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一类随机时滞捕食模型持久性与渐近稳定性的研究（英文）.docx

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一类随机时滞捕食模型持久性与渐近稳定性的研究（英文） ResearchonPersistenceandAsymptoticStabilityofaClassofStochasticDelayedPredator-PreyModels Predator-preyinteractionsareubiquitousinnatureandplayacrucialroleinmaintainingecologicalbalance.Inrecentyears,therehasbeengrowinginterestinstudyingstochasticdelayedpredator-preymodels,whichincorporaterandomnessandtimedelays,asthesemodelscanbettercapturethecomplexityanduncertaintiesofrealecologicalsystems.Inthispaper,wefocusonaclassofstochasticdelayedpredator-preymodelsandinvestigatetheirpersistenceandasymptoticstability. Thestochasticdelayedpredator-preymodelofinterestcanbeexpressedasfollows: dx(t)=[r1x(t)-a1x(t)y(t)-c1(x(t))-d1(x(t),t)]dt+σx(t)dW1(t) dy(t)=[-r2y(t)+a2x(t)y(t)-c2(y(t))-d2(y(t),t)]dt+σy(t)dW2(t) wherex(t)andy(t)denotethepopulationsizesofpredatorandprey,respectively,attimet,r1andr2aretheintrinsicgrowthratesofpredatorandprey,a1anda2arethecapturecoefficients,c1(x(t))andc2(y(t))arethedensity-dependentdeathratesofpredatorandprey,respectively,andd1(x(t),t)andd2(y(t),t)arethetime-delayeddeathsofpredatorandprey,respectively.Therandomtermsσx(t)dW1(t)andσy(t)dW2(t)representthestochasticfluctuationsofpredatorandpreypopulations,respectively,withdW1(t)anddW2(t)beingthestandardWienerprocesses. Tostudythepersistenceofthestochasticdelayedpredator-preymodel,weusethedefinitionofpersistenceproposedbyCushingetal.(1995),whichstatesthatastochasticmodelispersistentifthereexistsapositiveprobabilitythatthepopulationsofpredatorandpreyremainboundedawayfromextinctionforallsufficientlylongtimeperiods.Usingthetheoryofstochasticdifferentialequations,weobtaintheconditionsforthepersistenceofthemodel,whicharegivenby: r1>c1(0)+d1(0,0)andr2>c2(0)+d2(0,0) wherec1(0),c2(0),d1(0,0)andd2(0,0)denotetheinitialdensitiesofpredatorandprey,andtheirrespectivetimedelayfunctionsevaluatedattime0. Toinvestigatetheasymptoticstabilityofthestochasticdelayedpredator-preymodel,weemploythetheoryofstochasticLyapunovfunctions.Specifically,weconstructastochasticLyapunovfunctionV(x(t),y(t))forth