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Coalescence and sampling distributions for Feller diffusions

Author

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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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