Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

John R. King

doi:10.3934/nhm.2013.8.343

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2013, Volume 8, Issue 1: 343-378. Doi: 10.3934/nhm.2013.8.343

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Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity

John R. King^1,

1.
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD

Received: March 2012

Revised: February 2013

Published: March 2013

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Abstract

We adapt (ray-based) geometrical optics approaches to encompass the formal asymptotic analysis of front propagation in a Fisher-KPP equation with slowly varying spatial inhomogeneities. The wavespeed is shown to be selected by two distinct (and fully constructive) mechanisms, depending on whether the source term is an increasing or decreasing function of the spatial variable. Canonical inner problems, analogous to those arising in the geometrical theory of diffraction, are formulated to give refined expressions for the wavefront location. Additional phenomena, notably the initiation of new fronts and the transitions that occur when the source term is a non-monotonic function of space, are shown to be amenable to the same asymptotic approaches.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35C20.

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