A Galton–Watson process with a threshold

Abstract

In this paper we study a special class of size dependent branching processes. We assume that for some positive integer $K$ as long as the population size does not exceed level $K$, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level $K$, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time $T$. The question of when the mean value $\mathbb{E}(T)$ is finite or infinite is also addressed.