Reduced critical Bellman–Harris branching processes for small populations

A critical Bellman-Harris branching process $\left\{ Z(t), t\geq 0\right\} $ with finite variance of the offspring number is considered. Assuming that $0<Z(t)\leq \varphi (t)$, where either $\varphi (t)=o(t)$ as $t\rightarrow \infty $ or $\varphi (t)=at,\, a>0$, we study the structure of the process $ \left\{ Z(s,t),0\leq s\leq t\right\} ,$ where $Z(s,t)$ is the number of particles in the initial process at moment $s$ which either survive up to moment $t$ or have a positive number of descendants at this moment.