The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

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 article, 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 range of parameters. It is proposed that the probability 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 article.

An Asymptotic Result in a Pure Birth Stochastic Process

“…The first six generalized Bernoulli polynomials are given below. = Γ( / + 1) − / −1 (1 − ) − / −1 (1 + ( −1 )) The last equation confirms the asymptotic result obtained in [1,6]. However, (3.13) can be used to any degree of accuracy.…”

“…When = 1 we have (1) ( ) = ( ), the Bernoullia polynomials, and when = 1 and = 0 we have the Bernoulli numbers (0) = . The first six generalized Bernoulli polynomials are given below.…”

“…In a recent article [1], the researchers considered the number of individuals in a genetic lineage (family size or genus size) under a time homogeneous process that arises from a birth and death stochastic process [8]. The latter is a well-known process but we state the fundamentals here for clarity [9].…”

“…However, when the process leads to several sub-lineages of trees that originated at different times from the origin, then the distribution of the size of the population commonly referred to as taxon size distribution of the genus size must be weighted by the different times that the genera originate. It was shown in [1], that under the exponential distribution for time t the taxon distribution may be ex-pressed as an F 1 2 ( , ; + ; ) hypergeometric function. This form has not been exploited in the literature.…”

An Asymptotic Result in a Pure Birth Stochastic Process

“…We now return to the distribution given in (2.6). Note that the gamma ratio Γ( )/Γ( + ) cancels out with its inverse inside the series, and reduces to a single series involving the = Γ( / + 1) − / −1 (1 − ) − / −1 (1 + ( −1 )) The last equation confirms the asymptotic result obtained in [1,6]. However, (3.13) can be used to any degree of accuracy.…”