Summary: This paper is devoted to the study of Nicholson’s blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed
is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as
, but it can be large in other locations. The result develops and improves the previous wave stability obtained by M. Mei
et al. [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 3, 579–594 (2004; Zbl 1059.34019
)]. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.