Stability Analysis For Some Kinds Of Reaction Diffusion Equations With Delays

The system described by the delay reaction diffusion equations not only depends on the present states, but also the past states, and it is because the existence of the time delay, what makes it more objectively in describing the actual problem, and more accurately in the solution of the problem. Delving into the dynamic characteristics of the delay reaction diffusion system has important significance to understand these equation itself, and can promote the research of other field. In summary, studying the delay reaction diffusion equations has important theoretical and practical value.This paper study two kinds of different types of delayed reaction diffusion equation with the linear theory and Hassard standard theory, specification of the center manifold theory and Wu Jianhong’s thoughtway. The results were obtained as follows on the two kinds of model research:We study the stability and Hopf bifurcation of autocatalytic reaction diffusion equation with time delay in the Neumann boundary value conditions in chemistry. We obtain the conditions when the stability and the Hopf bifurcation occurs. We use the theory of center manifold and canonical form to discuss the branching direction, stability of periodic solutions of a branch and the branch cycle of change rule. Finally, numerical simulation is carried out. We also analyze a degenerate epidemic model with time delay by Neumann boundary conditions. By considering time delay as bifurcation parameter, the periodic solution occurs when delay passes through a sequence of critical values. Furthermore, the direction and stability of the Hopf bifurcation are discussed by Wu Jianhong’s method. Finally, numerical simulation is carried out. Graphics are in accordance with analysis results.