logo Journal de l'École Polytechnique
Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
[Des modèles stochastiques d’évolution aux équations de Hamilton-Jacobi]
Nicolas Champagnat1 ; Sylvie Méléard2 ; Sepideh Mirrahimi3 ; Viet Chi Tran4
1 Université de Lorraine, CNRS, Inria, IECL F-54000 Nancy, France
2 École Polytechnique & Institut Universitaire de France, CNRS, Institut polytechnique de Paris route de Saclay, 91128 Palaiseau Cedex, France
3 Institut Montpelliérain Alexander Grothendieck, Univ. Montpellier, CNRS Place Eugène Bataillon, 34090 Montpellier, France
4 LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS F-77454 Marne-la-Vallée, France
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1247-1275.

Nous considérons un modèle stochastique pour l’évolution d’une population discrète structurée en trait à valeurs dans une grille finie du tore, avec mutation et sélection. On se place dans une limite d’échelle de grande population, de petites mutations (mais pas rares), et où le maillage tend vers zéro. En temps long, la contribution de petites sous-populations peut fortement influencer la dynamique. Nous montrons que dans ce cadre, le processus stochastique discret converge sur une échelle logarithmique vers la solution de viscosité d’une équation de Hamilton-Jacobi. La preuve fait appel à un principe du maximum presque-sûr et à des estimées fines des parties martingales.

We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh is much smaller than the size of mutation steps. When considering the evolution of the population in long time scales, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We establish that under our rescaling, the stochastic discrete process converges to the viscosity solution of a Hamilton-Jacobi equation. The proof makes use of almost sure maximum principles and careful control of the martingale parts.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.244
Classification : 92D25, 92D15, 60J80, 60F99, 35F21
Keywords: Stochastic birth death models, large population approximation, selection, mutation, viscosity solution, maximum principle, Hamilton-Jacobi equation
Mot clés : Processus de naissances et morts, processus aléatoire, approximation grande population, sélection, mutation, solution de viscosité, principe du maximum, équation de Hamilton-Jacobi
Nicolas Champagnat 1 ; Sylvie Méléard 2 ; Sepideh Mirrahimi 3 ; Viet Chi Tran 4

1 Université de Lorraine, CNRS, Inria, IECL F-54000 Nancy, France
2 École Polytechnique & Institut Universitaire de France, CNRS, Institut polytechnique de Paris route de Saclay, 91128 Palaiseau Cedex, France
3 Institut Montpelliérain Alexander Grothendieck, Univ. Montpellier, CNRS Place Eugène Bataillon, 34090 Montpellier, France
4 LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS F-77454 Marne-la-Vallée, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2023__10__1247_0,
     author = {Nicolas Champagnat and Sylvie M\'el\'eard and Sepideh Mirrahimi and Viet Chi Tran},
     title = {Filling the gap between individual-based evolutionary models and {Hamilton-Jacobi} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1247--1275},
     publisher = {\'Ecole polytechnique},
     volume = {10},
     year = {2023},
     doi = {10.5802/jep.244},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.244/}
}
TY  - JOUR
AU  - Nicolas Champagnat
AU  - Sylvie Méléard
AU  - Sepideh Mirrahimi
AU  - Viet Chi Tran
TI  - Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2023
SP  - 1247
EP  - 1275
VL  - 10
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.244/
DO  - 10.5802/jep.244
LA  - en
ID  - JEP_2023__10__1247_0
ER  - 
%0 Journal Article
%A Nicolas Champagnat
%A Sylvie Méléard
%A Sepideh Mirrahimi
%A Viet Chi Tran
%T Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
%J Journal de l’École polytechnique — Mathématiques
%D 2023
%P 1247-1275
%V 10
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.244/
%R 10.5802/jep.244
%G en
%F JEP_2023__10__1247_0
Nicolas Champagnat; Sylvie Méléard; Sepideh Mirrahimi; Viet Chi Tran. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1247-1275. doi : 10.5802/jep.244. https://jep.centre-mersenne.org/articles/10.5802/jep.244/

