Traveling fronts for a delayed reaction–diffusion system with a quiescent stage

In this paper, we study the traveling wave fronts of a delayed reaction–diffusion system with a quiescent stage for a single species population with two separate mobile and stationary states. By transforming the corresponding wave system into a scalar delayed differential equation with an integral term, we establish the existence of the minimal wave speed cmin, and the asymptotic behavior, monotonicity and uniqueness (up to a translation) of the traveling wave fronts. In particular, the effects of the delay and transfer rates on the minimal wave speed are studied.

Research highlights
► We study a delayed reaction–diffusion model with a quiescent stage.
► We investigate the phenomenon of biological invasion using traveling wave theory.
► The effects of the delay and transfer rates on the invasion speed are studied.