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具有三时滞的捕食-被捕食系统的稳定性和分支分析(英文).docx

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具有三时滞的捕食-被捕食系统的稳定性和分支分析(英文) Introduction Predator-preyinteractionsisoneofthekeyecologicalmechanismsthatregulatepopulationdynamics.Itiswidelyunderstoodthatpredator-preysystemsarepronetoexhibitcomplexbehaviorsuchasoscillations,chaoticbehavior,andcoexistenceofmultipleequilibriumpoints.Spatiallydistributedmodelsofprey-predatorinteractionshavebeenpopularinstudyingtheimpactofmovement,dispersal,andotherecologicalfactorsonthestabilityandcoexistenceofthepopulations.Inthispaper,weconsiderasimplemodelofathreetime-delayedpredator-preysystemandanalyzeitsstabilityproperties. ModelFormulation Ourmodelconsistsofthreemainvariables:thepreypopulation(x),thepredatorpopulation(y),andadelayedpredatorpopulation(z).Thedynamicsofthemodelaregivenbythefollowingdifferentialequations: dx/dt=rx(1-x/k)-axy-bxz(t-d1) dy/dt=bxy-cy-dyz(t-d2) dz/dt=eyz(t-d3)-fz Thefirstequationdescribesthegrowthofthepreypopulationwithlogisticgrowth(rx(1-x/k)).Thesecondequationdescribestheinteractionbetweenthepreyandthepredatorpopulations,wherethepredationrateisproportionaltotheproductofthepreyandpredatorpopulations.Thethirdequationdescribesthedelayedresponseofthepredatorpopulationonthepreypopulationwithtimedelay(d1).Thefourthequationdescribesthedelayedresponseofthepredatorpopulationonthepredatorpopulationwithtimedelay(d2).Thefifthequationdescribesthedelayedresponseofthepredatorpopulationonthepreypopulationwithtimedelay(d3). Theparametersofthemodelareasfollows:ristheintrinsicgrowthrateofthepreypopulation,kisthecarryingcapacityofthepreypopulation,aistheattackrateofthepredatorontheprey,bistheconversionrateofpreybiomasstopredatorbiomass,cisthemortalityrateofthepredatorpopulation,disthedecayrateofthedelaypredatorpopulation,eistheconversionrateofpredatorbiomasstodelayedpredatorbiomass,andfisthedecayrateofthedelayedpredatorpopulation. StabilityAnalysis Toinvestigatethestabilityofthesystem,webeginbyfindingtheequilibriumpointsofthesystembysettingthetimederivativesofallvariablestozero: x*=k(1-d1b/f)/(r+ay*+bz(d1)/f) y*=(c+dy*z