Nonlocal delay driven spatiotemporal patterns in a single-species reaction–diffusion model

This paper presents Turing–Hopf bifurcation analysis and the resulting spatiotemporal dynamics in a single-species reaction–diffusion model with nonlocal delay. A linear stability analysis is performed to find the conditions for the occurrence of Turing–Hopf bifurcation. The weakly nonlinear analysis is then employed to derive the amplitude equations which describe the slow-time evolution of the critical Turing and Hopf modes. By the utilization of amplitude equations, stability conditions for different pattern formations can be obtained, helping to reveal the affluent spatiotemporal patterns near the Turing–Hopf bifurcation and predict when and where these patterns can emerge. Also, our analysis shows the role of the choice of initial conditions in pattern formations. Numerical simulations are given to verify the theoretical results.