Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration

Emeric Bouin; Vincent Calvez; Nicolas Meunier; Sepideh Mirrahimi; Benoît Perthame; Gaël Raoul; Raphaël Voituriez

doi:10.1016/j.crma.2012.09.010

Partial Differential Equations

Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration
[Fronts dʼinvasion avec motilité variable : Répartition des phénotypes et accélération de lʼonde]

Présenté par : Pierre-Louis Lions

Emeric Bouin ¹ ; Vincent Calvez ¹ ; Nicolas Meunier ² ; Sepideh Mirrahimi ³ ; Benoît Perthame ⁴ ; Gaël Raoul ⁵ ; Raphaël Voituriez ⁶

¹ Ecole normale supérieure de Lyon, CNRS UMR 5669 UMPA, INRIA project NUMED, 46, allée dʼItalie, 69364 Lyon, France
² Université Paris Descartes, CNRS UMR 8145 MAP5, 45, rue des Saints-Pères, 75270 Paris, France
³ Ecole polytechnique, CNRS UMR 7641 CMAP, INRIA project MAXPLUS, route de Saclay, 91128 Palaiseau, France
⁴ Université Pierre et Marie Curie, CNRS UMR 7598 LJLL, INRIA projet BANG, 4, place Jussieu, 75005 Paris, France
⁵ Centre dʼEcologie Fonctionnelle et Evolutive, CNRS UMR 5175, 1919 route de Mende, 34293 Montpellier, France
⁶ Université Pierre et Marie Curie, CNRS UMR 7600 LPTMC, 4, place Jussieu, 75252 Paris, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 761-766.

Résumé
Abstract

Les fronts dʼinvasion en écologie ont été largement étudiés. Cependant peu de résultats mathématiques existent pour le cas dʼun coefficient de motilité variable (à cause des mutations). A partir dʼun modèle minimal de réaction–diffusion, nous expliquons le phénomène observé dʼaccélération du front (lorsque la motilité nʼest pas bornée), et nous démontrons lʼexistence dʼondes progressives ainsi que la sélection des individus les plus motiles (lorsque la motilité est bornée). Le point clé pour la construction des fronts est la relation de dispersion qui relie la vitesse de lʼonde avec la décroissance en espace. Lorsque la motilité nʼest pas bornée nous montrons que la position du front suit une loi dʼéchelle en $t^{3 / 2}$ . Lorsque le taux de mutation est faible, nous montrons que, dans notre contexte, lʼéquation canonique pour la dynamique du meilleur trait est une EDP. Cʼest une équation de type Burgers avec terme source.

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction–diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other qualitative results, such as the existence of traveling waves and the selection of the most motile individuals (when the motility is bounded). The key argument for constructing and analysing the traveling waves is the derivation of a dispersion relation linking the wave speed and the spatial decay. When the motility is unbounded we show that the position of the front scales as $t^{3 / 2}$ . When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with a source term.

Reçu le : 2012-07-10
Accepté le : 2012-09-17
Publié le : 2012-08-01

DOI : 10.1016/j.crma.2012.09.010

Affiliations des auteurs :

Emeric Bouin ¹ ; Vincent Calvez ¹ ; Nicolas Meunier ² ; Sepideh Mirrahimi ³ ; Benoît Perthame ⁴ ; Gaël Raoul ⁵ ; Raphaël Voituriez ⁶

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     author = {Emeric Bouin and Vincent Calvez and Nicolas Meunier and Sepideh Mirrahimi and Beno{\^\i}t Perthame and Ga\"el Raoul and Rapha\"el Voituriez},
     title = {Invasion fronts with variable motility: {Phenotype} selection, spatial sorting and wave acceleration},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {761--766},
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     language = {en},
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Emeric Bouin; Vincent Calvez; Nicolas Meunier; Sepideh Mirrahimi; Benoît Perthame; Gaël Raoul; Raphaël Voituriez. Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 761-766. doi : 10.1016/j.crma.2012.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.010/

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