Spatiotemporal patterns behind propagating fronts in reaction-diffusion systems and the complex Ginzburg-Landau equation

Abstract: In oscillatory systems, 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. I will describe this phenomenon in detail, and will explain the concept of absolute stability of wavetrains, which is central to a full understanding of the behaviour. 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. I will describe a new method for calculating this width, based on the absolute stability of the wavetrain in moving frames of reference. I will illustrate the work via two examples: the generation of wavetrains in the wake of the invasion of a prey population by predators, and spatiotemporal patterning behind propagating fronts in the complex Ginzburg-Landau equation. The work that I will describe in this talk has been done in collaboration with Matthew Smith (Microsoft Research, Cambridge) and Jens Rademacher (CWI, Amsterdam).