Spatial dynamics in a predator-prey model with herd behavior

In this paper, a spatial predator-prey model with herd behavior in prey population and quadratic mortality in predator population is investigated. By the linear stability analysis, we obtain the condition for stationary pattern. Moreover, using standard multiple-scale analysis, we establish the amplitude equations for the excited modes, which determine the stability of amplitudes towards uniform and inhomogeneous perturbations. By numerical simulations, we find that the model exhibits complex pattern replication: spotted pattern, stripe pattern, and coexistence of the two. The results may enrich the pattern dynamics in predator-prey models and help us to better understand the dynamics of predator-prey interactions in a real environment.