Branching process

In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers.

The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation

[1] 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.

Using Wald's equation, it can be shown that starting with one individual in generation zero, the expected size of generation n equals μn where μ is the expected number of children of each individual.

If μ < 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability 1 by Markov's inequality.

If μ = 1, then ultimate extinction occurs with probability 1 unless each individual always has exactly one child.

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

Let Zn denote the state in period n (often interpreted as the size of generation n), and let Xn,i be a random variable denoting the number of direct successors of member i in period n, where Xn,i are independent and identically distributed random variables over all n ∈{ 0, 1, 2, ...} and i ∈ {1, ..., Zn}.

Alternatively, the branching process can be formulated as a random walk.

Let Si denote the state in period i, and let Xi be a random variable that is iid over all i.

In general, the waiting time is an exponential variable with parameter λ for all individuals, so that the process is Markovian.

The process can be analyzed using the method of probability generating function.

Let dm be the extinction probability by the mth generation, starting with a single individual (i.e.

(See the black curve in the graph) In case 1, the ultimate extinction probability is strictly less than one.

By observing that h′(1) = p1 + 2p2 + 3p3 + ... = μ is exactly the expected number of offspring a parent could produce, it can be concluded that for a branching process with generating function h(z) for the number of offspring of a given parent, if the mean number of offspring produced by a single parent is less than or equal to one, then the ultimate extinction probability is one.

If the mean number of offspring produced by a single parent is greater than one, then the ultimate extinction probability is strictly less than one.

For Athreya, the central parameters are crucial to control if sub-critical and super-critical unstable branching is to be avoided.

One specific use of simulated branching process is in the field of evolutionary biology.

[5][6] Phylogenetic trees, for example, can be simulated under several models,[7] helping to develop and validate estimation methods as well as supporting hypothesis testing.

In multitype branching processes, individuals are not identical, but can be classified into n types.

, a random vector representing the numbers of children in different types, satisfies a probability distribution on

[8] For multitype branching processes that the populations of different types grow exponentially, the proportions of different types converge almost surely to a constant vector under some mild conditions.

This is the strong law of large numbers for multitype branching processes.

For continuous-time cases, proportions of the population expectation satisfy an ODE system, which has a unique attracting fixed point.

This fixed point is just the vector that the proportions converge to in the law of large numbers.

The monograph by Athreya and Ney [9] summarizes a common set of conditions under which this law of large numbers is valid.

Branching processes where particles have to work (contribute resources to the environment) in order to be able to reproduce, and live in a changing society structure controlling the distribution of resources, are so-called resource-dependent branching processes.

The scaling limit of near-critical branching processes can be used to obtain superprocesses.