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The process of most recent common ancestors in an evolving coalescent

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  • Pfaffelhuber, P.
  • Wakolbinger, A.

Abstract

Consider a haploid population which has evolved through an exchangeable reproduction dynamics, and in which all individuals alive at time t have a most recent common ancestor (MRCA) who lived at time At, say. As time goes on, not only the population but also its genealogy evolves: some families will get lost from the population and eventually a new MRCA will be established. For a time stationary situation and in the limit of infinite population size N with time measured in N generations, i.e. in the scaling of population genetics which leads to Fisher-Wright diffusions and Kingman's coalescent, we study the process whose jumps form the point process of time pairs (E,B) when new MRCAs are established and when they lived. By representing these pairs as the entrance and exit time of particles whose trajectories are embedded in the look-down graph of Donnelly and Kurtz [P. Donnelly, T.G. Kurtz, Particle representations for measure-valued population models, Ann. Probab. 27 (1) (1999) 166-205] we can show by exchangeability arguments that the times E as well as the times B form a Poisson process. Furthermore, the particle representation helps to compute various features of the MRCA process, such as the distribution of the coalescent at the instant when a new MRCA is established, and the distribution of the number of MRCAs to come that live in today's past.

Suggested Citation

  • Pfaffelhuber, P. & Wakolbinger, A., 2006. "The process of most recent common ancestors in an evolving coalescent," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1836-1859, December.
  • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1836-1859
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    References listed on IDEAS

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    1. Erik M. Rauch & Yaneer Bar-Yam, 2004. "Theory predicts the uneven distribution of genetic diversity within species," Nature, Nature, vol. 431(7007), pages 449-452, September.
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    Cited by:

    1. Wojdyła, Tomasz & Kimmel, Marek & Bobrowski, Adam, 2011. "Time to the MRCA of a sample in a Wright–Fisher model with variable population size," Theoretical Population Biology, Elsevier, vol. 80(4), pages 265-271.
    2. Wirtz, Johannes & Wiehe, Thomas, 2019. "The Evolving Moran Genealogy," Theoretical Population Biology, Elsevier, vol. 130(C), pages 94-105.
    3. Kukla, Jonas & Möhle, Martin, 2018. "On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 939-962.

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