中文摘要：

研究了一个带扩散作用的Lotka-Volterra捕食模型。使用椭圆方程的相关理论、非线性泛函分析的指数理论和度理论，在一定条件下，获得稳态模型正解的存在性和非存在性，即物种共存是否存在。

英文摘要：

A Lotka-Volterra prey-predator model with diffusion terms is discussed. By using some theories of the elliptic equations, the index theory and the degree theory of the nonlinear functional analysis, the existence and non-existence of its positive solutions under some conditions are obtained, i.e. whether or not has the species coexistence.