Cross-diffusion induced spatiotemporal patterns in a predator–prey model with herd behavior

In this paper, we investigate the phenomena of pattern formation for a predator–prey model with herd behavior and cross-diffusion. We give the conditions for cross-diffusion induced instability in detail, and have derived amplitude equations for the excited modes. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits complex pattern replication: spotted pattern, stripe pattern, and coexistence of the two by increasing the control parameter ββ and cross-diffusion coefficient d21d21 respectively. Meanwhile, we also obtain the other patterns: “labyrinth” patterns and “black-eye” patterns. These results indicate that the cross-diffusion plays an important role in the pattern selection. The spatiotemporal patterns induced by cross-diffusion may enrich the pattern dynamics for diffusive predator–prey model and be helpful for the biologists to understand the phenomenon in which a flux of one species is induced by a gradient of another species.