Branching Processes With Immigration

A basic (Galton-Watson) branching process starts with some individuals, each of whom independently produces a random number of "children" for the next generation, and so on; if the average number of children per individual is below one, the population is guaranteed to eventually die out. Branching processes with immigration add one ingredient: independent of how the existing population reproduces, a random number of new individuals arrive from outside — "immigrants" — into the population each generation, on top of whatever offspring were produced internally.

That one addition changes the long-run behaviour qualitatively. Even if the internal reproduction rate is below one, so the population would go extinct on its own, a steady trickle of immigration can keep the population from ever dying out, instead settling into a stable long-run distribution of population sizes. This mirrors, for example, a trading desk where existing strategies occasionally spin off related sub-strategies (the branching part) while the firm also periodically hires teams bringing entirely new, unrelated strategies (the immigration part) — even if individual strategy lineages tend to shrink and die out over time, the desk as a whole persists because new arrivals keep replenishing it.