Poisson(μ) offspring — extinction probability q = 1 if μ ≤ 1, q < 1 if μ > 1
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Each individual independently produces Poisson(μ) offspring. Phase transition at μ = 1:
For μ ≤ 1: extinction probability q = 1 (certain extinction).
For μ > 1: q = smallest positive root of G(s) = s, where G(s) = e^{μ(s−1)} (Poisson pgf).
Population grows as μ^n on average when μ > 1.