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 generalize Bellman-Harris processes as well as customary demographic processes where individuals give birth during their lives but in a purely age-determined manner. Although the total population size of such a process is not Markovian 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 suitable way (with weights according to Fisher's reproductive value). The approach so far is that of stochastic calculus. General supercritical asymptotics then follows from a combination of L2 arguments and Tauberian theorems. It is shown that when the reproduction and life span parameters stabilise suitably 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 theory and produces a number of martingales.

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