Branching process

Branching process

In probability theory, a branching process is a Markov process that models a population in which each individual in generation "n" produces some random number of individuals in generation "n" + 1, according to a fixed probability distribution that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor.

A central question in the theory of branching processes is the probability of ultimate extinction, where no individuals exist after some finite number of generations. It is not hard to show that, starting with one individual in generation zero, the expected size of generation "n" equals μ"n" where μ is the expected number of children of each individual. If μ is less than 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability 1 by Markov's inequality. Alternatively, if μ is greater than 1, then the probability of ultimate extinction is less than 1 (but not necessarily zero; consider a process where each individual either dies without issue or has 100 children with equal probability). If μ is equal to 1, then ultimate extinction occurs with probability 1 unless each individual always has exactly one child.

See also Galton–Watson process and also [Bisexual] Galton–Watson process.

In theoretical ecology, the parameter μ of a branching process is called the basic reproductive rate.

References

* G. R. Grimmett and D. R. Stirzaker, "Probability and Random Processes", 2nd ed., Clarendon Press, Oxford, 1992. Section 5.4 discusses the model of branching processes described above. Section 5.5 discusses a more general model of branching processes known as age-dependent branching processes, in which individuals live for more than one generation.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • branching — noun A process of forming a branch …   Wiktionary

  • Galton–Watson process — The Galton–Watson process is a stochastic process arising from Francis Galton s statistical investigation of the extinction of surnames. History There was concern amongst the Victorians that aristocratic surnames were becoming extinct. Galton… …   Wikipedia

  • Direct process — The Direct Process, also called the Direct Synthesis, Rochow Process, and Müller Rochow Process is the most common technology for preparing organosilicon compounds on an industrial scale. It was first reported independently by Eugene G. Rochow… …   Wikipedia

  • Many-worlds interpretation — The quantum mechanical Schrödinger s cat paradox according to the many worlds interpretation. In this interpretation every event is a branch point; the cat is both alive and dead, even before the box is opened, but the alive and dead cats are in… …   Wikipedia

  • Mammary gland — in a human female Cross section of the breast of a human female …   Wikipedia

  • Transitional fossil — The London specimen of Archaeopteryx, discovered only two years after the publication of On the Origin of Species Missing link redirects here. For other uses, see Missing Link. A transitional fossil is any fossilized remains of a lifeform that… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • List of stochastic processes topics — In the mathematics of probability, a stochastic process can be thought of as a random function. In practical applications, the domain over which the function is defined is a time interval ( time series ) or a region of space ( random field… …   Wikipedia

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

  • Jean-François Le Gall — is a French mathematician working in areas of probability theory such as Brownian motion, Lévy processes, superprocesses and their connections with partial differential equations, the Brownian snake, random trees, branching processes, stochastic… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”