increase65ddd.com

Notes2208Ecology,84(8),2003,pp.
2208–22132003bytheEcologicalSocietyofAmericaAUTOCORRELATEDEXOGENOUSFACTORSANDTHEDETECTIONOFDELAYEDDENSITYDEPENDENCELINJIANG1ANDNANSHAODepartmentofEcology,Evolution,andNaturalResources,CookCollege,RutgersUniversity,NewBrunswick,NewJersey08901-8551USAAbstract.
Coupledtrophicinteractionswithspecialistpredatorsorresourcesarethoughttobeprimarilyresponsibleforgeneratingdelayeddensitydependence.
Previoustheoreticalstudiessuggestthatautocorrelationinexogenousfactorscouldgenerateapparentnegativedelayeddensitydependenceinpopulationsregulatedonlybydirectdensityde-pendence.
Usingbothlinearandnonlinearmodels,weshowthatautocorrelatedexogenousfactorscangeneratethespuriousappearanceofnotonlynegative,butalsopositive,delayeddensitydependenceinpopulationsregulatedonlybydirectdensitydependence.
Evidencefornegativedelayeddensitydependenceisfoundmainlyinpopulationsexhibitingmono-tonicdeterministicstability,whereasevidenceforpositivedelayeddensitydependenceisfoundmainlyinpopulationsexhibitingdampedorpersistenttwo-pointcycles,ormorecomplexdeterministicdynamics.
Wearguethatuctuatingresources(e.
g.
,mastseeding)inbottom-upcontrolledcommunitiescouldqualifyasautocorrelatedexogenousfactorsandcauseapparentdelayeddensitydependenceinthepopulationoftheirconsumers.
Keywords:autocorrelation;bottom-up;delayeddensitydependence;directdensitydependence;exogenousfactors;uctuatingresources;time-seriesmodels;trophicinteractions.
INTRODUCTIONAcentralprobleminpopulationecologyistoiden-tifythemechanismsbehindtheuctuatingpopulationdynamicsexhibitedbymanyspeciesinnature.
Oneapproach,adoptedbymanyecologists,istouseresultsfromtime-seriesanalysestoinferpossiblemechanismsaboutpopulationregulation.
Delayeddensitydepen-dence(hereafterDDD),animportantconceptrecentlydrawingattentionfrompopulationecologists,referstothetime-delayedregulatoryeffectofpastpopulationdensitiesonthereproductionandsurvivalofindivid-uals.
DDDhasbeenfoundinmanycyclicpopulationssincetheseminalworkofTurchin(1990)(e.
g.
,TurchinandTaylor1992,Hornfeldt1994,Bjrnstadetal.
1995,Stensethetal.
1996,Saitohetal.
1997,1999,Hansenetal.
1999a,b,Limaetal.
1999).
Whenmakingin-ference,ecologistsroutinelyinterpretDDDasasignoftrophicinteractions,becauselaggedfeedbackcanreadilyarisefromspecialistpredator–preyorconsum-er–resourceinteractions.
Mostecologicaltime-seriesstudieshaveemployedstatisticalproceduresbasedonautoregressivemodelsManuscriptreceived11September2002;revised29January2003;accepted7February2003.
CorrespondingEditor:A.
R.
Ives.
1E-mail:ljiang@rci.
rutgers.
edutodetectDDD.
AutoregressivemodelsusuallytaketheformNf(N,NN,)tt1t2tkt(1)whereNisthepopulationsize,kistheorderofthedensity-dependentprocess,fisalinearornonlinearfunctionofpastpopulationsizesandrepresentstheeffectsofendogenous(density-dependent)factors,andisarandomvariablerepresentingtheeffectsofex-ogenous(density-independent)factors(Royama1992,Turchin1995,BerrymanandTurchin2001).
Becausehigh-orderprocesses(k2)aregenerallyoflittleeco-logicalsignicance,primaryinterestisonrst-order(k1)directdensitydependence,whichcharacterizesalmostallanimaltimeseries,andsecond-order(k2)DDD.
