Brownian bees in the infinite swarm limit

TL;DR

This paper studies a particle system where particles move randomly and are selectively removed to maintain a constant population, showing that as the system size grows, its behavior converges to a solution of a specific free boundary problem.

Contribution

It proves the hydrodynamic limit of the Brownian bees model and establishes the convergence of the equilibrium density to the free boundary problem's steady state.

Findings

Hydrodynamic limit of the particle system is characterized by a free boundary problem.

Equilibrium densities converge to the steady state solution of the free boundary problem.

The selection principle holds in the infinite population limit.

Abstract

The Brownian bees model is a branching particle system with spatial selection. It is a system of $N$ particles which move as independent Brownian motions in $\mathbb{R}^d$ and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that as $N \to \infty$ the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds, i.e. that as $N \to \infty$ the equilibrium density of the particle system converges to the steady state solution of the free boundary problem.