The genealogy of an exactly solvable Ornstein-Uhlenbeck type branching process with selection

TL;DR

This paper analyzes a solvable population model with selection, showing that its genealogical trees converge to Beta coalescents as population size grows, providing insights into the genealogy of branching processes with selection.

Contribution

It introduces a new exactly solvable model of branching with selection and proves convergence of its genealogy to Beta coalescents as population size increases.

Findings

Genealogies converge to Beta coalescents as N increases

Model serves as a toy version of continuous-time branching with selection

Provides analytical insights into genealogical structures under selection

Abstract

We study the genealogy of a solvable population model with $N$ particles on the real line which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost new-born children are selected to form the next generation. We show that the genealogical trees of the process converge to those of a Beta coalescent as $N \to \infty$. The process we consider can be seen as a toy-model version of a continuous-time branching process with selection, in which particles move according to independent Ornstein-Uhlenbeck processes. The parameter $a$ is akin to the pulling strength of the Ornstein-Uhlenbeck motion.