Abstract: The predator-prey model with cross-diffusion is very important to describe the interaction between the species, and its dynamics behavior strongly depends on the properties of its positive solutions. In this paper, we consider the properties of positive solutions of predator-prey model with cross diffusion and Monod-Haldane type functional response function by using the principle of maximum value, local bifurcation theory and global bifurcation theory. Furthermore, one of the sufficient conditions for the existence of positive solutions is proved and the results indicate that the predator and prey can coexist under certain conditions.

Key words: predator-prey model, Monod-Haldane type, cross diffusion, global bifurcation theory