Extinction And Permanence Of The Non-autonomous Predator-prey System With Impulses

Population dynamics is an important branch of biological mathematics. The main research of population dynamics is the characteristics of species and the intra-relations with the other species during its evolution. We establish the mathematical models which reflect their dynamical characteristics. These models can describe the evolution of species quantitatively. Further, we use the mathematical theories and methods to analyze the dynamical behaviors of populations, which shows and predicts the evolution laws of population, and supply the theoretical references when the policy of protecting the species and maintaining the balance of biological system will be enacted.In nature, population birth rate, mortality rate, intra-relation and inter-relation among populations are often influenced by season, climate and human activities. As a result, in the ecological models, we often use variable coefficient differential equations to describe evolution process of population. However, many processes are often dis-turbed at certain moments of time, which leads to instantaneous and great changes of population density in the form of perturbations. These perturbations are not negligible. Therefore, we may deal with the short time interference phenomenon as impulses. It is of more practical significance studying the variable coefficient differential equation with impulsive effects. The richer theory of impulsive differential equations have been ob-tained. Therefore, applying the theory to various field and investigating the dynamical behaviors of systems are more valuable.In this paper, we study the dynamical behaviors for three classes of non-autonomous predator-prey system with impulses. We obtained the sufficient conditions which guar-antee the permanence and extinction of the system.In chapter1, we introduce the developments of population dynamics system and research value of predator-prey system, and give the basic definitions and theories of this paper.In chapter2, we assume that the predator-prey response function is the ratio-dependent functional response. We establish a non-autonomous ratio-dependent predator-prey system with impulses and obtain the extinction and permanence of the system with the qualitative theory and stability of ordinary differential equations.In chapter3, based on the Holling functional predator-prey reaction function, we established the Holling Ⅲ functional reaction of nonautonomous predator-prey system with impulses. We use impulsive differential comparison inequality and the comparison theorem to discuss the conditions of the extinction and permanence of the system. Based on the permanence of the system, we discuss the two situations when p>1and p=1respectively, and we obtain the situations of permanence of the system in two cases.In chapter4, we summarize this paper and point out the shortcomings and the direction of future work.