中文摘要：

本文在齐次Neumann边界条件下研究了一类捕食-食饵模型正平衡解的稳定性与存在性。首先，我们利用算子谱理论得到了正常数平衡解的一致渐近稳定性，其次，运用最大值原理和Harnack不等式，我们给出了正平衡解的先验估计，再次，利用积分的性质并结合ε-Young不等式和Poincar′e不等式，文中证明了非常数正平衡解的不存在性，最后，利用Leray-Schauder度理论证明了非常数正平衡解的存在性，并且给出了正平衡解存在的充分条件。研究结果表明，当参数满足一定条件时，两物种可以共存。

英文摘要：

The stability and existence of positive steady-state solutions for a predator-prey model are studied under homogeneous Neumann boundary condition. Firstly, the global asy-mptotic stability of positive constant steady-state solution is obtained by means of spectrum theory. Secondly, the priori estimates of positive steady-state solutions are given by applying the maximum principle and the Harnack inequality. Thirdly, the non-existence of the non-constant positive steady-state solutions is proved through the integral property,ε-Young inequality and Poincar′e inequality. Lastly, the existence of non-constant positive steady-state solutions is investigated with the help of the priori estimates and Leray-Schauder degree theory. Moreover, the su？cient conditions for the existence of positive steady-state solutions are obtained. The results show that when the parameters satisfy certain conditions, two species will coexist.