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A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit

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  • Etheridge, Alison M.
  • Griffiths, Robert C.
  • Taylor, Jesse E.

Abstract

The genealogical structure of neutral populations in which reproductive success is highly-skewed has been the subject of many recent studies. Here we derive a coalescent dual process for a related class of continuous-time Moran models with viability selection. In these models, individuals can give birth to multiple offspring whose survival depends on both the parental genotype and the brood size. This extends the dual process construction for a multi-type Moran model with genic selection described in Etheridge and Griffiths (2009). We show that in the limit of infinite population size the non-neutral Moran models converge to a Markov jump process which we call the Λ-Fleming–Viot process with viability selection and we derive a coalescent dual for this process directly from the generator and as a limit from the Moran models. The dual is a branching-coalescing process similar to the Ancestral Selection Graph which follows the typed ancestry of genes backwards in time with real and virtual lineages. As an application, the transition functions of the non-neutral Moran and Λ-coalescent models are expressed as mixtures of the transition functions of the dual process.

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  • 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.
    • Handle: RePEc:eee:thpobi:v:78:y:2010:i:2:p:77-92
    DOI: 10.1016/j.tpb.2010.05.004
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    References listed on IDEAS

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    1. Matthew Stephens & Peter Donnelly, 2003. "Ancestral Inference in Population Genetics Models with Selection (with Discussion)," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 45(4), pages 395-430, December.
    2. Durrett, Rick & Schweinsberg, Jason, 2005. "A coalescent model for the effect of advantageous mutations on the genealogy of a population," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1628-1657, October.
    3. Schweinsberg, Jason, 2003. "Coalescent processes obtained from supercritical Galton-Watson processes," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 107-139, July.
    4. Etheridge, A.M. & Griffiths, R.C., 2009. "A coalescent dual process in a Moran model with genic selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 320-330.
    5. Sargsyan, Ori & Wakeley, John, 2008. "A coalescent process with simultaneous multiple mergers for approximating the gene genealogies of many marine organisms," Theoretical Population Biology, Elsevier, vol. 74(1), pages 104-114.
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    1. Griffiths, Robert C. & Jenkins, Paul A. & Lessard, Sabin, 2016. "A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning," Theoretical Population Biology, Elsevier, vol. 112(C), pages 126-138.
    2. Desai, Michael M. & Nicolaisen, Lauren E. & Walczak, Aleksandra M. & Plotkin, Joshua B., 2012. "The structure of allelic diversity in the presence of purifying selection," Theoretical Population Biology, Elsevier, vol. 81(2), pages 144-157.
    3. Möhle, Martin, 2024. "On multi-type Cannings models and multi-type exchangeable coalescents," Theoretical Population Biology, Elsevier, vol. 156(C), pages 103-116.
    4. Malaguti, Giulia & Singh, Param Priya & Isambert, Hervé, 2014. "On the retention of gene duplicates prone to dominant deleterious mutations," Theoretical Population Biology, Elsevier, vol. 93(C), pages 38-51.
    5. Der, Ricky & Epstein, Charles L. & Plotkin, Joshua B., 2011. "Generalized population models and the nature of genetic drift," Theoretical Population Biology, Elsevier, vol. 80(2), pages 80-99.
    6. Kon Kam King, Guillaume & Pandolfi, Andrea & Piretto, Marco & Ruggiero, Matteo, 2024. "Approximate filtering via discrete dual processes," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    7. Bjarki Eldon, 2023. "Viability Selection at Linked Sites," Mathematics, MDPI, vol. 11(3), pages 1-23, January.
    8. Koskela, Jere & Šatuszyński, Krzysztof & Spanò, Dario, 2024. "Bernoulli factories and duality in Wright–Fisher and Allen–Cahn models of population genetics," Theoretical Population Biology, Elsevier, vol. 156(C), pages 40-45.

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