Dynamical Behaviors Of Five-Order Stably Dissipative Lotka-Volterra Systems

This paper considers the Lotka-Volterra system which consists of the following ordinary differential equations:The application of the above system covers a wide range of areas. At present, it not only is widely used in many fields such as physics, chemistry, biology, evolutionary game theory, economics and other social sciences, but also plays a vital role in many popular subjects as neural networks, biochemical reactions, cell evolution, resource management and epidemiology. What's more, it also becomes one of the important equation models in applied mathematics.As is well-known, there is a close relationship between the dynamics properties of Lotka-Volterra system and the algebraic properties of their interaction matrix A= (αjk). Hence, according to the different properties of the interaction matrix A=(αjk), the Lotka-Volterra system can be classified as follows:cooperative(resp. competitive) ifαjk≥0(resp.αjk≤0) for all j≠k;conservative 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, in the sense of quadratic forms, AD≤0. In the practical application of the Lotka-Volterra systems, because the date of inter-action matrix cannot be completely accurate, it's necessary to consider adding the small perturbation to the interaction matrix. Hence, the stably dissipative concept comes into being. In comparison with the research on other classes of systems, however, study-ing the stably dissipative ones is relatively rare, and particularly the high situational one is even rarer. Therefore, this paper mainly discusses the algebraic necessary and sufficient condition and dynamic property analysis of the five-order stably dissipative Lotka-Volterra systems.Firstly, we review the relevantly previous knowledge and results of the stably dis-sipative Lotka-Volterra systems and generalized Hamilton systems. Then using the scholars'criteria, we describe the algebraic necessary and sufficient condition of the stably dissipative matrices corresponding to 27 classes of the maximum stably dissipa-tive graph.Based on the scholars'research to the dynamics behavior of five-order stably dissi-pative Lotka-Volterra systems, which have been divided into four categories, and type I has already been discussed clearly. Therefore, this paper mainly aims at discussing the latter three types, especially type IV, of which the dynamics properties are quite com-plex. Existence of periodic solution, invariant torus and Hamiltonian chaos are more deeply analyzed and studied mainly by these theories and methods such as Lyapunov sub-center theorem, perturbation theory, Poincare section, Lyapunov exponents and so on.