Brownian bees in the infinite swarm 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 ${\mathbb{R}}^{\mathit{d}}$ 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 $\mathit{N}\to \infty$, 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 (Trans. Amer. Math. Soc. 374 (2021) 6269–6329)), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds; that is, that as $\mathit{N}\to \infty$, the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem.