Global bifurcation of a predator-prey model with cross-diffusions

The existence of the positive solutions of a predator-prey model with cross-diffusions is studied. Firstly, by using the integral method and the maximum principle, we give the prior estimate of the positive solution. Secondly, by means of the Crandall-Rabinowtz bifurcation theory and taking a as the bifurcation parameter, it is showed the local bifurcation can be generated at the constant solution. Finally, making use of global bifurcation theory, we prove that the local bifurcation can be extend to global bifurcation, and the continuum joins up with infinity.