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.
@article{SLSEDP_2013-2014____A19_0, author = {Henri Berestycki and Anne-Charline Coulon and Jean-Michel Roquejoffre and Luca Rossi}, title = {Speed-up of reaction-diffusion fronts by a line of fast diffusion}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:19}, pages = {1--25}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2013-2014}, doi = {10.5802/slsedp.62}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.62/} }
TY - JOUR AU - Henri Berestycki AU - Anne-Charline Coulon AU - Jean-Michel Roquejoffre AU - Luca Rossi TI - Speed-up of reaction-diffusion fronts by a line of fast diffusion JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:19 PY - 2013-2014 SP - 1 EP - 25 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.62/ DO - 10.5802/slsedp.62 LA - en ID - SLSEDP_2013-2014____A19_0 ER -
%0 Journal Article %A Henri Berestycki %A Anne-Charline Coulon %A Jean-Michel Roquejoffre %A Luca Rossi %T Speed-up of reaction-diffusion fronts by a line of fast diffusion %J Séminaire Laurent Schwartz — EDP et applications %Z talk:19 %D 2013-2014 %P 1-25 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.62/ %R 10.5802/slsedp.62 %G en %F SLSEDP_2013-2014____A19_0
Henri Berestycki; Anne-Charline Coulon; Jean-Michel Roquejoffre; Luca Rossi. Speed-up of reaction-diffusion fronts by a line of fast diffusion. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 19, 25 p. doi : 10.5802/slsedp.62. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.62/
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