# American Institute of Mathematical Sciences

August  2015, 20(6): 1785-1803. doi: 10.3934/dcdsb.2015.20.1785

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

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

Received  October 2013 Revised  March 2014 Published  June 2015

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.
Citation: Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
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