The effect of a line with non-local diffusion on Fisher-KPP propagation

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 en-hances 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 com-pletely the invasion in the upper half-plane. This work is a non-local version of the model introduced in [16], where the line had a strong but local diffusion described by the classical Laplace operator. Contents