoflag1densitydependencewhenthetruemodelmay haveahigherdimensionality.Ourresultsshowthat ö ! dependedonboththemodellag structureandthetruevalueof ! ,withthebiasincreasingathighermagnitudesof ! andasafunctionofthetrueunderlyingmodellag(FigureS1).Overall,thisapproach appearedtopresentreliableestimatesofthePEAforthemoderatelevelsof ! that wouldbeexpectedinrealpopulations(FigureS1). InoursecondMSAbiaswaspositiveforlowandmoderatePEAlevels,suggesting thatwemayoverestimatetheautocorrelationusingthisapproach(FigureS2a,S2c). Lessbiaswaspresentin ö ! atlowtomoderatelevelsof ! whenusingtheARMA(1,1) modelcomparedtotheAR(1)model.However,whenhighPEAwaspresentthe AR(1)modeltendedtoperformbetterthantheARMA(1,1)model.ForbothAR(1)and ARMA(1,1)modelsthevariancein ö ! tendedtoincreasewithincreasedlagsinthe generatingmodelandwiththemagnitudeofPEA, ! (FigS2b,S2d). 96

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FigureB-1.Biasandvariancein ö ! fordatageneratedfromagivenlagwiththe Gompertzmodelwhenperformingselectionbetweendensityindependence, andGompertzandRickerlag1modelsandwithanARMA(1,1)error structure. a) Thebiaspresentin ö ! ,and b) thevariancein ö ! .Similarresults areobtainedwhensimulatingfromtheRickermodel. 97

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FigureB-2.Biasandvariancein ö ! fordatageneratedfromagivenlagwiththe GompertzmodelandwithanARMA(1,1)errorstructure. a) Thebiaspresent whenÞttingonlytheAR(1)modeland, b) thecorrespondingvarianceinthe estimatorof ! . c) ThebiaspresentwhenÞttingtheARMA(1,1)model,and b) thevariancein ö ! .ResultsfromthesimulatedRickermodelareconsistent withtheseresults. 98

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APPENDIXC GPDDCOVARIATEANALYSIS Wetestedtheexplanatorypowerofasuiteofcovariates,availableintheGPDD, on ö ! .Weexploredwhetherenvironmentalfactors,speciesinformation,anddata qualitymetricscouldexplainvariationin ö ! .SpeciÞcally,wetestedwhethertimeseries length,latitudeandlongitudeofthedatacollectionlocation,GPDDreliabilityindex,and samplingfrequencyweresigniÞcantpredictorsof ö ! atthe ) =0.10 level.Thearctanh transformationwasperformedon ö ! ,deÞnedby Z =arctan( ö ! ) ,inordertosatisfythe assumptionofnormallydistributedresiduals.Werananadditionalregressionlookingat speciestaxaaspredictorsof ö ! atthe ) =0.10 level.Wealsotestedwhetherspecies statusasspecialistsorgeneralistscouldexplainvariationin ö ! .WeuseddeÞnitionsfrom Murdoch etal. ( 2002 )todeterminespecialist/generaliststatusin48timeoftheGPDD series.Wepredictedthatgeneralists,withpotentiallymoretrophicinteractionswould havelowerlevelsofPEAthanspecialistswhomayhavefewer,butstronger,trophic interactionsconsistentwithourresultsfromtheSLLMmodelinthemaintext. Results Ourexploratoryregressionanalysiswasusedtodeterminewhethercovariates couldexplainvariationin ö ! .Weperformedatestonallsamplingunitsreportedinthe GPDD.WefoundthatonlytheBreedingfemalescategorywasastatisticallysigniÞcantly predictorof ö ! (t-test,p-value=0.0293).Thetwospeciesinthiscategorywere Tringa nebularia and Melospizamelodia .ItisunclearwhythesebirdsshouldhavesigniÞcantly highervaluesof ö ! thantheotherpopulationsexploredhere,thoughitmayhaveto dowithbreedingstatusbeingmorestronglycoupledtoenvironmentalconditions thanotherabundancemeasures.WealsoexaminedwhethersamplesizeandGPDD reliabilityindexinßuencedestimates.Wefoundthat ö ! tendedtodecreasewithincreases insamplesizeandthereliabilityindex,thoughtheseeffectswerenotstatistically signiÞcant. 99

