Amoeba Extinction Probability

A branching-process visualization for finding the probability that the population eventually disappears.

1. One amoeba produces…

💀
0
25%
🦠
1
25%
🦠🦠
2
25%
🦠🦠🦠
3
25%

Average offspring: (0 + 1 + 2 + 3)/4 = 1.5. Since this is greater than 1, extinction is not certain.

2. Define the extinction probability

Let q be the probability that a population starting from one amoeba eventually becomes extinct.

q = ¼·1 + ¼·q + ¼·q² + ¼·q³

Why q²? If there are two offspring, both independent family lines must eventually die out. Likewise, three offspring gives q³.

3. Watch the fixed point appear

Starting with q₀ = 0, repeatedly apply f(q) = (1 + q + q² + q³)/4. The values converge to the extinction probability.

y = q y = f(q) 0 1 1 q
Current iteration
q₀ = 0
Interpretation
Probability the process is extinct by generation 0.
Limit
√2 − 1
≈ 0.4142

4. Solve the fixed-point equation

Start q = (1 + q + q² + q³)/4
Multiply by 4 4q = 1 + q + q² + q³
Rearrange q³ + q² − 3q + 1 = 0
Factor (q − 1)(q² + 2q − 1)=0
q = 1   or   q = −1 + √2

For a branching process, the extinction probability is the smallest fixed point in [0,1].

Probability of eventual extinction
√2 − 1 ≈ 41.42%
Probability of surviving forever ≈ 58.58%
Galton–Watson branching process • offspring distribution {0,1,2,3} with equal probability