Branching processes with dependence but homogeneous growth

A (general) branching process, where individuals need not reproduce indepen dently, satisfies a homogeneous growth condition if, vaguely, one would not expect the progeny from any one individual to make out more than its prope r fraction of the whole population at any time in the future. This notion i s made precise, and it is shown how it entails classical Malthusian growth in supercritical cases, in particular for population size-dependent Bienaym e-Galton-Watson and Markov branching processes, and for nondecreasing age-d ependent processes with continuous life span distributions.