( 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. (Likewise, if mitochondrial transmission is analyzed, only women need to be considered, since only females transmit their mitochondria to descendants.)Ī model more closely following actual sexual reproduction is the so-called "bisexual Galton–Watson process", where only couples reproduce. This effectively means that reproduction can be modeled as asexual. In the classical family surname Galton–Watson process described above, only men need to be considered, since only males transmit their family name to descendants. Suppose the number of a man's sons to be a random variable distributed on the set For a detailed history see Kendall (19).Īssume, for the sake of the model, that surnames are passed on to all male children by their father. Galton and Watson appear to have derived their process independently of the earlier work by I. Together, they then wrote an 1874 paper titled "On the probability of the extinction of families" in the Journal of the Anthropological Institute of Great Britain and Ireland (now the Journal of the Royal Anthropological Institute). Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of The Educational Times, and the Reverend Henry William Watson replied with a solution. There was concern amongst the Victorians that aristocratic surnames were becoming extinct.
4 Extinction criterion for Galton–Watson process. The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. But the probability of survival of a new type may be quite low even if λ > 1 and the population as a whole is experiencing quite strong exponential increase. For λ ≤ 1, eventual extinction will occur with probability 1. In multitype branching processes, individuals are not identical, but can be classified into n types.Galton–Watson survival probabilities for different exponential rates of population growth, if the number of children of each parent node can be assumed to follow a Poisson distribution. Phylogenetic trees, for example, can be simulated under several models, helping to develop and validate estimation methods as well as supporting hypothesis testing. One specific use of simulated branching process is in the field of evolutionary biology. In this example, we can solve algebraically that d = 1/3, and this is the value to which the extinction probability converges with increasing generations.īranching processes can be simulated for a range of problems. Taking as example probabilities for the numbers of offspring produced p 0 = 0.1, p 1 = 0.6, and p 2 = 0.3, the extinction probability for the first 20 generations is as follows: For the ultimate extinction probability, we need to find d which satisfies d = p 0 + p 1d + p 2 d 2. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation n 
The random variables of a stochastic process are indexed by the natural numbers. In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. For the process in representation theory, see Restricted representation § Classical branching rules.
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