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 $\Lambda $-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 $\Lambda $-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 $\Lambda $-Wright–Fisher SDE of [1].

Keywords: Moran model with selection, Bolthausen–Sznitman’s coalescent, $\Lambda $-Wright–Fisher SDE

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@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, Volume 12 (2023) no. 1, pp. 87-116. doi : 10.5802/msia.33. https://msia.centre-mersenne.org/articles/10.5802/msia.33/

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