Dissipation And Dynamics Of The Lotka-Volterra Systems And Generalized Lotka-Volterra Systems

Lotka-Volterra systems have been one of the most important applied mathematical models. They are widely used in many fields such as physics, chemistry, biology, economics and other social sciences, etc. They also play a role in such diverse topics of current interest as neural networks, biochemical reactions, cell evolution, resource management and epidemiology. When they are studied, they can be classified three types: cooperative(resp. competitive), conservative and dissipative. For each of them, many results have been gained. Stably dissipative system is more important at the real meaning . The sufficient and necessary conditions for a LV system being stably dissipative and its characters have been discussed by many people. One of the important characters is that the stably dissipative LV system has a global attractor. If the attractor is a equilibrium point, the system is global asymptotically stable. On the basis of the results gained by predecessors, we will restrict our attention to problems on classification and dynamics of six-dimensional stably dissipative Lotka-Volterra systems. Also, we will extend the dissipation to the generalized Lotka-Volterra system. This paper is composed of three parts. The first part is the preface, some background knowledge of LV systems and related prepare knowledge is introduced. The second part is the first chapter and the third chapter, a graph-theoretic classification method for stably dissipative matrices are proposed, based on which, all six-dimensional stably dissipative matrices are classified completely as 73 classes. For the other classes of six-order stably dissipa- tive matrices, their graphs can be obtained from the above 73 classes by removing some links. We gain the reduced graphs of all the 73 classes by the reduce rules, and discuss the dynamics of the corresponding Lotka-Volterra system for each class. The third part is the forth chapter, it is discussed and presented the criterias for the dissipative and stable dissipative of the generalized Lotka-Volterra system. In other words, we discuss what conditions the matrices A and B are satisfied that the generalized Lotka-Volterra systems are dissipative or stably dissipative. Moreover, two examples were showed to illustrate that how the system parameters were specified so that the systems were dissipative. Compared with the dissipation of Lotka-Volterra system, the conditions of matrix A can be weaken in the generalized Lotka-Volterra system. It means that we can make some nondissipative Lotka-Volterra systems become dissipative systems by adding some parameters. There are examples to interpret it.