Problem 4.2 ā¢ Galton-Watson process - will the bacteria lineage survive or go extinct?
š¦ Scenario: A single bacterium produces offspring according to a probability distribution.
Each offspring independently reproduces with the same distribution.
ā Question: What is the probability that the lineage eventually goes extinct?
This is a classic Galton-Watson branching process problem.
š³ Branching Tree Simulation
0
Generation
1
Population
Alive
Status
š Offspring Distribution
Each bacterium produces 0, 1, 2, or 3 offspring with these probabilities:
0
25%
1
25%
2
25%
3
25%
P(0 offspring)0.25
P(3 offspring)0.25
*P(1) and P(2) are balanced to sum to 1
š Extinction Equation
q = P(extinction)
q = pā + pāq + pāqĀ² + pāqĀ³
Solve for smallest non-negative root
Mean offspring (Ī¼):
1.50
If Ī¼ ā¤ 1: extinction is certain (q = 1)
If Ī¼ > 1: survival is possible (q < 1)