Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay
Journal
Journal of Differential Equations
Journal Volume
259
Journal Issue
4
Pages
1503-1541
Date Issued
2015
Author(s)
Abstract
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) exist for all c≥c* and are exponentially stable for all wave speed c>c* [13], where c* is called the critical wave speed. In this paper, we prove that the critical traveling waves 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.
Type
journal article