Abstract: The existence of positive solutions to a predator-prey model with cross-diffusion is considered. First, we derive some estimates which are independent of cross-diffusion by the maximum principle; second, the limiting behavior of positive solutions for large cross-diffusion is established; finally, we show the existence of local bifurcation solutions of the limiting system near the semi-trivial solution by the local bifurcation theorem, and we extend the local bifurcation solutions to the global bifurcation solutions by the global bifurcation theorem, and we show that the global bifurcation solutions can extend to infinity as the bifurcation parameter approaches infinity. The results demonstrate that the two species can coexist when the cross-diffusion is large.

Key words: predator-prey model, cross-diffusion, positive solutions, existence