The Nature Of The Two Biological Model Of Coexistence Solutions

Reaction-diffusion system has now been widely used in biology, it often used to explain and predict ecological phenomena through research mathematical models, and thus on the ecological provide a reasonable solution to the problem program. It has a very important practical significance to equilibrium of ecology, ecological environments preservation and even saving the rare and precious creature on the brink of extinction. This paper is divided into two parts, mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equations. The existence of solutions and stability issues of two types of biological reaction system were discussed in this paper, one is competition model,and the other is a predator-prey-mutualist model.In section 1,we discuss the following competition model: By bifurcation theory and perturbation theory of linear operators,the local bifur-cation from the semitrivial solution(u*,0) is obtained by treating d as a bifurcation parameter. Moreover, global bifurcation can be also obtained. In other hand, some stability results of the local bifurcation solutions are obtained. Some sufficient con-ditions for the existence of positive solution are given.Moreover, stability of the local bifurcation solutions are obtained when I>O, on the contrary,unstability of the local bifurcation solutions can be obtained.In section 2,we discuss the following predator-prey-mutualist model First,the local stability of positive steady-solution is given by Maximum principle. Furthermore, the non-existence of non-constant positive steady state is analyzed. In the end, the conditions of existence of non-constant positive solutions for the steady-state system are proved through the degree theory and the sufficient conditions of existence of non-constant positive equilibrium solution is obtained by bifurcation theory.