Travelling waves, spreading and extinction for Fisher–KPP propagation driven by a line with fast diffusion

In an earlier work (Berestycki et al., 2013), we introduced a parabolic system to describe biological invasions in the presence of a line (the “road”) with a specific diffusion, included in a domain (the “field”) subject to Fisher–KPP propagation. We determined the asymptotic spreading speed in the direction of the road, w∗w∗, in terms of the various parameters. The new result we establish here is the existence of travelling fronts for this system for any speed c⩾w∗c⩾w∗. In addition, we show that no front can travel with a speed c