Hopf Bifurcation For A Class Of Three Dimensional Predator And Prey System

Innature,thereexistsawayinwhichdiferentspeciesareindependentofeachoth-er,and can interact each other as well. For example,population A lives on the abundantnaturalresources,whilepopulationBlivesofthepredationofA.ThenwecallAapreywhile B a predator in the ecology, and the coexistence of the two forms a predator-preysystem.Hsu，Hubbell and Waltman have studied a class of three-dimensional predator-prey system that is composed of two predators and one prey,in which the two predatorseat the prey but have no direct interaction with each other.Besides,the functional re-sponse is supposed to be the type of Holling II,and then a model of three dimensionaldiferential equations is given,and if the model has a positive equilibrium, then it willhave infnite ones.On the basis of their work, we consider the system that consists of two kinds ofpredators and one prey from the two perspectives on biology and mathematics. Fromthe point of view on biology,the two kinds of predators is considered to have direc-t interaction,while on mathematics we consider the disturbance of the original threedimensional diferential equations and study the existence of the infnitely many pos-itive equilibriums under the perturbation as well as the dynamic behavior of the threepopulations.In one case,we prove that only one will be remained of the infnite positive equi-libriums of the original system under the disturbance.By the calculation of focus valueand center manifold of the unique positive equilibrium,we fnd that the unique posi-tive equilibrium can experience Hopf branch in its own center manifold, and a stableperiodic orbit will appear.But it is presented as an unstable periodic orbit in the3D s- pace,whichshowsthatthethreepopulationscancoexistinaperiodicformapproachingthe positive equilibrium on some conditions.