Galton–Watson process
Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of The Educational Times:[4]A large nation, of whom we will only concern ourselves with adult males, N in number, and who each bear separate surnames colonise a district.
Instead of the extinction of family names, he studied the probability for a mutant gene to eventually disappear in a large population.
In the analogy with family names, Xn can be thought of as the number of descendants (along the male line) in the nth generation, and
If the number of children ξ j at each node follows a Poisson distribution with parameter λ, a particularly simple recurrence can be found for the total extinction probability xn for a process starting with a single individual at time n = 0: giving the above curves.
(Likewise, if mitochondrial transmission is analyzed, only women need to be considered, since only females transmit their mitochondria to descendants.)
A model more closely following actual sexual reproduction is the so-called "bisexual Galton–Watson process", where only couples reproduce.
[citation needed] (Bisexual in this context refers to the number of sexes involved, not sexual orientation.)
In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called "mating function" determines how many couples will form in a given generation.
Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as is the case in the classical Galton–Watson process.
[citation needed] However, excluding the non-trivial case, the concept of the averaged reproduction mean (Bruss (1984)) allows for a general sufficient condition for final extinction, treated in the next section.
If in the non-trivial case the averaged reproduction mean per couple stays bounded over all generations and will not exceed 1 for a sufficiently large population size, then the probability of final extinction is always 1.
Citing historical examples of Galton–Watson process is complicated due to the history of family names often deviating significantly from the theoretical model.
Examples include: On the other hand, some examples of high concentration of family names are not primarily due to the Galton–Watson process: Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or the chances of extinction of small population of organisms.
In the late 1930s, Leo Szilard independently reinvented Galton-Watson processes to describe the behavior of free neutrons during nuclear fission.
This work involved generalizing formulas for extinction probabilities, which became essential for calculating the critical mass required for a continuous chain reaction with fissionable materials.
