The almost sure growth behavior of some time-homogeneous Markov chains is studied. They generalize the ordinary Galton-Watson processes with regard to allowing state-dependent offspring distributions and also to controlling the number of reproducing individuals by a random variable that depends on the state of the process. The main assumption is that the mean offspring per individual is nonincreasing while the state increases. These controlled Galton-Watson processes can be included in a general growth model whose divergence rate is determined. In case of processes that differ from the Galton-Watson process only by the state dependence of the offspring distributions, a necessary and sufficient moment condition for divergence with "natural" rate is obtained generalizing the $(x \log x)$ condition of Galton-Watson processes. In addition, some criteria are given when a state-dependent Galton-Watson process behaves like an ordinary supercritical Galton-Watson process.
Petra Kuster. "Asymptotic Growth of Controlled Galton-Watson Processes." Ann. Probab. 13 (4) 1157 - 1178, November, 1985. https://doi.org/10.1214/aop/1176992802