The Problem Of The Existence Of Positive Periodic Solutions Of Lotka-Volterra Ecosystem And The Problem Of Solutions Of D-Operator Neutral Functional Differential System

American mathematician Lotka(in 1925) and Italian mathematician Volterra(in 1926)put forward the famous Lotka-Volterra model. Their first model was put forward based on two specises predator-prey situation. With functional differential equations in the wide application of mathe-matical biology and the needs of ecology, Lotka-Volterra model with de-lays was established. According to the ecological meaning, Lotka-Volterra model can be divided into three classes:competition model, predator-prey model and mutually beneficial cooperation model. This paper we will con-sider those three kinds model's positive periodic solution of existence and stability. For further study of multi-dimensional D-operator type neutral functional differential equation, there are some literatures discussed the existence of periodic solutions of it, but most of them are based on that multi-dimensional differential operator D is stable to get the sufficient con-ditions for existence of periodic solutions of that. This paper study proper-ties of unstable multi-dimensional D-operator by using auxiliary operator; then using the Mawhin's coincidence degree theorem to obtain the exis-tence of periodic solution on the situation of multi-dimensional operator is unstable.This paper mainly research the problem of existence of positive peri-odic solutions to a 3-specise predator-prey Lotka-Volterra model, existence and stability of positive periodic solutions to n specises Lotka-Volterra model and new properties of D-operator and its applications on the prob- lem of periodic solutions to neutral functional differential system basing on Mawhin's coincidence degree theory, matrix theory and Lyapunov func-tional. Full text is divided into four chapters.In the first chapter, we introduce the background knowledge of Lotka-Volterra ecosystem and the current progress of neutral functional differ-ential equation. In addition, the main work and the main lemmas of this paper are presented.In the second chapter, by application of Mawhin's coincidence de-gree theorem, existence of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments is studied. It presents the new sufficient condition to it.In the third chapter, by using Mawhin's coincidence degree theorem, appropriate matrix theory and Lyapunov functional, we study new con-ditions on the existence and stability of positive periodic solutions for n-species Lotka-Volterra systems with deviating arguments, which is pop-ularized the previous work.In the last chapter, we first get the new properties of unstable D-operator by a auxiliary operator and then applicate it on the problem of periodic solutions to neutral functional differential system.