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Mutation in populations governed by a Galton–Watson branching process

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  • Burden, Conrad J.
  • Wei, Yi

Abstract

A population genetics model based on a multitype branching process, or equivalently a Galton–Watson branching process for multiple alleles, is presented. The diffusion limit forward Kolmogorov equation is derived for the case of neutral mutations. The asymptotic stationary solution is obtained and has the property that the extant population partitions into subpopulations whose relative sizes are determined by mutation rates. An approximate time-dependent solution is obtained in the limit of low mutation rates. This solution has the property that the system undergoes a rapid transition from a drift-dominated phase to a mutation-dominated phase in which the distribution collapses onto the asymptotic stationary distribution. The changeover point of the transition is determined by the per-generation growth factor and mutation rate. The approximate solution is confirmed using numerical simulations.

Suggested Citation

  • Burden, Conrad J. & Wei, Yi, 2018. "Mutation in populations governed by a Galton–Watson branching process," Theoretical Population Biology, Elsevier, vol. 120(C), pages 52-61.
    • Handle: RePEc:eee:thpobi:v:120:y:2018:i:c:p:52-61
    DOI: 10.1016/j.tpb.2017.12.001
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    References listed on IDEAS

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    1. Burden, Conrad J. & Simon, Helmut, 2016. "Genetic drift in populations governed by a Galton–Watson branching process," Theoretical Population Biology, Elsevier, vol. 109(C), pages 63-74.
    2. Cyran, Krzysztof A. & Kimmel, Marek, 2010. "Alternatives to the Wright–Fisher model: The robustness of mitochondrial Eve dating," Theoretical Population Biology, Elsevier, vol. 78(3), pages 165-172.
    3. Durrett, Richard & Moseley, Stephen, 2010. "Evolution of resistance and progression to disease during clonal expansion of cancer," Theoretical Population Biology, Elsevier, vol. 77(1), pages 42-48.
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    Cited by:

    1. Burden, Conrad J. & Soewongsono, Albert C., 2019. "Coalescence in the diffusion limit of a Bienaymé–Galton–Watson branching process," Theoretical Population Biology, Elsevier, vol. 130(C), pages 50-59.

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