中文摘要：

在齐次Dirichlet边界条件下，研究一类低密度食饵下，捕食者具有自控能力的捕食模型平衡态正解存在性。通过连续延拓意义下建立的连续算子，利用度理论给出了平衡态正解存在的充分条件，并对理论结果进行数值模拟。研究结果表明，只要捕食者和食饵的生长率适当大，则捕食者和食饵可以共存。

英文摘要：

The existence of steady-state positive solutions for a predator-prey model with low density prey and self-limitedpredator is studied under the homogeneous Dirichlet boundary conditions. Using the continuous operators established bycontinuous extension, a sufficient condition for the existence of steady-state positive solutions is given by the degree theory.Furthermore, the theoretical results are simulated by numerical method. The research shows that, the predator and preycan coexist as long as the growth rates of the predator and prey are large suitably.