Existence and asymptotics of traveling wave fronts for a delayed nonlocal diffusion model with a quiescent stage

• In this paper, we propose a delayed nonlocal diffusion model with a quiescent stage.
• We establish the existence of traveling wave fronts.
• We discuss the asymptotic behavior of traveling wave fronts and then get the nonexistence of traveling wave fronts.

In this paper, we propose a delayed nonlocal diffusion model with a quiescent stage and study its dynamics. By using Schauder’s fixed point theorem and upper-lower solution method, we establish the existence of traveling wave fronts for speed c⩾c∗(τ)c⩾c∗(τ), where c∗(τ)c∗(τ) is a critical value. With the method of Carr and Chmaj (PAMS, 2004), we discuss the asymptotic behavior of traveling wave fronts and then get the nonexistence of traveling wave fronts for c