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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,PRCHINAWANGLINGSHU126COMGUANGHUIFENGDEPARTMENTOFMATHEMATICS,SHIJIAZHUANGMECHANICALENGINEERINGCOLLEGE,SHIJIAZHUANG,PRCHINAFENGGUANGHUI126COMABSTRACTADELAYEDANDSTAGESTRUCTUREDPREDATORPREYSYSTEMWITHHOLLINGTYPEIIFUNCTIONALRESPONSEISDISCUSSEDBYUSINGTHENORMALFORMTHEORYANDCENTERMANIFOLDTHEOREM,THELINEARSTABILITYOFTHESYSTEMISINVESTIGATEDANDHOPFBIFURCATIONSAREESTABLISHEDFORMULADETERMININGTHEDIRECTIONOFBIFURCATIONSANDTHESTABILITYOFBIFURCATINGPERIODICSOLUTIONSAREGIVENNUMERICALSIMULATIONSARECARRIEDOUTTOILLUSTRATETHETHEORETICALRESULTSKEYWORDSTIMEDELAY;STAGESTRUCTURE;PREDATORPREYSYSTEM;STABILITY;HOPFBIFURCATIONIINTRODUCTIONTHEPREDATORPREYSYSTEMISVERYIMPORTANTINPOPULATIONMODELSANDHASBEENSTUDIEDBYMANYAUTHORSSEE,FOREXAMPLE,1,2,3ITISGENERALLYRECOGNIZEDTHATSOMEKINDSOFTIMEDELAYSAREINEVITABLEINPOPULATIONINTERACTIONSANDTENDTOBEDESTABILIZINGINTHESENSETHATLONGERDELAYSMAYDESTROYTHESTABILITYOFPOSITIVEEQUILIBRIUMTIMEDELAYDUETOGESTATIONISACOMMONEXAMPLE,BECAUSEGENERALLYTHECONSUMPTIONOFPREYBYTHEPREDATORTHROUGHOUTITSPASTHISTORYGOVERNSTHEPRESENTBIRTHRATEOFTHEPREDATORRECENTLY,GREATATTENTIONHASBEENRECEIVEDANDALARGEBODYOFWORKHASBEENCARRIEDOUTONTHEEXISTENCEOFHOPFBIFURCATIONSINDELAYEDPOPULATIONMODELSSEE,FOREXAMPLE2,5ANDREFERENCESCITEDTHEREINTHESTABILITYOFPOSITIVEEQUILIBRIUMANDTHEEXISTENCEANDTHEDIRECTIONOFHOPFBIFURCATIONSWEREDISCUSSEDRESPECTIVELYINTHEREFERENCESMENTIONEDABOVEINTHENATURALWORLD,THEREAREMANYSPECIESWHOSEINDIVIDUALSPASSTHROUGHTWOSTAGESIMMATUREANDMATUREPREDATORPREYSYSTEMSWHEREONLYIMMATUREINDIVIDUALSARECONSUMEDBYTHEIRPREDATORSAREWELLKNOWNTOTHISAIM,WECONSIDERTHEFOLLOWINGDELAYDIFFERENTIALEQUATIONS1211121221122222TAXT11XTBXTBRXTAXTYTXTBXTRXMXTAXTYTYTRYTMXTΤΤΤ⎧⎪−⎪⎪−⎨⎪⎪−−−⎪−⎩1WHERE1XTAND2XTREPRESENTTHEDENSITIESOFTHEIMMATUREANDTHEMATUREPREYATTIMET,RESPECTIVELY;YTREPRESENTSTHE1THEAUTHORWASSUPPORTEDBYTHENATIONALNATURALSCIENCEFOUNDATIONOFCHINANO10926064ANDTHESCIENTIFICRESEARCHFOUNDATIONOFHEBEIEDUCATIONDEPARTMENTNO2009114DENSITYOFTHEPREDATORATTIMET;0Τ≥REPRESENTINGATIMEREPRESENTINGATIMEDELAYDUETOTHEGESTATIONOFTHEPREDATOR;THEPARAMETERS12112,,,,,,,,AAABBMRRRAREPOSITIVECONSTANT;/1XMXISTHEHOLLINGTYPEIIRESPONSEFUNCTIONIISTABILITYANDHOPFBIFURCATIONINTHISSECTION,WEDISCUSSTHESTABILITYOFTHEPOSITIVEEQUILIBRIUMANDTHEEXISTENCEOFHOPFBIFURCATIONSFOR1WITHTIMEDELAYΤASAPARAMETERASSUME1221211110,HAMRAMRBBRBRARBR−−−ITISEASYTOCHECKTHAT1HASAPOSITIVEEQUILIBRIUM12,,,EXXYWHERE2221122212112,,BXABXRXAXRXXYBRAMRAR−−−LET111222,,,XXXXXXYYY−−−DROPPINGTHEBARS,SYSTEM1BECOMES121112112112222222222222222222TYTAXT111111XTBXTBRXTARAXTBXTXAMXMXXTYTMYXTMXMXTMXXTYTMYXTYTAMXMXMXTΤΤΤΤ−−∂−−−−−−22221AYXTRYTRYTMXΤΤ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪−⎪−−⎪⎩2WHERE21221RAAYMX∂THECHARACTERISTICEQUATIONOFSYSTEM2ATTHEORIGINIS32221021003PPPQQQEΛΤΛΛΛΛΛ−WHERE0111,PRBRBBR∂−211PBRR∂,111111PRBRBRB∂∂−,11101221ARYBRQBRMX∂9781424447138/10/25002010IEEE1111222,1ARYQRBRQRMX−∂−WHEN0,ΤEQUATION3BECOMES3222110004PQPQPQΛΛΛASSUME121112201ARYHBRBRBMX∂∂∂−THISASSUMPTIONIMPLIESTHAT22000,0,PQPQ2211000PQPQPQ−BYHURWITZCRITERION,WEKNOWTHATALLROOTSOF4HAVENEGATIVEREALPARTWHEN0,ΤNOTINGTHAT0IΩΩISAROOTOF3IFANDONLYIF6422100,5HHHΩΩΩWHERE22220001110202,22,HPQHPQPPQQ−−−2222212HPQP−−ASSUME111311222201ABRYHBBRBRMX−∂THENWECANOBTAIN11111101222222011ABRRYABRRYHBRMXMX−−∂221111112222224011ARYAYHBRBRMXMX∂−∂22211120HBRB∂HENCE,5HASONLYONEPOSITIVEREALROOT0ΩLET532021120010010242220201ARCSIN22,0,1,2,6JQPQPQQPQPQQQQQJJΩΩΩΤΩΩΩΠ−−−−THEN3HASAPAIROFPURELYIMAGINARYROOTS0IΩLEMMA1FOREQUATION3,IF1H,2HAND3HHOLDS,THENWEHAVETHEFOLLOWINGTRANSVERSALCONDITION0RE0IDDΛΩΛΤ⎛⎞⎜⎟⎟⎜⎠⎝PROOFDIFFERENTIATINGBOTHSIDESOF3ABOUTΤYIELDS111210202PQDDPPQΛΛΤΤΛΛΛΛΛΛ−⎛⎞−−⎜⎟⎝⎠012201022102RE0IHDDQQΛΩΩΛΩΤ−⎛⎞⎜⎟⎝⎠THEREFORE,001RERE0IIDDSIGNSIGNΛΩΛΩΛΛΤΤ−⎧⎫⎧⎫⎪⎪⎪⎪⎛⎞⎛⎞⎨⎬⎨⎬⎜⎟⎜⎟⎝⎠⎝⎠⎪⎪⎩⎭⎩⎭FROMLEMMA1ANDTHERESULTSIN4,WEHAVETHEOREM1FORSYSTEM12,IF1H,2HAND3HARESATISFIED,THENTHEFOLLOWINGRESULTSHOLDIWHEN00,,ΤΤ∈THEZEROSOLUTIONISASYMPTOTICALLYSTABLE;IIWHEN0,ΤΤTHEZEROSOLUTIONISUNSTABLE;III0,1,JJΤARETHEVALUESOFHOPFBIFURCATIONSIIIDIRECTIONANDSTABILITYOFHOPFBIFURCATIONINTHISSECTION,WESTUDYTHEDIRECTIONOFBIFURCATIONSANDTHESTABILITYOFBIFURCATINGPERIODICSOLUTIONSTHEMETHODWEUSEDHEREISBASEDONTHENORMALFORMTHEORYANDCENTERMANIFOLDTHEORYINTRODUCEDBYHASSARDETALIN5NOW,WERESCALETHETIMEBY0,TSΤΤΤ,THENSYSTEM1CANBEREWRITTENAS102111212012222221222222022TYTAXT111111XTBXTBRXTARXTBXTXAMXXTYTMYXTAMXMXMXTAYXTYTRYTRYTMXΤΤΤ−−∂−−−−2222222221111111MXXTYTMYXTAMXMXMXT⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪−−−⎪−⎪⎩7FOR3012,,1,0,,TCRΦΦΦΦ∈−DEFINEOPERATOR01201,LBBΦΤΦΦ−8111112000BRBARBBARΑ⎛⎞−⎜⎟−−−⎝⎠222200000001BAYRMX⎛⎞⎜⎟⎜⎟⎝⎠2222320212222222322222201000,01101111111MXMYFAAMXMXMMXMYAMXMXMΦΦΦΦΤΦΦΦΦΦΦ⎛⎞⎜⎟⎜⎟⎜⎟−−−⎜⎟⎜⎟⎜⎟−−−−⎜⎟⎜⎟−⎝⎠BYTHERIESZREPRESENTATIONTHEOREM,THEREEXISTSAMATRIXWHOSECOMPONENTSAREBOUNDEDVARIATIONFUNCTIONS3,1,0,RΗΘ−→SUCHTHAT01,LDΦΗΘΦΘ−∫DEFINE01,1,0,,0PDSSΦΘΘΦΗΦΘ−∈⎧⎪⎨⎪⎩∫,0,1,0,,0RFΘΦΦΘ∈⎧⎨⎩HENCE,SYSTEM7CANBEREWRITTENASPURU9TTTUWHERE12UX,X,YTFOR10,1,CΨ∈DEFINE01,1,0,0,0TSSPSDTTSΨΨΗΨ−−∈⎧⎪⎨−⎪⎩∫FOR30,1,,CCΦ∈30,1,,CCΨ∈DEFINE010,00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