Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves ϕ(x+ct)ϕ(x+ct) exist for all c≥c⁎c≥c⁎ and are exponentially stable for all wave speed c>c⁎c>c⁎[13], where c⁎c⁎ is called the critical wave speed. In this paper, we prove that the critical traveling waves ϕ(x+c⁎t)ϕ(x+c⁎t) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.