Autoregressivemodelsfocusontheeffectsofendogenousfactors,withtheassumptionthatex-ogenousfactorscanalwaysbemodeledasindepen-dentlydistributedrandomvariables.
NonrandomexogenousfactorsmaycomplicatethetestsforDDD.
Royama(1981)reportedthatapparentsecond-orderDDDcouldariseinlinearrst-ordermod-elswithautocorrelatedexogenousfactors.
WilliamsandLiebhold(1995)demonstratedthatthefrequencyofapparentsecond-orderDDDincreasedwiththede-greeofautocorrelationintheexogenousfactorsandAugust20032209NOTESdecreasedwiththerateofpopulationgrowthinRickerlogisticmodels.
Suchapparentsecond-orderDDDre-sultsfromimposingrst-orderexogenousdynamicsuponrst-orderendogenousdynamics,andlacksthedynamicalfeedbackprocessesthatcharacterizeregularDDD(primarilyduetocoupledtrophicinteractions).
ThesendingshavecastashadowontheprevailingpracticeofinferringDDDastheresultoftrophicin-teractions.
ThispaperseekstoprovideafullinvestigationofapparentDDDcausedbyexogenousautocorrelation.
Royama(1981)didnotreportwhataffectedthefre-quencyofapparentDDDinhismodels.
WilliamsandLiebhold(1995)onlystudiedhowapparentDDDaroseinpopulationswithmonotonicdeterministicstability.
Inthisstudy,weinvestigatehowapparentDDDcouldariseinpopulationswithavarietyofdeterministicdy-namics.
Wedemonstratethatexogenousautocorrela-tioncouldleadtothedetectionofnotonlyapparentnegative(standard)DDD,butalsopositive(inverse)DDD.
Weconsiderbothlinearandnonlinearmodels.
THELINEARMODELWerststudiedalinearpopulationmodel.
Weac-knowledgethatlinearmodelsmaynotbeadequatetocapturethecomplexandoftennonlinearpopulationdynamicsformanyspecies.
However,themainpurposeofthelinearmodelistoillustratethatapparentDDDduetoexogenousautocorrelationisacommonphe-nomenon,regardlessofthestructureofthedetermin-isticskeleton(i.
e.
,themodelwith0).
Thesimplelinearmodelalsohastheadvantageofbeinganalyti-callytractable.
Weconsiderasimplelinearrst-orderdensity-de-pendentmodelN(1a)Nat11t0(2)whereNtisthepopulationdensityattimet,a1measuresthestrengthofdirectdensitydependence,anda0isaconstantthataffectspopulationsizebutnotpopulationdynamics(thesteadystatepointN*a0/a1).
ThedynamicsofEq.
2areentirelydeterminedbya1(Royama1992).
Intheabsenceofnoise,Ntwouldconvergemono-tonicallytothesteadystatepointN*when1a10,andwouldconvergetoN*withoscillatorytwo-pointdynamicswhen2a11.
Werestrictourattentiontothesestablecases,becauseunstablepopulations(2a1ora10)donotpersist.
Addinganexogenousfactor(noise)etintoEq.
2,weobtainthestochasticmodelN(1a)Nae.
t11t0t(3)Tosimulateautocorrelatednoise,wemodeletasarst-orderautoregressiveprocessebett1t(4)wherebistheautoregressivecoefcient,andtisdrawnfromastandardnormaldistribution.
Welimitourat-tentiontopositivelyautocorrelatednoise(i.
e.
,0b1),whichismuchmorecommonthannegativecor-relationinnature.
Afteralgebraicmanipulations,wecanrewriteEq.
3asN(1ab)Nt11tb(1a)Na(1b).
(5)1t10tSimilarformsofEq.
5aredescribedinRoyama(1981:Appendix1)andinIves(1995).
ItcanbeshownthatEq.
5depictsastationarysecond-orderprocesswhenbothEqs.
3and4arestationary,i.
e.
,if2a10and1b1.
Weusedthepartialautocorrelationcoefcientfunction(PACF)tocharacterizeDDD(BoxandJenkins1970,Turchin1990).
