Asymptotic behavior of solutions for a system of reaction-diffusion equations with density-dependent diffusion.

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In this research, we consider a system of reaction-diffusion equations with density-dependent diffusion, which describes the dynamics of the population for two competing species community, and we intend to understand the mechanism of the coexistence by studying the existence and stability of stationary solutions for the system.In general, we comparatively easily determine the global attractor for the scalar reaction-diffusion equation by using the comparison principle and the energy. function, so that we can understand the precise asymptotic behavior of solutions for the equation. However, since the comparison principle does not always hold for the system, we have the considerable complexity for discussing the bifurcation structure of stationary solutions for the system.In this research, we consider a system such that the comparison principle holds at some values of a certain parameter, and we study the global bifurcation structure of stationary solutions with respect to such a parameter by employing the mathematical method such as the comparison principle and the bifurcation theory, and the numerical verification method such as the interval arithmetic built into Mathematica. As a result, it is shown that the global bifurcation structure of stationary solutions for the system under a certain condition is similar to that for the scalar reaction-diffusion equation shown by Chafee and Infante (1974/75). Moreover throughout this research, it is recognized again that the numerical verification is effective for analyzing the property of the bifurcation equation.Since the numerical verification does not succeed in some of regions, we have not determined the bifurcation structure of stationary solutions in such regions so far. In the future, it will be necessary to improve the algorithm for the numerical verification and its programming.