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

Abstract

We study the genealogy of an exactly 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 newborn children are selected to form the next generation. We show that the genealogy of the process converges toward 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 process.