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