Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion

In this paper, we consider a predator-prey model with herd behavior and cross-diffusion subject to homogeneous Neumann boundary condition. Firstly, the existence and priori bound of a solution for the model without cross-diffusion are shown. Then, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we find a wealth of spatiotemporal dynamics near the Turing-Hopf bifurcation point under suitable conditions. Furthermore, some numerical simulations to illustrate the theoretical analysis are carried out.