Large deviations for the maximum of a reducible two-type branching Brownian motion

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 motions and create offspring at a 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, Belloum and Mallein in \cite{BeMa21} showed that the maximum position $M_t$ of all particles alive at time $t$, suitably centered by a deterministic function $m_t$, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as $t\rightarrow\infty$, $${\mathbb P}(M_t\geq \theta m_t),\quad \theta>1.$$ We shall show that the decay rate function exhibits phase transitions depending on certain relations between $\theta$, the variance of the underlying Brownian motion and the branching rate.