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∗, 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∗. In addition, we show that no front can travel with a speed c < w∗. We further extend the results to a new model where there is Fisher-KPP growth on the road, embedded in an elsewhere unfavourable plane. This is relevant to discuss certain invasions that have been observed along roads. We establish sharp conditions for extinction/invasion and describe the invasion dynamics.