A Metapopulation Model with Explicit Local Populat

AMetapopulationModelwithExplicitLocalPopulationDynamicsZhilanFengDepartmentofMathematics,PurdueUniversity,WestLafayette,IN47907,USA,e-mail:[e-mail]ntofForestryandNaturalResources,PurdueUniversity,WestLafayette,IN47907,USA,email:[e-mail]ematics,GeorgiaInstituteofTechnology,Atlanta,GA30332,USA,e-mail:[e-mail]ofMathematicsandStatistics,YorkUniversity,Toronto,ON,Canada,M3J1P3,e-mail:[e-mail]19,2003AbstractManypatch-basedmetapopulationmodelsassumethatthelocalpopulationwithineachpatchisatitsequilibriumandindependentofchangesinpatchoc-cupancy.Westudiedametapopulationmodelthatexplicitlyincorporatesthelocalpopulationdynamicsoftwocompetingspecies.Thesingularperturbationmethodisusedtoseparatethefastdynamicsofthelocalcompetitionandtheslowprocessofpatchcolonizationandextinction.Ourresultsshowthatthecoupledsystemleadstomuchmorecomplexoutcomesthansimplepatchmodelsthatdonotincludeexplicitlocaldynamics.Wealsodiscussimplicationsofthemodelforecologicalsystemsinfragmentedlandscapes.1Keywords:Metapopulationmodel{fast-slowdynamics{localcompeti-tion{coexistence1IntroductionDestructionandfragmentationofnativehabitatsarewidespreadandviewedasthemostimportantthreatstobiodiversityworldwide(WilcoxandMurphy[31]).Agricul-ture,urbansprawl,deforestation,andotherhumanactivitieschangethecompositionandphysiognomyoflandscapes,oftenalteringindividualbehavior(SheperdandSwi-hart[24];Zollner[33]),populationdynamics(Hanski[5]),geneticstructure(Gainesetal.[4]),andcommunitycomposition(Wrightetal.[32])oforganisms.Metapopulationmodelshavebeenusedextensivelytostudytheconservationimplicationsofhabitatlossandfragmentation.Ametapopulationconsistsofasetofdiscretelocalpopulationswithindependentinternaldynamicsthatarelinkedbydispersal(Hanski[5]).Metapopula-tionsexistwithinanetworkofidealizedhabitatpatches(fragments),occupyingsomeproportionpofthese.Theoriginalsingle-speciesmetapopulationmodelofLevins[13])assumedthatchangesinpatchoccupancywerefunctionssolelyofcolonizationratesofemptypatches(c)andextinctionratesofoccupiedpatches(e).Althoughoverlysimplis-tic,theLevinsmodelprovidedanessentialframeworkforstudiesofspatiallystructuredsubpopulationlinkedbydispersal.Inadditiontohabitatdestructionandfragmentation,interspeciccompetitioncanbeapowerfulforcestructuringlocalcommunities(Hopfetal.[11];McIntosh[15];Schoener[23]).Thejointeectsoftheseforcesoncommunitystructureisofconsiderableinterest,becauseasymmetriceectsofhabitatdestructionandfragmentationactingonspecieshavethepotentialtoalteroutcomesofinteractionsforcompetingspecies.Theoreticalmodelsofvarioustypespredictthathabitatfragmentationmaypromotecoexistenceofcompetingspeciesbypermittinginferiorcompetitorstoescapespatiallybyvirtueofgreaterdispersalability(HolmesandWilson[10];MoilanenandHanski[16];NeeandMay[17]).Unfortunately,metapopulationmodelsgeneralizedtomultiplespecies(e.g.,Swihartetal.[26];Taneyhill[28];Wangetal.[29])havefailedtoincorporateexplicitlythelocaldynamicsofspeciesineachpatch.AnimportantexceptionwasthemodelofHanskiandZhang[7],inwhichlocalandmetapopulationdynamicswereexplicitlycoupledtoenableexaminationoftheeectofmigrationonmetapopulationpersistence.Theydemonstratedthattheuseofcoupledmodelscanprovideinsightsintoconditionsformetapopulationpersistencewhichcannotbeobtainedfromsimplepatchmodels.Inthispaper,wegeneralizethemodelofHanskiandZhang[7]byincludingthelocaldynamicsoftwoweaklycompetingspecies.Sincelocaldynamicsoccuronamuchfastertimescalethanchangesinpatchoccupancy,wecanuseasingularperturbationargumenttoseparatethemodeldynamicsintotwotimescales.Ouranalysesoftheslowsystemshowthatitispossibleforthesystemtohavemultipleinteriorequilibriaaswellasauniqueglobalinteriorattractor.Whenmultipleinteriorequilibriaarepresent,bi-stabilitymayoccur,inwhichcasethecompetingspeciesmaystabilizeeitherataninteriorequilibrium2(bothspeciesstablycoexist)orataboundaryequilibrium(onespeciesexcludestheotherspecies).Finally,weapplythemodeltoacompetitiveinteractioninafragmentedagroecosystemanddiscusstheimplicationsofourndingsforcommunitystructureandspeciesconservation.2ThemodelanditsfastandslowdynamicsTheLevinsmodelhastheformdpdt=cp(1p)ep;(2.1)wherepdenotestheproportionoftheoccupiedpatches.Itsfocusisonextinctioneandcolonizationcrates,withnoconsiderationgiventotheeectofmigrationonlocaldynamics.Suchanomissionmaybereasonablewhenmigrationrateislow,butifmigrationrateishigh,failuretoconsiderlocaldynamicsmayproducemodelsthatpredictbiasedresults(Hanski[6]).Tostudythepopulation-levelconsequencesoflocaldynamicswhenmigrationratesarehighHanksiandZhang[7]proposedthefollowingpatch-basedmetapopulationmodel:8:dNdt=rN1NK!mN+mNp;dpdt=mNp(1p)ep;(2.2)wherep2[0;1]isthefractionoftheoccupiedhabitatpatches,N2[0;1)istheaveragesizeofexistinglocalpopulations,r0istheaveragepercapitagrowthrateduetolocalbirthsanddeaths,K0istheaverageper-patchcarryingcapacity,m0isthepercapitaemigrationrate,0isthefractionofmigratingindivi