On the Role of Variable Generation Time in the Development of a Stochastic Birth Process

Abstract

The "birth-and-death" stochastic process introduced by W. Feller (Acta Biotheoretica, 1939) describes growth of a population under simplified laws of reproduction and mortality. One simplification is that chances of both are completely independent of the previous history of the individual, including elapsed time since his own birth. This assumption implies that the stochastic dependence of population size on time is given by a discontinuous Markoff process. The author claims, however, that this assumption is unrealistic. For the case where mortality is neglected, generation time is distributed as X2 with 2 degrees of freedom under Feller''s assumption. The author examines a modified birth process in which generation time is distributed as X2 with 2k d.f., where k is an integer > 1. When k = 1, we have the Feller birth process, while for k approaching infinity, the population doubles at regular intervals. For intermediate values, we have "multiple-phase" stochastic birth processes. It is suggested that the coeffs. of variation of population size and generation time are approx. equal for such processes.