This paper examines the size, $Z(t)$, of a population as a function of time. $Z(t)$ is just like the ordinary Bellman-Harris age dependent branching process except that the number of daughters born to an individual in the $n$th generation is allowed to depend on $n$. The renewal theory of William Feller and Laplace transform theory are used to obtain the behavior of $EZ(t)$ as $t$ approaches infinity, and the convergence of $Z(t)/E(Z(t))$ in quadratic mean.
"Supercritical Age Dependent Branching Processes with Generation Dependence." Ann. Probab. 4 (1) 27 - 37, February, 1976. https://doi.org/10.1214/aop/1176996178