Spatiotemporal dynamics of a general reaction–diffusion model with time delay and nonlocal effect

Abstract Due to the combination of time delay and nonlocal reaction, which is more consistency with the reality of the interaction, we propose and investigate the spatiotemporal dynamics of the delayed reaction–diffusion system with nonlocal effect. We first study the stability and the bifurcations of the constant steady state induced by diffusion and nonlocality. Meanwhile, we show the sufficient conditions that can lead to the occurrence of Hopf bifurcations, Turing bifurcations, double Hopf bifurcations, and Turing–Hopf bifurcations. Then we discuss the impact of the time delay on stability and bifurcation. By the method of a transcendental polynomial equation with second-order transcendental term we get the conditions of Hopf and Turing–Hopf bifurcations with respect to delay parameter. Subsequently, we give the normal form expression for Turing–Hopf bifurcation in the general delayed reaction–diffusion system with nonlocal effect. Finally, we apply the obtained results to the model and numerically illustrate the spatiotemporal patterns induced by Turing–Hopf bifurcation.