A Leslie-Gower model with diffusion under homogeneous Neumann boundary condition 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 maximum 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.