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Statusasspecialist( ö ! =0.11 )orgeneralist( ö ! =0.03 )wasnotfoundtobea signiÞcantexplanatoryvalueof ö ! atthe ) =0.1 level(p-value=0.44).Howeverthe resultswereconsistentwithourpredictionswiththespecialisthavingslightlyhigher meanlevelsofautocorrelation,aspredictedbytheSLLMmodel.Giventhelowsample size(n=48)usedtotestthishypothesisitmaybeworthrevisitinginthefuturewithmore samples. 100

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APPENDIXD DERIVATIONOFTHEABUNDANCEMEANANDVARIANCE Herewederivethemeanandvarianceofpopulationabundancesusingastochastic processapproachsimilartopreviousworkby Engen etal. ( 1998 )and Melbourne& Hastings ( 2008 ).Lettheoffspringforindividual i inapopulationbegivenby X i .We assumethat E [ X i ]= & and V [ X i ]= ! ( & ) 2 .Thenumberofindividualsbornforthenext timestepofapopulationwithabundance N t " 1 isgivenby Y t = N t ! 1 ! i =1 X i ,wherethe X i 's areidenticallyandindependentlydistributed.Itfollowsthat E [ Y t | N t " 1 = n t " 1 ]= & n t " 1 and V [ Y t | N t " 1 ]= n t " 1 ! ( & ) 2 .Thesurvivalofoffspringisgivenby p t = p 0 p ( n t ) ,where p 0 isthedensity-independentcomponentofsurvivaland p ( n t ) isthedensity-dependent componentofsurvival.Thenumberofsurvivorsoutofthe Y t bornisgivenby N t andwe assumeabinomialdistributionofsurvival, ( N t | N t " 1 = n t " 1 , Y t ) ) BIN( Y t , p t " 1 ) .This variationinbirthsandsurvivalisthedemographicvariability. Wecanusethepropertiesofexpectedvaluestoaverageovertheunobserved randomvariable Y t togetthenextgenerationsabundances, E [ N t | N t " 1 = n t " 1 ]= E [ E [ N t | N t " 1 = n t " 1 , Y t ] ] = E [ Y t p t " 1 ] = & n t " 1 p t " 1 . (DÐ1) Similarly,forthevariance, V [ N t | N t " 1 = n t " 1 ]= E [ V [ N t | N t " 1 = n t " 1 , Y t ]]+ V [ E [ N t | N t " 1 = n t " 1 , Y t ]] = E [ Y t p t " 1 (1 $ p t " 1 )]+ V [ Y t p t " 1 ] = E [ Y t ] p t " 1 (1 $ p t " 1 )+ p 2 t " 1 V [ Y t ] = & n t " 1 p t " 1 (1 $ p t " 1 )+ p 2 t " 1 n t " 1 ! ( & ) 2 = & n t " 1 p t " 1 + n t " 1 p 2 t " 1 ( ! ( & ) 2 $ & ). (DÐ2) 101

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Thedensitydependencecanalsobeplacedinthefecundityterm,withsurvivalasa constant,leadingtoasimilarformforthemeanandvarianceterms.Itisalsoimportant tonotethattheterms & and p 0 arenotidentiÞableunderthePoissonassumptionfor demographicstochasticity, ! ( & ) 2 = & .UnderthePoissonassumption V [ N t | N t " 1 = n t " 1 ]= & n t " 1 p t " 1 = & p 0 n t " 1 p ( n t " 1 ) .Therefore,weexploredwhethertheparameter p 0 wasidentiÞableinourdatawhenthePoissonassumptionisrelaxedbyÞttingthe joint-likelihoodproÞleof p 0 and & fortheÞrstmicrocosmexperimentaldataset(the likelihoodsarefullydescribedinAppendixC).Asinthemainmanuscriptweletthe demographicstochasticityinreproductionbeaconstant, ! ( & ) 2 = ! 2 .Wefoundthat & and p 0 werecollinear(Figure D-1 )showingthatitisnotpossibletojointlyestimateboth parametersfromabundancedata.Thereforewefollow Melbourne&Hastings ( 2008 ) andtreat & p 0 asasingleparameter r .Thisleadstothefollowingapproximationfor thedemographicvariancewhichignoresthedensityindependentcontributioninthe binomialvarianceterm: V [ N t | N t " 1 = n t " 1 ] % rn t " 1 p ( n t " 1 )+ n t " 1 p ( n t " 1 ) 2 ( ! ( & ) 2 $ r ). (DÐ3) Thisapproximationcanleadtoestimatesthatpredicthigherorlowervariancethanthe truevaluedependingontheabundanceandtheunderlyingparametervalues,however, simulationsindicatethattheapproximationisreasonableoverarangeofvaluesfor p 0 . Weincludedrandomenvironmentalcontributionsinreproductionbyallowing r tovaryfromyeartoyear.ThisisdonebydeÞning W t tobethedistributionofthe population'sbirthrate,wherethebirthratecannowvaryfromyeartoyearwithamean & andvariance ' 2 .Then E [ X i | N t " 1 = n t " 1 , W t = w t ]= w t and V [ X i | N t " 1 = n t " 1 , W t = w t ]= ! ( w t ) 2 ,while E [ Y i | N t " 1 = n t " 1 , W t = w t ]= w t n t " 1 and V [ Y i | N t " 1 = n t " 1 , W t = w t ]= n t " 1 ! ( w t ) 2 .Asabove,assumingthatsurvivalisgivenbyaconditionalbinomial distribution, ( N t | N t " 1 = n t " 1 , Y t , W t = w t ) ) BIN( Y t , p t " 1 ) ,nowthemeanandvariance 102

