A Galton-Watson process with a threshold at 1 and an immigration at 0

In this paper we consider a special class of population-size-dependent branching processes, which can also be seen as an extension of Galton-Watson processes with state-dependent immigration (GWPSDI). The model is formulated as follows. Let Zn be the size of individuals belonging to the nth generation of a population. If Zn > 1, the population evolves as a critical Galton-Watson process with finite variance; if Zn = 1, the population evolves as another Galton-Watson process; if Zn = 0, Zn+1 is drawn from a fixed immigration distribution. Based on the technical routes in Foster (1971) and Pakes (1971), some asymptotic results identical with those of GWPSDI are obtained by detailed computations. & COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates a special class of population-size-dependent branching processes, which can be viewed as an extension of Galton-Watson processes with state-dependent immigration. The model considers the size of individuals in each generation and evolves according to different Galton-Watson processes or fixed immigration distributions. By using technical results from Foster (1971) and Pakes (1971), asymptotic results similar to those of Galton-Watson processes with state-dependent immigration are obtained through detailed computations.