Qualitative Analysis On A Class Of Reaction Diffusion Equations

Since the20th century, Biological model was widespread concern. Lots ofresearchers paid special attention to a kind of predator-prey model. The interactionsof predator-prey model are always a focus of the research work. Biology is thebackground of the reaction diffusion equations. It is common to explain biologicalphenomenon with mathematics method. Reaction diffusion equations have been animportant branch of PDE in the field of nonlinear partial differential equations. It isimportant to research the stability of the normal balance solution of a parabolicequation and the existence of the coexistence solution of elliptic equations. Theresearch has important practical significance. This paper mainly concerns thequalitative property for some nonlinear differential equations with Beddington-DeAngelis functional response. The main work includes the stability of the odemodel, the stability of the normal balance solution of the reaction diffusionequations and the existence of the coexistence solution of elliptic equations. weintroduced the background and history about the related work and summarized themain work，and discussed the balance of stability solution of the ordinarydifferential equation, especially the stability of the positive equilibrium solution. Weconsidered the parabolic equations with the homogeneous Neumann boundarycondition in turn. We used iteration method and Lyapunov function to prove thestability of the normal balance solution. At last we discussed the elliptic systemswith the homogeneous Dirichlet boundary condition. By virtue of topologicaldegree method and the supper and lower solution method, we got the coexistencesolution of elliptic equations.