Speed-up of reaction-diffusion fronts by a line of fast diffusion

In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line.