一类具有Allee效应的偏害系统的稳定性分析-豆柴文库

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一类具有Allee效应的偏害系统的稳定性分析
Introduction:
Alleeeffectreferstothephenomenonofdecreasedpopulationgrowthrateorextinctionthresholdatlowpopulationdensities.Thiseffectarisesfromchangesinecologicalinteractionsorfeedbackprocessesthatcreateapositivefeedbackloopleadingtoextinctionofthepopulation.Thiseffecthasbeenobservedinmanyspecies,includingplants,animals,andmicroorganisms.Inthispaper,wewilldiscussthestabilityanalysisofatypeofpartiallypredator-preysystemwithAlleeeffect.
Concepts:
Theproposedmodelconsistsoftwospecies,apredatorandaprey,interactingwitheachother.Thepreypopulationgrowslogistically,whilethepredatorpopulationispartiallydependentontheprey.Theinteractionbetweenthetwospeciesisbasedonpredator-preydynamicequations.TheAlleeeffectisincludedinthemodelbyassumingthatthepreygrowthratedecreaseswithdecreasingpreydensity.
Model:
LetusconsidertwovariablesXandYrepresentingpredatorandpreypopulation,respectively.Themodelequationforthepredatorpopulationcanbewrittenas:
dX/dt=mXY/(a+Y)-dX
Wheremisthemaximumpredatorgrowthrate,aisthepredatorpopulationdensityatwhichthepredatorgrowthrateishalfofthemaximumvalue,anddisthepredatormortalityrate,whichisassumedtobedensity-independent.
Similarly,themodelequationforthepreypopulationcanbewrittenas:
dY/dt=rY(1-Y/K)-hY/(1+bY)
Whereristheintrinsicgrowthrateofthepreypopulation,Kisthecarryingcapacityoftheenvironment,histhepreyharvestrate,andbistheAlleethresholddensity,belowwhichthepreygrowthratedecreasessharply.
StabilityAnalysis:
Toestablishthestabilityofthesystem,weneedtofindtheequilibriumpointsofthemodel.BysettingthederivativeofbothXandYtozero,weobtainthefollowingequations:
mXY/(a+Y)-dX=0...(1)
rY(1-Y/K)-hY/(1+bY)=0...(2)
Solvingequations(1)and(2),wegettwoequilibriumpoints,P1(0,0)andP2((aKry)/(my+bh),K(ry-d)/(hy+bm)).
Toestablishthestabilityoftheseequilibriumpoints,weneedtocalculatetheJacobianmatrixofthesystemandfinditseigenvalues.TheJacobianmatrixisgivenby:
J=[[mY/(a+Y),mX/(a+Y)],[-rh/(1+bY)^2,h(K-bY)/(1+bY)^2-r/K]]
AtequilibriumpointP1(0,0),theJacobian