CrumpModeJagersprosesser

Crump-Mode-Jagers processes, often abbreviated as CMJ processes, are a broad class of general branching processes with age structure. In these models, each individual lives for a random duration and, during its lifetime, gives birth to new individuals according to a stochastic reproduction process. Lifetimes and reproduction patterns are typically independent across individuals, and the population state at time t is the number of individuals alive then. The full historical information of births is captured by a reproduction measure that describes when after birth each new individual is produced.

History and scope: CMJ processes were developed to generalize classical Galton-Watson branching processes by incorporating arbitrary lifetimes and fertility patterns. They were studied and expanded in the 1960s and 1970s, notably by Crump, Mode, and Jagers, leading to a flexible framework for age-dependent branching phenomena.

Key concepts: A central object is the Malthusian parameter α, determined by the reproduction measure, typically via a condition like the Laplace transform E[∫0^∞ e^{-α t} Z(dt)] = 1. When α > 0 the process is supercritical and population size N(t) grows roughly at rate e^{α t}. Under mild regularity conditions, there exists a random variable W with N(t) e^{-α t} → W almost surely, yielding a law of large numbers for the growth, and further results include central limit theorems and asymptotic descriptions of the age distribution among living individuals.

Variants and extensions: CMJ processes have many extensions, including immigration (external entrants into the population), multi-type versions (different classes of individuals), and measure-valued formulations to handle more complex life histories and fertility patterns.

Applications: They are used in population biology to model age-structured populations, in epidemiology to study spread with varying infectious periods, in ecology and forestry, and in reliability theory and genealogy studies where age and reproduction timing are essential factors.