Each individual independently produces a random number of offspring. If mean offspring μ ≤ 1, extinction is certain. If μ > 1, there's a positive survival probability q satisfying q = G(q) where G is the probability generating function.
The extinction probability q is the smallest fixed point of G(s) = Σ pₖsᵏ in [0,1]. For Poisson(μ): G(s)=e^(μ(s−1)), so q satisfies q=e^(μ(q−1)). When μ≤1, q=1. For μ>1, q<1 — the probability of indefinite survival is 1−q (the Bienaymé-Galton-Watson theorem, 1874).