中文摘要：

研究了一类具有扩散的捕食食饵模型的平衡态正解。利用极值原理得到正解的先验估计。通过局部分歧理论给出了局部分歧解的存在性。运用全局分歧理论证明局部分歧解可以延拓为全局分歧解,并得到了全局分歧解的走向,从而得到了正解存在的充要条件。利用稳定性理论研究了局部分歧解的稳定性。最后通过数值模拟验证和完善已得到的理论结果。

英文摘要：

The steady state positive solutions of a predator-prey model with diffusion are studied. A priori estimate for positive solutions is obtained by using maximum principle. Applying to the local bifurcation theory,the existence of local bifurcation solutions is given. It follows from the global bifurcation theory that the local bifurcation solution can be extended to global bifurcation solution and the trend is obtained.Then a necessary and sufficient condition for the existence of positive solutions is obtained. By the stability theory,the stability of local bifurcation solutions is investigated. Finally,the theoretical results are verified and complemented by the numerical simulation.