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FigureD-1.Contourlikelihoodofthemean(Equation DÐ1 )andvariance(Equation DÐ2 ) assumingalognormaltransitiondistribution.Themaximumlikelihood estimateislabeledwiththeredpoint.ThelackofawelldeÞnedpeak indicatesthattheparameters r and p 0 arecollinearandarenot independentlyidentiÞable. conditionalonthestateoftheenvironment, W t ,are: E [ N t | N t " 1 = n t " 1 , W t = w t ]= w t n t " 1 p ( n t " 1 ) V [ N t | N t " 1 = n t " 1 , W t = w t ]= w t " p ( n t " 1 )(1 $ p ( n t " 1 ))+ n t " 1 p ( n t " 1 ) 2 ! ( w t ) 2 # . 103

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Finally,averagingovertheenvironmentalstate W t givesamodelwithbothdemographic andenvironmentalstochasticity.Themeanisgivenby E [ N t | N t " 1 = n t " 1 ]= E [ E [ N t | N t " 1 , W t ]] = E [ W t ] n t " 1 p t " 1 = rn t " 1 p ( n t " 1 ), andthevarianceby V [ N t | N t " 1 = n t " 1 ]= E [ V [ N t | N t " 1 , W t ]]+ V [ E [ N t | N t " 1 , W t ]] = n t " 1 E [ W t ] p t " 1 (1 $ p t " 1 )+ n t " 1 E [ ! ( W t ) 2 ] p 2 t " 1 + n 2 t " 1 p 2 t " 1 V [ W t ] % n t " 1 " rp ( n t " 1 )(1 $ p ( n t " 1 ))+ p ( n t " 1 ) 2 E [ ! ( W t ) 2 ] # + n 2 t " 1 p ( n t " 1 ) 2 ' 2 , (DÐ4) whereweusedtheapproximationforthedemographictermdescribedabove. Forestimationpurposesinthemanuscript,weassumedthatthedemographic varianceinreproductionisproportionaltotheyearlyrealizedreproductivevalue, V [ X i | W t = w t ]= * w t .Fortheterm E [ ! ( W t ) 2 ] inEquation DÐ4 ,wecanthenplugin *& . Thevarianceisthengivenby, V [ N t | N t " 1 = n t " 1 ]= n t " 1 E [ W t ] p t " 1 (1 $ p t " 1 )+ n t " 1 E [ ! ( W t ) 2 ] p 2 t " 1 + n 2 t " 1 p 2 t " 1 V [ W t ] % n t " 1 rp ( n t " 1 )(1 $ p ( n t " 1 ))+ n t " 1 * rp ( n t " 1 ) 2 + n 2 t " 1 p ( n t " 1 ) 2 ' 2 , wherewehaveusedtheapproximationduetothenonidentiÞabilityofthedensity independentsurvivalthatwasusedabove.Additionally, * and r inthedemographic stochasticityarenon-identiÞable.ThereforewedeÞnetheterm, ! 2 ( r * .Wethenget theformofthevarianceusedinthemanuscript, V [ N t | N t " 1 = n t " 1 ]= n t " 1 " rp ( n t " 1 )(1 $ p ( n t " 1 ))+ p ( n t " 1 ) 2 ! 2 # + ' 2 n 2 t " 1 p ( n t " 1 ) 2 . (DÐ5) 104

