Spatio-temporal dynamics of a reaction-diffusion equation with the nonlocal spatial average and delay

In this paper we investigate the influence of the delay on the dynamics of a single species model with the nonlocal spatial average. Especially, the joint interaction of the nonlocal spatial average and delay results in the occurrence of the steady-state-Hopf bifurcation, which is the source of the spatio-temporal patterns. Applying the obtained results to the Logistic growth model subject to an Allee effect, the stability region, steady state bifurcation and steady-state-Hopf bifurcation are specifically determined in the plane of the diffusion coefficient and delay. Stable spatial patterns and the spatio-temporal patterns near the steady-state-Hopf bifurcation point are numerically obtained. (c) 2020 Elsevier Ltd. All rights reserved.