BellmanHarris

BellmanHarris, also known as the Bellman–Harris branching process, is a continuous-time stochastic model in probability theory used to describe age-structured populations. It generalizes the Galton-Watson process by incorporating random lifetimes and births that occur at death rather than at fixed generations.

In the Bellman–Harris framework, each individual lives for a random time drawn from a lifetime distribution G. At the end of its life, the individual reproduces, producing a random number X of offspring according to a fixed offspring distribution with mean m. Offspring behave independently and identically to their predecessors, and each new individual begins its own life cycle. The process is typically denoted by Z(t), the number of individuals alive at time t, starting from an initial population Z(0).

Key concepts include the Malthusian parameter α, defined as the positive solution to m ∫_0^∞ e^{-α t} dG(t) = 1 (when such a solution exists). If α > 0 and certain regularity conditions hold, Z(t) grows roughly like a constant times e^{α t} as t grows large. The extinction probability depends on the offspring mean and the lifetime distribution, with the supercritical case (m > 1 in a suitable sense) leading to a positive chance of persistence.

The Bellman–Harris process is a specific instance of the Crump–Mode–Jagers branching framework and reduces to a Galton–Watson process in the degenerate case where lifetimes are negligible or births occur at time zero. It has applications in population biology, epidemiology, and reliability theory, where age structure and time to reproduction influence growth dynamics. See also Galton-Watson process; Crump–Mode–Jagers process.