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 Z n be the size of individuals belonging to the n th generation of a population. If Z n > 1 , the population evolves as a critical Galton–Watson process with finite variance; if Z n = 1 , the population evolves as another Galton–Watson process; if Z n = 0 , Z n + 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. [ABSTRACT FROM AUTHOR]