Coalescence and sampling distributions for Feller diffusions

2022/10/24 by Burden, Conrad J., Griffiths, Robert C.
#FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR)

paper · doi:10.48550/arxiv.2210.12894

Abstract

Consider the diffusion process defined by the forward equation u_t(t, x) = \tfrac12\x u(t, x)\_xx - α\x u(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 A_n(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 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.

Related

• The Feller diffusion conditioned on a single ancestral founder
• Coalescence in the diffusion limit of a Bienayme-Galton-Watson branching\n process
• The stationary distribution of a sample from the Wright-Fisher diffusion\n model with general small mutation rates
• Coalescent processes emerging from large deviations
• The transition distribution of a sample from a Wright-Fisher diffusion with general small mutation rates