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. Prom 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 μ1 < μ ≤ μ2, the steady state is the only stable solution of the model; for μ2 < μ ≤ μ4, by increasing the control parameter μ, the sequence Hπ-hexagons → Hπ-hexagon-stripe mixtures → stripes → H0-hexagon-stripe mixtures → H0-hexagons is observed; for μ > μ4, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator—prey model. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)