Forma, Vol. 11 (No. 1), pp. 61-80, 1996
Original Paper

Periodic Waves in Reaction-Diffusion Models of Oscillatory Biological Systems

Jonathan A. Sherratt

Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.

(Received July 26, 1995; Accepted December 20, 1995)

Keywords: Periodic Plane Waves, Spatiotemporal Chaos, Travelling Waves, Chaotic Wakes, Ecology, Calcium

Abstract. Many systems in biology and chemistry have been successfully modelled by oscillatory reaction-diffusion equations. In such equations, periodic waves are a fundamental solution type, and have been extensively studied. In this paper, the author discusses ways in which periodic waves can arise naturally from simple initial conditions. The mechanisms by which a particular wave is selected by the details of the initial data are explained, and formation of periodic waves behind invasive transition fronts is discussed. In some cases, the selection mechanism can cause an unstable wave to form, in which case it degenerates into spatiotemporal irregularities. This process is described and numerical evidence is given which suggests that these irregularities are genuinely chaotic. The various results have applications to a number of real oscillatory systems, and applications to intracellular calcium signalling and predator-prey invasion are discussed in detail.