Abstract: In this paper，existence of positive steady-state solutions for a predator-prey system with predator cannibalism under Neumann boundary condition is discussed.By means of calculating the indices of fixed points of compact operators in cones，in combination with the maximum principles and spectrum analysis of operators，the sufficient conditions for nonexistence and existence of positive steady-state solution are obtained.Finally，the bifurcation from positive constant solution is considered with diffusion coefficient d2 as bifurcation parameter.

摘要： 运用谱分析、拓扑度理论和分歧理论的方法，在齐次Neumann边界条件下，研究了一类具有种内相食现象的捕食模型非常数正解的存在性。讨论了正常数解的稳定性，给出了正解的先验估计，讨论了非常数正解的存在性，以扩散系数d₂为分歧参数，讨论了发自正常数解的分歧。