Galton–Watson process

Galton–Watson process, also known as the branching process, is a probabilistic model used to study the growth of populations, particularly focusing on the extinction probabilities of family names or genetic traits. This stochastic process was named after Francis Galton and Henry Watson, who studied the extinction of family names to understand how aristocratic surnames disappear over generations.

Overview

The Galton–Watson process begins with a single ancestor in generation 0. In each subsequent generation, each individual in the population produces some number of offspring according to a fixed probability distribution, independently of the other individuals. The process is iterated, creating a tree-like structure of descendants. The main question of interest in the Galton–Watson process is under what conditions the population will eventually become extinct, meaning that there will be a generation with no offspring.

Mathematical Definition

Let \(Z_n\) represent the number of individuals in the \(n\)-th generation. The process starts with \(Z_0 = 1\), indicating a single ancestor. The number of offspring of each individual is a random variable with a common probability distribution \(p_k\), where \(k\) can take non-negative integer values. The distribution \(p_k\) represents the probability that an individual has \(k\) offspring. The sequence \(\{Z_n\}\) defines the Galton–Watson process.

Extinction Probability

The extinction probability, denoted by \(q\), is the probability that the population eventually dies out. It can be shown that \(q\) is the smallest non-negative root of the generating function \(f(s) = \sum_{k=0}^{\infty} p_k s^k\), where \(s\) is a dummy variable. If the mean number of offspring per individual, \(\mu\), is less than or equal to 1 (\(\mu \leq 1\)), the population will almost surely become extinct.

Applications

The Galton–Watson process has applications beyond the original question of family name extinction. It is used in epidemiology to model the spread of diseases, in ecology to study species survival, and in nuclear physics to describe chain reactions. Additionally, it serves as a fundamental concept in the theory of random graphs and complex networks.

Limitations

While the Galton–Watson process provides valuable insights into population dynamics, it has limitations. The assumption of a fixed probability distribution for offspring number and the independence of individuals' reproduction events are simplifications that may not hold in real-world populations. Environmental factors, resource limitations, and changing social structures can significantly influence reproduction rates and dependencies.

See Also

• Stochastic process
• Probability distribution
• Epidemiology
• Ecology