Galton-Watson Extinction

Worked Example

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.

The probability generating function for the offspring distribution is:

G(s) = (1-p) + ps2 = 0.25 + 0.75s2

The extinction probability by generation n, denoted q_n, is calculated recursively: q₀ = 0 and q_n+1 = G(q_n).

1. Generation 0: q₀ = 0
2. Generation 1: q₁ = 0.25 + 0.75(0)² = 0.25 (or 1/4)
3. Generation 2: q₂ = 0.25 + 0.75(0.25)² = 0.25 + 0.75(0.0625) = 0.25 + 0.046875 = 0.296875 (or 19/64)

The ultimate extinction probability is the smallest fixed point s in [0, 1] of G(s) = s:

0.25 + 0.75s2 = s

The roots are s = 1 and s = (1-p)/p = 0.25/0.75 = 1/3. Since p > 0.5, the smallest root is 1/3, which is the ultimate extinction probability.

The expected population size at generation 2 is E[Z₂] = z * (2p)² = 1 * (1.5)² = 2.25 (or 9/4).

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.

Limitations

This module is designed for a specific offspring distribution (0 or 2 children). A real Galton-Watson process can have any discrete probability distribution for offspring (e.g., Poisson or Geometric). The exact generating function calculation becomes mathematically more complex for general distributions and might not have simple closed-form roots.

Offspring are independent across all individuals and generations and use one unchanging distribution. Growing expectation is not survival of every realization: at p = 0.75 the mean grows while one lineage still dies out with probability 1/3. At the critical p = 0.5 the mean stays 1 but extinction is certain in the limit, not at every finite generation. The red cobweb shows exactly n iterations starting at zero; the finite-generation bars use z independent ancestors. Large exact fractions remain available in scrollable result fields; decimals round only for display.

References

The properties of generating functions, extinction probabilities, and expected population sizes can be studied further in the following resources:

• Introduction to Probability and Statistics - Branching Processes (Cornell University)