We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift −µ and killed upon reaching 0, starting with N particles. More precisely, particles branch at rate ρ/2 in the interval [0, 1], for some ρ > 1, and at rate 1/2 in (1, +∞). The drift µ(ρ) is chosen in such a way that, heuristically, the system is critical in some sense: the number of particles stays roughly constant before it eventually dies out. This particle 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 from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi-pushed and pushed fronts. Here, we rigorously verify and make precise this classification and focus on the semi-pushed regime. More precisely, we prove the existence of two critical values 1 < ρ 1 < ρ 2 such that for all ρ ∈ (ρ 1 , ρ 2 ), there exists α(ρ) ∈ (1, 2) such that the rescaled number of particles in the system converges to an α-stable continuous-state branching process on the time scale N α−1 as N goes to infinity. This complements previous results from Berestycki, Berestycki and Schweinsberg for the case ρ = 1. Contents

We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale 1/ε in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient. . . ). The model incorporates another parameter, K, which is a measure of the local population density. We study the model in the limit when first ε → 0 and then K → ∞. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit K → ∞. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits ε → 0 and K → ∞ do not commute in general. We conjecture that an interpolating regime appears when log K and 1/ε are of the same order.