  We propose here a new model of accelerating fronts, consisting of one
equation with non-local diffusion on a line, coupled via the boundary condition
with a reaction-diffusion equation in the upper half-plane. The underlying
biological question is to understand how transportation networks may enhance
biological invasions. We show that the line accelerates the propagation in the
direction of the line and enhances the overall propagation in the plane and
that the propagation is directed by diffusion on the line, where it is
exponentially fast in time. We also describe completely the invasion in the
upper half-plane. This work is a non-local version of the model introduced in
[Berestycki-Roquejoffre-Rossi 2013], where the line had a strong but local
diffusion described by the classical Laplace operator.