[1] V. Bansaye & S. Méléard - Stochastic models for structured populations. Scaling limits and long time behavior, Math. Biosciences Institute Lect. Series. Stochastics in Biological Systems, vol. 1.4, Springer, Cham, 2015 | DOI

[2] G. Barles - Solutions de viscosité des équations de Hamilton-Jacobi, Math. & Applications, vol. 17, Springer-Verlag, Paris, 1994

[3] G. Barles, S. Mirrahimi & B. Perthame - “Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result”, Methods Appl. Anal. 16 (2009) no. 3, p. 321-340 | DOI | MR | Zbl

[4] J. Berestycki, É. Brunet, J. W. Harris, S. C. Harris & M. I. Roberts - “Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential”, Stochastic Process. Appl. 125 (2015) no. 5, p. 2096-2145 | DOI | MR | Zbl

[5] J. D. Biggins - “Branching out”, in Probability and mathematical genetics, London Math. Soc. Lecture Note Ser., vol. 378, Cambridge University Press, Cambridge, 2010, p. 113-134 | DOI | MR | Zbl

[6] S. Billiard, P. Collet, R. Ferrière, S. Méléard & V. C. Tran - “Stochastic dynamics for adaptation and evolution of microorganisms”, in European Congress of Math., European Mathematical Society, Zürich, 2018, p. 525-550 | DOI | Zbl

[7] J. Blath, T. Paul & A. Tóbiás - “A stochastic adaptive dynamics model for bacterial populations with mutation, dormancy and transfer”, ALEA Lat. Am. J. Probab. Math. Stat. 20 (2023) no. 1, p. 313-357 | DOI | MR | Zbl

[8] A. Bovier, L. Coquille & C. Smadi - “Crossing a fitness valley as a metastable transition in a stochastic population model”, Ann. Appl. Probab. 29 (2019) no. 6, p. 3541-3589 | DOI | MR | Zbl

[9] A. Callegaro & M. I. Roberts - “A spatially- dependent fragmentation process”, 2021 | arXiv

[10] V. Calvez, S. Figueroa Iglesias, H. Hivert, S. Méléard, A. Melnykova & S. Nordmann - “Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches”, in CEMRACS 2018, ESAIM Proc. Surveys, vol. 67, EDP Sci., Les Ulis, 2020, p. 135-160 | DOI | MR | Zbl

[11] N. Champagnat - “A microscopic interpretation for adaptive dynamics trait substitution sequence models”, Stochastic Process. Appl. 116 (2006) no. 8, p. 1127-1160 | DOI | MR | Zbl

[12] N. Champagnat & B. Henry - “A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources”, Ann. Appl. Probab. 29 (2019) no. 4, p. 2175-2216 | DOI | MR | Zbl

[13] N. Champagnat & S. Méléard - “Polymorphic evolution sequence and evolutionary branching”, Probab. Theory Relat. Fields 151 (2011) no. 1-2, p. 45-94 | DOI | MR | Zbl

[14] N. Champagnat, S. Méléard & V. C. Tran - “Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer”, Ann. Appl. Probab. 31 (2021) no. 4, p. 1820-1867 | DOI | MR | Zbl

[15] L. Coquille, A. Kraut & C. Smadi - “Stochastic individual-based models with power law mutation rate on a general finite trait space”, Electron. J. Probab. 26 (2021), article ID 123, 37 pages | DOI | MR | Zbl

[16] U. Dieckmann & R. Law - “The dynamical theory of coevolution: a derivation from stochastic ecological processes”, J. Math. Biol. 34 (1996) no. 5-6, p. 579-612 | DOI | MR | Zbl

[17] O. Diekmann, P.-E. Jabin, S. Mischler & B. Perthame - “The dynamics of adaptation: An illuminating example and a Hamilton–Jacobi approach”, Theoret. Population Biol. 67 (2005) no. 4, p. 257-271 | DOI | Zbl

