Study On Predator-Prey Systems With Functional Response

This paper consists of three parts:Firstly, we consider a Leslie-Gower system with time delay and Holling III functional response function. Through the differ-ential inequality, the upper and lower bounds of the system is obtained. According to Lyapunov stability theory, we construct the Lyapunov function and the sufficient condition of global attraction at positive solutions is obtained. Secondly, a predator-prey system of Holling’s type IV functional response with Semi-ratio-dependent is studied, we mainly analyse the nature of equilibrium points. We adopt the method of perturbation on the system to study the higher order singular point. We discuss the fine focus with the form series method and Mathmatica software to calculate the focus quantities and the positive equilibrium can be a fine focus of order2. At the same time we also study the problem about Hopf bifurcation under some given conditions and analyse the number of limit cycles near the positive equilibrium. We judge the nonexistence of limit cycles with Dulac function and obtain the condition of the limit cycle’s nonexistence. We also explain the biological sense of positive equilibrium. Finally, a quadratic perturbed Hamilton system with two parameters is studied.Study the permanence and the global stability of the system’s positive equilibrium, and calculate the Picard-Fuchs equation which the defined Abel inte-gral meets. The Mel’nikov function of order one and order two is calculated too. It can be studied that the quality of limit cycles of the system under the disturbance of four times polynomial perturbations by calculation of Mel’nikov function.