Galton-Watson Process

Summary

Given a population where each individual reproduces independently with the same offspring distribution, will the population eventually die out, and how do we characterize that mathematically?

Body

• We start with a root node (generation 0). The number of children this root node has is a random variable $\xi$ (offspring distribution).
• Plain-language note: a Random Variable is the “number-generating machine,” and the distribution is the recipe for the odds of each number (e.g., 0 kids 30%, 1 kid 40%, 2 kids 20%, 3 kids 10%).

What does it mean when “each individual reproduces independently with the same offspring distribution”?

• suppose we have a root node. It reproduces and has three kids. These three kids now reproduce.
• The probability these kids have some number of children, is the same (i.e. idendically distributed). They are also independed of each other.

#Definition The probability of having $i$ children, with $i \geq 0$, is

$$p_i = \mathbf{P}\{\xi = i\}, \quad i \geq 0$$

• e.g. $p_0 = \mathbf{P}\{\xi = 0\}$ is the probability of having no children.
• If $p_0 = 1$, then $\xi = 0$ almost surely and the population dies out immediately.

#Definition Let $Z_n$ be the number of individuals in generation $n$

• $Z_0 = 1$
• the next generation, $Z_{n+1}$, is the sum of all offspring of the current generation. It is characterized below:

$$Z_{n+1} = \sum_{k=1}^{Z_n} \xi_{n,k}$$

• for $k=1$ up to $Z_n$, i.e. iteratre over all nodes in the current generation.
• we sum up the number of children produced by each $k$, i.e. $\xi_{n,k}$.
  • the reproduction factor is i.i.d. copies of $\xi$ (branching property).

#Definition Malthusian Parameter Mean offspring number $m = \mathbb{E}[\xi] = \sum_i i p_i$ controls growth:

• $m < 1$ (subcritical): extinction occurs with probability 1 and $\mathbb{E}[Z_n] = m^n$ decays.
• $m = 1$ (critical): extinction occurs with probability 1 (unless $\xi \equiv 1$), but more slowly.
• $m > 1$ (supercritical): positive probability of survival.

Connections

• Markov chains: $(Z_n)$ is a Markov chain on $\mathbb{N}_0$ with absorbing state 0.
• Random trees: The family tree of the process is a random rooted tree.
• Phase transitions: The critical value $m=1$ separates almost-sure extinction from possible survival.
• Reproduction Generating Function (RGF)

Sources

• Bienayme (1845)
• Galton and Watson (1874)
• Jagers (2009)