Moran model with simultaneous strong and weak selections: convergence towards a Λ-Wright–Fisher SDE
François G. Ged1
1 Chair of Statistical Field Theory, École Polytechnique Fédérale de Lausanne, Switzerland
MathematicS In Action, Tome 12 (2023) no. 1, pp. 87-116.

We establish a connection between two population models by showing that one is the scaling limit of the other, as the population grows large. In the infinite population model, individuals are split into two subpopulations, carrying either a selective advantageous allele, or a disadvantageous one. The proportion of disadvantaged individuals in the population evolves according to the Λ-Wright–Fisher stochastic differential equation (SDE) with selection, and the genealogy is described by the so-called Bolthausen–Sznitman coalescent. This equation has appeared in the Λ-lookdown model with selection studied by Bah and Pardoux [1]. Schweinsberg in [16] showed that in a specific setting, due to the strong selection, the genealogy of the so-called Moran model with selection converges to the Bolthausen–Sznitman coalescent. By splitting the population into two adversarial subgroups and adding a weak selection mechanism, we show that the proportion of disadvantaged individuals in the Moran model with strong and weak selections converges to the solution of the Λ-Wright–Fisher SDE of [1].

Publié le :
DOI : 10.5802/msia.33
Classification : 60J80, 92D15, 92D25, 60H10
Mots clés : Moran model with selection, Bolthausen–Sznitman’s coalescent, $\Lambda $-Wright–Fisher SDE
François G. Ged 1

1 Chair of Statistical Field Theory, École Polytechnique Fédérale de Lausanne, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{MSIA_2023__12_1_87_0,
     author = {Fran\c{c}ois G. Ged},
     title = {Moran model with simultaneous strong and weak selections: convergence towards a $\Lambda ${-Wright{\textendash}Fisher} {SDE}},
     journal = {MathematicS In Action},
     pages = {87--116},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {12},
     number = {1},
     year = {2023},
     doi = {10.5802/msia.33},
     language = {en},
     url = {https://msia.centre-mersenne.org/articles/10.5802/msia.33/}
}
TY  - JOUR
AU  - François G. Ged
TI  - Moran model with simultaneous strong and weak selections: convergence towards a $\Lambda $-Wright–Fisher SDE
JO  - MathematicS In Action
PY  - 2023
SP  - 87
EP  - 116
VL  - 12
IS  - 1
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://msia.centre-mersenne.org/articles/10.5802/msia.33/
DO  - 10.5802/msia.33
LA  - en
ID  - MSIA_2023__12_1_87_0
ER  - 
%0 Journal Article
%A François G. Ged
%T Moran model with simultaneous strong and weak selections: convergence towards a $\Lambda $-Wright–Fisher SDE
%J MathematicS In Action
%D 2023
%P 87-116
%V 12
%N 1
%I Société de Mathématiques Appliquées et Industrielles
%U https://msia.centre-mersenne.org/articles/10.5802/msia.33/
%R 10.5802/msia.33
%G en
%F MSIA_2023__12_1_87_0
François G. Ged. Moran model with simultaneous strong and weak selections: convergence towards a $\Lambda $-Wright–Fisher SDE. MathematicS In Action, Tome 12 (2023) no. 1, pp. 87-116. doi : 10.5802/msia.33. https://msia.centre-mersenne.org/articles/10.5802/msia.33/

[1] Boubacar Bah; Etienne Pardoux The Λ-lookdown model with selection, Stochastic Processes Appl., Volume 125 (2015) no. 3, pp. 1089-1126 | MR

[2] Nathanael Berestycki Recent Progress in Coalescent Theory, Ensaios Matematicos, 16, Sociedade Brasileira de Matemática, 2009 | DOI | MR

[3] Jean Bertoin; Jean-François Le Gall Stochastic flows associated to coalescent processes, Probab. Theory Relat. Fields, Volume 126 (2003) no. 2, pp. 261-288 | DOI | MR | Zbl

[4] Jean Bertoin; Jean-François Le Gall Stochastic flows associated to coalescent processes II: Stochastic differential equations, Annales de l’Institut Henri Poincare (B) Probability and Statistics, Volume 41 (2005) no. 3, pp. 307-333 | DOI | Numdam | MR | Zbl

[5] Patrick Billingsley Convergence of probability measures, John Wiley & Sons, 2013

[6] Erwin Bolthausen; A.-S. Sznitman On Ruelle’s probability cascades and an abstract cavity method, Commun. Math. Phys., Volume 197 (1998) no. 2, pp. 247-276 | DOI | MR | Zbl

[7] Éric Brunet; Bernard Derrida; Alfred H. Mueller; Stéphane Munier Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization, Phys. Rev. E, Volume 76 (2007) no. 4, p. 041104 | DOI | MR

[8] Donald A. Dawson; Zenghu Li Stochastic equations, flows and measure-valued processes, Ann. Probab., Volume 40 (2012) no. 2, pp. 813-857 | MR | Zbl

[9] Michael M. Desai; Aleksandra M. Walczak; Daniel S. Fisher Genetic diversity and the structure of genealogies in rapidly adapting populations, Genetics, Volume 193 (2013) no. 2, pp. 565-585 | DOI

[10] Jean Jacod; Albert Shiryaev Limit theorems for stochastic processes, 288, Springer, 2013

[11] Thomas G. Kurtz Equivalence of stochastic equations and martingale problems, Stochastic analysis 2010, Springer, 2011, pp. 113-130 | DOI | Zbl

[12] Martin Möhle Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models, Adv. Appl. Probab., Volume 32 (2000) no. 4, pp. 983-993 | DOI | MR | Zbl

[13] Martin Möhle The coalescent in population models with time-inhomogeneous environment, Stochastic Processes Appl., Volume 97 (2002) no. 2, pp. 199-227 | DOI | MR | Zbl

[14] Richard A. Neher; Oskar Hallatschek Genealogies of rapidly adapting populations, Proc. Natl. Acad. Sci. USA, Volume 110 (2013) no. 2, pp. 437-442 | DOI

[15] Jason Schweinsberg Rigorous results for a population model with selection I: evolution of the fitness distribution, Electron. J. Probab., Volume 22 (2017), pp. 1-94 | MR | Zbl

[16] Jason Schweinsberg Rigorous results for a population model with selection II: genealogy of the population, Electron. J. Probab., Volume 22 (2017), pp. 1-54 | MR | Zbl

Cité par Sources :