Traveling Wave Solutions Of The Reaction Diffusion System Estimates Of The Wave Speed

In the developing of modern science and technology, disciplinarian veracity is an important assurance for their development. The disciplinarian veracity is realised often by establishing mathematical models. Reaction diffusion equations are a common model in the description of nonlinear systems such as physical phenomena, chemical processes, ecological systems. At the present time a large number of papers are devoted to reaction diffusion equations and this number continues to increase. In the research of reaction-diffusion equations, the existence, uniqueness, stability, and estimate of wave speed of the traveling wave solutions have developed.This article discusses mostly the existence of traveling wave solutions in some coupled reaction-diffusion models. We used the main method which is the upper and lower solutions method (or monotone method). The monotone method is a very effective method in dealing with the existence of traveling wave solutions in some material reaction-diffusion equations. And the key of monotone method is constructing appropriate upper and lower solutions. The reaction-diffusion systems which can be dealed with monotone method have following feactures, their reaction terms must have some quasi-monotonicity. As we know the traveling wave solutions exist in the systems with quasi-monotone increased reaction terms. Using this theorem, we proved the existence of traveling wave solutions in cooperative Lotka-Voterra system with stage structure. But if the reaction terms are not all quasi-monotone increased, it equate that some reaction terms are quasi-monotone increased, and others are quasi-monotone decreased, the problem will be complicated. In this paper we use the monotone method, and the iterative initial values are chosen as the appropriate upper and lower solutions . Then, we pass the limits. It can be proved the limits are quasi-solutions of the system. Though quasi-solutons are not real solutions, under some conditions there is at least a real solution between the quasi-solutions.