MATH 6241 Section 8 Interactive — Branching Processes

1. Learning Objectives

Define Galton–Watson

Model $X_n$ as the population size in generation $n$ with independent offspring counts.

Use PGFs

Compute $G(s)=E[s^Z]$ and use fixed points of $G$ to find extinction probabilities.

Classify regimes

Use the mean offspring number $\mu=E[Z]$ to classify subcritical, critical, and supercritical behavior.

2. Galton–Watson Branching Process

A Galton–Watson process models a population by generations. Each individual independently produces a random number of children with common offspring distribution $Z$.

$$X_{n+1}=\sum_{i=1}^{X_n} Z_{n,i},\qquad n=0,1,2,\ldots.$$

Here $X_n$ is the number of individuals in generation $n$, and the $Z_{n,i}$ are independent copies of $Z$.

The state $0$ is absorbing. Once the population goes extinct, it stays extinct forever.

3. Interactive Offspring Distribution

Choose probabilities for $Z=0,1,2,3$. The page normalizes them and computes the mean, variance, PGF, and regime.

Set probabilities

$P(Z=0)$: 0.25

$P(Z=1)$: 0.25

$P(Z=2)$: 0.40

$P(Z=3)$: 0.10

Distribution summary

Mean $\mu$0.000

Variance0.000

Regime—

4. Simulate Branching Process Paths

Simulate many population paths and see how extinction and growth depend on $\mu$.

initial population $X_0$: 1 generations: 30 sample paths: 30

Extinct by final generation0.000

Mean final population0.000

Max final population0

5. Moment Formulas

If $X_0=1$ and the offspring mean and variance are $\mu=E[Z]$ and $\sigma^2=\operatorname{Var}(Z)$, then

$$E[X_n]=\mu^n.$$

The variance is

$$\operatorname{Var}(X_n)= \begin{cases} \sigma^2\mu^{n-1}\dfrac{\mu^n-1}{\mu-1}, & \mu\ne1,\\[6pt] n\sigma^2, & \mu=1. \end{cases}$$

max generation: 25

6. Extinction Probability

Let $\rho$ be the probability that the population eventually goes extinct starting from one ancestor. It is the smallest solution in $[0,1]$ of

$$\rho=G(\rho),\qquad G(s)=E[s^Z].$$

Fixed-point visualization

starting ancestors $X_0$: 1

Monte Carlo extinction estimate

runs: 3000 generation cutoff: 80

7. PGF Iteration

The probability of extinction by generation $n$ starting from one ancestor is

$$P(X_n=0\mid X_0=1)=G_n(0),$$

where $G_n$ is the $n$-fold composition $G\circ G\circ\cdots\circ G$.

iterations: 30

8. Quick Check

Question 1

In a Galton–Watson process, what does $X_n$ represent?

Question 2

If $\mu=E[Z]<1$, the process is:

Question 3

The extinction probability starting from one ancestor is:

Question 4

If extinction probability from one ancestor is $\rho$, then from $k$ independent ancestors it is: