Abstract: We consider a diffusive Leslie-Gower predator-prey model with Crowley-Martin type functional response in the spatially heterogeneous environment. First, the stability of the semi-trivial solutions is discussed by using the method of linearization analysis. The results show that the stability of the semi-trivial solution for extinction of prey is changes as the diffusion rate varies in the spatially heterogeneous environment. Secondly, the existence of coexistence solution is obtained by using the local bifurcation theory, and the direction of bifurcation and the stability of the local bifurcation solution are discussed.