The effects of obstacle size on periodic travelling waves in oscillatory reaction-diffusion equations

Matthew J. Smith, Jonathan A. Sherratt, Nicola J. Armstrong

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Many natural populations undergo multi-year cycles, and field studies have shown that these can be organized into periodic travelling waves (PTWs). Mathematical studies have shown that large-scale landscape obstacles represent a natural mechanism for wave generation. Here, we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size. We firstly consider a large circular obstacle in an infinite domain for a reaction-diffusion system of '?-?' type. We use perturbation theory to derive a leading order approximation to the wave generated by the obstacle. This shows the dependence of the wave properties on both parameter values and obstacle size. We find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge, rather than exponentially in the case of a flat boundary. We use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator-prey system, via a reduction of the predator-prey model to normal form close to Hopf bifurcation. Our predictions compare well with numerical simulations. We also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries. © 2007 The Royal Society.

Original languageEnglish
Pages (from-to)365-390
Number of pages26
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume464
Issue number2090
DOIs
Publication statusPublished - 8 Feb 2008

Keywords

  • Hypergeometric function
  • Multi-annual population cycles
  • Parabolic partial differential equations
  • Wave train

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