Pattern selection in a predation model with self and cross diffusion

In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator--prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for \mu_1<\mu\leq\mu_2, the steady state is the only stable solution of the model; for \mu_2<\mu\leq\mu_4, by increasing the control parameter \mu, the sequence H_\pi-hexagons \rightarrow H_\pi-hexagon-stripe mixtures \rightarrow stripes \rightarrow H_0-hexagon-stripe mixtures \rightarrow H_0-hexagons is observed; for \mu>\mu_4, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator--prey model.