Consider supercritical general branching processes, where, however,
individual reproduction may be influenced by the history of the population, in
particular by the total population size. Assume that reproductions approach
those of a classical, possibly multitype, supercritical process either from
above or from below as the population grows. Conditions are established for the
population-history dependent populations to display balanced exponential growth
with the same Malthusian parameter as the limiting population. Applications are
made to a tumor model with quiescence.