Particle configurations for branching Brownian motion with an inhomogeneous branching rate

TL;DR

This paper analyzes the distribution of fitness levels in a population modeled by branching Brownian motion with location-dependent birth and death rates, revealing a traveling wave behavior related to the Airy function.

Contribution

It provides explicit formulas for the asymptotic fitness distribution and its empirical density, extending previous work with a detailed description of the distribution's evolution.

Findings

Distribution converges to a traveling wave profile.

Asymptotic density relates to the Airy function.

Results complement and extend prior research.

Abstract

Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distribution. We show that after a sufficiently long time, the fitness distribution from the lowest to the highest fitness levels approximately evolves as a traveling wave with a profile which is asymptotically related the the Airy function. Our work complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.