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Duality, ancestral and diffusion processes in models with selection

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  • Mano, Shuhei

Abstract

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.

Suggested Citation

  • Mano, Shuhei, 2009. "Duality, ancestral and diffusion processes in models with selection," Theoretical Population Biology, Elsevier, vol. 75(2), pages 164-175.
    • Handle: RePEc:eee:thpobi:v:75:y:2009:i:2:p:164-175
    DOI: 10.1016/j.tpb.2009.01.007
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    Cited by:

    1. Kobayashi, Yutaka & Wakano, Joe Yuichiro & Ohtsuki, Hisashi, 2018. "Genealogies and ages of cultural traits: An application of the theory of duality to the research on cultural evolution," Theoretical Population Biology, Elsevier, vol. 123(C), pages 18-27.
    2. Pokalyuk, Cornelia & Pfaffelhuber, Peter, 2013. "The ancestral selection graph under strong directional selection," Theoretical Population Biology, Elsevier, vol. 87(C), pages 25-33.
    3. González Casanova, Adrián & Miró Pina, Verónica & Pardo, Juan Carlos, 2020. "The Wright–Fisher model with efficiency," Theoretical Population Biology, Elsevier, vol. 132(C), pages 33-46.
    4. Bossert, S. & Pfaffelhuber, P., 2018. "The fixation probability and time for a doubly beneficial mutant," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4018-4050.
    5. 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.
    6. Lenz, Ute & Kluth, Sandra & Baake, Ellen & Wakolbinger, Anton, 2015. "Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution," Theoretical Population Biology, Elsevier, vol. 103(C), pages 27-37.
    7. Wakeley, John & Sargsyan, Ori, 2009. "The conditional ancestral selection graph with strong balancing selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 355-364.
    8. 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.
    9. Cordero, Fernando, 2017. "Common ancestor type distribution: A Moran model and its deterministic limit," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 590-621.

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