On the formulation and analysis of general determi

OntheFormulationandAnalysisofGeneralDeterministicStructuredPopulationModels.II.NonlinearTheoryDedicatedtoProfessorWilliJagerontheoccasionofhis60thbirthdayOdoDiekmannMatsGyllenbergHaiyangHuangMarkusKirkilionisJ.A.J.MetzHorstR.ThiemeTurkuCentreforComputerScienceTUCSTechnicalReportsNo339,May2000OntheFormulationandAnalysisofGeneralDeterministicStructuredPopulationModels.II.NonlinearTheoryDedicatedtoProfessorWilliJagerontheoccasionofhis60thbirthdayOdoDiekmannDepartmentofMathematics,UniversityofUtrecht,P.O.Box80010,3580TAUtrecht,TheNetherlands.MatsGyllenbergDept.ofMath.,Univ.ofTurku,20014Turku,Finland.HaiyangHuangDepartmentofMathematics,BeijingNormalUniversity,Beijing100875,P.R.ofChina.MarkusKirkilionisUniversityofHeidelberg,InterdisziplinaeresInst.f.wiss.Rechnen,ImNeuenheimerFeld368,69120Heidelberg,GermanyJ.A.J.MetzInstituteforEvolutionaryandEcologicalSciences,LeidenUniversity,Kaiserstraat63,NL-2311GPLeiden,TheNetherlandsandAdaptiveDynamicsNetwork,IIASA,A-2361Laxenburg,Austria.HorstR.ThiemeDepartmentofMathematics,ArizonaStateUniversity,Tempe,AZ85287-1804,USA.TurkuCentreforComputerScienceTUCSTechnicalReportNo339May2000ISBN952-12-0640-3ISSN1239-1891AbstractThispaperisasmuchaboutacertainmodellingmethodology,asitisabouttheconstructivedenitionoffuturepopulationstatesfromadescrip-tionofindividualbehaviourandaninitialpopulationstate.Thekeyideaistobuildanonlinearmodelintwosteps,byexplicitlyintroducingtheen-vironmentalconditionviatherequirementthatindividualsareindependentfromoneanother(andhenceequationsarelinear)whenthisconditionisgiven(prescribed)asafunctionoftime.Alinearphysiologicallystructuredpopulationmodelisdenedbytworules,oneforreproductionandonefordevelopmentandsurvival,bothde-pendingontheinitialindividualstateandtheprevailingenvironmentalcon-dition.InPartIweshowedhowonecanconstructivelydenefuturepopu-lationstateoperatorsfromthesetwoingredients.Anonlinearmodelisalinearmodeltogetherwithafeedbacklawthatdescribeshowtheenvironmentalconditionatanyparticulartimedependsonthepopulationsizeandcompositionatthattime.Whenappliedtothesolutionofthelinearproblem,thefeedbacklawyieldsaxedpointproblem.Thiswesolveconstructivelybymeansofthecontractionmappingprinciple,foranygiveninitialpopulationstate.Usingsubsequentlythisxedpointasinputinthelinearpopulationmodel,weobtainapopulationsemiow.Wethensaythatwesolvedthenonlinearproblem.Thepaperisorganizedinatop-downspirit:Wedescribeageneralab-stractsettingrstandthenspecialise,whilebecomingmoretechnical.Theresultsarenotrestrictedtoasinglepopulationbutalsocovertheinteraction(includingpredation)ofseveralstructured(andunstructured)populations.Keywords:populationdynamics,physiologicalstructure,nonlinear,feed-backviatheenvironment,deterministicatpopulationlevel,cannibalismTUCSResearchGroupBiomathematicalResearchGroup1Anintroductoryexample:amodelinvolv-ingcannibalisticbehaviourConsiderapopulationof,say,shandassumethatthebehaviourofindi-viduals(notablyreproduction,survival,foodconsumption)dependsontheirsizex.Letbirthsizebexedatxbandletgrowthbedeterministicwitharategthatdependsonsizexandthex-specicenergyintakerate.Likewise,reproductionisdescribedbyspecifyinghowtherateofproducingospringdependsonsizexandthex-specicenergyintakerate.Assumingthatpre-dationistheonlycauseofdeath,wedescribesurvivalintermsofadeathrate,whichwecallthex-specicpredationpressure.Nextwehavetodescribehowenergyintakerateandpredationpressurearethemselvesdetermined.Forthepurposeofexpositionweassumethat,apartfromeectsduetocannibalism,bothfoodconcentrationandpredationpressureareconstantintime.Inotherwords,cannibalismistheonlyfeedbackloopbywhichtheindividualsinuencetheenvironmentalconditionsofeachother.Ifweneglecttheeectsofhandlingtimes,satiationetc.,wecandescribethepredationprocessintermsofattackratesandconcentrations(therateofeatingbeingbydenitiontheproduct,inthissimplecaseofalinearfunctionalresponse).AssumethatindividualsofsizexhaveaccesstoafoodsourcewithconcentrationZ(x),whichtheyattackatrateC(x)andwhichhasenergeticvalueE(x).Moreover,theyattackconspecicsofsizeywithratec(x;y)andthesehaveenergeticvaluee(y).Letmtbethemeasuredescribingthepopulationsizeandcompositionattimet(somt(!)isthenumber(orratherspatialconcentration)ofindividualswithsizein!attimet).ThentheenergyintakerateofanindividualofsizexisgivenbyI1(t;x)=F1(O1(t;x);x)=E(x)C(x)Z(x)+O1(t;x)(1)withO1(t;x)=Z[xb;1)e(y)c(x;y)mt(dy)(2)whilethepredationpressureexertedonindividualsofsizexisgivenbyI2(t;x)=F2(O2(t;x);x)=(x)+O2(t;x)(3)withO2(t;x)=Z[xb;1)c(y;x)mt(dy)(4)wheredenotesthex-specicnon-cannibalisticpredationpressure.1Oncewenowalsospecifygand,weobtainanonlinearstructuredpopulationmodel,allinteractionsbeingduetocannibalism.Wewanttoshowconstructivelythatonecanassociateinameaningfulwayadynamicalsystemwithsuchamodeldescription.Inotherwords,wewantaconstructionthatforgivenm0yieldsmt,atleastfortsucientlysmall,andthenwewanttoestablishuniquenessinordertoconcludethatmtqualiesasthepopulationstate(p-state)attimet.Todoso,werstcutthefeedbackloopandthenre-establishitas