Existence Of Periodic Solutions For Impulsive Ecosystem

The theory of impulsive differential is one of important branches of differential equations and mathematics theory. In the field of modern applied mathematics, it has made considerable head way in recent years, because all the structure of its emergence has deep physical background. As an important branch of impulsive differential theory, periodic solutions and periodic boundary value problems have always been a very active field which attract many scholars’researching, and have acquired many copious results.This thesis of Master is composed of three chapters, which mainly studied the existence of periodic solution for nonlinear impulsive differential equation. Using two methods, one is Poincare map method, another is bifurcation method.In the first chapter, we first introduce the development of impulsive differ-ential theory and the latest advancement, and study some results of impulsive periodic boundary value problems, which given by some experts in the inner and outer country.In the second chapter, we discuss Holling-IV predator-prey system. The sufficient conditions of existence and stability of semi-trivial solution and positive period-1 solution are obtained by using the Poincare map and analogue of the Poincare criterion.In the third chapter, we use bifurcation methods research the existence of periodic solution for nonlinear Holling-Ⅱ predator-prey system. Through the s-tandard bifurcation theorems, we can out of branches of solutions. Then we use the homotopy analysis method, we can get the existence of periodic solution.In the last, we make a conclusion of this paper and give a plan of the future work.