[18] R. Durrett & J. Mayberry - “Traveling waves of selective sweeps”, Ann. Appl. Probab. 21 (2011) no. 2, p. 699-744 | DOI | MR | Zbl

[19] R. Forien, J. Garnier & F. Patout - “Ancestral lineages in mutation selection equilibria with moving optimum”, Bull. Math. Biol. 84 (2022) no. 9, article ID 93, 43 pages | DOI | MR | Zbl

[20] P.-E. Jabin - “Small populations corrections for selection-mutation models”, Netw. Heterog. Media 7 (2012) no. 4, p. 805-836 | DOI | MR | Zbl

[21] J. Jacod & A. N. Shiryaev - Limit theorems for stochastic processes, Grundlehren Math. Wissen., vol. 288, Springer-Verlag, Berlin, 2003 | DOI

[22] A. Jakubowski - “On the Skorokhod topology”, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) no. 3, p. 263-285 | Numdam | MR | Zbl

[23] A. Joffe & M. Métivier - “Weak convergence of sequences of semimartingales with applications to multitype branching processes”, Adv. in Appl. Probab. 18 (1986) no. 1, p. 20-65 | DOI | MR | Zbl

[24] P. J. Keeling & J. D. Palmer - “Horizontal gene transfer in eukaryotic evolution”, Nat. Rev. Genet. (2008) no. 8, p. 605-618 | DOI

[25] B. R. Levin & F. M. Stewart - “The population biology of bacterial plasmids: a priori conditions for the existence of mobilizable nonconjugative factors”, Genetics 94 (1980) no. 2, p. 425-443 | DOI | MR

[26] A. Lorz, S. Mirrahimi & B. Perthame - “Dirac mass dynamics in multidimensional nonlocal parabolic equations”, Comm. Partial Differential Equations 36 (2011) no. 6, p. 1071-1098 | DOI | MR | Zbl

[27] P. Maillard, G. Raoul & J. Tourniaire - “Spatial dynamics of a population in a heterogeneous environment”, 2021 | arXiv

[28] B. Mallein - “Maximal displacement of a branching random walk in time-inhomogeneous environment”, Stochastic Process. Appl. 125 (2015) no. 10, p. 3958-4019 | DOI | MR | Zbl

[29] J. Metz, S. Geritz, G. Meszéna, F. Jacobs & J. V. Heerwaarden - “Adaptative dynamics: a geometrical study of the consequences of nearly faithful reproduction”, in Proc. Colloquium (Amsterdam, Jan. 1995) (S. V. Strien & S. Verduyn Lunel, eds.), Stochastic and Spatial Structures of Dynamical Systems, vol. 45, North-Holland, 1996, p. 183-231

[30] S. Mirrahimi, G. Barles, B. Perthame & P. E. Souganidis - “A singular Hamilton-Jacobi equation modeling the tail problem”, SIAM J. Math. Anal. 44 (2012) no. 6, p. 4297-4319 | DOI | MR | Zbl

[31] H. Ochman, J. G. Lawrence & E. A. Groisman - “Lateral gene transfer and the nature of bacterial innovation”, Nature 405 (2000), p. 299-304 | DOI

[32] B. Perthame & G. Barles - “Dirac concentrations in Lotka-Volterra parabolic PDEs”, Indiana Univ. Math. J. 57 (2008) no. 7, p. 3275-3301 | DOI | MR | Zbl

[33] B. Perthame & M. Gauduchon - “Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations”, Math. Med. Biol. 27 (2010) no. 3, p. 195-210 | DOI | MR | Zbl

[34] D. Waxman & S. Gavrilets - “20 Questions on adaptive dynamics”, J. Evol. Biol. 18 (2005) no. 5, p. 1139-1154 | DOI

Cité par Sources :