# The influence of a line with fast diffusion on Fisher-KPP propagation

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## Abstract

We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.

## Keywords

KPP equations Reaction-diffusion system Fast diffusion on a line Asymptotic speed of propagation## Mathematics Subject Classification

35K57 92D25 35B40 35K40 35B53## Notes

### Acknowledgments

This study was supported by the French “gence Nationale de la Recherche” through the project PREFERED (ANR 08-BLAN-0313). H.B. was also supported by an NSF FRG grant DMS-1065979. L.R. was partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.

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