  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.
