Classification Algorithm Of N-Dimensional Stably Dissipative Lotka-Volterra Systems

Lotka-Volterra system is one of the most important model in applied math-ematics. It’s widely used in physics,biology, chemistry, economics and other social sciences. Nowdays,the system and the research methods are often applied in neural networks, biochem-istry reactions, cell evolutions and so on. In this thesis,The Lotka-Volterra system is classified into three types:cooperative, conservative, dissipative.This thesis is concerned with the classification algorithm and the dynamical properties of the maximum stably dissipative system. It is divided into four chapters.In the first chapter some background and research results of Lotka-Volterra system’s some properties of stably dissipative matrix are introduced, and proofs of related theories and lemmas are given in chapter2.In the third chapter by using the improved AHU isomorphism algorithm which is on the basis of the graph theory, the classification algorithm of the maximum stably dissipative graph to the n-dimensional Lotka-Volterra system is obtained.When n=6,there are76types of the maximum stably dissipative graph, which corrects the result,73types in the literature [2].When n=7, there are235types of the maximum stably dissipative graph, which corrects the result,229types in the literature [28].When n=8, there are769types. The fourth chapter discusses the asymptotic stability of the Lotka-Volterra systems for the maximum stably dissipative graph as n≤7, and defines the minimum stably dissipative graph, classifies the dynamical properties of the Lotka-Volterra systems which is related to the minimum stably dissipative graph. In the end, there comes to the conclusion,if a system reduced is a non-chain system of (·) type, then the corresponding minimum stably dissipative system is also a (·) type after reduced.