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

考虑一类具有密度依赖扩散的竞争扩散系统，该系统描述了两个竞争种群群落的种群密度动力学行为，研究了当群落的栖息地在某个球的内部时该系统的径向对称定态解的分支结构.此时，定常解周围的局部分歧结构可以由被积函数为带正权函数的第一类Bessel函数的立方的积分的值来确定。在本研究中，当生境的维数不大于3时，我们利用数学方法和数值验证的方法来确定积分值的符号。本文的结果不仅适用于竞争扩散系统，而且也适用于反应扩散系统的定常解附近的局部分歧结构。

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.