Birth-death process

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The birth-death process is a special case of Continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths. Birth-death processes have many application in demography, queueing theory, or in biology, for example to study the evolution of bacteria.

When a birth occurs, the process goes from state n to n+1. When a death occurs, the process goes from state n to state n-1. The process is specified by birth rates $\{\lambda_{i}\}_{i=0..\infty}$ and death rates $\{\mu_{i}\}_{i=1..\infty}$ .

[edit] Examples of birth-death processes

A pure birth process is a birth-death process where $μ i = 0$ for all $i \ge 0$ .

A pure death process is a birth-death process where $λ i = 0$ for all $i \ge 0$ .

A (homogeneous) Poisson process is a pure birth process where $λ i = λ$ for all $i \ge 0$

[edit] Limit behaviour

In a small time $Δ t$ , only three types of transitions are possible: one death, or one birth, or no birth nor death. If the rate of occurrences (per unit time) of births is $λ$ and that for deaths is $μ$ , then the probabilities of the above transitions are $λΔ t$ , $μΔ t$ , and $1 - (λ + μ)Δ t$ respectively. For a population process, "birth" is the transition towards increasing the population by 1 while "death" is the transition towards decreasing the population size by 1.