Turing pattern of a diffusive predator-prey model with nonlocal delay and herd behavior

Abstract

In this paper, we introduce nonlocal time delay into a diffusive predator-prey system with herd behavior. Special attention is paid to the modeling of the time delays to incorporate associated nonlocal spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. We first discuss the stability of the positive equilibrium of system without diffusion and give the necessary and sufficient conditions for the stability of positive equilibrium of diffusion system. Then we investigate Turing instability and deduce the amplitude equation of Turing patterns by using multiple scale method. The numerical simulations show the spots pattern and stripes pattern and verify the results of theoretical analysis. Our research reveals that nonlocal time delay plays an important role in the formation of pattern.