Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering two different prey behaviours' transition

In this paper, we study the Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analysing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by calculating and analysing the normal form on the centre manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.