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Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion

Jonathan A. Sherratt, Matthew J. Smith, and Jens D.M. Rademacher

Abstract

In systems with cyclic dynamics, invasions often generate periodic spatiotemporal oscillations, which undergo a subsequent transition to chaos. The periodic oscillations have the form of a wavetrain and occur in a band of constant width. In applications, a key question is whether one expects spatiotemporal data to be dominated by regular or irregular oscillations, or to involve a significant proportion of both. This depends on the width of the wavetrain band. Here, for the first time, we present mathematical theory that enables the direct calculation of this width. Our method synthesises recent developments in stability theory and computation. It is developed for only one equation system, but because this is a normal form close to a Hopf bifurcation, the results can be applied directly to a wide range of models. We illustrate this by considering a classic example from ecology: wavetrains in the wake of the invasion of a prey population by predators.

Details

Publication typeArticle
Published inProceedings of that National Academy of Sciences of the United States of America, 106(27), pp. 10890-10895
URLhttp://www.pnas.org/content/106/27/10890.full.pdf+html
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