The Analysis On Some Lotka-Volterra Facultative Mutualism Systems With Different Delays

The persistence, existence, stability and global attractivity of periodic solutions of Lotka-Volterra systems are an important direction of mathematical ecology research. It has been an important topic to give a definite criterion of the properties of a complicated ecological system. Many ecologist and mathematician have paid much attention in this field. The study of the quality property of the solution for the nonautonomous facultative mutualism has a wide application.For ecological system, we should consider not only the growing principle of the species (i.e influence of the delay), but also the influence of the environment to the ecological system (i.e the coefficient is the function of time). Thus, we will study the nonautonomous ecological systems with delay.Using quality theory of differential equation, M-matrix theory, nonlinear method etc, we focus on some kinds of nonautonomous Lotka-Volterra facultative mutualism systems with delay, and discuss the system's persistence of solutions; the existence, the global attractivity of periodic solution in this paper.The paper is organized as follows:In the first chapter, we introduce the background of this article and its practical meaning.In the second chapter, we discuss the persistence, bound of solution about a kind of two dimensions system with different delays, the existence, the global attraction of periodic solution and give some sufficient conditions.In the third chapter, we discuss the persistence, bound of solution about a kind of three dimensions system with different delays and the existence, the global attraction of periodic solution.In the fourth chapter, we study the dynamical property of a kind of three dimension nonautonomous Lotka-Volterra facultative mutualism system with different delays and prove the existence of positive periodic solution of system.In the fifth chapter, we discuss a kind of n dimensions nonautonomous Lotka-V olterra facultative mutualism with distributional delay and give the sufficient conditions of the global attraction of positive periodic solution of system.