Branching Process

A branching process tracks the population size $Z_n$ across discrete generations $n=0,1,2,\dots$, where $Z_0$ is the initial population and each individual in generation $n$ independently produces a random number of offspring (according to a common offspring distribution) that together make up generation $n+1$. Branching process is the general umbrella term for this family of models; the classical case with i.i.d. offspring counts across all individuals and generations is specifically called the Galton-Watson process. The defining question for any branching process is whether the population goes extinct ($Z_n=0$ eventually) with probability 1, or survives with positive probability, and this dichotomy is governed by the mean offspring number $\mu$: subcritical ($\mu <1$), critical ($\mu =1$), and supercritical ($\mu >1$) branching processes have qualitatively different extinction behavior.

• The mean offspring number alone determines the qualitative fate of the population: subcritical ($\mu <1$) processes go extinct with probability 1, critical ($\mu =1$) processes also go extinct with probability 1 (but more slowly), and only supercritical ($\mu >1$) processes have a genuine positive probability of surviving forever.
• Branching process theory extends well beyond biological populations, epidemic outbreak modeling (each infected person 'branches' into a random number of new infections, connecting directly to the epidemiological basic reproduction number $R_0$) and nuclear chain reactions are both classic applied branching-process settings.
• More elaborate variants generalize the basic model considerably, multitype branching processes (different types of individuals with different offspring distributions), continuous-time branching processes, and branching processes with immigration all extend the same core generation-by-generation logic.