We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate $r(x)$, negative drift $-\mu$, and killed upon reaching $0$. More precisely, the particles branch at rate $\rho/2$ in $[0,1]$, for some $\rho\geq 1$, and at rate $1/2$ in $(1+\infty)$. The drift $\mu=\mu(\rho)$ is chosen in such a way that the system is critical.
This system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population.
Recent studies by Birzu, Hallatschek and Korolev on the noisy FKPP equation with Allee effect suggest the existence of three classes of fluctuating fronts: pulled, semi-pushed and fully-pushed fronts.
In this talk, we will focus on the pushed regime. We will show that the BBM exhibits the same phase transitions as the noisy FKPP equation. We will then use this particle system to explain how the internal mechanisms driving the invasion shape the genealogy of an expanding population.

This talk is based on joint work with Félix Foutel-Rodier and Emmanuel Schertzer.