On transition phenomena in Bellman–Harris branching processes

Let $\mu(t)=(\mu_1(t),\dots,\mu_r(t))$ be a vector of number of particles in Bellman–Harris branching process with several types of particles. In this article the asymptotic behaviour of the probability
$$ \mathbf P\{a^{-1}\otimes\mu(t)\ge x\mid\mu(0)=e_i\},\qquad x>0, $$
when $t\to\infty$, $R\to 1$ in a compact processes class $\mathscr H$, is investigated.