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

Aser Cortines,Bastien Mallein

doi:10.1214/18-ecp197

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.

Highlights

A branching–selection particle system is a Markov process of particles on the real line that evolves through the repeated application of the two following steps: Branching step: each particle currently alive in the process independently gives birth to children according to a point process whose law might depend on the position of the particle
From a biological perspective, such models can be thought of as toy-models for the competition between individuals in a population evolving in an environment with limited resources: natural selection
The prototypical example of such systems is the so-called N -branching random walk, which was introduced by Brunet and Derrida in [10]

Summary

Introduction

A branching–selection particle system is a Markov process of particles on the real line that evolves through the repeated application of the two following steps: Branching step: each particle currently alive in the process independently gives birth to children according to a point process whose law might depend on the position of the particle. Other examples for which above conjecture has been verified consist in models where the system becomes exactly solvable [11, 13, 14] This is in particular the case for the so-called exponential model [11]: it consists in a discrete-time N -branching random walk where particles reproduce according to independent Poisson point processes with intensity e−xdx. Proposition 1.2 shows that the cloud of particle in the (N, a)-exponential model is roughly of size log N , which is typical in many branching selection particle systems The proofs of both Proposition 1.2 and Theorem 1.1 rely on the observation that the distribution of the children at time n + 1 is a Poisson point process with exponential intensity around the position of a unique fictitious particle.

Proofs of main results

Findings

Branching Ornstein–Uhlenbeck process with selection