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Coalescence and sampling distributions for Feller diffusions.

Conrad J Burden1, Robert C Griffiths2

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This study analyzes Feller diffusion, a model for population growth, by examining coalescent processes. Researchers calculated ancestor distributions and coalescent tree properties for both subcritical and supercritical population dynamics.

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Area of Science:

  • Stochastic processes
  • Population dynamics
  • Mathematical biology

Background:

  • Feller's diffusion equation models population growth with independent individuals.
  • Coalescent processes are crucial for understanding population history and genetic diversity.
  • Previous work established Feller's solution but lacked detailed coalescent analysis.

Purpose of the Study:

  • To investigate coalescent processes associated with Feller's diffusion model.
  • To calculate the distribution of the number of ancestors for a sample from a Feller diffusion.
  • To characterize population and sample coalescent trees in subcritical and supercritical diffusions.

Main Methods:

  • Analysis of the forward diffusion equation u_t(t,x) = 1/2{xu(t,x)}_xx - α{xu(t,x)}_x.
  • Calculation of the distribution of A_n(s;t), the number of ancestors at time s for a sample of size n at time t.
  • Conditional analysis of coalescent trees in subcritical diffusions (non-extinction as t→∞).
  • Construction of a single-founder coalescent tree and derivation of coalescent times in supercritical diffusions.

Main Results:

  • The distribution of the number of ancestors A_n(s;t) was derived for Feller diffusion.
  • For subcritical diffusions, distributions of population and sample coalescent trees were found, conditional on non-extinction.
  • A coalescent tree with a single founder was constructed for supercritical diffusions, with coalescent times derived.

Conclusions:

  • The study provides a comprehensive analysis of coalescent phenomena within Feller's population growth model.
  • Understanding these coalescent processes is vital for inferring population history from genetic data.
  • The findings offer new insights into the structure of ancestral lineages in branching processes.