Coexistence states of a predator-prey model with cross-diffusion

This paper is concerned with positive solutions of a predator-prey model with cross-diffusion. By virtue of the Leray-Schauder degree theory and the bifurcation theory, some sufficient conditions for the existence of positive solutions of the system are established. In particular, we derive the multiplicity results when some parameters are suitably large. Moreover, the existence and stability of nonconstant solutions are studied by the techniques of space decomposition and the implicit function theorem. Finally, the uniqueness of positive solution is studied when the spatial dimension is one.