Stability And Birurcation Analysis For A Predator-frey Model With Time Delay

Presently, the influence of time delay on biological populations is attracted as an important issue by biologists.In the process of the activities of the population, the delay will appear, the change of the growth rate of population density will not occur immediately.And it is related to their life activities.Time delay occurs in the digestion of food in animals.So people began to pay attention to the study of time-delay systems It also became a research focus in the nearly twenty years and a hot research object in the field. The studyt with time delay on the dynamic behavior has become the important and meaningful topic.This paper mainly considers the time-delay dynamic model.For the prey-predator system with time-delay model,this paper introduces the background and the classification of Lotka-Volterra model,as well as the development and the status quo.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 and the stability of the hopf branch.At the same time discusses the direction of the Hopf branch and stability. This paper discusses the direction of the periodic solution and stability with center popular theorem and enrich the research contents of population dynamics with the numerical simulation.The study of ecosystem models has important significance.