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

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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.