Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity
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Received:
01 March 2012
Revised:
01 February 2013
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Primary: 35K57; Secondary: 35C20.
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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.
Citation: John R. King. Wavespeed selection in the heterogeneous Fisher equation: Slowly varying inhomogeneity[J]. Networks and Heterogeneous Media, 2013, 8(1): 343-378. doi: 10.3934/nhm.2013.8.343
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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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This article has been cited by:
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