A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

TL;DR

This paper proves that in a population model with position-dependent birth and death rates, the distribution of individuals' positions becomes approximately Gaussian over time, supporting biological observations of fitness evolution as a Gaussian wave.

Contribution

It provides a rigorous mathematical proof that the empirical distribution in an inhomogeneous branching Brownian motion converges to a Gaussian, connecting stochastic processes with biological fitness models.

Findings

Distribution of particles becomes Gaussian over time

Supports biological models of fitness evolution as Gaussian waves

Provides rigorous mathematical justification for empirical observations

Abstract

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.