Qualitative Analysis On A Class Of Predator-prey System With Holling Type-Ⅳ Functional Response

In this paper, we study a class of predator-prey system with Holling type-IV functional response as follows: For the biological signification , we only study system (*) in the region (R+)|-² ={(x,y)|x≥0,y≥0}. In the first section , we study the quality of the equilibria of system (*), especially make global qualitative analysis on the positive equilibra. In the second section , we use the method of Poincare to calculate the order of the weak-focus, and get a result that the positive equilibrium ( x₁ ,y₁) is a stable weak-focus of order at most 2. In the third section , we show the condition of the parameter of the globally asymptotically stable equilibrium ( x₁ ,y₁),and also prove that the positive solutions of system (*) are all bounded; and when the equilibrium ( x₁ ,y₁) is unstable ,system (*) has at least one stable limit cycle in (R+)|-² ={(x,y)|x≥0,y≥0}. In the fourth section , we consider the Bifurcation of system (*): 1.Through the analysis of the graph, we can describe the process of the change(from weak-focus of order 2 to weak-focus of order 1) of the positive equibrium ( x₁ ,y₁) clearly; we also use the theory of bifurcation of Hopf to consider the problem of limit cycles ,and abtain that the system at least has two limit cycles under certain conditions . 3. Also we study the bifurcation of the degenerated equilibrium .