Allee效应下的一类捕食-食饵系统的动力分析-豆柴文库

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Allee效应下的一类捕食-食饵系统的动力分析
Introduction
Alleeeffectreferstoaphenomenoninpopulationecology,wherethepopulationsizeofacertainspeciesispositivelyaffectedbyanincreaseinpopulationdensityuptoacertainthreshold(knownastheAlleethreshold),afterwhichitisnegativelyaffectedbyfurtherincreasesindensity.TheAlleeeffecthasbeenobservedinawiderangeofspecies,includingpredatorsandprey.Inthispaper,wewillanalyzethedynamicsofapredator-preysystemunderAlleeeffect.
Model
Weconsiderasimplepredator-preymodel,wherethepreypopulationisdenotedbyNandthepredatorpopulationisdenotedbyP.Thedynamicsofthesystemaregivenbythefollowingsetofdifferentialequations:
dN/dt=rN(1-N/K)-cNP/(aN+P)
dP/dt=acNP/(aN+P)-dP
Whereristheintrinsicgrowthrateofprey,Kisthecarryingcapacityoftheenvironmentforprey,cisthepredationefficiencyofpredators,aisthecaptureefficiencyofpredators,anddisthenaturalmortalityrateofpredators.
ToincorporatetheAlleeeffect,weintroduceathresholdpopulationsizeforprey,belowwhichthepopulationexperiencesadeclineduetoreducedreproductivesuccess.ThisthresholdpopulationsizeisdenotedbyN0,andwemodifythepreygrowthrateterminthefirstequationtoreflectthis:
dN/dt=rN(1-N/K)(N>N0)-cNP/(aN+P)
dN/dt=0(N<=N0)-cNP/(aN+P)
Thus,whenthepreypopulationfallsbelowN0,itsgrowthratebecomeszero,andthepopulationstartstodeclineduetopredation.ThismodificationtothemodelisconsistentwiththeAlleeeffect,whichpredictsadeclineinreproductivesuccessatlowpopulationdensities.
Analysis
Toanalyzethedynamicsofthemodel,wefirstexaminetheequilibriumpointsofthesystem.TheequilibriumpointforPisgivenbyP=0,whichrepresentstheabsenceofpredators.ThepreyequilibriumpointcanbefoundbysettingdN/dt=0,whichyields:
rN(1-N/K)(N>N0)=cNP/(aN+P)
Thisequationhastworoots:N=0,whichrepresentstheabsenceofprey,andN=N*,givenby:
N*=K(2-c/d(aN0/P0+1))
WhereP0istheinitialpredatorpopulation,andN0istheAlleethresholdpopulationforprey.Theexistenceoftworootsimpliestheexistenceofastablelimitcycle,whichdescribestheoscillatorybehaviorofthesystem.
Tofurtherinvestigatethedynamicsofthesystem,weperf