Stability and pattern formation driven by top-hat kernel and delay in the non-local reaction-diffusion equation

In this paper, we study the stability and spatiotemporal patterns of a non-local reaction-diffusion equation with delay. The non-locality is characterized by the top-hat kernel function. It has been shown that the joint effect of the spatial non-locality and the delay can lead to the complex spatiotemporal patterns and patterns transition. Finally, we apply our theoretical results to a delayed logistic model with non-local reaction terms and find that the spatial inhomogeneous periodic patterns caused by the interaction of Turing bifurcation and Hopf bifurcation, and the transition mechanism of different spatiotemporal patterns. The equation considered in this paper can establish the bridge between different classical well-known equations.