Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

Abstract

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .