Reduced critical Bellman-Harris branching processes for small populations

TL;DR

This paper investigates the structure of critical Bellman-Harris branching processes with finite variance under small population constraints, focusing on the process's behavior when populations are limited by functions growing slower than or linearly with time.

Contribution

It introduces a reduced model for critical Bellman-Harris processes under small population conditions, analyzing the process's structure with specific growth constraints.

Findings

Characterization of process structure under small population limits

Asymptotic behavior when population constraints grow slower than time

Analysis of process with linear population constraints

Abstract

Let $\left\{ Z(t), t\geq 0\right\} $ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. 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 process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment.