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外文资料--Qualitative Analysis of a Delayed and Stage-Stuctured Predator-Prey System (1).PDF

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外文资料--Qualitative Analysis of a Delayed and Stage-Stuctured Predator-Prey System (1).PDF

QualitativeAnalysisofaDelayedandStageStucturedPredatorPreySystemLingshuWang1SchoolofMathematicsandStatistics,HebeiUniversityofEconomicsBusinessShijiazhuang,P.R.Chinawanglingshu126.comGuanghuiFengDepartmentofMathematics,ShijiazhuangMechanicalEngineeringCollege,Shijiazhuang,P.R.Chinafengguanghui126.comAbstractAdelayedandstagestructuredpredatorpreysystemwithHollingtypeIIfunctionalresponseisdiscussed.Byusingthenormalformtheoryandcentermanifoldtheorem,thelinearstabilityofthesystemisinvestigatedandHopfbifurcationsareestablished.Formuladeterminingthedirectionofbifurcationsandthestabilityofbifurcatingperiodicsolutionsaregiven.Numericalsimulationsarecarriedouttoillustratethetheoreticalresults.KeywordstimedelaystagestructurepredatorpreysystemstabilityHopfbifurcation.I.INTRODUCTIONThepredatorpreysystemisveryimportantinpopulationmodelsandhasbeenstudiedbymanyauthorssee,forexample,1,2,3.Itisgenerallyrecognizedthatsomekindsoftimedelaysareinevitableinpopulationinteractionsandtendtobedestabilizinginthesensethatlongerdelaysmaydestroythestabilityofpositiveequilibrium.Timedelayduetogestationisacommonexample,becausegenerallytheconsumptionofpreybythepredatorthroughoutitspasthistorygovernsthepresentbirthrateofthepredator.Recently,greatattentionhasbeenreceivedandalargebodyofworkhasbeencarriedoutontheexistenceofHopfbifurcationsindelayedpopulationmodelssee,forexample2,5andreferencescitedtherein.ThestabilityofpositiveequilibriumandtheexistenceandthedirectionofHopfbifurcationswerediscussedrespectivelyinthereferencesmentionedabove.Inthenaturalworld,therearemanyspecieswhoseindividualspassthroughtwostagesimmatureandmature.Predatorpreysystemswhereonlyimmatureindividualsareconsumedbytheirpredatorsarewellknown.Tothisaim,weconsiderthefollowingdelaydifferentialequations1211121221122222taxt11xtbxtbrxtaxtytxtbxtrxmxtaxtytytrytmxtτττ⎧⎪−⎪⎪−⎨⎪⎪−−−⎪−⎩1where1xtand2xtrepresentthedensitiesoftheimmatureandthematurepreyattimet,respectivelyytrepresentsthe1TheauthorwassupportedbytheNationalNaturalScienceFoundationofChinaNo.10926064andtheScientificResearchFoundationofHebeiEducationDepartmentNo.2009114.densityofthepredatorattimet0τ≥representingatimerepresentingatimedelayduetothegestationofthepredatortheparameters12112,,,,,,,,aaabbmrrrarepositiveconstant/1xmxistheHollingtypeIIresponsefunction.II.STABILITYANDHOPFBIFURCATIONInthissection,wediscussthestabilityofthepositiveequilibriumandtheexistenceofHopfbifurcationsfor1withtimedelayτasaparameter.Assume1221211110,Hamramrbbrbrarbr−−−Itiseasytocheckthat1hasapositiveequilibrium12,,,Exxywhere2221122212112,,bxabxrxaxrxxybramrar−−−Let111222,,,xxxxxxyyy−−−droppingthebars,system1becomes121112112112222222222222222222tytaxt111111xtbxtbrxtaraxtbxtxamxmxxtytmyxtmxmxtmxxtytmyxtytamxmxmxtττττ−−∂−−−−−−22221ayxtrytrytmxττ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪−⎪−−⎪⎩2where21221raaymx∂.Thecharacteristicequationofsystem2attheoriginis32221021003pppqqqeλτλλλλλ−where0111,prbrbbr∂−211pbrr∂,111111prbrbrb∂∂−,11101221arybrqbrmx∂9781424447138/10/25.00©2010IEEE1111222,1aryqrbrqrmx−∂−When0,τequation3becomes3222110004pqpqpqλλλAssume121112201aryHbrbrbmx∂∂∂−Thisassumptionimpliesthat22000,0,pqpq2211000.pqpqpq−ByHurwitzcriterion,weknowthatallrootsof4havenegativerealpart.When0,τnotingthat0iωωisarootof3ifandonlyif6422100,5hhhωωωwhere22220001110202,22,hpqhpqppqq−−−2222212.hpqp−−Assume111311222201abryHbbrbrmx−∂thenwecanobtain11111101222222011abrryabrryhbrmxmx−−∂22211120hbrb∂Hence,5hasonlyonepositiverealroot0.ωLet532021120010010242220201arcsin22,0,1,2,6jqpqpqqpqpqqqqqjjωωωτωωωπ−−−−then3hasapairofpurelyimaginaryroots0.iω±Lemma1Forequation3,if1H,2Hand3Hholds,thenwehavethefollowingtransversalcondition0Re0.iddλωλτ⎛⎞⎜⎟⎟⎜⎠⎝ProofDifferentiatingbothsidesof3aboutτyields111210202pqddppqλλττλλλλλλ−⎛⎞−−⎜⎟⎝⎠012201022102Re0ihddqqλωωλωτ−⎛⎞⎜⎟⎝⎠Therefore,001ReRe0iiddsignsignλωλωλλττ−⎧⎫⎧⎫⎪⎪⎪⎪⎛⎞⎛⎞⎨⎬⎨⎬⎜⎟⎜⎟⎝⎠⎝⎠⎪⎪⎩⎭⎩⎭.FromLemma1andtheresultsin4,wehaveTheorem1Forsystem12,If1H,2Hand3Haresatisfied,thenthefollowingresultsholdiwhen00,,ττ∈thezerosolutionisasymptoticallystableiiwhen0,ττthezerosolutionisunstableiii0,1,jjτarethevaluesofHopfbifurcations.III.DIRECTIONANDSTABILITYOFHOPFBIFURCATIONInthissection,westudythedirectionofbifurcationsandthestabilityofbifurcatingperiodicsolutions.ThemethodweusedhereisbasedonthenormalformtheoryandcentermanifoldtheoryintroducedbyHassardetal.in5.Now,werescalethetimeby0,tsτττµ,thensystem1canberewrittenas102111212012222221222222022tytaxt111111xtbxtbrxtarxtbxtxamxxtytmyxtamxmxmxtayxtytrytrytmxτµτµτµ−−∂−−−−2222222221111111mxxtytmyxtamxmxmxt⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪−−−⎪−⎪⎩7For3012,,1,0,,TCRϕϕϕϕ∈−defineoperator01201,LBBµϕτµϕϕ−8111112000brbarBbarα⎛⎞−⎜⎟−−−⎝⎠222200000001Bayrmx⎛⎞⎜⎟⎜⎟⎝⎠2222320212222222322222201000,01101111111mxmyfaamxmxmmxmyamxmxmϕϕϕµϕτµϕϕϕϕϕϕ⎛⎞⎜⎟⎜⎟⎜⎟−−−⎜⎟⎜⎟⎜⎟−−−−⎜⎟⎜⎟−⎝⎠BytheRieszrepresentationtheorem,thereexistsamatrixwhosecomponentsareboundedvariationfunctions3,1,0,Rηθµ−→suchthat01,Ldµϕηθµϕθ−∫.Define01,1,0,,0Pdssϕθθµϕηµϕθ−∈⎧⎪⎨⎪⎩∫,0,1,0,,0Rfθµϕµϕθ∈⎧⎨⎩Hence,system7canberewrittenaspURU9tttUµµwhere12Ux,x,y.TFor10,1,Cψ∈define01,1,0,0,0TssPsdttsψψηψ−−∈⎧⎪⎨−⎪⎩∫For30,1,,CCϕ∈30,1,,CCψ∈define010,00TTddθθξψϕψϕψξθηθϕξξ−−−∫∫where,0.ηθηθThen,0PPandPareadjointoperators.BydiscussioninSectionIIandtransformation,tsτweknowthat00iτω±areeigenvaluesofP.Thus,theyarealsoeigenvaluesof.PDirectcomputationyieldsthefollowingresult.Lemma200231,,iTqqqeτωθθand00231,,isTqsDqqeτωareeigenvectorsofPandPcorrespondingto00iτωand00iτω−,respectively,and,1,,0,qqqqθθθθwhere1102briqbω,212011031abbaibriqarbωω−∂1102briqbω−,0022111003211imxbbbriiqeabyτωωω−−−∂12223303232211ayDqqqqqqrqmxτ−Nowwecomputethecoordinatestodescribethecentermanifold0Cat0.