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APPENDIXE ARMAMODELDERIVATION Thedynamicsofasinglespeciesinalinearinteractingsystemcanbere-expressed throughacombinationoflaggedobservationsofthepopulationabundanceand laggederrorterms( Royama 1981 ).Hereweshowhowtoobtainthesedynamics inatwo-speciessystembysubstitutingthedynamicsofonespeciesintotheother. Beginningwiththestochasticdiscrete-timeGompertzmodel, N ( t )= N ( t $ 1)exp[ c + a ln N ( t $ 1)+ + ( t )] ,wecanre-expressthedynamicsasalinearfunctionwiththe substitution X ( t )=ln N ( t ) thentakethenaturallogofbothsidesoftheGompertz equation.Thisgives X ( t )= X ( t $ 1)+ c + aX ( t $ 1)+ + ( t ) or X ( t )= c + bX ( t $ 1)+ + ( t ) were b =1+ a .Thestrengthofdensitydependenceisthengivenbytheparameter a whichforastationaryprocesscangobetween $ 1 and 1 ,with a =0 correspondingto arandomwalk.Therangefortheparameter b isthengivenby 0 ' b ' 2 .Considera linearinteractingstochasticsystemwiththisformulation,writtenas: X 1 ( t )= c 1 + b 11 X 1 ( t $ 1)+ b 12 X 2 ( t $ 1)+ + 1 ( t ) (EÐ1) X 2 ( t )= c 2 + b 21 X 1 ( t $ 1)+ b 22 X 2 ( t $ 1)+ + 2 ( t ) (EÐ2) Theerrortermsareindependentnonidentical-normaldistributionssuchthat: + n ( t ) ) Norm(0, # 2 n ( t )) .Weremovetheconstants c 1 and c 2 andreintroducethemlater inordertosimplifythealgebra. Royama ( 1981 )and Abbott etal. ( 2009 )haveshown thatwecanexpressthedynamicsof X 1 ( t ) or X 2 ( t ) throughasubstitutionprocess.The substitutionfor X 1 ( t ) isasfollows,ÞrstsolveEquation EÐ1 for X 2 ( t $ 1) intermsofthe X 1 ( t ) 'sandplugthisintoEquation EÐ2 : X 2 ( t $ 1)= 1 b 12 ( X 1 ( t ) $ b 11 X 1 ( t $ 1) $ + 1 ( t ) ) X 2 ( t )= b 21 X 1 ( t $ 1)+ b 22 b 12 ( X 1 ( t $ 1) $ b 11 X 1 ( t $ 2) $ + 1 ( t $ 1) ) + + 2 ( t $ 1) 105

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Notethatthisprocedureimpliesthat b 12 * =0 orelsethereisnosolution.Now, pluggingthisresultbackintoEquation EÐ1 ,thencollectingtermsgives X 1 ( t )= b 11 X 1 ( t $ 1)+ b 12 . b 21 X 1 ( t $ 1)+ b 22 b 12 ( X 1 ( t $ 1) $ b 11 X 1 ( t $ 2) $ + 1 ( t $ 1) ) + + 2 ( t $ 1) / + + 1 ( t ) X 1 ( t )=( b 11 + b 22 ) X 1 ( t $ 1) 0 12 3 densitydependence +( b 12 b 21 $ b 11 b 22 ) X 1 ( t $ 2) 0 12 3 AR + + 1 ( t ) $ b 22 + 1 ( t $ 1)+ b 12 + 2 ( t $ 1) 0 12 3 MA , (EÐ3) whereeachmodelcomponentislabeledinEquation EÐ3 .Ifweincludeconstantgrowth termsthentheconstantgrowthrate, c 1 (1 $ b 22 )+ c 2 b 12 ,needstobeaddedonto Equation EÐ3 .Theerrortermcanberewrittenasamovingaverageprocess,butitcan beseenimmediatelyfromEquation EÐ3 thatthemovingaverageparameteriscontrolled by $ b 22 ,whichpropogatestheerrorinspecies1attheprevioustimesteps, + 1 ( t $ 1) , backinto X 1 ( t ) ,whileastronginteractionterm b 12 propogatesvariabilityfromspecies2 attime t ,whichtendstomovethemovingaveragetowards0.Usingtheresultthatthe sumoftwoindependentmovingaverageprocessesisanothermovingaverageprocess whoseorderisthesameasthatofthehighestorderofthecomponentprocesses(result from Box&Jenkins ( 1970 )),theprocesscanberewrittenas: + 1 ( t ) $ b 22 + 1 ( t $ 1)+ b 12 + 2 ( t $ 1)= , ( t )+ $ ( t ) , ( t $ 1). (EÐ4) TherighthandsideofEquation EÐ4 isanewmovingaverageprocess, , ,with unknownmovingaverageparameter, $ ( t ) andvariance, # 2 ' ( t $ 1) .Thevariance andautocovarianceofleft-handsideofEquation EÐ4 areknownandgivenby 0 =(1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t ) 1 = $ b 22 # 2 1 ( t $ 1), 106

