一类具有交叉扩散的捕食-食饵模型正解的存在性

本文研究一类带有交叉扩散的捕食-食饵模型正解的存在性．首先，利用最大值原理得到了与交叉扩散系数无关的正解的先验估计；其次，建立了当交叉扩散系数充分大时的极限系统；最后，利用局部分歧理论得到了极限系统在半平凡解附近的局部分歧解的存在性，借助全局分歧理论说明了该极限系统的局部分歧解可以延拓为全局分歧解，并且该全局分歧解随着分歧参数在正椎内延伸至无穷．结论表明：当交叉扩散系数充分大时，两物种可以共存．

Existence of Positive Solutions for a Predator-prey Model with Cross-diffusion

The existence of positive solutions to a predator-prey model with cross-diffusion is considered. First, we derive some estimates which are independent of cross-diffusion by the maximum principle; second, the limiting behavior of positive solutions for large cross-diffusion is established; finally, we show the existence of local bifurcation solutions of the limiting system near the semi-trivial solution by the local bifurcation theorem, and we extend the local bifurcation solutions to the global bifurcation solutions by the global bifurcation theorem, and we show that the global bifurcation solutions can extend to infinity as the bifurcation parameter approaches infinity. The results demonstrate that the two species can coexist when the cross-diffusion is large.