µLettUbethesolutionof9when0,µanddefine,,Wt,2Re{.ttztqUUztqθθθ−10On0C,wehaveWt,,,Wztztθθ,where22201102,,zzWzzWWzzWθθθθ11zandzarelocalcoordinatesfor0Cinthedirectionofqandq.Forthesolution0tUC∈,since0µ,then0000,.ztiztqfzzτω12Werewrite12as00,ztiztgzzτωwith22220110221,zzgzzggzzgg13Hence0,0,00,tgzzqfzzqfU.SubstitutetUθintoaboveandcomparingwith13,weget0022232231220032222121iqmxqmyqaqeaqgDmxaqqτωτ−−−−23122232322113222021212aqaqmxqqqqmyqqgmxaqqqDτ−−−002223122302032222121iqmxqmyqaqeaqgDmxaqqτωτ−−−000022210202112232221202122223322202113112023322202112120201102012100002121iigDaqqWqWmxaqmyqWqWmxqWqWqWqWaqmyqWeqWeτωτωτ−−−−−−000000002233220211311322011211112114iiiimxqWeqWeqWeqWeτωτωτωτω−−−−−−−Now,wecompute20Wθand11Wθ.By9,weget0002Re{0},1,02Re{0},0tWUzqzqAWqFqAWqFqFθθθθ−−⎧−∈⎪⎨−⎪⎩2201102AWHz,z,AWh22zzhzzhθθθθ15For1,0θ∈−,wecanget0020201111A2iW,AW.16hhτωθθθθ−−By15,wecanget0,,2Re{0},,HzzqFqgzzqgzzqθθθθ−−−Comparingthecoefficientswith13,wecanobtain202002111111,h.hgqgqgqgqθθθθθθ−−−−Ontheotherhand,by16,weget200020202WiWhθτωθθ−.Solvingit,wehave00000022002200000003iiiigigWqeqeEeτωθτωθτωθθτωτω−−Similarly,wecanget0000111111000000iiiggWqqeFiτωθτωθθτωτω−−Inwhatfollows,weseekappropriateEandF.ThedefinitionofAand16implythat0200020201200dWiWhηθθτω−−∫0111110.dWhηθθ−−∫Bythedefinitionof,,Hzzθin15,wehave20200232023111101100,,00,,TTHgqgqohhHgqgqohhθτ−−−−where2122322232121aqmxqmyqhaqmx−−−12232322222321221amxqqqmyqqhaqmx−−−0022223233211iaqmxqmyqehmxτω−−22232322332121amxqqmyqhmx−Substituting20Wθand11Wθintoaboveequations,wehave12311111,,EΔΔΔΔ,12322221,,FΔΔΔΔ,where0021110010211102222221iibriibrirearybriemxωωωωωω−−Δ∂−−0211201322/ihbrireabhraωω−Δ−−02211102031222/ibrihrireharaωωω−Δ−−0023221110301322221ihayebrihibhmxτωωω−Δ∂−221112/1arybrmxΔ−,13212/abrhaΔ−2321112/arbrhaΔ−3221112112/1hbrbayhbrmxΔ∂−Basedontheanalysisabove,wecancomputethefollowingquantities2221120110200102,23giCggggτω−−212Re0,Cβ120Re0,ReCµλτ−120200Im0Im.Ctµλττω−Fromtheexpressionof10C,itiseasytogetthevaluesof22,µβand2.tOntheotherhand,weknowthat2µdeterminesthedirectionoftheHopfbifurcationif200µThisindicatesthatitisasupercriticalHopfbifurcation.NumericalsimulationsarepresentedinFig.1andFig.2.FromFig.1,itiscleartheoriginisasymptoticallystablewith00.1ττseeFig.2.REFERENCES1C.S.Holling,Thefunctionalresponseofpredatorstopreydensityanditsroleinminicryandpopulationregulation,Mem.Entomolog.Soc.Can.451965360.2C.Sun,M.HanandY.Lin,AnalysisofstabilityandHopfbifurcationforadelayedlogisticequation,Chaos,SolitonsFractals,312007672682.3W.Wang,L.Chen,Apredatorpreysystemwithstagestructureforpredator,Comput.Math.Appl.3319978391.4K.Cooke,Z.Grossman,Discretedelay,distributeddelayandstabilityswitches,J.Math.Anal.Appl.861982592627.5B.Hassard,N.Kazarinoff,Y.H.Wan,TheoryandApplicationsofHopfBifurcation,LondonMathSoc.Lect.Notes,Series,41.CambridgeCambridgeUniv.Press,1981.−0.4−0.200.20.4−0.3−0.2−0.100.10.2−0.15−0.1−0.0500.050.10.150.2x__1x2y−2−10123−1−0.500.511.5−0.8−0.6−0.4−0.200.20.40.60.8x1x2yFig.1phaseportraitwith0.1τFig.2phaseportraitwith1.8τ

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