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whilethevarianceandautocovarianceoftherighthandsideisgivenby 0 =(1+ $ ( t ) 2 ) # 2 ' ( t ) 1 = $ $ ( t ) # 2 ' ( t $ 1). Wecanthenequatetheautocovariancestogive $ $ ( t ) # 2 ' ( t $ 1)= $ b 22 # 2 1 ( t $ 1) # 2 ' ( t $ 1)= b 22 $ ( t ) # 2 1 ( t $ 1). Pluggingthisinfor # 2 ' ( t ) inthevariances, (1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t )=(1+ $ ( t ) 2 ) # 2 ' ( t ) (1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t )=(1+ $ ( t ) 2 ) b 22 $ ( t ) # 2 1 ( t ) $ ( t ) " (1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t ) # =(1+ $ ( t ) 2 ) b 22 # 2 1 ( t ) $ ( t ) " (1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t ) # =(1+ $ ( t ) 2 ) b 22 # 2 1 ( t ) $ $ ( t ) 2 b 22 # 2 1 ( t )+ $ ( t ) " (1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t ) # $ b 22 # 2 1 ( t )=0 Wecanthensolvefor $ ( t ) usingthequadraticformula, $ ( t )= B ( t ) ± 4 B ( t ) 2 $ 4( b 22 # 2 1 ( t )) 2 2 b 22 # 2 1 ( t ) , (EÐ5) where B ( t )=(1+ b 2 22 ) # 2 1 ( t )+ b 2 12 # 2 2 ( t ) .Forestimationpurposesweassume $ ( t )= $ . Thisislikelytobeareasonableassumptionwhendemographicstochasticityislowbut becomesworseasitincreases.Atime-dependencyalsoentersthevarianceofthe process # 2 ' ( t ) whichisalsodependenton # 2 1 ( t ) and $ ( t ) .Whenusingaformofdensity dependenceotherthantheGompertztheinteractionequationsarenolongerlinear. However,theRickermodelwasstudiedby Abbott etal. ( 2009 )whofoundthatestimated numberofARandMAtermswasconsistentinmodelsimulations.Basedonourthese 107

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assumptionsweapplythismodelasarough,butpotentiallyuseful,approximationtothe complexstochasticdynamicsthatcanbeinducedthroughpredator-preyinteractions. 108

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APPENDIXF LIKELIHOODS WeÞtgamma,log-normal,andnegativebinomialdistributionstothedatabysetting meanandvarianceforthedistributionstoexpressionsderivedinAppendixA.Herewe usesomeshortcutnotationforconvenience, E[ N t | N t " 1 = n t " 1 , N t " 2 = n t " 2 ] ( µ t = rn t " 1 p ( n t " 1 , n t " 2 ) (FÐ1a) Var[ N t | N t " 1 = n t " 1 , N t " 2 = n t " 2 ] ( # 2 t = n t " 1 [ p ( n t " 1 , n t " 2 )(1 $ p ( n t " 1 , n t " 2 )) r + p ( n t " 1 , n t " 2 ) 2 ! 2 # + ' 2 n 2 t " 1 p ( n t " 1 ) 2 . (FÐ1b) WeÞtallmodelsbymatchingthemeanandvariancetermsinEquation FÐ1 tothe distributionmeanandvariance.Forthelognormaldistributionthisleadsto: ( N t | N t " 1 , N t " 2 ) ) LN . µ ln =log( µ t ) $ 1 2 # 2 ln , # 2 ln =ln( # 2 t /µ 2 t +1) / . (FÐ2) SimilarlyweÞtthegammadistributiontotheabundancetransitionswithshape ( k ) and scale ( $ ) parametersgivenby ( N t | N t " 1 , N t " 2 ) ) Gamma . k = µ 2 t # 2 t , $ = # 2 t µ t / . ThenegativebinomialdistributionwasalsoÞttotheabundancedata.Thismodelwas parameterizedbythemean ( µ NB ) andsize ( k ) parameters, ( N t | N t " 1 , N t " 2 ) ) NBin . µ NB = µ t , k = ' 2 r 2 / . Inthenegativebinomialthedemographicvarianceisassumedtobeequaltothegrowth rate, ! ( & ) 2 = r ,whilethesizeparameter, k ,isrelatedtheenvironmentalvariance. Weusedtheone-stepformulationsoftheAR,MA,andARMAprocesses( Shumway &Stoffer 2006 ).Anexploratoryanalysissuggestedthattheautocorrelationprocesses operatedonthescaleofthepopulationgrowthrate, ln & N t n t ! 1 ' ,thereforeweincluded theautocorrelationasmultiplicativeprocessessuchthattheywereadditivetothe 109

