The influence of nonlocal exchange terms on Fisher-KPP propagation driven by a line of fast diffusion

A new model to describe biological invasion influenced by a line with fast diffusion has been introduced by H. Berestycki, J.-M. Roquejoffre and L. Rossi in 2012. The purpose of this article is to present a related model where the line of fast diffusion has a nontrivial range of influence, i.e. the exchanges between the line and the surrounding space has a nontrivial support. We show the existence of a spreading velocity depending on the diffusion on the line. Two intermediate model are also discussed. Then, we try to understand the influence of different exchange terms on this spreading speed. We show that various behaviour may happen, depending on the considered exchange distributions.