Kinetic Study Of Two Types Of Biological Population Reaction Diffusion Models

Diffusion and spatial environment have a significant influence on the total species, even re-lated to the survival of the species. The first model introduced in this paper is about the classical Lotka-Volterra competition-diffusion system. We mainly study the combined effects of diffu-sion, spatial environment and competition ability on the model. In fact, the competition ability of species will also change under different environments. So, spatial relativity is merged into com-petition ability. We aim at generalizing the constant competition ability to non-constant ability in previous models. Our results confirm that, under the condition of weak competition, assume that at least one of the spatial environment is homogeneous, the species can always overcome the other species when the infimum of the competition ability reached a critical value; When spatial heterogeneity with two species is included in the model, if the species can completely prevail over its competitor, the infimum of the competition ability reached a critical value is needed, and also needs its competitor is less competitive than a critical value.The flow of people have a great impact on the disappearance and spread of infectious dis-eases. In addition, timely treatment contributes to preventing the spread of diseases, but the treat-ment of an infectious disease is limited in each region. Therefore, diffusion and the saturated treatment are introduced into this model. Based on this, the second model introduced in this paper is the reaction-diffusion equation with saturated treatment under the heterogeneous space.Through analysis, we finally get the existing uniqueness of the disease-free equilibrium. And we also get the threshold of the epidemics through the principal eigenvalue, which is defined as basic reproduction number, used for discussing the stability of equilibrium solution. It shows that if R0 < 1, the disease-free equilibrium is locally stable. If R0 > 1, the disease-free equilibrium is unstable and there is a local endemic equilibrium.