一类带交叉扩散项的捕食-食饵模型全局分歧研究

Research of Global Bifurcation of a Predator-Prey Model with Cross-Diffusion

研究了一类具有交叉扩散项的捕食-食饵模型在齐次Dirichlet边界条件下分歧正解的存在性.首先利用极大值原理得到正解的先验估计;接着,借助Crandall-Rabinowitz分歧理论,得到局部分歧正解的存在性;再将局部分歧延拓为全局分歧,从而给出捕食者与食饵在一定条件下可以共存的条件;最后,利用谱分析的方法得到了关于分歧解稳定性的一个条件.

In this paper,the existence of positive solution of the steady-state system for the predator-prey model with cross-diffusion is studied under the homogeneous Dirichlet boundary condition.First,by means of maximum principle,a priori estimate is established.Next,by the Crandall-Rabinowitz local bifurcation theory,the existence of local bifurcation solution is obtained.Then,resorting to the global bifurcation theory,the local bifurcation solution is extended to the global bifurcation solution,and the conditions under which the predator and the prey can co-exist are given.Finally,a condition for the stability of bifurcation solution is obtained by spectral analysis.