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IMA Journal of Applied Mathematics Advance Access published online on June 23, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxn015
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction–diffusion equations

Jonathan A. Sherratt{dagger}

Department of Mathematics and Maxwell Institute for Mathematical Sciences,Heriot-Watt University, Edinburgh EH14 4AS, UK

{dagger} Email: jas{at}ma.hw.ac.uk

Received on April 25, 2007; Revision received January 23, 2008. Accepted on April 29, 2008

Periodic travelling waves are an important solution form in oscillatory reaction–diffusion equations. I have shown previously that such waves arise naturally near a boundary at which a Dirichlet condition is applied. This result has applications in ecology, providing a potential explanation for the periodic waves seen in a number of natural populations. However, in ecological applications the Dirichlet boundary condition typically arises as a simple approximation to a more realistic Robin condition. In this paper, I consider the generation of periodic travelling waves by Robin boundary conditions and how the wave amplitude compares with that arising from Dirichlet conditions. I study a ‘{lambda}–{omega}’ system of equations, which is the normal form of an oscillatory reaction–diffusion system with scalar diffusion matrix close to a Hopf bifurcation. I consider a Robin boundary condition close to the Dirichlet limit, with proximity measured by a small parameter {epsilon}, and I study the equations as a perturbation problem in this small parameter. I show that the perturbation is singular and that although the solution itself changes at O({epsilon}), the amplitude of the periodic travelling wave which this solution approaches far from the boundary is unchanged at both O({epsilon}) and O({epsilon}2). This provides strong justification for the use of the Dirichlet approximation to the Robin condition when studying periodic travelling wave generation in equations of {lambda}–{omega} type. Finally, I discuss the ecological applications of the results.

Keywords: oscillatory system; perturbation theory; reaction-diffusion; wavetrain.


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