Galton-Watson Extinction

Consider a Galton-Watson branching process where an individual has 0 children with probability 1-p or 2 children with probability p. Let p = 0.75 and assume we start with z = 1 ancestor.

Experiment

You can use the interactive module (Galton-Watson Extinction) to explore how the two-child probability p affects the extinction of a family line:

1. Set the Two-child probability p slider to 0.75.
2. Set the Initial ancestors z slider to 1.
3. Set the Generation n slider to 2.
4. Observe that the Ultimate Extinction matches 0.333333 and the Extinction by Gen n is 19/64 (approximately 0.296875). The cobweb plot visually shows the sequence q₀, q₁, q₂ converging to the smallest fixed point.
5. Change the Two-child probability p slider to 0.25 (a subcritical process). Notice that the only valid root in [0, 1] is 1, meaning ultimate extinction is certain. The cobweb plot now converges towards 1.
6. Restore Two-child probability p to 0.75, then increase Initial ancestors z to 2. The ultimate extinction probability becomes (1/3)² = 1/9 for p=0.75.

At Two-child probability p = 0 and Initial ancestors z = 2, set Generation n to 0: the population is exactly 2 and extinction by generation 0 is zero. Then set n to 1: extinction is certain and the expected population is zero. At p = 1, z = 1, n = 3, the population is deterministically 8 and extinction is impossible.