Six-order Stably Dissipative Matrix And The Dynamics Of The Corresponding Lotka-Volterra Systems

In this paper,we study the famous Lotka—Volterra system with the following form:(?)From a biological point of view,as a connection model of biological species of n,xi ? 0 usually refers to the population density of the ith species,aij denotes action coefficient,?i indicates parameters of environmental correlation,A =(aij)is called the action ma-trix of Lotka—Volterra system.The application of the Lotka-Volterra system is much wide,and the system becomes one of the important equation models in the field of applied mathematics.Lotka-Volterra system have been tightly integrated with many popular disciplines,such as neural networks,biological response,cell evolution,resource management and the spread of the virus.Despite researches on low-dimensional Lotka—Volterra system and particular types of Lotka—Volterra systems achieved fruitful results,according to present situation,knowledge on general Lotka—Volterra system is not enough,their dynamics are far from being a complete understanding.According to the literature,we know that algebraic properties of the action matrix A ?(aij)determine the dynamic properties of the Lotka—Volterra systems.Due to the different algebraic properties of the action matrix A =(aij),Lotka—Volterra sys-tem can be divided into the following three categories:competitive(resp.cooperative),conservative and dissipative.At present,domestic and foreign scholars have made a wealth of research results on cooperation competition or conservative Lotka—Volterra system.But,the research results is relatively few to the dissipative systems,particular-ly for less research of stability dissipative systems.Under normal circumstances,in the practical application of the Lotka—Volterra systems the interaction matrix A =(aij)data acquisition is not completely accurate,hence one needs to consider the interaction matrix data error,thus,new concepts of perturbation for interaction matrix and stably dissipative matrix should be introduced.This master degree thesis focuses on the six-order algebraic necessary and sufficient condition under which a matrix is stably dissipative and dynamical properties of the cor-responding stably dissipative Lotka—Volterra systems.The thesis is divided into five chapters:Chapter ? and chapter ? introduce the research background and some preliminary knowl-edge to be used in the thesis.Chapter ?,based on the predecessors' study of six order maximum stable dissipative graphs,some faults in previous literatures are corrected,76 kinds of complete classification for six order maximum stable dissipative graphs are obtained,and algebraic necessary and sufficient condition under which six-order matrix is stably dissipative are given.Chapter IV,on the basis of the third chapter,for each of 76 types maximum stable dissipative matrices,we study on dynamics of corresponding Lotka—Volterra system.According to different situation of their reduced system,these Lotka—Volterra system can be divided into four classes,among them,properties of class ? and class ? has been discussed in literatures.Hence,this chapter focuses on the dynamic properties of the class ?.Chapter ?,for systems S(1),S(2)and S(5)of class ?,the authors of literature[29]only made a simple discussion,this chapter study in detail the dynamical behavior of S(1),S(2)and S(5).In addition,for the rest system S(4)and S(6)of class ?,their dy-namical properties,existence of periodic orbit and invariant torus,are also discussed by using Lyapunov center theorem and perturbation theory.