Beta-coalescents when sample size is large

TL;DR

This paper extends models of sweepstakes reproduction to include an upper bound on offspring number, analyzing how different bounds influence the convergence to Beta-coalescents and their impact on genetic diversity patterns.

Contribution

It introduces a biologically motivated upper bound on offspring number in coalescent models and analyzes its effects on convergence and genetic diversity.

Findings

Convergence to Beta-coalescents can be slow with certain bounds.

Small sample sizes may invalidate coalescent approximation.

Site-frequency spectrum aligns with the limiting tree predictions under Beta-coalescents.

Abstract

Sweepstakes reproduction refers to a highly skewed individual recruitment success without involving natural selection and may apply to individuals in broadcast spawning populations characterised by Type III survivorship. We consider an extension of the model of sweepstakes reproduction for a haploid panmictic population of constant size $N$; the extension also works as an alternative to the Wright-Fisher model. Our model incorporates an upper bound on the random number of potential offspring (juveniles) produced by a given individual. Depending on how the bound behaves relative to the total population size, we obtain the Kingman coalescent, an incomplete Beta-coalescent, or the (complete) Beta-coalescent. We argue that applying such an upper bound is biologically reasonable. Moreover, we estimate the error of the coalescent approximation. The error estimates reveal that convergence can be slow, and small sample size can be sufficient to invalidate convergence, for example if the stated bound is of the form $N/\log N$. We use simulations to investigate the effect of increasing sample size on the site-frequency spectrum. When the limit is a Beta-coalescent, the site frequency spectrum will be as predicted by the limiting tree even though the full coalescent tree may deviate from the limiting one. When in the domain of attraction of the Kingman coalescent the effect of increasing sample size depends on the effective population size as has been noted in the case of the Wright-Fisher model. Conditioning on the population ancestry (the random ancestral relations of the entire population at all times) may have little effect on the site-frequency spectrum for the models considered here (as evidenced by simulation results).