Stability And Birurcation Analysis For A Predator-frey Model With Holling Type ?

In the ecological system,the system between predator and prey populations are widely attracted by biologists and mathematicians in recent years.The stability and bifurcation of a predator prey model with functional response function are studied by biologists and mathematicians, and the research about population dynamics is enriched by numerical simulation.Consider with the complexity and diversity of ecological system,the dynamic behavior of differential equations with impulsive and delayis more abundant and practical than ordinary differential equations.This paper mainly considers the time-delay and impulse predator-prey model with Holling-II type functional response.For the prey-predator system with functional response of Holling-II type,this paper introduces current research progress of the prey-predator system with functional responsefunction,the basic theory and methods are given first.Then discusses the existence and stability of the equilibrium points,boundedness of solutions with positive initial conditions.By constructing appropriate Dulac function and the uniqueness theory of Zhang Zhifen,the existence? uniqueness and stabitiiy of the system are obtained.For the prey and predator system with time-delay,time delay is added respectively to the predator and the prey,with the theory of time-delay differential equations,the sufficient condition of stability and Hopf bifurcation about the equilibrium points is obtained.Finally,with the theory of impulsive differential equation and Floquent,the paper introduces and anslysis the system with impulse, obtains sufficient conditions on the stability and persistence of the impulsive system.