CrumpModeJagersprocessen

Crump-Mode-Jagers process, often abbreviated CMJ process, is a general age-dependent branching process in which individuals have random lifetimes and give birth at random times during their lives. It extends the classical Bienaymé-Galton-Watson process by incorporating aging and a flexible temporal pattern of reproduction.

In the standard construction, the process starts with a single ancestor born at time zero. Each individual lives for a random time L with a given distribution, and during its life it produces offspring according to a general point process on the positive real line, with birth times measured from the individual’s birth. The births form independent copies of the same process, so every newborn starts its own life at its birth time. The population is the union of all individuals born over time.

A central object is the reproduction measure μ, which describes the expected number of births at different ages during an individual’s life. The CMJ framework allows non-Markovian dynamics because births can occur at arbitrary ages, and lifetimes and reproduction patterns can be highly heterogeneous across individuals.

Key results describe growth and structure under suitable regularity conditions. In the supercritical regime, there exists a Malthusian parameter α > 0 solving ∫0^∞ e^{-α t} μ(dt) = 1, and the population typically grows roughly like e^{α t} as time progresses, with a stable age distribution in the limit. Analyses often employ characteristic methods and renewal equations to study quantities such as the number of individuals alive over time or total births.

CMJ processes have applications in ecology, epidemiology, cell biology, and other areas where population dynamics depend on ages and variable reproduction schedules. They are named after Crump, Mode, and Jagers, who developed foundational theory in the late 1960s and 1970s.