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densitydependenceonthelog-scale.Thisanalysesalsosuggestedthatlog-population abundancesperformedbetterthanabundancessoallautocorrelationmodelsare formulatedwiththistransformation.AllmodelscontainedanAR(1)contributionthrough populationgrowthandregulationprocessescontainedin rn t " 1 p ( n t " 1 ) .TheAR(2) processwasmodeledasacontributiontothepopulationgrowthrategivenby . ln( n t " 2 ) . TheMA(1)processwasalsomodeledasacontributiontothepopulationgrowthrate, thistermwasmodeledas, $ ( ln( n t " 1 ) $ ln( µ t " 1 ) ) .Themeanoftheprocessgivenin Equation FÐ1 withthefullARMAprocessthereforebecomes µ t = rn t " 1 p ( n t " 1 )exp[ . ln( n t " 2 )+ $ (ln( n t " 1 $ µ t " 1 )]. (FÐ3) TheARMAprocessesareadditivecombinationsoftheARandMAcomponentssowe canremovetheAR(2)termbysetting . =0 andremovetheMA(1)termbysetting $ =0 toexaminemodelsnestedtotheARMA(2,1).Althoughevenhigherorder ARMAprocessescanbeÞttothedatausingsimilarmethods,welimitedourselvesto ARMA(2,1)processesduetothepatternsobservedintheempiricalautocorrelation functionofthepopulationgrowthrate(Figure F-1 ). Thelikelihoodsoftheabundancesforthesedatawerecalculatedusingthe transitionprobabilitieswiththetransitiondistributionsdeÞnedabove: L ( $ ; N t )= t max $ t =3 P ( N t | N t " 1 , N t " 2 ; $). (FÐ4) Here, $ isthesetofparameterstooptimizeover,and t iteratesoverallobservations uptothetimeoftheÞnalobservationat t max .Forallmodelsweconditiononthe Þrsttwoobservationssotheactualnumberoftransitionscalculatedisgivenby t max $ 2 .Asanexample,considerÞttingtheRickermodelwithanARMA(2,1) autocorrelationstructureandalognormaltransitiondistribution.Themeanandvariance foreachtimestepgiveninEquation FÐ1 aredeterminedbysetting p ( n t " 1 , n t " 2 )= exp [ $ bn t " 1 + . ln( n t " 2 )+ $ (ln( n t " 1 $ µ t " 1 ) ] ,thenpluggingintheresultingvaluesfor µ t 110

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FigureF-1.Darkbarsaretheempiricallyrealizedvaluesfortheautocorrelationateach lag.Horizontalbluelineistheapproximate95%conÞdenceinterval. and # 2 t intoEquation FÐ2 togettheone-stepprobabilitiesforallobservedabundances exceptfortheÞrsttwo,whicharebeingconditionedon.Inthismodeltheparametersto beoptimizedoverare $ =( r , b , . , $ , ! 2 , ' 2 ) .Theseprobabilityvaluesareplugged intoEquation FÐ4 togettheresultinglikelihoodofalltheobservations.Wemaximized thelog-likelihoodasafunction $ usingtheRgenoudpackage( Mebane&Sekhon 2011 ) inR( RDevelopmentCoreTeam 2012 ).Wealsonotethatthelogisticformofdensity dependence, p ( n t " 1 )=1 $ bn t " 1 ,wasconstrainedintheoptimizationbyapplying aconstrainttothelikelihoodsuchthat 0 < p ( n t " 1 ) < 1 .The ! AICvaluesforthe abundancesarepresentedinTable F-1 . 111

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TableF-1. ! AICvaluesforeachmodelÞttothepopulationabundancesbycommunity type.Note-ThenumberofparametersusedintheAICcalculationisgiven by k andboldnumbersrepresentthebestmodelwithinasetofcomparisons. Model M k SimplecommunityComplexcommunity Ricker 4 21.77 9.02 Beverton-Holt 4 14.32 8.15 Logistic 4 12.73 10.92 Gompertz 4 3.62 5.76 Exponential 3 66.91 16.97 Log-normal 4 6.65 10.41 Negativebinomial 4 63.97 54.17 Gamma 4 21.77 9.02 MA 5 0.68 0.00 Nocorrelation 4 21.77 9.02 ARMA 6 0.00 4.32 AR 5 17.81 8.78 DemographicandEnvironmental 4 21.77 9.02 Nodensitydependenceinvariance 4 72.91 22.97 Environmentalonly 3 72.42 81.39 Demographiconly 3 54.68 64.64 112

