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

7 July 2009

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.

In Proceedings of that National Academy of Sciences of the United States of America, 106(27), pp. 10890-10895

Type | Article |

URL | http://www.pnas.org/content/106/27/10890.full.pdf+html |

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