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

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Abstract

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 '? - ?' 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 e, 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(e), the amplitude of the periodic travelling wavewhich this solution approaches far from the boundary is unchanged at both O(e) and O(e2). This provides strong justification for the use of the Dirichlet approximation to the Robin condition when studying periodic travelling wave generation in equations of ? - ? type. Finally, I discuss the ecological applications of the results. © The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Original languageEnglish
Pages (from-to)759-781
Number of pages23
JournalIMA Journal of Applied Mathematics
Volume73
Issue number5
DOIs
Publication statusPublished - 2008

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Periodic Traveling Waves
Robin Boundary Conditions
Reaction-diffusion Equations
Dirichlet Boundary Conditions
Dirichlet conditions
Small Parameter
Dirichlet
Perturbation
Periodic Wave
Ecology
Approximation
Reaction-diffusion System
Justification
Hopf Bifurcation
Normal Form
Proximity
System of equations
Scalar

Keywords

  • Oscillatory system
  • Perturbation theory
  • Reaction-diffusion
  • Wavetrain

Cite this

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title = "A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction-diffusion equations",
abstract = "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 '? - ?' 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 e, 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(e), the amplitude of the periodic travelling wavewhich this solution approaches far from the boundary is unchanged at both O(e) and O(e2). This provides strong justification for the use of the Dirichlet approximation to the Robin condition when studying periodic travelling wave generation in equations of ? - ? type. Finally, I discuss the ecological applications of the results. {\circledC} The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.",
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