HerethePACFtwotimestepsapart(Royama1992)isPACF[2]b(1a).
1(6)SeveralimportantconclusionscanbedrawnfromEq.
6.
First,givenconstanta1,themagnitudeofPACF[2]shouldincreasewithb,implyingthatapparentDDDshouldbemorefrequentlydetectedasautocorrelationintheexogenousfactorsbecomesstronger.
Second,onlynegativeDDDemergesifpopulationsareregu-latedbyweakdirectdensitydependence(1a10),i.
e.
,whenthepopulationmonotonicallyapproachesN*intheabsenceofnoise.
OnlypositiveDDDispresentifdirectdensitydependenceisstrong(2a11),i.
e.
,whenthepopulationapproachesN*withtwo-pointcyclesintheabsenceofnoise.
Finally,givenconstantb,thefrequencyofnegativeDDDdeclinesasthestrengthofdirectdensitydependenceincreases(1←a1),andthefrequencyofpositiveDDDincreasesasthestrengthofdirectdensitydependenceincreasesfur-ther(2←a1).
THENONLINEARMODELThemeritsofthelinearmodelsjustanalyzedlieintheirsimplicity,butintheabsenceofnoise,theylacktheabilitytoproducedynamicsotherthanstableequi-libria.
Bycontrast,nonlineardeterministicmodelscanyieldavarietyofdynamics,includingstableequilibria,limitcycles,andchaos(May1981).
WebuildontheworkofWilliamsandLiebhold(1995)tostudyhowexogenousautocorrelationcangiverisetoapparentDDDinnonlinearmodels.
WilliamsandLiebhold's(1995)resultswerebasedonRickerlogisticmodels,whichimplicitlyassumethatstrongdirectdensitydependenceoperatesinpopula-tions.
Theiranalyseswerealsorestrictedtopopulationswithdeterministicmonotonicstability(i.
e.
,popula-tionswithsmallgrowthrates).
Hence,theirresults2210NOTESEcology,Vol.
84,No.
8←FIG.
1.
ThevalueofPACF[2](lag-2partialautocorre-lationcoefcientfunction)atthreelevelsofautocorrelationstrength()asafunctionoftheeigenvalueofthenonlinearEq.
7fordifferentcombinationsof(thestrengthofdensitydependence)and(thegeometricgrowthrate).
Eqs.
7–9wereusedtogeneratethetimeseries.
TheconstantforcarryingcapacityisK02inEq.
8;theconstantdeningnoisevar-iationamplitudeisc1inEq.
9.
wasvariedfrom0.
2to10inallsimulations,butonlythreevaluesofareshownforclarity:undercompensation(0.
2,solidcircles),exactcompensation(1,openinvertedtriangles),andovercom-pensation(8,solidsquares).
Foreachvalueof,wasincreasedfrom2to20,causingthereductionoftheeigenvalue.
Notethateigenvalueschangedlittleinthecaseof0.
2,indicatedbythesinglelleddot.
Theverticallinesseparateregionswithdifferentstabilityproperties.
ThetwohorizontaldottedlinesrepresenttheBartlettbandsoutsidewhichDDD(delayeddensitydependence)issignicant.
mightnotbeapplicabletopopulationscharacterizedbyotherdynamics.
Moregeneralconclusionscanbeachievedbyusingmoregeneralizedmodelsthatallowboththestrengthofdensitydependenceandtherateofpopulationgrowthtobevaried.
Here,westudysuchamodel,originallyformulatedbyHassell(1975):NtN.
(7)t1(1N/K)ttHere,isthegeometricgrowthrate,affectsthestrengthofdirectdensitydependence,andKtisasur-rogateforcarryingcapacity.
Weaddedanexogenousfactor(noise)toKt,asinWilliamsandLiebhold(1995).
Ateachtimestep,KKt0t(8)whereK0isaconstant,andtheexogenousfactor,t,wasdrawnfromarst-orderautoregressiveprocess.
