Algebraic Conditions And Dynamical Properties Of The Seven Dimensional Stably Dissipative Systems

Lotka-Volterra systems are one of the most classical and important systems in mathematical biology research field, which were initially independently proposed in the 1920s by the American population expert Lotka when researching chemical re-action and the Italian mathematician Volterra when studying fish competition. After decades of research and development, the application of Lotka-Volterra systems has al-ready been extended to different fields, not only applied in physics, chemistry, biology, population, economics and other social science, but also widely used in many popular subjects, such as neural networks, biological reaction, cell evolution, resource manage-ment and epidemiology. In research field, Lotka-Volterra systems are often divided into three categories, namely:cooperative type (or competitive type), conservative type and dissipative type. We have made a lot of achievements for each kind of system. Rela-tively, researches for dissipative systems are less than that for the other two, and as data are uncertain in practical applications, many scholars have transferred their research di-rections to stably dissipative systems. One of the important characters is that if a stably dissipative LV system has an equilibrium point then it has a global attractor. People have drawn a lot of conclusions for the sufficient and necessary conditions of lower di-mension stably dissipative Lotka-Volterra systems, but for high dimensional condition, which are comparatively less. As the dimension increasing, LV systems become more and more complex, and the contents for research will become increasingly richer. In order to study the systems of relatively high dimension many old methods for lower di-mension case must be amended and some new methods should be established. A brief discussion and analysis of the properties of the seven dimensional stably dissipative systems will be made in this paper, and more research work expect to be finished by the scholars in the similar field. The whole paper is composed of four parts:The first part is the introduction of the whole paper, in which some specific mathematical models for Lotka-Volterra systems are introduced, and several commonly used concepts and conclusions are also stated in order to prepare for the detailed discussion of the next three parts. In the the second part, according to the nature of stably dissipative matrix and the definition of maximal stably dissipative graph, and with careful screening, we deal with the topological classification for all of the seven dimensional stably dissipa-tive Lotka-Volterra systems using maximal stably dissipative graph, and we gain the first important conclusion of this paper. In the third part, based on the topological clas-sification of the former part, we analyze the algebraic conditions for the Lotka-Volterra systems which corresponds to some kind of maximal stably dissipative graph being sta-bly dissipative. The main thought is that according to the number of black vertex, we divide the maximal stably dissipative graphs into several types for research, and draw the corresponding conclusion. In the fourth part,by using the principle of reducing we transform the maximal stably dissipative graphs got in the second part into their respec-tive final style. According to the characteristics of the reduced graphs, we divide them into four categories and give the macroscopic conclusion. Then, with the assumption that there is a positive equilibrium point in the system, we choose several systems to which the maximal stably dissipative graph corresponds and make detailed analysis of dynamical properties.