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APPENDIXG DERIVATIONOFTHEPOPULATIONABUNDANCEMEANANDVARIANCEWITH DEMOGRAPHICANDENVIRONMENTALVARIATION Environmentalvariationistypicallyassumedtooperateonthemaximumrateof reproduction, & ,howeverthedensitydependenceparameter, b ,canalsobeaffectedby theenvironment.Thiscanoccur,forexample,whenthecarryingcapacityisafunctionof theenvironment. Westartwithapopulationmodeloftheform, N t = N t " 1 e A t ! 1 " B t ! 1 X t ! 1 ,where X t " 1 = f ( N t " 1 ) .Weletthepopulationparameters A and B berandomvariables thatvarythroughtimeduetoenvironmentalßuctuations.Weassumethesefollow normaldistributions,thoughwenotethatthescalingsweÞndareconsistentwith derivationsundermoregeneralconditions( Ferguson&Ponciano 2014 ).Let A t " 1 ) N ( a =ln( & ), # 2 ( ) and B t " 1 ) N ( b , # 2 b ) .Then A t " 1 $ B t " 1 X t " 1 ) N( a $ bX t " 1 , # 2 ( + X 2 t " 1 # 2 b $ 2 / X t " 1 # ( # b ) wherethecorrelation, / ,describeshowßuctuationsinAandB canpotentiallybesynchronous.Notethatpositivecorrelationsbetween & and b lead toreductionsintheoverallvariance.Weassumethat / =0 fortheremainderofthe derivationbutitiseasytoincorporatenonzerocorrelationintotheÞnalvariance. Wealsoincludedemographicvariationinthemodel.Thedemographicvarianceis thevarianceconditionaltheenvironmentandisgivenby Var[ N t | N t " 1 , A t " 1 , B t " 1 ]= N t " 1 e A t ! 1 " B t ! 1 X t ! 1 (1 $ e " B t ! 1 X t ! 1 )+ # 2 dem e " B t ! 1 X t ! 1 ). (GÐ1) Nowusingthefactthatthat A t " 1 $ B t " 1 ) N( a $ bX t " 1 , # 2 ( + # 2 b X t " 1 ) wecanusethe doubleexpectationanddoublevarianceformulastogettheexpectedvalueandvariance notconditionalontheenvironment.Firsttogettheexpectedvalueofthepopulationat time t . E [ N t | N t " 1 ]= E [ E [ N t | N t " 1 , A t " 1 , B t " 1 ]] (GÐ2) = E [ N t " 1 e A t ! 1 " B t ! 1 X t ! 1 ] (GÐ3) 113

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= N t " 1 e a " bX t ! 1 + " 2 " / 2+ " 2 b X 2 t ! 1 / 2 (GÐ4) Nowthevarianceisgivenbythelawoftotalexpectation, V [ N t | N t " 1 ]= V [ E [ N t | N t " 1 , A t " 1 , B t " 1 ]]+ E [ V [ N t | N t " 1 , A t " 1 , B t " 1 ]] (GÐ5) = V " N t " 1 e A t ! 1 " B t ! 1 X t ! 1 # + (GÐ6) E " N t " 1 e A t ! 1 " B t ! 1 X t ! 1 (1 $ e " B t ! 1 X t ! 1 )+ # 2 dem N t " 1 e " 2 B t ! 1 X t ! 1 ) # (GÐ7) = N 2 t " 1 & e " 2 " + " 2 b X 2 t ! 1 $ 1 ' e " 2 " + " 2 b X 2 t ! 1 +2 a " 2 bX t ! 1 + (GÐ8) N t " 1 e a " bX t ! 1 + " 2 " / 2+ " 2 b X 2 t ! 1 / 2 $ N t " 1 e " 2 bX t ! 1 +2 " 2 b X 2 t ! 1 + (GÐ9) N t " 1 # 2 dem e a " 2 bX t ! 1 +2 " 2 b X 2 t ! 1 . (GÐ10) Whentheßuctuationsarerelativelysmall,astheyappeartobeinallthecaseswehave examined,themeanandvariancearewellapproximatedby E [ N t | N t " 1 ] % N t " 1 e a " bX t ! 1 (GÐ11a) V [ N t | N t " 1 ] % N 2 t " 1 5 # 2 ( + # 2 b X 2 t " 1 6 e 2 a " 2 bX t ! 1 + (GÐ11b) N t " 1 e a " bX t ! 1 $ N t " 1 e " 2 bX t ! 1 + N t " 1 # 2 dem e " 2 bX t ! 1 . (GÐ11c) Line GÐ11b givesthecontributionoftheenvironmentalcontribution,while GÐ11c gives thedemographiccontributionofthetotalvariance.ThePoissonassumption,thatthe demographicvarianceisequaltothemean,isgivenbythespecialcase # 2 dem = r .These quantitiesareequivalenttothesemi-parametricderivationsgiveninourpreviouswork ( Ferguson&Ponciano 2014 ),exceptforthenewterm # 2 b whichbehavesasexpected. 114

