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

在这项研究中，我们考虑了一个具有密度相关扩散的反应扩散方程组，它描述了两个相互竞争的物种群落的种群动力学，我们试图通过研究该系统的定态解的存在性和稳定性来理解共存的机制，通常，我们利用比较原理和能量相对容易地确定了标量反应扩散方程的全局吸引子。函数，这样我们就可以了解方程解的精确渐近性态。然而，由于比较原理并不总是适用于系统，因此讨论系统的定常解的分叉结构具有相当的复杂性.在本研究中，我们考虑了比较原理在某个参数的某个值成立的系统，并利用比较原理和分叉理论等数学方法和数学中内置的区间算法等数值验证方法研究了关于该参数的定常解的全局分叉结构.结果表明，在一定条件下，该系统定常解的全局分支结构类似于Chafee和Infante(1974/75)所给出的标量反应扩散方程的全局分支结构。此外，在整个研究过程中，我们再次认识到数值验证对于分析分支方程的性质是有效的，由于数值验证在一些区域没有成功，我们到目前为止还没有确定这些区域内定常解的分支结构。在今后的工作中，还需要对算法进行进一步的改进，以便于数值验证和编程实现。

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.