The Toological Classification Of Two-Dimensional Lotka-Volterra System And The Global Behavior Of Nonautonomous/Random Monotone Systems

In the thesis, we give the topological classification on the global phase portraits of two-dimensional Lotka-Volterra system, and study the asymptotic behavior for almost periodic systems comparable to strongly monotone systems and the global attractivity of monotone random dynamical systems.The paper is organized as follows:In the introduction, the origin and main studying contents of Lotka-Volterra system and monotone dynamical systems are presented. We mainly introduce the development of the research on almost periodic systems, random systems and the global behavior of monotone dynamical systems.In Chapter 2, we first give the sufficient and necessary conditions for two-dimensional Lotka-Volterra systemto have closed orbits, and discuss the normal forms for Lotka-Volterra system. Then, using Poincare compactification, we study the properties of infinite critical points and give a classification on them. In the following, based on the results of the sufficient and necessary conditions for Lotka-Volterra system with closed orbits, the properties of finite critical points are discussed. All global phase portraits are finally classified by combining all local information.In Chapter 3, Firstly, we prove that on the state space possessing lattice structure, if every forward orbit of the strongly monotone order-compact skew-product semiflow has compact closure and is uniformly stable, then on any fiber the set of all 1-covers of the base space is either a singleton or homeomorphic to [0,1], [0,1), (0,1] or R, and the homeomorphism is order-preserving. Furthermore, if on each fiber the set of all 1-covers is homeomorphic to R, then we get that any minimal set of the comparable skew-product semiflow is also a 1-cover of the base space.In Chapter 4, we study the global attractivity of monotone random dynamical systems. Firstly, we study the properties ofω-limit set of the pull back trajectories, and show that it is an invariant random compact set measurable with respect to F~u. Basing on this, we prove that the unique equilibrium of the system is globally attractive if every pull back trajectory has compact closure. Finally, the result is used to study a class of sublinear monotone RDS and random parabolic equations of Fisher type.