Anomalous spreading in reducible multitype branching Brownian motion

We consider a two-type reducible branching Brownian motion, defined as a two type branching particle system on the real line, in which particles of type $1$ can give birth to particles of type $2$, but not reciprocally. This process has been shown by Biggins to exhibit an anomalous spreading behaviour under specific conditions: in that situation, the rightmost particle at type $t$ is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type $1$ or of type $2$. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer. The aim of this article is to refine the previous results and study the asymptotic behaviour of the extremal process of the two-type reducible branching Brownian motion. If the branching Brownian motion exhibits an anomalous spreading behaviour, its asymptotic differs from what it typically expected in branching Brownian motions.