Speed-up of reaction-diffusion fronts by a line of fast diffusion

Henri Berestycki[1]; Anne-Charline Coulon[2]; Jean-Michel Roquejoffre[2]; Luca Rossi[3]

  • [1] École des Hautes Études en Sciences Sociales CAMS 54, bd Raspail F-75270 Paris France
  • [2] Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne F-31062 Toulouse Cedex 4 France
  • [3] Dipartimento di Matematica Università degli Studi di Padova Via Trieste, 63 35121 Padova Italy

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

  • Volume: 25, Issue: 13, page 1-25
  • ISSN: 2266-0607

Abstract

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In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. 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. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line.

How to cite

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Berestycki, Henri, et al. "Speed-up of reaction-diffusion fronts by a line of fast diffusion." Séminaire Laurent Schwartz — EDP et applications 25.13 (2013-2014): 1-25. <http://eudml.org/doc/275817>.

@article{Berestycki2013-2014,
abstract = {In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. 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. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line.},
affiliation = {École des Hautes Études en Sciences Sociales CAMS 54, bd Raspail F-75270 Paris France; Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne F-31062 Toulouse Cedex 4 France; Institut de Mathématiques de Toulouse Université Paul Sabatier 118 route de Narbonne F-31062 Toulouse Cedex 4 France; Dipartimento di Matematica Università degli Studi di Padova Via Trieste, 63 35121 Padova Italy},
author = {Berestycki, Henri, Coulon, Anne-Charline, Roquejoffre, Jean-Michel, Rossi, Luca},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {spreading rate; accelerating fronts; Fisher-KPP dynamics; fractional diffusion; line of fast diffusion},
language = {eng},
number = {13},
pages = {1-25},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Speed-up of reaction-diffusion fronts by a line of fast diffusion},
url = {http://eudml.org/doc/275817},
volume = {25},
year = {2013-2014},
}

TY - JOUR
AU - Berestycki, Henri
AU - Coulon, Anne-Charline
AU - Roquejoffre, Jean-Michel
AU - Rossi, Luca
TI - Speed-up of reaction-diffusion fronts by a line of fast diffusion
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2013-2014
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 25
IS - 13
SP - 1
EP - 25
AB - In these notes, we discuss a new model, proposed by H. Berestycki, J.-M. Roquejoffre and L. Rossi, to describe biological invasions in the plane when a strong diffusion takes place on a line. This model seems relevant to account for the effects of roads on the spreading of invasive species. In what follows, the diffusion on the line will either be modelled by the Laplacian operator, or the fractional Laplacian of order less than 1. Of interest to us is the asymptotic speed of spreading in the direction of the line, but also in the plane. 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. When the diffusion is the Laplacian, the global asymptotic speed of spreading on the line grows as the square root of the diffusion. In the other directions, the line of strong diffusion influences the spreading up to a critical angle, from which one recovers the classical spreading velocity. When the diffusion is the fractional Laplacian, the spreading on the line is exponential in time, and propagation in the plane is equivalent to that of a one-dimensional infinite planar front parallel to the line.
LA - eng
KW - spreading rate; accelerating fronts; Fisher-KPP dynamics; fractional diffusion; line of fast diffusion
UR - http://eudml.org/doc/275817
ER -

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