Population-size-dependent and age-dependent branching processes

Supercritical branching processes are considered which are Markovian in the age structure but where reproduction parameters may depend upon population size and even the age structure of the population. Such processes generali ze Bellman-Harris processes as well as customary demographic processes wher e individuals give birth during their lives but in a purely age-determined manner. Although the total population size of such a process is not Markovi an the age chart of all individuals certainly is. We give the generator of this process, and a stochastic equation from which the asymptotic behaviour of the process is obtained, provided individuals are measured in a suitabl e way (with weights according to Fisher's reproductive value). The approach so far is that of stochastic calculus. General supercritical asymptotics t hen follows from a combination of L-2 arguments and Tauberian theorems. It is shown that when the reproduction and life span parameters stabilise suit ably during growth, then the process exhibits exponential growth as in the classical case. Application of the approach to, say, the classical Bellman- Harris process gives an alternative way of establishing its asymptotic theo ry and produces a number of martingales. (C) 2000 Elsevier Science B.V. All rights reserved.