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October  2005, 13(5): 1235-1246. doi: 10.3934/dcds.2005.13.1235

Front propagation for a jump process model arising in spacial ecology

Benoît Perthame 1, and  P. E. Souganidis 2,

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.
Keywords: jump processes, ecology, Front propagation, homogenization., Hamilton-Jacobi equation.
Mathematics Subject Classification: 35B25, 35B27, 45G10, 70H20, 92D4.
Citation: Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
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