中文摘要：

利用偏微分方程研究生物种群动力学,已成为非线性偏微分方程研究领域中的一个重要研究方向.针对具体的捕食模型,一个关键因素是响应函数.主要考虑了一类带有Sigmoidal型响应函数的捕食模型的Dirichlet边值问题,首先利用上下解方法给出了正解的先验估计,进而借助于锥上的拓扑度理论和极值原理,讨论了正平衡态解的存在性,并且得出了共存解存在的一个充分必要条件.另外,还应用分支理论研究了共存解的分支.

英文摘要：

The dynamics of biological models have received intensive study,and it has been an important aspect in the field of nonlinear partial differential equations.In the specific predator-prey model,a key factor is the response function.In this paper,we present a predator-prey model with Sigmoidal functional response under homogeneous Dirichlet boundary.First,we give a priori estimates of positive steady-states solutions by using the method of upper and lower solutions,and then by means of the topological degree theory in cones,combing with maximum principle,we obtain some results on the existence of positive steady states.Moveover,a necessary and sufficient condition for the existence of positive solutions is given.Besides,the bifurcation of positive solutions is investigated with bifurcation theory.