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The Quarterly Journal of Mechanics and Applied Mathematics 2005 58(2):257-268; doi:10.1093/qjmamj/hbi012
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Q. Jl Mech. Appl. Math. (2005) 58 (2), 257–268 © Oxford University Press 2005; all rights reserved.

Linear stability of travelling fronts in an age-structured reaction–diffusion population model

S. A. Gourley *

( Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH )

This paper is concerned with the equation

{partial}u(x, t)/{partial}t = d({partial}2u(x, t)/{partial}x2) + {alpha}e–{gamma}{tau} u(x, t – {tau}) – ßu2(x, t)

which models the evolution of an adult population in a situation where the juveniles do not disperse. For this equation it is known that, for any delay {tau} >= 0, monotone travelling wave front solutions, connecting the two uniform equilibria, exist for any speed c exceeding some critical ({tau}-dependent) minimum value.

In this paper we study the linear stability of these travelling fronts using weighted energy norms. We prove that, if the delay {tau} satisfies 4{alpha}{tau}e–{gamma}{tau} < cosh–1 (2), then a given travelling front of speed c is linearly stable if the initial data is sufficiently close (in a c-dependent sense) to the front at infinity. Our main theorem includes Fisher's equation as a special case and our findings for this case are discussed in relation to the existing literature on wave speed selection and initial data.

Received 19 December 2003. Revise 16 December 2004. Accepted 10 January 2005.

* <s.gourley{at}surrey.ac.uk>


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