一类交叉扩散系统定态解的分歧与稳定性

为得到一类在交叉扩散效应下两种群相互竞争的生物数学模型的正定态解的分歧和稳定性，运用谱分析方法和分歧理论，首先对半平凡定态解的稳定性作出了分析，然后分别以生长率a和b为分歧参数，得到发自半平凡定态解的非平凡定态正解的存在性和稳定性．将以上结论用于具体的生物模型，发现当a和b在某个具体范围时，分别存在非平凡正定态解，文中同时证明了其渐进稳定的充要条件。

Bifurcation and Stability of the Steady-State Solutions to a System with Cross-Diffusion Effect

The aim of this paper is to investigate the bifurcation and stability of the positive steady-state solutions to a mathematical biology model of two competition species. These two species interact with each other under the cross-diffusion effect. In this investigation, the spectral analysis method and the bifurcation theory are employed to analyze the stability of the semitrivial steady-state solutions. Then, by respectively using the growth rates a and b as bifurcation parameters, the existence and stability of the nontrivial positive steady-state solutions from the semitrivial steady-state solutions are obtained. The above-mentioned results are finally applied to a specific biology model, with the conclusion that there are nontrivial positive steady-state solutions when a and b lie in some specific ranges. The necessary and sufficient conditions for the asymptotical stability of the solutions are also proved.