Theautoregressiveprocesswascreatedseriallyas2c(1)(9)tt1twheredenesthestrengthofautocorrelationandrangesfrom0to1(againwefocusonpositiveauto-correlation),tisastandardnormalvariable,andcisaconstantdeningtheamplitudeofnoisevariation(RipaandLundberg1996).
WesubjectedalltherawnoiseseriescreatedbyEq.
9toaprocedurecalledspectralmimicry(Cohenetal.
1999,Petchey2000)toensurethatallnoiseserieshadidenticalvarianceandallKtremainedpositive.
Thenonlinearityinthemodelspreventedanalyticalsolutions,soweusednumericalsimulation.
Foreachcombinationofparametervalues,weiteratedthemodelfor2000timesteps.
Suchlongtimeseries,presumablyapproximatingastationarydistribution,allowedustoestimatethestatisticalproperties(i.
e.
,thestructureandstrengthofdensitydependence)ofthesimulatedpro-August20032211NOTEScesswithmoreprecision.
Wediscardedtherst1000pointstoremovetransientdynamics,andlog-trans-formedthedatabeforestatisticalanalysis.
TotestforDDD,weappliedbothPACFandlinearautoregressivemodelstosimulateddata(Turchin1990).
WereporttheresultsonlyfromPACFbecausethetwoapproachesyieldedverysimilarresults.
SignicantDDDwasin-dicatedbylag-2partialautocorrelationcoefcients(PACF[2])outsidetheBartlettbands(Royama1992).
BothapparentnegativeandpositiveDDDemerged.
Asinthelinearmodels,thestrengthofapparentDDDincreasedasautocorrelationintheexogenousfactorincreased(Fig.
1).
Whenexogenousautocorrelationwasabsent(0),PACF[2]valueslaywellwithintheBartlettbandsandthereforenosignicantDDDwaspresent(Fig.
1a).
SignicantDDD,especiallyneg-ativeDDD,emergedwithmoderateexogenousauto-correlation(0.
4;Fig.
1b).
FurtherincreasesinthedegreeofexogenousautocorrelationresultedinmorefrequentoccurrenceofbothnegativeandpositiveDDD(0.
8;Fig.
1c).
TheincidenceofapparentpositiveandnegativeDDDalsodependsontheparametervaluesofand(Fig.
1).
Forpopulationsregulatedbyundercompen-satory(0.
2)orexact-compensatory(1)directdensitydependence,onlyapparentnegativeDDDwaspresentregardlessof.
Incontrast,forpopulationsregulatedbyovercompensatory(8)directdensitydependence,bothapparentnegativeandpositiveDDDweredetected,andincreasingtendedtoreducetheincidenceofnegativeDDD,buttoincreasetheinci-denceofpositiveDDD.
ItcanbeshownthatthestabilityofthenonlinearEq.
7dependsononlyand.
Specically,popula-tionsdescribedbyEq.
7arealwaysstableandreturnmonotonicallytoequilibriumstatesafterperturbation(i.
e.
,eigenvaluebetween0and1)iftheyaregovernedbyexact-orundercompensatorydirectdensitydepen-dence(i.
e.
,1),regardlessofthevalueof.
Con-versely,populationsregulatedbyovercompensation(1)canshowvarioustypesofdynamics,includingmonotonicdamping(eigenvaluebetween0and1),os-cillatorydamping(eigenvaluebetween1and0),andunstablepopulationoscillationsuchaslimitcyclesorchaos(eigenvaluelessthan1).
AsimplepatternemergedwhenthevaluesofPACF[2]wereplottedagainsttheeigenvaluesofthenonlinearmodelfordif-ferentcombinationsofand:apparentnegativeDDDwasonlyfoundintheregionofmonotonicdamping,andapparentpositiveDDDwasonlyfoundinregionsofoscillatorydampingorunstabledynamics(Fig.
1).
Thissimplerelationshipappliestoseveralothernon-linearmodelsthatweexplored,andisconsistentwiththendingfromthelinearmodel,Eq.
3,forwhichtheeigenvalueequals1a1.
