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Common ancestor type distribution: A Moran model and its deterministic limit

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  • Cordero, Fernando

Abstract

In a Moran model with population size N, two types, mutation and selection, let hkN be the probability that the common ancestor is fit, given that the current number of fit individuals is k. First, we express hkN in terms of the tail probabilities of an appropriate random variable LN. Next, we show that, when N tends to infinity (without any rescaling of parameters or time), LN converges to a geometric random variable. We also obtain a formula for h(x), the limit of hkN when k/N tends to x∈(0,1). In a second step, we describe two ways of pruning the ancestral selection graph (ASG) leading to the notions of relevant ASG and of pruned lookdown ASG (pruned LD-ASG). We use these objects to provide graphical derivations of the aforementioned results. In particular, we show that LN is distributed as the asymptotic number of lines in the relevant ASG and as the stationary number of lines in the pruned LD-ASG. Finally, we construct an asymptotic version of the pruned LD-ASG providing a graphical interpretation of the function h.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:2:p:590-621
    DOI: 10.1016/j.spa.2016.06.019
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    References listed on IDEAS

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    1. Mano, Shuhei, 2009. "Duality, ancestral and diffusion processes in models with selection," Theoretical Population Biology, Elsevier, vol. 75(2), pages 164-175.
    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. 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.
    4. 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.
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    Cited by:

    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.

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