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

本研究考虑了一个具有密度依赖扩散的反应扩散方程组，该方程组描述了两个竞争物种群落的种群动态，我们打算通过研究该系统的平稳解的存在性和稳定性来了解共存的机制。一般来说，利用比较原理和能量可以比较容易地确定标量反应扩散方程的全局吸引子。函数，使我们能够精确地理解方程解的渐近行为。然而，由于比较原理并不总是适用于系统，因此我们讨论系统平稳解的分岔结构具有相当的复杂性。在本研究中，我们考虑了在某一参数的某些值处比较原理成立的系统，并利用比较原理和分岔理论等数学方法，以及Mathematica内置的区间算法等数值验证方法，研究了该参数下平稳解的全局分岔结构。结果表明，在一定条件下，系统平稳解的全局分岔结构与Chafee和Infante（1974/75）所示的标量反应扩散方程的分岔结构相似。此外，在整个研究过程中，再次认识到数值验证对于分析分岔方程的性质是有效的。由于在某些区域的数值验证不成功，我们到目前为止还没有确定这些区域的定态解的分岔结构。在未来，有必要对数值验证算法及其编程进行改进。

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.