Rare Mutations Limit of a Steady State Dispersal Evolution Model

B. Perthame¹ ; P. E. Souganidis²

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, UMR 7598 Laboratoire Jacques-Louis Lions Équipe MAMBA, 4, place Jussieu 75005, Paris, France
² Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

Mathematical modelling of natural phenomena, Tome 11 (2016) no. 4, pp. 154-166

Résumé

The evolution of a dispersal trait is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and, more recently, for traveling waves in the full space.In the present study, we reformulate the problem in the context of adaptive evolution. We consider a population structured by space and a genetic trait acting directly on the dispersal (diffusion) rate under the effect of rare mutations on the genetic trait. We show that, as in simpler models, in the limit of vanishing mutations, the population concentrates on a single trait associated to the lowest dispersal rate. We also explain how to compute the evolution speed towards this evolutionary stable distribution.The mathematical interest stems from the asymptotic analysis which requires a completely different treatment for each variable. For the space variable, the ellipticity leads to the use the maximum principle and Sobolev-type regularity results. For the trait variable, the concentration to a Dirac mass requires a different treatment. This is based on the WKB method and viscosity solutions leading to an effective Hamiltonian (effective fitness of the population) and a constrained Hamilton-Jacobi equation.

Détail

DOI : 10.1051/mmnp/201611411

Bibliographie

[1] A. Arnold, L. Desvillettes, C. Prévost Existence of nontrivial steady states for populations structured with respect to space and a continuous trait Commun. Pure Appl. Anal. 83 96 2012

[2] M. Baar, A. Bovier, N. Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics - in one step. http://arxiv.org/abs/1505.02421 arXiv:1505.02421, 2015. To appear in Ann. Appl. Probab.

[3] H. Berestycki, B. Nicolaenko, B. Scheurer Traveling wave solutions to combustion models and their singular limits SIAM J. Math. Anal. 1985 1207 1242

[4] H. Berestycki, T. Jin, L. Silvestre. Propagation in a nonlocal reaction diffusion equation with spatial and genetic trait structure. http://arxiv.org/abs/1411.2019 arXiv:1411.2019, 2014. To appear in Nonlinearity.

[5] N. Berestycki, C. Mouhot, G. Raoul. Existence of self-accelerating fronts for a non-local reaction-diffusion equation. http://arxiv.org/abs/1512.00903 ArXiv:1512.00903, (2015).

[6] E. Bouin, V. Calvez Travelling waves for the cane toads equation with bounded traits Nonlinearity 2233 2014

[7] E. Bouin, V. Calvez Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts Archive for Rational Mechanics and Analysis 571 617 2015

[8] E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul, R. Voituriez Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration C. R. Math. Acad. Sci. Paris 761 766 2012

[9] E. Bouin, S. Mirrahimi A Hamilton-Jacobi limit for a model of population structured by space and trait Comm. Math Sci. 1431 1452 2015

[10] F. Campillo, C. Fritsch Weak convergence of a mass-structured individual-based model Applied Mathematics & Optimization 2015 37 73

[11] R. S. Cantrell, C. Cosner, Y. Lou Approximating the ideal free distribution via reaction-diffusion-advection equations J. Differential Equations 2008

[12] N. Champagnat, R. Ferrière, S. Méléard Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models Th. Pop. Biol. 297 321 2006

[13] C. Cosner, J. Dávila, S. Martínez Evolutionary stability of ideal free nonlocal dispersal J. Biol. Dyn. 2012

[14] J. Coville, L. Dupaigne Propagation speed of travelling fronts in non local reaction-Diffusion equations Nonlin. Anal. 2005 797 819

[15] L. Desvillettes, P.-E. Jabin, S. Mischler, G. Raoul On selection dynamics for continuous structured populations Comm. Math. Sci. 729 747 2008

[16] O. Diekmann A beginner's guide to adaptive dynamics Mathematical modeling of population dynamics. Banach Center Publications 2004 47 86

[17] O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach Th. Pop. Biol. 257 271 2005

[18] J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski The evolution of slow dispersal rates: a reaction Diffusion model J. Math. Biol. 1998 61 83

[19] L. C. Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc. Roy. Soc. Edinburgh 1989 359 375

[20] S. A. H. Geritz, J. A. J. Metz, E. Kisdi, G. Meszena Dynamics of adaptation and evolutionary branching Physical Review Letters 1997 2024 2027

[21] R. Hambrock, Y. Lou The evolution of conditional dispersal strategies in spatially heterogeneous habitats Bull. Math. Biol. 2009

[22] A. Hastings Can spatial variation alone lead to selection for dispersal? Theoret. Popul. Biol. 1983 244 251

[23] J. Hofbauer, K. Sigmund, Evolutionary games and population dynamics. London Mathematical Society, Student texts 7. Cambridge University Press (2002).

[24] V. Hutson, S. Martínez, K. Mischaikow, G. T. Vickers The evolution of dispersal J. Math. Biol. 2003 483 517

[25] P.-E. Jabin, G. Raoul Selection dynamics with competition J. Math. Biol. 2011 493 517

[26] K.-Y. Lam Y. Lou. A mutation-selection model for evolution of random dispersal. http://arxiv.org/pdf/1506.00662.pdf ArXiv 1506.00662, 2015.

[27] P.-L. Lions Neumann type boundary conditions for Hamilton-Jacobi equations Duke Math. J. 1985 793 820

[28] A. Lorz, S. Mirrahimi, B. Perthame Dirac mass dynamics in multidimensional nonlocal parabolic equations Comm. Partial Differential Equations 1071 1098 2011

[29] J. Maynard Smith The theory of games and the evolution of animal conicts J. Theor. Biol. 1974 209 221

[30] S. Mirrahimi, B. Perthame Asymptotic analysis of a selection model with space J. Math. Pures et Appl 2015 1108 1118

[31] S. Mirrahimi, J.-M. Roquejoffre Uniqueness in a class of Hamilton-Jacobi equations with constraints Comptes Rendus Ac. Sc. Paris, Mathematiques 2015 489 494

[32] S. Mirrahimi, J.-M. Roquejoffre A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach Journal of Differential Equations 4717 4738

[33] K. Parvinen, U. Dieckmann, M. Gyllenberg, J. A. J. Metz Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity J. Evol. Biol. 2003 143 53

[34] B. Perthame, G. Barles Dirac concentrations in Lotka-Volterra parabolic PDEs Indiana Univ. Math. J. 3275 3301 2008

[35] A. Potapov, U. E. Schläagel, M. A. Lewis Evolutionary stable diffusive dispersal DCDS(B) 2014 3319 3340

[36] O. Ronce How does it feel to be like a rolling stone? Ten questions about dispersal evolution Annu. Rev. Ecol. Evol. Syst. 2007 231 253

[37] O. Turanova On a model of a population with variable motility Mathematical Models and Methods in Applied Sciences 1961 2014 2015