中文摘要：

本文研究了一类具有扩散的Leslie-Gower模型.利用谱分析和稳定性理论得到了两个正常数平衡态的局部稳定性;利用最大值原理、Harnack不等式和能量积分的方法得到了正稳态解的上下界估计和非常数正解的存在性;利用单特征值分歧理论研究了系统发自两个正常数平衡态的解分支,得到了非常数正解的存在性;利用Hopf分歧理论,得到了在平衡解处Hopf分歧的存在性.

英文摘要：

A Leslie-Gower model with diffusion under homogeneous Neumann boundary con- dition is investigated in this paper. Firstly, the local stability of the constant steady-state is obtained by using the theory of spectral analysis and stability; secondly, by using the maxi- mum principle, Harnack inequalities and energy method, the estimate of the upper and positive lower bound and nonexistence of nonconstant positive steady-state solution are obtained; by further applying simple eigenvalue bifurcation theory, the local bifurcations from the two constant solu-tions and the existence of nonconstant steady-state are given; finally, the constant solution where Hopf bifurcation occurs is obtained by using Hopf bifurcation theory.