DISCUSSIONInadditiontoapparentnegativeDDD,ourinvesti-gationclearlydemonstratedthatautocorrelationinex-ogenousfactorscanalsoleadtothespuriousappear-anceofpositiveDDD,inpopulationsregulatedonlybydirectdensitydependence.
EvidenceforapparentnegativeDDDwasprimarilyobservedinpopulationswithdeterministicmonotonicstability,andapparentpositiveDDDwasfoundinpopulationswithdeter-ministicdampedorsustainedtwo-pointoscillations,ormorecomplexdynamics.
Theresultsofourmodelsdepend,ofcourse,uponthecriticalquestion:doautocorrelatedexogenousfac-torsexistintherealworldWewishtofocusonter-restrialenvironmentsandunivoltinespecies,forwhichmostpopulationtimeserieshavebeencollected.
Itisimportanttonote,however,thattemporalautocorre-lationinenvironmentalvariables(e.
g.
,temperature)oftenexistsinmarinehabitats,duetothelargethermalcapacityoftheocean(Steele1985).
Forterrestrialen-vironments,WilliamsandLiebhold(1995,1997)ar-guedthatweatherandgeneralistpredatorsmightserveasautocorrelatedexogenousfactors,whereasBerry-manandTurchin(1997)contendedthatannualweatherpatternsshowlittleautocorrelationbetweenyearsandthatgeneralistpredatorsaremorelikelytoimposedi-rectdensitydependenceontheirpreypopulations.
WeagreewithBerrymanandTurchinintheirassertions.
Weathervariablesaregenerallyautocorrelatedonverylargetimescales,butseemuncorrelatedonthetimescaleof50years(Steele1985,Halley1996).
Becausemostecologicaltimeseriesareshorterthan50years,weatherseemsunlikelytocauseapparentDDDinmostobservedtimeseriesonunivoltinespecies.
Generalistpredatorsoftenrespondtopreydensityinstantaneouslyviaswitchingormigratorybehavior,resultingindirectdensitydependenceratherthandelayeddensitydepen-dence(Hassell1978,2000,Hanskietal.
1991,2001).
Plausibleautocorrelatedexogenousfactorsinter-restrialhabitats,however,arelikelytobebioticfactorsthatareinherentlyautocorrelatedandyetdynamicallyindependent.
Forexample,forspeciesinvolvedinasymmetriccompetition,thelittle-affectedspeciesmayberegardedastheautocorrelatedexogenousfactorforthespeciesmoreaffectedbycompetition.
Herewewishtoemphasizethatimportantautocorrelatedexogenousfactorsinterrestrialenvironmentsareuctuatingbioticresourcesinbottom-upcontrolledcommunities.
Com-munitiescharacterizedbymastseedingorpulsedpri-maryproductionareoftenbottom-upcontrolled(Ost-feldandKeesing2000).
Inthesecommunities,thepro-ductionofbioticresources(i.
e.
,seedcrops,plantgrowth)oftendominantlydetermines,butseemsun-affectedby,thedynamicsofspeciesthatconsumethe2212NOTESEcology,Vol.
84,No.
8resources(Wolff1996,McShea2000,OstfeldandKeesing2000).
Temporalautocorrelationoftenexistsinresourceproductionbetweenyearsinthesebottom-upcommunities(e.
g.
,Sorketal.
1993,KoenigandKnops2000).
WethushypothesizethatapparentDDDmayfrequentlyariseinconsumersresidinginbottom-upcommunitieswithvariableresourceproduction.
Onepossibleexampleistheleaf-earedmouse(Phyllotisdarwini)insemiaridChile,whereprimaryproductionvariesconsiderablyfromyeartoyear.
Limaetal.
(1999)detectedDDDintherodenttimeseriesandspeculatedthattrophicinteractionswereitssources.
We,however,wouldliketoarguethattherodent'suc-tuatingresources,astheautocorrelatedexogenousfac-tor,mighthavecausedthespuriousappearanceofDDD.
Weawaitfurtherstudiestounravelthemecha-nismsbehindDDDintherodentpopulations.
ThendingofapparentpositiveDDDdeservesfur-therattention.
