Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle

$N$-Brownian bees is a branching-selection particle system in $\mathbb{R}^d$ in which $N$ particles behave as independent binary branching Brownian motions, and where at each branching event, we remove the particle furthest from the origin. We study a variant in which $d=1$ and particles have an additional drift $\mu\in\mathbb{R}$. We show that there is a critical value, $\mu_c^N$, and three distinct regimes (sub-critical, critical, and super-critical) and we describe the behaviour of the system in each case. In the sub-critical regime, the system is positive Harris recurrent and has an invariant distribution; in the super-critical regime, the system is transient; and in the critical case, after rescaling, the system behaves like a single reflected Brownian motion. We also show that the critical drift $\mu_c^N$ is in fact the speed of the well-studied $N$-BBM process, and give a rigorous proof for the speed of $N$-BBM, which was missing in the literature.