A multi-epoch model for the number of species within genera

An early question in evolutionary theory asked why frequency distributions of taxonomic group sizes exhibit "hollow curves" so frequently. An answer to this question was provided by G. Udny Yule's seminal contribution introducing a discrete model for those distributions. But Yule observed that the fit of his model to observed distributions was sometimes imperfect, in particular for the class of reptiles. The present study introduces a multi-epoch extension of the discrete Yule model that accounts for unobserved extinction of ancient lineages. The multi-epoch model is described as a Pòlya urn embedded in a continuous-time branching process with an harmonic sequence of diversification rates. The main results include equivalent descriptions of multi-epoch models, their probability distributions, expected values, tail behavior and a self-similarity property. As an illustration of the theory, the multi-epoch model is applied to study the taxonomic diversity of reptile species, and provides a much better fit to the observed distribution of species than the original discrete Yule model.