<p>This paper is concerned with the effects of the nonlinear diffusion and spatial heterogeneity of the environment on the dynamics of a Leslie–Gower predator–prey model. The linear stability of semi-trivial solutions is obtained by analyzing the sign of the associated principal eigenvalues. The existence of positive steady-state solution is shown by employing the degree theory on a positive cone. These existence results can be reconsidered from the viewpoint of bifurcation theory. The direction of bifurcation of positive solutions and their stability are characterized in detail. Moreover, we obtain the existence and stability of positive steady states bifurcate from two semi-trivial steady states with large cross-diffusion by using Lyapunov–Schmidt reduction. Finally, we present some numerical simulations to illustrate our main results.</p>