Smaller population size at the MRCA time for stationary branching processes

TL;DR

This paper analyzes a stationary continuous state branching process, deriving distributions and asymptotics for population size, MRCA timing, and genealogical structure, revealing a mild bottleneck effect and explicit results for quadratic mechanisms.

Contribution

It provides explicit formulas and asymptotic results for the genealogical structure and population sizes in stationary branching processes, including the quadratic case.

Findings

Population size at MRCA is smaller than current population.

Explicit distribution of time to MRCA and population sizes.

Quadratic case shows mean population at MRCA is 1/3 less than current.

Abstract

We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individuals involved in the last coalescent event of the genealogical tree. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, less by 1/3 than size of the current population size. We also provide in this case the fluctuations for the renormalized number of ancestors.