Front propagation for a jump process model arising in spacial ecology

We study the propagation of a front arising as the asymptotic (macroscopic) limit of a model in spatial ecology in which the invasive species propagate by "jumps". The evolution of the order parameter marking the location of the colonized/uncolonized sites is governed by a (mesoscopic) integro-differential equation. This equation has structure similar to the classical Fisher or KPP - equation, i.e., it admits two equilibria, a stable one at $k$ and an unstable one at $0$ describing respectively the colonized and uncolonized sites. We prove that, after rescaling, the solution exhibits a sharp front separating the colonized and uncolonized regions, and we identify its (normal) velocity. In some special cases the front follows a geometric motion. We also consider the same problem in heterogeneous habitats and oscillating habitats. Our methods, which are based on the analysis of a Hamilton-Jacobi equation obtained after a change of variables, follow arguments which were already used in the study of the analogous phenomena for the Fisher/KPP - equation.