  We consider a two-type reducible branching Brownian motion, defined as a
particle system on the real line in which particles of two types move according
to independent Brownian motion and create offspring at constant rate. Particles
of type $1$ can give birth to particles of types $1$ and $2$, but particles of
type $2$ only give birth to descendants of type $2$. Under some specific
conditions, Biggins (2012) shows that this process exhibit an anomalous
spreading behaviour: the rightmost particle at time $t$ 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 (2014,2016). The aim of this article is to study the
asymptotic behaviour of the position of the furthest particles in the two-type
reducible branching Brownian motion, obtaining in particular tight estimates
for the median of the maximal displacement.
