Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave

Periodic wave trains are the generic solution form for oscillatory reaction-diffusion equations in one space dimension. It has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion [Sherratt, Physica D 70 (1994) 370-382]. In this paper, the mechanism of wave train generation is studied in detail for systems of two reaction-diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. A combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient, in which phase singularities occur periodically. This calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion. Copyright © 1998 Elsevier Science B.V.