How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

Gregoire  Nadin

doi:10.3934/dcdsb.2015.20.1785

Article Contents

2015, Volume 20, Issue 6: 1785-1803. Doi: 10.3934/dcdsb.2015.20.1785

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How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

Gregoire Nadin^1,

1.
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris

Received: September 30, 2013

Revised: February 28, 2014

Published: July 2015

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Abstract

We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. A classical result of Freidlin and Gartner [16] yields that the solutions of the initial value problems associated with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in [8] in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null average or rescaling the coefficients by the change of variables $x\to x/L$, with $L>1$, speeds up the propagation. From a modelling point of view, these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.

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Mathematics Subject Classification: 34F05, 35B40, 35K57, 35P15, 92D25.

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