## 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,wepresent mathematical theory that enables the direct calculation of this width. Our method synthesizes recent developments in stability theory and computation. It is developed for only 1 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.

Original language | English |
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Pages (from-to) | 10890-10895 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences |

Volume | 106 |

Issue number | 27 |

DOIs | |

Publication status | Published - 7 Jul 2009 |

## Keywords

- Absolute stability
- Complex Ginzburg-Landau equation
- Periodic travelling wave
- Predator-prey
- Reaction-diffusion