Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On travelling wavefronts of Nicholson's blowflies equation with diffusion

Chi-Kun Lina1 and Ming Meia2

a1 Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan, (cklin@math.nctu.edu.tw)

a2 Department of Mathematics, Champlain College Saint-Lambert, Saint-Lambert, Quebec J4P 3P2, Canada, and Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3G 1M8, Canada, (mmei@champlaincollege.qc.ca; mei@math.mcgill.ca)

Abstract

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 c > c* (c* 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 x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. 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.

(Received July 23 2008)

(Accepted April 28 2009)