The Dynamics Of Cooperation Between Two Species Under The Influence Of Infectious Diseases

This dissertation is concerned with the dynamics of cooperation between two species under the influence of infectious diseases. It is well known that we have a relatively complete understanding to the dynamics of two-dimensional cooperation or competition between two species. Various models have been established to research them. But the understanding to the dynamics of high-dimensional systems with time delay factors is less than that of two-dimensional because of the complexity of its dynamic behavior. The basic theory and method also need to be improved and enhanced. In this dissertation, we obtain some dynamic conclusions of high-dimensional systems through studying the dynamic behavior of the slow variable systems.The high-dimensional system is divided into the fast variable system and the slow variable system in the dissertation. The fast variable system is composed of the number of species and the slow variable system is composed of the proportion of the occupied patches. Supposed that the manifold determined by the fast variable system is normally hyperbolic,we can obtain some dynamic conclusions of high-dimensional systems through studying the dynamic behavior of the slow variable systems on the manifold by using geometric singular perturbation theory and methods of dynamics.This dissertation is divided into four chapters.The first chapter introduces the development and meaning of ecological stability and metapopulation and the works done in the thesis.Chapter two, we introduce the main theories and research methods.Chapter three, we study a model of cooperation between two species under the influence of infectious diseases. We discuss the equilibrias on the coordinate axes and when it is the origin. A necessary condition is given when there are four interior equilibrias. The conditions that two equilibrias are nodes are also given. We prove that there are neither periodic orbits nor homoclinic loops in the interior and give several phase portraits.Chapter four, some valuable significance is obtained based on the theoretical results.The research method used in this dissertation is that: we translate high-dimensional systems into two-dimensional systems by using geometric singular perturbation theory and obtain some dynamic conclusions of high-dimensional systems through studying the dynamic behavior of the slow variable systems.