On the genealogy of a population of biparental individuals

TL;DR

This paper analyzes the genealogical structure of a finite, panmictic population, revealing how ancestral repetitions stabilize over time and how individual genealogies converge after a certain number of generations.

Contribution

It provides a mathematical characterization of ancestor repetitions and the convergence of genealogical trees in finite populations, extending to growing populations.

Findings

Ancestor repetition distribution reaches stationarity after ~log N generations.

Approximately 80% of the ancestral population is included in an individual's genealogy.

Genealogical trees of individuals become identical after ~log N generations.

Abstract

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic closed population of constant size N can be calculated. We show that the distribution of the repetitions of ancestors reaches a stationary shape after a small number Gc ~ log N of generations in the past, that only about 80% of the ancestral population belongs to the tree (due to coalescence of branches), and that two trees for individuals in the same population become identical after Gc generations have elapsed. Our analysis is easy to extend to the case of exponentially growing population.