中文摘要：

文中旨在研究一类带有交叉扩散项的捕食一食饵模型在齐次Dirichlet边界条件下局部分歧正解的存在性．利用极大值原理得到了正解的先验估计．采用Crandall—Rabinowitz分歧理论，得到了局部分歧正解的存在性．引理推导结果表明：模型正解不存在的充分条件是确定的，在一定条件下模型的正解是有界的，并给出了局部分歧正解存在的充分条件．

英文摘要：

The existence of positive solutions for a kind of predator-prey model with cross-diffusion under Dirichlet boundary condition is studied. The priori estimate of the positive solutions is obtained by using the maximum principle, and the existence of positive solutions is obtained with the help of Crandall-Rabinowitz bifurcation theory. The result shows that the sufficient conditions for the inexistence of positive solutions of the model are definited and that under certain conditions, the positive solution of the model is bounded. And also the sufficient conditions for the positive solutions emanated from the local bifurcation solutions are derived .