FewbiologicalmechanismsareknownforpositiveDDD.
HerewereportthatautocorrelatedexogenousfactorscouldgeneratetheappearanceofpositiveDDD.
Saitohetal.
(1999)comparedthepatternofdensitydependencebetweenthegrey-sidedvole(Clethrionomysrufocanus)andtwowoodmousespe-cies(ApodemusspeciosusandApodemusargenteus)inHokkaido,Japan.
Thegrey-sidedvolesfeedprimarilyongreenplantsandarevulnerabletopredation,where-aswoodmicefeedmainlyoninsectsandseeds,andarelessvulnerabletopredationbecauseoftheirgreatermobility(Saitohetal.
1999).
TheincidenceofnegativeDDDwassignicantlyhigherinthevolepopulationsthaninthemousepopulations.
Interestingly,Saitohetal.
(1999)found27outof28signicantDDDsinthewoodmicetimeseriestobepositive(P107,as-sumingthatpositiveandnegativeDDDareequallylikely),whichtheyattributedtotheinabilityoftheirstatisticalteststocompletelypartialouttheeffectsofdirectdensitydependence.
Ourmodelssuggest,how-ever,thatseedsandinsectsmighthaveservedasau-tocorrelatedexogenousfactors,producingapparentpositiveDDD.
TheapparentDDDhypothesisgainsfur-thersupportfromthendingthatpositiveDDDwasgenerallyassociatedwithmousepopulationsregulatedbystrongdirectdensitydependence.
OursimulationsdemonstratethatapparentnegativeDDDisassociatedwithpopulationscharacterizedwithmonotonicstability,whereasapparentpositiveDDDisassociatedwithpopulationscharacterizedbyoscilla-toryorcomplexdynamics.
TheseresultssuggestthatknowledgeofdeterministicdynamicsisimportantforpredictingthesignofapparentDDDforpopulationsresidinginautocorrelatedenvironments.
Inaclassicstudy,Hasselletal.
(1976)examined24eldinsectpopulations,usingEq.
7,andfoundthemajorityofthemtobemonotonicallystable.
Ifthemodelsdidcorrectlycapturetheessenceofpopulationdynamicsforthesespecies,wewouldexpecttondapparentnegativeDDDintheirpopulationswhensubjectedtoautocorrelatedexogenousenvironments.
Inpractice,apparentDDD(negativeorpositive)mayeitherstrengthenorweakenDDD(negative)duetotrophicinteractions.
Underthiscircumstance,con-foundingthetwotypesofDDDandfailingtorecognizetheirrelativecontributionstoobservedpopulationdy-namicsmayresultinerroneousinterpretationofpop-ulationregulationmechanisms.
AlthoughthesignofapparentDDDcanbepredicted,withoutanypriorknowledgeaboutexogenousfactors,itisextremelydif-culttostatisticallydistinguishtheeffectsofendog-enousfactorsfromautocorrelatedexogenousfactors(Ives1995,Jonzenetal.
2002).
Consequently,wecau-tionagainstanuncriticalinterpretationofndingsofDDDinnaturalpopulationsastheresultofcoupledtrophicinteractions.
We,inagreementwithotherau-thors(BerrymanandTurchin1997,2001),advocatethatthebestuseoftime-seriesmodelsshouldbeasdiagnostictoolstogeneratemechanistichypotheses,whichcanthenbetestedbymanipulativeexperiments(Turchinetal.
1999,Korpimakietal.
2002).
ACKNOWLEDGMENTSWethankA.
Berryman,T.
Casey,A.
Ives,C.
Kaunzinger,Z.
Long,T.
McPhearson,P.
Morin,O.
Petchey,J.
Price,J.
Ripa,E.
Schauber,P.
Smouse,C.
Steiner,H.
Stevens,P.
Tur-chin,andoneanonymousreviewerfortheirhelpfulcommentsonthemanuscript.
WearegratefultoP.
Turchinforhisadviceontheanalyses.
ThisresearchwassupportedinpartbyRut-gersBevierdissertationfellowshipawardedtoL.
Jiang.
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