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Coalescence and sampling distributions for Feller diffusions

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  • Burden, Conrad J.
  • Griffiths, Robert C.

Abstract

Consider the diffusion process defined by the forward equation ut(t,x)=12{xu(t,x)}xx−α{xu(t,x)}x for t,x≥0 and −∞<α<∞, with an initial condition u(0,x)=δ(x−x0). This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller’s solution. For any α and x0>0 we calculate the distribution of the random variable An(s;t), defined as the finite number of ancestors at a time s in the past of a sample of size n taken from the infinite population of a Feller diffusion at a time t since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time t back, conditional on non-extinction as t→∞. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.

Suggested Citation

  • Burden, Conrad J. & Griffiths, Robert C., 2024. "Coalescence and sampling distributions for Feller diffusions," Theoretical Population Biology, Elsevier, vol. 155(C), pages 67-76.
    • Handle: RePEc:eee:thpobi:v:155:y:2024:i:c:p:67-76
    DOI: 10.1016/j.tpb.2023.12.001
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    References listed on IDEAS

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    1. Ignatieva, Anastasia & Hein, Jotun & Jenkins, Paul A., 2020. "A characterisation of the reconstructed birth–death process through time rescaling," Theoretical Population Biology, Elsevier, vol. 134(C), pages 61-76.
    2. 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.
    3. Wiuf, Carsten, 2018. "Some properties of the conditioned reconstructed process with Bernoulli sampling," Theoretical Population Biology, Elsevier, vol. 122(C), pages 36-45.
    4. Crespo, Fausto F. & Posada, David & Wiuf, Carsten, 2021. "Coalescent models derived from birth–death processes," Theoretical Population Biology, Elsevier, vol. 142(C), pages 1-11.
    5. 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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