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

Emeric Bouin; Vincent Calvez; Nicolas Meunier; Sepideh Mirrahimi; Benoit Perthame; Gael Raoul; Raphael Voituriez

hal-00716349, version 1

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

Emeric Bouin () ¹, Vincent Calvez () ¹, Nicolas Meunier () ², Sepideh Mirrahimi () ³, Benoit Perthame () ^4, 5, Gael Raoul () ⁶, Raphael Voituriez () ⁷

(2012-07-10)

Abstract: 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 quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave 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 source term.

1: Unité de Mathématiques Pures et Appliquées (UMPA-ENSL)
CNRS : UMR5669 – École Normale Supérieure - Lyon
2: Mathématiques appliquées Paris 5 (MAP5)
CNRS : UMR8145 – Université Paris V - Paris Descartes
3: Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
Polytechnique - X – CNRS : UMR7641
4: BANG (INRIA Rocquencourt)
INRIA – Laboratoire Jacques-Louis Lions
5: Laboratoire Jacques-Louis Lions (LJLL)
CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI
6: Centre d'écologie fonctionnelle et évolutive (CEFE)
CNRS : UMR5175 – Université Montpellier II - Sciences et techniques – Université Montpellier I – Université Paul Valéry - Montpellier III – Ecole Pratique des Hautes Etudes
7: Laboratoire de Physique Théorique de la Matière Condensée (LPTMC)
CNRS : UMR7600 – Université Pierre et Marie Curie [UPMC] - Paris VI

hal-00716349, version 1
http://hal.archives-ouvertes.fr/hal-00716349
oai:hal.archives-ouvertes.fr:hal-00716349
From: Vincent Calvez <>
Submitted on: Tuesday, 10 July 2012 14:03:35
Updated on: Tuesday, 10 July 2012 15:59:34

See detailed view

Associated documents

PDF :

arXiv: 1207.2355

Export

Bibtex EndNote TEI RefWorks

%% hal-00716349, version 1
%% http://hal.archives-ouvertes.fr/hal-00716349
@unpublished{bouin:hal-00716349,
    hal_id = {hal-00716349},
    url = {http://hal.archives-ouvertes.fr/hal-00716349},
    title = {{Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration}},
    author = {Bouin, Emeric and Calvez, Vincent and Meunier, Nicolas and Mirrahimi, Sepideh and Perthame, Benoit and Raoul, Gael and Voituriez, Raphael},
    abstract = {{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 quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave 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 source term.}},
    keywords = {Reaction-Diffusion Equation, WKB limit, Front invasion, Front acceleration},
    language = {English},
    affiliation = {Unit{\'e} de Math{\'e}matiques Pures et Appliqu{\'e}es - UMPA-ENSL , Math{\'e}matiques appliqu{\'e}es Paris 5 - MAP5 , Centre de Math{\'e}matiques Appliqu{\'e}es - Ecole Polytechnique - CMAP , BANG - INRIA Rocquencourt , Laboratoire Jacques-Louis Lions - LJLL , Centre d'{\'e}cologie fonctionnelle et {\'e}volutive - CEFE , Laboratoire de Physique Th{\'e}orique de la Mati{\`e}re Condens{\'e}e - LPTMC},
    note = {7 pages },
    year = {2012},
    month = Jul,
    pdf = {http://hal.archives-ouvertes.fr/hal-00716349/PDF/cane-toads-CRAS-v6.pdf},
}

<?xml version="1.0" encoding="UTF-8"?>
<listBibl xmlns="http://www.tei-c.org/ns/1.0" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<listBibl type="article"><head>Documents without publication reference (Preprint)</head>
<biblStruct type="article" xml:id="hal-00716349"><analytic>
<author><persName><forename>Emeric</forename><surname>Bouin</surname></persName>
<affiliation>
<orgName type="laboratory">UMPA-ENSL</orgName>
<orgName type="team">Unité de Mathématiques Pures et Appliquées</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR5669</orgName>
<orgName type="institution">École Normale Supérieure - Lyon</orgName>
</affiliation>
</author>
<author><persName><forename>Vincent</forename><surname>Calvez</surname></persName>
<affiliation>
<orgName type="laboratory">UMPA-ENSL</orgName>
<orgName type="team">Unité de Mathématiques Pures et Appliquées</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR5669</orgName>
<orgName type="institution">École Normale Supérieure - Lyon</orgName>
</affiliation>
</author>
<author><persName><forename>Nicolas</forename><surname>Meunier</surname></persName>
<affiliation>
<orgName type="laboratory">MAP5</orgName>
<orgName type="team">Mathématiques appliquées Paris 5</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR8145</orgName>
<orgName type="institution">Université Paris V - Paris Descartes</orgName>
</affiliation>
</author>
<author><persName><forename>Sepideh</forename><surname>Mirrahimi</surname></persName>
<affiliation>
<orgName type="laboratory">CMAP</orgName>
<orgName type="team">Centre de Mathématiques Appliquées - Ecole Polytechnique</orgName>
<country>FR</country>
<orgName type="institution">Polytechnique - X</orgName>
<orgName type="institution">CNRS : UMR7641</orgName>
</affiliation>
</author>
<author><persName><forename>Benoit</forename><surname>Perthame</surname></persName>
<affiliation>
<orgName type="laboratory">INRIA Rocquencourt</orgName>
<orgName type="team">BANG</orgName>
<country>FR</country>
<orgName type="institution">INRIA</orgName>
<orgName type="institution">Laboratoire Jacques-Louis Lions</orgName>
</affiliation>
<affiliation>
<orgName type="laboratory">LJLL</orgName>
<orgName type="team">Laboratoire Jacques-Louis Lions</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR7598</orgName>
<orgName type="institution">Université Pierre et Marie Curie [UPMC] - Paris VI</orgName>
</affiliation>
</author>
<author><persName><forename>Gael</forename><surname>Raoul</surname></persName>
<affiliation>
<orgName type="laboratory">CEFE</orgName>
<orgName type="team">Centre d'écologie fonctionnelle et évolutive</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR5175</orgName>
<orgName type="institution">Université Montpellier II - Sciences et techniques</orgName>
<orgName type="institution">Université Montpellier I</orgName>
<orgName type="institution">Université Paul Valéry - Montpellier III</orgName>
<orgName type="institution">Ecole Pratique des Hautes Etudes</orgName>
</affiliation>
</author>
<author><persName><forename>Raphael</forename><surname>Voituriez</surname></persName>
<affiliation>
<orgName type="laboratory">LPTMC</orgName>
<orgName type="team">Laboratoire de Physique Théorique de la Matière Condensée</orgName>
<country>FR</country>
<orgName type="institution">CNRS : UMR7600</orgName>
<orgName type="institution">Université Pierre et Marie Curie [UPMC] - Paris VI</orgName>
</affiliation>
</author>
<title type="main" level="a">Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration</title></analytic><monogr><imprint/></monogr><idno type="HALid">http://hal.archives-ouvertes.fr/hal-00716349</idno><idno type="HALFile">http://hal.archives-ouvertes.fr/hal-00716349/PDF/cane-toads-CRAS-v6.pdf</idno></biblStruct>
</listBibl>
</listBibl>