Bifurcation and spatiotemporal patterns of a density-dependent predator-prey model with Crowley-Martin functional response

In this paper, we consider a diffusive density-dependent predator-prey model with Crowley-Martin functional responses subject to Neumann boundary condition. We analyze the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions through the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold theory. Finally, numerical simulations are given to verify our theoretical analysis.