Hopf bifurcation and global stability of a diffusive Gause-type predator-prey models

This paper mainly provides Hopf bifurcation formulas for a general Gause type predator-prey system with diffusion and Neumann boundary condition by using the center manifold theory and normal form method, where the spectral and stability analysis around an equilibrium is addressed, and our results can be applied to the case without diffusion. As an application of these results, we give a complete and rigorous analysis of the global dynamics of a diffusive predator-prey model with herd behavior, especially, the Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions.