Bifurcation structure of positive stationary solutions for a competition-diffusion system and its numerical verification

我们试图通过研究描述两种群竞争种群密度动态的竞争扩散系统正定态解的存在性和稳定性来理解共存的机制。在以前的研究中，对于群落的栖息地是区间的情况，我们研究了正平稳解的空间轮廓和特征值的分布，然后建立了系统正平稳解的全局分支结构。为此，我们采用了分叉理论和比较原理等数学方法，以及数值计算和数值验证等数值方法。到目前为止，似乎还没有得到足够的结果，因为栖息地一般是二维或三维的区域。在本研究中，我们假设栖息地是球的内部，并试图研究该系统的径向对称正平稳解的分支结构，但这种情况下正平稳解的性质很难被研究，因此关于正平稳解集的信息不足以建立该分支结构。然而，可以证明单调正定解的集合被表示为某一函数相对于球原点的正定解的值的图形。此外，从数值验证中可以看到，系统发生了鞍结型二次分叉。这些事实为我们理解分叉结构提供了线索。在未来，需要解决一些开放的问题，如每个正平稳解具有什么样的空间轮廓，存在什么样的完整正平稳解等。

We intend to understand the mechanism of the coexistence by studying the existence and stability of positive stationary solutions for a competition-diffusion system, which describes the dynamics of the population density for a two competing species community. In earlier studies, for the case where the habitat of the community is an interval, we investigate the spatial profile and the distribution of eigenvalues to the positive stationary solution, and then we establish the global bifurcation structure of positive stationary solutions for the system. To do this, we employ mathematical methods such as the bifurcation theory and the comparison principle, and numerical methods such as the numerical computation and the numerical verification. It seems that enough results have not been obtained so far, because the habitat is in general a two- or three-dimensional region. In this research, we assume that the habitat is the inside of a ball, and try to study the bifurcation structure of radially symmetric positive stationary solutions for the system.It is hard to investigate the property of positive stationary solutions for this case, so that the information on the set of positive stationary solutions is not obtained enough to establish the bifurcation structure. However, it could be shown that the set of monotone positive stationary solutions is represented as the graph of a certain function with respect to the value of the positive stationary solution at the origin of the ball. Moreover, we see from the numerical verification that the secondary bifurcation of saddle-node type occurs. These facts give us a clue to understand the bifurcation structure.In the future, it will be necessary to solve some open problems such as what spatial profile each positive stationary solution has, what kind of entire positive stationary solution exists, and so on.