Galton–Watson process

The Galton -Watson process, named after the British scientist Francis Galton (1822-1911) and his compatriot, the mathematician Henry William Watson (1827-1903), is a special stochastic process that is used to determine the numerical evolution of a population of self-replicating individuals to model mathematically. He is sometimes referred to as a Galton -Watson Bienaymé - process, in honor of the French Irenee -Jules Bienaymé (1796-1878), who had already processed the same problem a long time ago.

History

In England during the Victorian age, the aristocracy was increasingly concerned about the fact that time and again extinct noble families for lack of male offspring and thus disappeared more and more traditional names from the aristocratic society. Galton, who was himself a mathematician, published in 1873 in the scientific journal Educational Times, the question of the likelihood of such cancellation and got an answer from Watson. The following year, their community work On the probability of extinction of families in which they presented a stochastic approach, which is known as a Galton -Watson process today appeared. The conclusion to which they came was that at a constant population over time all the names would become extinct except one. Apparently this work was done in ignorance of the results of Bienaymé.

First was the problem of the extinction surname the only the Galton the Watson- concept has been applied. But soon began biologists, so the spread of organisms to model. Today, the process is used in diverse areas of queuing theory to the distribution of computer viruses and chain letters.

Mathematical modeling

In the mathematical model of the process are the number of individuals at time t on, and the start value is fixed. The distribution, which only takes values ​​in, specifies how many offspring are produced a single individual (in the original case, the male offspring of a male heir ). It is believed that the individuals their offspring independently of the other individuals and even die after a period, so that the distribution of the process by the transition probability

Optionally, wherein the n-fold convolution of the distribution refers to ( the zero -fold convolution of a distribution is defined by a distribution which is almost certain to zero).

Thus X is a (time- homogeneous ) Markov chain in which the ( countably infinite ) transition matrix by

Is given. As is, a process that has once reached the value 0, in the future do not accept any value other than 0 (extinct populations can not be resurrected ). 0 is thus the state of an absorbent, and generally also the only one. An exception is the ( uninteresting ) special case where is that so each individual almost certainly has exactly one offspring. Then the number of individuals is of course almost surely constant and each state is absorbing.

The extinction

The question, were interested in the Galton and Watson was that the probability of extinction of a population, ie after the variable

First, we see immediately that, therefore, it is sufficient to consider populations that start with an individual.

The main result is the following: The probability of extinction is the smallest fixed point of the generating function of, ie the power series function

One implication states that the extinction probability is exactly then strictly smaller than 1, if the expected value of the distribution is strictly greater than 1, so if generated every individual on average more than one offspring. Again, the special case is an exception: although the expectation value is exactly 1, extinction is of course impossible. Ultimately, this is exactly the case when, ie when each individual is a descendant of at least guaranteed.

Another property of the Galton -Watson process can also be made ​​of the generating functions win: if almost surely, then for the generating function of the random variable:

Wherein the n-fold composition ( sequential execution ) of a function f denotes.

• Stochastic process
• Theoretical Biology