Taxon Size Distribution in a Time Homogeneous Birth and Death Process

TL;DR

This paper derives the distribution of lineage sizes in a birth-death process, representing it with hypergeometric functions, and provides approximations validated across various parameters, with applications to biological diversity and lineage coalescence.

Contribution

It introduces a novel analytical representation of lineage size distributions using hypergeometric functions and offers robust approximations validated through comparisons with asymptotic and numerical methods.

Findings

Distributions can be expressed by hypergeometric functions of the second kind.

Second-order approximations are accurate across a wide parameter range.

Results have applications in biological lineage coalescence and species number prediction.

Abstract

The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this paper, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide rane of parameters. It is proposed that the probability density distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this paper.