With Stage Structure, Density Restricted Predator - Prey Model

The aim of this work is to construct some predator-prey systems, to analyze the asymptotic behavior of these models and to study the effect of stage-structure and the density dependence on the populations .In chapter one, we classify the individuals of predator as belonging to either the immature or the mature and suppose that only mature population feeds on prey and immature population does not feed on prey. It is assumed that the transition rate from immature stage to mature stage depends on the density of immature individuals. We construct a predator-prey system with stage structure. Conditions for the permanence and the extinction of the predator are obtained. It is also found that the model admits an orbitally asymptotically stable periodic orbit. This suggests that stage structure may be a cause of periodic oscillation of populations and can make the behavior of population models more complex.In chapter two, we study a ratio-dependent predator-prey system. We consider the death rate of predator as the natural mortality rate and the other death rate caused by the intraspecies competition or other factors, i.e., we consider the predator density dependence. The qualitative behavior of the system at the origin in the interior of the first quadrant is studied. It is shown that the origin is indeed a critical point of higher order. There exsist different kinds of topological structures in a neighborhood of the origin. Conditions for the stability of system are obtained. The limit cycle is obtained by the bifurcation theory. The Bogdanov-Takens bifurcation is studied when there is a unique degenerate positive equilibrium. Lastly, we make a numerical analysis.In chapter three, firstly, it is assumed that the death rate of predator depends on the ratio of the predator density and the prey density. We consider the predator-prey models with the bilinear functional response and the ratio-dependence respectively. For the predator-prey models with the bilinear functional response, we obtained the global asympotical stability ofthe positive equlibrium. For the ratio-dependent predator-prey, the limit cycle is obtained bythe bifurcation theory.