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The ancestral selection graph for a Λ-asymmetric Moran model

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  • González Casanova, Adrián
  • Kurt, Noemi
  • Pérez, José Luis

Abstract

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

Suggested Citation

  • González Casanova, Adrián & Kurt, Noemi & Pérez, José Luis, 2024. "The ancestral selection graph for a Λ-asymmetric Moran model," Theoretical Population Biology, Elsevier, vol. 159(C), pages 91-107.
    • Handle: RePEc:eee:thpobi:v:159:y:2024:i:c:p:91-107
    DOI: 10.1016/j.tpb.2024.02.010
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    References listed on IDEAS

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    1. Baake, E. & Esercito, L. & Hummel, S., 2023. "Lines of descent in a Moran model with frequency-dependent selection and mutation," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 409-457.
    2. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    3. Bah, B. & Pardoux, E., 2015. "The Λ-lookdown model with selection," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 1089-1126.
    4. González Casanova, Adrián & Kurt, Noemi & Wakolbinger, Anton & Yuan, Linglong, 2016. "An individual-based model for the Lenski experiment, and the deceleration of the relative fitness," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2211-2252.
    5. Kluth, Sandra & Baake, Ellen, 2013. "The Moran model with selection: Fixation probabilities, ancestral lines, and an alternative particle representation," Theoretical Population Biology, Elsevier, vol. 90(C), pages 104-112.
    6. Pokalyuk, Cornelia & Pfaffelhuber, Peter, 2013. "The ancestral selection graph under strong directional selection," Theoretical Population Biology, Elsevier, vol. 87(C), pages 25-33.
    7. Etheridge, Alison M. & Griffiths, Robert C. & Taylor, Jesse E., 2010. "A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit," Theoretical Population Biology, Elsevier, vol. 78(2), pages 77-92.
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