Spatiotemporal patterns driven by the nonlocal advection and delay

In this paper, we investigate the spatiotemporal dynamics in the nonlocal advection–diffusion equation with delay. The nonlocal advection is characterized by the top-hat kernel and time delay measures the delay phenomenon in the reaction term. The joint effect of the nonlocal advection and delay on the stability of the steady state and spatiotemporal dynamics is investigated. The conditions for the occurrence of Turing bifurcation and Turing–Hopf bifurcation are determined. Our results show that the large perception range can stabilize the steady state, but a small perception range is more likely to make the system unstable, and negative feedback of delay is more easy to make system produce complex patterns. It has also been shown that spatially inhomogeneous oscillatory patterns are triggered by the joint interaction of nonlocal advection and delay, which can not occur only for one factor. • Find the new mechanism of spatiotemporal pattern formation in the advection–diffusion. • Investigate the joint effect of perception range and delay on the stability and possible bifurcation. • Obtain the absolutely stability region independent of the delay. • Find that the small perception range drive Turing bifurcation and large one stabilize the equation. • Find that the negative feedback of the delay and moderate perception range is the source of the oscillatory pattern.