Qualitative Analysis For Two Kinds Of Single Food Chain Models

The predator-prey system is a very important part in ecology, and the mathematical method has been well applied in the research. Recently, the single food chain system has received a great deal of attention of mathematicians and biologists. However, the single food chain system with density dependence received little attention. This thesis mainly studies the dynamics of the single food chain system with two kinds of different functional response based on density dependence. A series of results are obtained, which of them improve or extend the results in the literations.In this paper, the background and the recent works of the predator-prey system are introduced. In the second chapter, the mathematical knowledge which used in study of the predator-prey system is simply given. In the third chapter, an autonomous system of differential equations is dealed with and the existence-uniqueness of positive equilibrium point to the problem is obtained, and then the local stability and global asymptotical stability of positive equilibrium point are established by means of the stretching method and Liapunov function method, respectively. In the fourth chapter, hopf bifurcation analysis of a single food chain model with Monod type and Beddington-DeAngelis functional response is focused. The extinction conditions of the prey and predator are established. The existence and local stability of the semi-trivial equilibrium points and the existence of the positive equlibrium point are studied. Also, hopf bifurcation analysis of the semi-trivial equilibrium points and the positive equlibrium point is dealed with by using Routh-Hurwitz Stability Criterion and Hopf bifurcation theroy, respectively. Finally, the key work done in the paper is summarized and some outlook is given.