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.