Positive solutions of a predator-prey model with cross-diffusion

In this paper, we consider the positive solutions for a predator-prey model with cross-diffusion and Holling type II functional response. In particular, the existence of positive solutions can be established by the bifurcation theory. Moreover, the uniqueness and the exact number of positive solutions is studied when the parameter m is large. Furthermore, for large cross-diffusion rate α with the spatial dimension is less than 5, we can derive the corresponding limit systems and also study the asymptotic behavior of positive solutions. Finally, some numerical simulations are presented to supplement the analysis results in one dimension case. This results give us some important information on the structure of positive solutions.