Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction–diffusion systems

The purpose of this work is to study the spatial dynamics of some delayed nonlocal reaction–diffusion systems in whole space. We first establish a series of comparison theorems to investigate the global attractivity of the equilibria for a delayed nonlocal reaction–diffusion system with and without quasi-monotonicity. Then we show that the spreading speed of a general system without quasi-monotone conditions is coincident with the minimal wave speed. Applying a fluctuation method, we further provide some sufficient conditions to ensure the upward convergence of the spreading speed and traveling wave solutions. Finally, we point out the effects of the delay and nonlocality on the spreading speed of the non-quasi-monotone systems.