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

Abstract

This paper is concerned with the equation ∂u(x,t)/∂t = d∂ 2 u(x,t)/∂x 2 + αe -γτ u(x,t-t)-βu 2 (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 τ ≥ 0, monotone travelling wave front solutions, connecting the two uniform equilibria, exist for any speed c exceeding some critical (r-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 T satisfies 4ατe -γτ < 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.