一类具有Allee效应的Filippov害虫治理模型的研究-豆柴文库

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一类具有Allee效应的Filippov害虫治理模型的研究
Introduction
Inrecentyears,Filippovpestcontrolmodelshavegainedincreasinglyattentioninecologicalstudiesduetotheircapabilitytodescribebiologicalandchemicalcontrolofpestsinvolvedwithdiscontinuoustransitions.Alleeeffect,ontheotherhand,referstothepositivecorrelationbetweenpopulationdensityandpopulationgrowthrateatlowdensities.Inthisstudy,wefocusontheinvestigationofFilippovpestcontrolmodelsthatexhibitAlleeeffect.
Background
Alleeeffecthasbeenobservedinmanypopulationsoforganisms,includinginsects,mammals,andfish,andcanhaveasignificantimpactonpopulationdynamicsandcontrolstrategies.TheAlleethresholdistheminimumpopulationdensityrequiredforapopulationtomaintainitsgrowthrate,andwhenpopulationdensityfallsbelowthisthreshold,thepopulationgrowthratedecreasesrapidly.Thisphenomenonisattributedtoseveralfactors,suchassocialandecologicalfactorsthatpromotecooperation,matefindingandpredatoravoidance.Therefore,theexistenceofAlleeeffectleadstomorecomplexandunpredictablepopulationdynamics,andthiseffectisessentialtoconsiderinpestcontrolprograms.
Filippovequationsarediscontinuousorimpulsivesystemsthatdescribetheinteractionbetweenapopulationanditsenvironment.Theseequationsareparticularlyusefulinmodelingpestcontrolsystems,astheyallowincorporatingbiologicalandchemicalcontrolstrategiesaswellasstochasticfactors.ThemathematicalframeworkofFilippovequationsinvolvesdiscontinuousmaps,wherethestateofthepopulationchangesinstantaneouslyaccordingtothecontrolstrategiesimplemented.Consequently,thestabilityandbifurcationanalysisofsuchmodelsrequireamoresophisticatedapproach.
ModelDescription
Weconsideraone-dimensionalFilippovmodelthatdescribestheinteractionbetweenthepopulationofapestanditsenvironmentunderbiologicalandchemicalcontrolstrategies.Themodelisgivenby:
dx/dt=rx(1-x/K)-βxH(x)-ux
dx(t_k+)=g(x(t_k))
wherexisthepopulationdensity,ristheintrinsicgrowthrate,Kisthecarryingcapacity,βisthebiologicalcontrolcoefficient,H(x)istheHillfunctionthatdescribesthepesticidekillingeffect,uisther