Monostable traveling waves for a time-periodic and delayed nonlocal reaction–diffusion equation

This paper is concerned with a time-periodic and delayed nonlocal reaction–diffusion population model with monostable nonlinearity. Under quasi-monotone or non-quasi-monotone assumptions, it is known that there exists a critical wave speed $c_{\ast }>0$ such that a periodic traveling wave exists if and only if the wave speed is above $c_{\ast }$. In this paper, we first prove the uniqueness of non-critical periodic traveling waves regardless of whether the model is quasi-monotone or not. Further, in the quasi-monotone case, we establish the exponential stability of non-critical periodic traveling fronts. Finally, we illustrate the main results by discussing two types of death and birth functions arising from population biology.