Permanence And Periodic Solutions For Several Kinds Of Impulsive Differential Equations

Population ecology is an important branch of ecology science. Since the complexity of ecological relations, mathematical methods and results have been used in and have emerged from ecology. Now population ecology has become the branch that mathematics is most deeply applied in and which is the most systematic one. The relationship between predator and prey is one of the basicrelationships among species. Recently, because of its wide applications, the predator-prey system has received a great deal of attention of mathematicians and biologists. The predator-prey system are considered systemically in this paper.This paper is composed of four chapters, which mainly studied the permanence,extinction and periodicity of solutions for several kinds of impulsive equations.In chapter 1, the background and history of periodicity solution problems and permanence for impulsive differential equations are briefly addressed, and the main work of this paper are given.In chapter 2, we studied the existence of periodic solution of the cyclic and predator-prey system of three species with Rolling's Typeâ…¢functional response and impulses.By using the comparability theorem for inequality, the Brouwer fixed point theoremand Lyapunove method, we obtain the sufficient conditions of the global asymptotic stability and uniqueness of periodic solutions. In chapter 3, we study the extinction and permanence of a prey-dependent consumption system with impulsive effect.By using an impulsive differential inequality, comparability theorem and Floquet theorem,we obtain a comparison result ensuring the permanence and extinction of periodic solution of the impulsive differential equation with periodic coefficients. This result extends some of the existing literature.In the last chapter, we study the permanence and existence of periodic solutionof the dynamic complexities of a periodic Hollingâ…¢two-prey one-predator system with impulsive effect.By using an impulsive differential inequality, comparability theorem, coincide degreetheory, and Brouwer fixed point theorem, we obtain the existence of positive periodic solutions for the impulsive equations and some sufficient conditions are established.