Steady states of a diffusive predator-prey model with prey-taxis and fear effect

In this paper, a diffusive predator-prey system with prey-taxis response subject to Neumann boundary conditions is considered. First, the stability of the positive equilibrium and the conditions for the occurrence of Hopf bifurcation are provided. Second, the existence of nonconstant steady states is proved by applying the the bifurcation theory of Crandall and Rabinowitz, bifurcation types and stability of the bifurcation solutions are analyzed. Finally, numerical simulation results are carried out to verify the theoretical results.