Time to a single hybridization event in a group of species with unknown ancestral history

TL;DR

This paper models the timing of a single hybridization event in a species group with an unknown origin, combining a Yule process with hybridization, and derives the distribution of the hybridization time.

Contribution

It introduces a stochastic model integrating Yule process and hybridization, providing formulas for the hybridization time distribution and its asymptotic behavior.

Findings

Distribution of hybridization time derived

Distribution tends to exponential under certain conditions

Formulas for all moments of hybridization time

Abstract

We consider a stochastic process for the generation of species which combines a Yule process with a simple model for hybridization between pairs of co-existent species. We assume that the origin of the process, when there was one species, occurred at an unknown time in the past, and we condition the process on producing n species via the Yule process and a single hybridization event. We prove results about the distribution of the time of the hybridization event. In particular we calculate a formula for all moments, and show that under various conditions, the distribution tends to an exponential with rate twice that of the birth rate for the Yule process.