# American Institute of Mathematical Sciences

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Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle
October  2005, 13(5): 1235-1246. doi: 10.3934/dcds.2005.13.1235

## Front propagation for a jump process model arising in spacial ecology

 1 Ecole Normale Supérieure, DMA, UMR8553, 45 rue d'Ulm, 75230 Paris, France 2 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, United States

Received  November 2004 Revised  July 2005 Published  September 2005

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.
Citation: Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
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