On global bifurcation for a cross-diffusion predator-prey system with prey-taxis

This paper is concerned with a cross-diffusion predator-prey system with prey-taxis incorporating Holling type II functional response under homogeneous Neumann boundary condition. By employing global bifurcation theory, it is obtained that a branch of nonconstant solutions can bifurcate from the positive constant solution whenever the chemotactic is attractive or repulsive. Furthermore, by using perturbation of simple eigenvalues it is found that the bifurcating solutions are locally stable near the bifurcation point under suitable conditions. These results imply that cross-diffusion can create coexistence for the predators and preys under the above special case.