A two-type Bellman--Harris process initiated by a large number of particles

We investigate a two-type critical Bellman--Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order $o(t^{-2})$; the tail of the life-length distribution of the second type particles is regularly varying at infinity with index $-\beta$, $\beta \in (0,1]$; at time $t=0$ the process starts with a large number $N$ of the second type particles and no particles of the first type. It is shown that the time axis $0\leq t<\infty$ splits into several regions whose ranges depend on $\beta$ and the ratio $N/t$ within each of which the process at time $t$ exhibits asymptotics (as $N,t\rightarrow \infty$) which is different from those in the other regions.