The Study Of Lotka-Volterra Predator-Prey Model With A Variable Coefficient

The bio-mathematical models are established by some real problems in ecology,and it reflects the interdependent relationships between various species. Thus, the bio-mathematical models have its abundant real backgrounds and the extensive application values. Especially, for this reason of the global rapid development of the industrialization, the environment where the species are living on has suffered the fearful destruction and changes. The destruction and changes of the environment will impact on the natural growth rates of the species, the food sources and its distributions, the range of the activity of the species, and the interdependent relationships of the species. At the same time, tremendous international trade has also brought about a large number of alien species, and these alien species are likely to have many impact on native species. As a result, the bio-mathematical models have been widely concerned by biologists and mathematicians.In this paper, we investigate the Lotka-Volterra predator-prey model with variable coefficients. This model implies that the predator can be with crowding term in a sub-region, and in addition, this system is often used to study the control to the alien species. In the model, the prey is with homogeneous Robin boundary condition, and the predator is with homogeneous Neumann boundary condition.In this paper, we mainly study the effect of the degenerated sub-region on the positive solutions of the predator-prey model.This paper is mainly divided into the following five parts:Chapter 1, Introduction, which includes the significance and the current situation of the research for the bio-mathematical models, and the main research contents of this paper.Chapter 2 is some preliminary knowledge.Chapter 3 is the necessary conditions for the existence of positive solutions of the predator-prey model.Chapter 4 is the existence of positive solutions of the predator-prey model.Chapter 5 is the local stability of positive solutions of the predator-prey model.Chapter 6 is the dynamic behaviors of the corresponding parabolic system.Finally, it is summary and it is some problems to be studied.