Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 743–766 | Cite as

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

  • Henri BerestyckiEmail author
  • Jean-Michel Roquejoffre
  • Luca Rossi


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.


KPP equations Reaction-diffusion system Fast diffusion on a line  Asymptotic speed of propagation 

Mathematics Subject Classification

35K57 92D25 35B40 35K40 35B53 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Henri Berestycki
    • 1
    Email author
  • Jean-Michel Roquejoffre
    • 2
  • Luca Rossi
    • 3
  1. 1.Ecole des Hautes Etudes en Sciences Sociales, CAMSParisFrance
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 4France
  3. 3.Dipartimento di MatematicaUniversità degli Studi di PadovaPadovaItaly

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