Turing Pattern Induced by Cross-Diffusion in a Predator–Prey Model with Pack Predation-Herd Behavior

Abstract

In this paper, we propose a diffusive predator–prey model with herd behavior in prey and pack predation behavior in predator under the homogeneous Neumann boundary condition. Due to the pack predation-herd behavior, the predator–prey interaction occurs only at the outer edge of each group. First, we analyze the existence and local stability of the equilibria of temporal model using vector field analysis and characteristic equations. Second, we deduce the conditions under which Turing instability occurs with the help of linear stability analysis. Then, using standard multiple-scale analysis, we derive the amplitude equations for the excited modes. By numerical simulations, we find that the model exhibits complex pattern replication by varying the value of parameters [Formula: see text] and [Formula: see text], and study the relationships between the average density of both prey and predator and these two parameters.