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We consider a system of particles performing a one-dimensional dyadicbranching Brownian motion with space-dependent branching rate, negative drift$-\mu$ and killed upon reaching $0$, starting with $N$ particles. Moreprecisely, particles branch at rate $\rho/2$ in the interval $[0,1]$, for some$\rho>1$, and at rate $1/2$ in $(1,+\infty)$. The drift $\mu(\rho)$ is chosenin such a way that, heuristically, the system is critical in some sense: thenumber of particles stays roughly constant before it eventually dies out. Thisparticle system can be seen as an analytically tractable model for fluctuatingfronts, describing the internal mechanisms driving the invasion of a habitat bya cooperating population. Recent studies from Birzu, Hallatschek and Korolevsuggest the existence of three classes of fluctuating fronts: pulled,semi-pushed and pushed fronts. Here, we rigorously verify and make precise thisclassification and focus on the semi-pushed regime. More precisely, we provethe existence of two critical values $1<\rho_1<\rho_2$ such that for all$\rho\in(\rho_1,\rho_2)$, there exists $\alpha(\rho)\in(1,2)$ such that therescaled number of particles in the system converges to an $\alpha$-stablecontinuous-state branching process on the time scale $N^{\alpha-1}$ as $N$ goesto infinity. This complements previous results from Berestycki, Berestycki andSchweinsberg for the case $\rho=1$.

A branching particle system as a model of semi pushed fronts