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APPENDIXH UNGULATEBICTABLES Stage/SexYears ACF Var ( ( n t " 1 )Var b ( n t " 1 ) Var ( ( n t " 1 ) + Var b ( n t " 1 ) Var dem ( n t " 1 ) + Var ( ( n t " 1 ) Var dem ( n t " 1 ) + Var b ( n t " 1 ) Var dem ( n t " 1 ) + Var ( ( n t " 1 ) + Var ( ( n t " 1 ) Alladults 1956-2008MA(0) 779.82 778.46 782.42 783.79 782.41 786.38 Alladults 1956-2008MA(1) 780.31 778.86 782.83 784.28 782.83 786.80 Femaleadults1956-2008MA(0) 652.06 652.35 655.95 656.03 656.08 659.92 Maleadults1956-2008MA(0) 720.83 719.93 723.90 724.80 723.90 727.87 Yearlings 1956-2008MA(0) 601.91 608.61 604.53 605.84 603.80 607.74 Yearlings 1956-2008MA(1) 589.82 612.20 605.30 593.90 592.55 608.02 Yearlings 1956-2008MA(2) 540.13 536.98 540.38 544.08 540.93 548.03 Yearlings 1956-2008MA(3) 881.77 1,124.231041.27 944.30 1041.27 982.08 Kids 1956-2008MA(0) 656.22 668.58 660.19 658.77 658.39 662.36 Kids 1956-2008MA(1) 580.24 593.16 880.17 579.57 579.57 588.18 Alladults 1956-1985MA(0) 429.14 425.14 428.54 432.54 428.54 431.94 Alladults 1956-1985MA(1) 431.82 437.54 436.64 435.22 440.94 440.04 TableH-1.ModelselectiontablefortheAlpineibex. Model Years ACF Var ( ( n t " 1 )Var b ( n t " 1 ) Var ( ( n t " 1 ) + Var b ( n t " 1 ) Var dem ( n t " 1 ) + Var ( ( n t " 1 ) Var dem ( n t " 1 ) + Var b ( n t " 1 ) Var dem ( n t " 1 ) + Var ( ( n t " 1 ) + Var ( ( n t " 1 ) theta-Ricker breakpointmodel 1985-2010MA(0) 398.55 382.96 386.20 388.80 386.22 389.48 theta-Ricker breakpointmodel 1985-2010MA(1) 401.81 386.23 389.68 391.85 389.52 392.77 theta-Ricker 1995-2010MA(0) 234.57 230.56 234.13 237.34 233.34 235.84 theta-Ricker 1995-2010MA(1) 235.90 233.88 235.51 235.81 236.43 235.28 TableH-2.ModelselectiontablefortheSoaysheep. 115

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BIOGRAPHICALSKETCH JakeFergusonwasbornatseain1979.HereceivedhisBachelorofScience inphysicsfromtheUniversityofWashingtonin2002.Jakethenobtainedamaster's degreeinexperimentalcondensedmatterphysicsfromtheUniversityofMassachusetts in2005.Afterdecidinghewantedtogetmorehikinginatwork,heobtainedamaster's degreeintheEcologicalandEnvironmentalStatisticsprogramatMontanaState University,Bozeman,in2009.HereceivedhisPh.D.fromtheUniversityofFlorida inthesummerof2014continuingtheworkonthestatisticalanalysisofpopulation dynamicsbegunduringhismaster's.Hismainareasofacademicinterestareinbuilding modelstoaidinwildlifeconservationdecisionsandinunderstandinghowenvironmental ßuctuationsdriveanimalpopulationgrowth. 129



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