Two Classes With The Functional Response Function Prey - Predator System Qualitative Nature And The Hopf Bifurcation,

In this paper, two prey-predator systems with functional response are studied. By using the qualitative and stable theory and bifurcation methods, the dynamical behaviors of the models are discussed, The full-text is divided into four chapters. The main tasks of each section are as follows:The first chapter is introduction. Firstly, the development and research trend of the prey-predator system without time delay is introduced. Secondly, the development and research trend with a time delay of prey-predator system is introduced, and then the research content of this paper is given. Finally, the main research work and the arrangement of this paper is given.In chapter 2, a system of predator-prey of functional response without time delay is considered. By using the qualitative theory of ordinary equations, the quality of the equilibrium point for the system is analyzed and the boundary of the solutions, the sufficient conditions of nonexistence and existence of a limit cycle of the system are obtained. Finally, a summary of research and prospect of this chapter are given. And related results are improved and generalized.In chapter 3, a leslies prey-predator system with time delay is studied, by using the method of the qualitative theory and stability theory and bifurcation theory of ordinary differential equation. By analyzing the positive equilibrium, the sufficient condition of the global stability of positive equilibrium point E1,(x1,y1) is obtained whenτ=0. Using Cook k.. Lemma, a sufficient condition that can lead to the phenomenon of Hopf bifurcation is obtained whenτ≠0. Using center manifold theorem and normal form theory, the direction and the stability of Hopf bifurcation are established. As a further study, a summary of research and prospect of this chapter are given and some results of predecessors are extended.