Qualitative Analysis Of Three Kinds Of Predator-Prey System

In the world of today, our ability of changing the nature is greatly heightened. Because of over-exploitation, some species are close to extinction or already extinct. We destroy the ecological balance seriously. The system's asymptotic behavior includes the stability of the solutions, the attractivity of the solutions, the oscillation of the solutions and the Hopf-bifurcation , these behaviors reveal the long-term behavior of the species. Since a lot of biological regular and phenomenon in the biological were described by using differential equation, which drew the attention of a lot of experts and scholars, and formed many new topics that have strong practical background. We know that the study of the co-exist, stability, oscillation of the spices has very important practical meaning to keep ecological equilibration and protect ecological environment, even to save valuable and rare biologics which are on the verge of becoming extinct.Recognizing the reality , in this paper, we will investigate three mathematical models of a predator-prey system and discuss the biological significance of the models.In chapter 1, the background and the current status of research of the predator-prey model and several related definitions are introduced.In chapter 2, the basic knowledge of the next study are introduced.In chapter 3, a kind of predator-prey system is exploited. The paper mainly studied the behaviors of the equilibria and the stability. By constructing Dulac function and Piocare cure, a sufficient condition of the nonexistence of closed orbit was obtained. The existence and the uniqueness of limit cycle which depend on Hopf bifurcation theory and unique theorem was proved.In chapter 4, a kind of prey-predator system with sparssing effect is studied. By using qualitative theory of ordinary differential equations, we have analyzed the equilibrium points, obtained the parameter region of the existence, uniqueness and nonexistence of limit cycle of the above system.In chapter 5, a kind of predator-prey system with functional response is discussed. Sufficient conditions are obtained for the existence and stability of the positive equilibrium of the system, and the existence and uniqueness of the limit cycles are proved in this system.