Nonconstant Steady States in a Predator–Prey System with Density-Dependent Motility

We investigate the existence, structure and stability of the nonconstant steady states for a predator–prey system with density-dependent motility under the Neumann boundary condition. By applying the Leray–Schauder degree theory, we show that under certain conditions, a small prey diffusion rate can ensure the existence of the nonconstant steady states, which is verified by numerical simulations. Over 1D domain, we treat prey diffusion rate as a bifurcation parameter and obtain the local and global structure of steady states near the homogeneous steady states with the aid of bifurcation theory and index theory. Moreover, a stability criterion of the bifurcating steady states is also presented. Finally, we give the existence and stability of time-periodic nontrivial solutions.