Analysis Of The Solution Of The Two Types Of Biological Models

People usual solve practical problems by set up mathematical model.This method is the important direction of developing of applied mathematics.As the important branch of mathematics,by the research and exploration of lots scientists,partial dif-ferential equation become the important bridge of theory with practice,and,it is applied on many scientific system such as chemistry,physics and biologic. In the biological science,by constructing relationship of species to set up reaction-diffusion equation model to solve complex problems is applied very widely. This work studies two reaction-diffusion equations.In chapter l,we discuss the following diffusive pioneer-climax species model: whereΩis a bounded domain with smooth boundary.ai(i=1,2,3), ci(i=1,2) is all positive constant,and a3>a2.In the model,u and v is separate the growth density of pioneer and climax species.In the article,we will use diffusion coefficient as bifurcation parameter, the bifurcation at positive constant steady-state solution is obtained by bifurcation theory and degree theory,and the structure of solution near bifurcation point is obtained,and,it will discuss that the local bifurcation can be extended to global bifurcation.In chapter 2,we discuss the following diffusive HIV-1 model with homogeneous Neumann boundary condition:In the artical,uw/1+w is Holling-Ⅱ,and it is used to describe the growth rate of cell. In the work, the local stability of positive constant is obtained by Hurwitz theorem under some conditions, and when the number of free virus is more than enough, the global stability can be obtained by construct Lyapunov functional under some conditions.