Bifurcation structure of stationary solutions for a reaction-diffusion system with density-dependent diffusion

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We consider a competition-diffusion system with the density-dependent diffusion, which describes the dynamics of the population density for a two competing species community, and we study the bifurcation structure of radially symmetric stationary solutions of the system for the case where the habitat of the community is the inside of a certain ball. At this time, the local bifurcation structure around the constant stationary solution can be determined by the value of the integral whose integrand is the cubic of the Bessel function of the first kind with the positive weight function. In this research, when the dimension of the habitat is not bigger than 3, we determine the sign of the value of the integral by employing the mathematical method and the numerical verification method. The result of this research is applicable for determining the local bifurcation structure around the constant stationary solution to not only the competition-diffusion system but also the reaction-diffusion system.