Classification And Dynamics Of Four-Order Stably Dissipative Lotka-Volterra Systems

This paper discusses the Lotka-Volterra equations as follows Lotka-Volterra equations are widely used in many fields such as physics, chemistry, biology, evolutionary game theory, economics and other social sciences, and become a important model in applied mathematics. When they are studied, they always be classified three types according to their interaction matrix:·cooperative(resp.competitive) if aij≥0(resp.aij≤0) for all i≠j;·conservation if there exists a diagonal matrix D>0 such that AD is skew-symmetric;·dissipative if there exists a diagonal matrix D>0 such that AD≤0. when the Lotka-Volterra equations are used, the date of interaction matrix can not be precision, so the small perturbation should be considered. A matrix is stably dissipative, if it is dissipative, and remains dissiaptive at the small perturbation, and the Lotka-Volterra equation corresponding to the matrix is a stably dissipative system.The majority of the results are all about cooperative(competitive) systems, the disspative ones are relatively small, and the stably dissipative ones are few and far be-tween. This paper discusses classification and dynamics of four-oder stably dissipative Lotka-Volterra systems. First, we introduce the concept of the stably dissipative Lotka-Volterra equations and some former results. Then describe the Maximum stably dissipative graph by graph theory combined with the features of stably dissipative matrices, as well as the classi-fication graph methods of Maximum stably dissipative matrices. Based on this, for all four-order stably disspative matrices, the associated graphhs are classified completely as 11 topologically different graphs. Also, the necessary and sufficient condition of stably disspative matrices for each graph is given.Then the dynamics of the systems corresponding to the graphs are studied. The results show that according to the limit dynamics of the system, they can be divided into four categories:Global asymptotic stability to the equilibrium point; phase space with foliation and the solution of different foliation is asymptotic stability to the equilibrium point of the asymptotic stability to the equilibrium point of the foliation; phase space with foliation and the solution of different foliation is all periodic; periodic solutions, invariant torus and hamilton chaos coexist in the phase space.In oder to in-depth analysis the fourth category, a chain Lotka-Volterra system is detailed studied. Existence of periodic solution, invariant torus and hamilton chaos are more in-depth analysed by Lyapunov sub-center thorem, perturbation theory, poincare section, Lyapunov exponent and so on. the dynamics of this system is more completely understanded.