Partial Differential Equations
Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration
Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 761-766.

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 t3/2. 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.

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 t3/2. 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.09.010
Bouin, Emeric 1; Calvez, Vincent 1; Meunier, Nicolas 2; Mirrahimi, Sepideh 3; Perthame, Benoît 4; Raoul, Gaël 5; Voituriez, Raphaël 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
@article{CRMATH_2012__350_15-16_761_0,
     author = {Bouin, Emeric and Calvez, Vincent and Meunier, Nicolas and Mirrahimi, Sepideh and Perthame, Beno{\^\i}t and Raoul, Ga\"el and Voituriez, Rapha\"el},
     title = {Invasion fronts with variable motility: {Phenotype} selection, spatial sorting and wave acceleration},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {761--766},
     publisher = {Elsevier},
     volume = {350},
     number = {15-16},
     year = {2012},
     doi = {10.1016/j.crma.2012.09.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.09.010/}
}
TY  - JOUR
AU  - Bouin, Emeric
AU  - Calvez, Vincent
AU  - Meunier, Nicolas
AU  - Mirrahimi, Sepideh
AU  - Perthame, Benoît
AU  - Raoul, Gaël
AU  - Voituriez, Raphaël
TI  - Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 761
EP  - 766
VL  - 350
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.09.010/
DO  - 10.1016/j.crma.2012.09.010
LA  - en
ID  - CRMATH_2012__350_15-16_761_0
ER  - 
%0 Journal Article
%A Bouin, Emeric
%A Calvez, Vincent
%A Meunier, Nicolas
%A Mirrahimi, Sepideh
%A Perthame, Benoît
%A Raoul, Gaël
%A Voituriez, Raphaël
%T Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration
%J Comptes Rendus. Mathématique
%D 2012
%P 761-766
%V 350
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.09.010/
%R 10.1016/j.crma.2012.09.010
%G en
%F CRMATH_2012__350_15-16_761_0
Bouin, Emeric; Calvez, Vincent; Meunier, Nicolas; Mirrahimi, Sepideh; Perthame, Benoît; Raoul, Gaël; Voituriez, Raphaël. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.09.010/

[1] Arnold, A.; Desvillettes, L.; Prévost, C. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait, Commun. Pure Appl. Anal., Volume 11 (2012) no. 1, pp. 83-96

[2] Barles, G.; Perthame, B. Concentrations and constrained Hamilton–Jacobi equations arising in adaptive dynamics, Recent Developments in Nonlinear Partial Differential Equations, Contemp. Math., vol. 439, Amer. Math. Soc., Providence, RI, 2007, pp. 57-68

[3] O. Bénichou, V. Calvez, N. Meunier, R. Voituriez, Front acceleration by dynamic selection in Fisher population waves, preprint.

[4] Bouin, E.; Calvez, V. A kinetic eikonal equation, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 5–6, pp. 243-248

[5] Champagnat, N.; Méléard, S. Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., Volume 55 (2007) no. 2, pp. 147-188

[6] Champagnat, N.; Ferrière, R.; Méléard, S. From individual stochastic processes to macroscopic models in adaptive evolution, Stoch. Models, Volume 24 (2008) no. Suppl. 1, pp. 2-44

[7] Dieckmann, U.; Law, R. The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol., Volume 34 (1996) no. 5–6, pp. 579-612

[8] Diekmann, O.; Jabin, P.-E.; Mischler, S.; Perthame, B. The dynamics of adaptation: an illuminating example and a Hamilton–Jacobi approach, Theor. Popul. Biol., Volume 67 (2005) no. 4, pp. 257-271

[9] Dockery, J.; Hutson, V.; Mischaikow, K.; Pernarowski, M. The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol., Volume 37 (1998) no. 1, pp. 61-83

[10] Evans, L.C.; Souganidis, P.E. A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J., Volume 38 (1989) no. 1, pp. 141-172

[11] Freidlin, M.I. Geometric optics approach to reaction–diffusion equations, SIAM J. Appl. Math., Volume 46 (1986) no. 2, pp. 222-232

[12] Kokko, H.; López-Sepulcre, A. From individual dispersal to species ranges: perspectives for a changing world, Science, Volume 313 (2006) no. 5788, pp. 789-791

[13] Lorz, A.; Mirrahimi, S.; Perthame, B. Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differential Equations, Volume 36 (2011) no. 6, pp. 1071-1098

[14] Phillips, B.L.; Brown, G.P.; Webb, J.K.; Shine, R. Invasion and the evolution of speed in toads, Nature, Volume 439 (2006) no. 7078, p. 803

[15] Ronce, O. How does it feel to be like a rolling stone? Ten questions about dispersal evolution, Annu. Rev. Ecol. Syst., Volume 38 (2007), pp. 231-253

[16] Shine, R.; Brown, G.P.; Phillips, B.L. An evolutionary process that assembles phenotypes through space rather than through time, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 14, pp. 5708-5711

[17] Simmons, A.D.; Thomas, C.D. Changes in dispersal during speciesʼ range expansions, Amer. Nat., Volume 164 (2004), pp. 378-395

Cited by Sources: