# Birth process

A birth process, also called a pure birth process, is a continuous-time [Markov chain](https://www.edgepedia.org/markov-chain) whose state counts individuals or events and can only move upward, from state i to state i + 1, never downward or by more than one at a time.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup> It is the special case of a birth-death process in which all death rates are zero, and its name comes from biology, where the state is the size of a population of organisms that reproduce but do not die.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup><sup> • </sup><sup>[2](https://archive.nptel.ac.in/content/storage2/courses/117103017/module2/lec2/3.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A continuous-time Markov chain on \( \{0, 1, \ldots, n\} \) in which, for all i, only the transition from state i to i + 1 is possible; the positive transition rates are the q_{i,i+1}.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup> |
| Local behavior | From state k the process jumps to k + 1 with rate \( \lambda_k \) and otherwise stays put; in a small interval of length h, Pr(X(t+h) − X(t) = 1 \| X(t) = k) = \( \lambda_k h + o(h) \).<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> |
| Poisson special case | A Poisson process with rate r is a pure birth process with constant birth rate \( \alpha(x) = r \) in every state.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)</sup> |
| Yule process | The classical linear case \( \lambda_n = n\lambda \), in which each of n individuals splits independently at rate \( \lambda \); found by George Udny Yule in 1924.<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> |
| Holding times | The time spent in state i is exponentially distributed with rate \( \lambda_i \), so \( p_{ii}(t) = e^{-\lambda_i t} \).<sup>[5](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)</sup> |
| Explosion | If the rates grow fast enough that \( \sum_n \lambda_n^{-1} \) converges, infinitely many jumps can accumulate in finite time; the linear birth process is regular because \( \sum_n (\lambda n)^{-1} = \infty \).<sup>[6](https://www.fcfm.buap.mx/jzacarias/cursos/procesos/libros/book3pe.pdf)</sup><sup> • </sup><sup>[7](https://www.utwente.nl/en/eemcs/sor/boucherie/education/mqsn/mqsn-lnmb-chap-1.pdf)</sup> |
| Main uses | Population growth without mortality, arrival counting, queueing, reliability and inventory systems, and speciation-without-extinction models in macroevolution.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup><sup> • </sup><sup>[8](https://www.phyloeco.bio.ens.psl.eu/pdf/Morlon2014ELE.pdf)</sup> |

## Definition and postulates

Let X(t) be a Markov process taking values in the non-negative integers, started at X(0) = N. The pure birth postulates are that in a short interval of length h the state increases by one with probability \( \lambda_j h + o(h) \), stays the same with probability \( 1 - \lambda_j h + o(h) \), and never changes by more than one.<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> The sequence \( \{\lambda_k\} \) of birth rates determines the process: state k is held for an exponentially distributed time with rate \( \lambda_k \), after which the state jumps to k + 1.<sup>[7](https://www.utwente.nl/en/eemcs/sor/boucherie/education/mqsn/mqsn-lnmb-chap-1.pdf)</sup><sup> • </sup><sup>[5](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)</sup>

The state probabilities \( P_n(t) = \Pr(X(t) = n) \) satisfy the Kolmogorov forward equations of the pure birth chain,<sup>[5](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)</sup>

\[ P'_n(t) = -\lambda_n P_n(t) + \lambda_{n-1} P_{n-1}(t), \quad n \geq 1, \qquad P'_0(t) = -\lambda_0 P_0(t). \]

These can be solved recursively starting from \( P_0(t) = P_0(0) e^{-\lambda_0 t} \).<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup>

## Special cases

**Poisson process.** Setting every birth rate equal to a constant \( \lambda \) gives the Poisson process, the counting process in which events occur at a rate that does not depend on how many events have already occurred.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)</sup><sup> • </sup><sup>[9](https://www.sciencedirect.com/topics/mathematics/pure-birth-process)</sup> The pure birth process generalizes the Poisson process by letting the chance of an event at an instant depend on the number of events already counted.<sup>[9](https://www.sciencedirect.com/topics/mathematics/pure-birth-process)</sup>

**Yule process.** The Yule process, found by [George Udny Yule](https://www.edgepedia.org/george-udny-yule) in 1924, models a population in which each member independently splits into two at rate \( \lambda \), so the total birth rate in state n is \( \lambda_n = n\lambda \).<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> Starting from one individual, the differential equations solve iteratively to give

\[ P_n(t) = e^{-\lambda t} \left(1 - e^{-\lambda t}\right)^{n-1}, \]

a geometric form of the negative binomial distribution.<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> Yule developed the model in his 1925 paper on evolution to fit data on the number of species per genus, treating speciation as analogous to a birth.<sup>[10](https://assets.press.princeton.edu/chapters/s13_8458.pdf)</sup> In modern macroevolution, this pure birth (no extinction) model of diversification remains a baseline against which birth-death models that add extinction are compared.<sup>[8](https://www.phyloeco.bio.ens.psl.eu/pdf/Morlon2014ELE.pdf)</sup>

## Transition probabilities

For a pure birth process with distinct rates \( \lambda_i \neq \lambda_j \) for i ≠ j, the transition probabilities have an explicit form. The probability of still being in the starting state is \( p_{ii}(t) = e^{-\lambda_i t} \), and for i < j the probability of having reached j is a linear combination of exponentials \( e^{-\lambda_k t} \) for k from i to j, with coefficients built from products of the rates; the sum of independent exponential holding times with distinct rates has a hypoexponential distribution.<sup>[5](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)</sup><sup> • </sup><sup>[13](https://www.mdpi.com/2227-7390/8/12/2207)</sup> For the one-step case this reduces to

\[ p_{i,i+1}(t) = \frac{\lambda_i}{\lambda_{i+1} - \lambda_i} \left( e^{-\lambda_i t} - e^{-\lambda_{i+1} t} \right). \]

When rates are equal, sums of holding times produce Gamma densities instead; for two equal rates \( \lambda \), the density of the sum is \( \lambda^2 t e^{-\lambda t} \).<sup>[5](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)</sup>

## Explosion

A pure birth process need not be defined for all time. If the birth rates grow quickly enough, the holding times shrink so fast that infinitely many jumps accumulate within a finite interval, a failure called explosion, in which case the process is not properly defined for all t > 0 from its rate matrix.<sup>[7](https://www.utwente.nl/en/eemcs/sor/boucherie/education/mqsn/mqsn-lnmb-chap-1.pdf)</sup> The criterion is the divergence of the sum of expected holding times: the linear birth process is regular, meaning no explosion, precisely because \( \sum_n (\lambda n)^{-1} = \infty \), the harmonic series.<sup>[6](https://www.fcfm.buap.mx/jzacarias/cursos/procesos/libros/book3pe.pdf)</sup> A pure birth process explodes if and only if the sum of the reciprocals of its birth rates, \( \sum_n 1/\lambda_n \), converges; rates growing faster than linearly do not by themselves guarantee explosion.<sup>[6](https://www.fcfm.buap.mx/jzacarias/cursos/procesos/libros/book3pe.pdf)</sup><sup> • </sup><sup>[14](https://arnavs1ngh.github.io/notes/docs/Year3/ap/)</sup>

## Applications

[Birth and death processes](https://www.edgepedia.org/birth-death-process), of which the pure birth process is the no-death case, are standard tools for modeling queueing, reliability, and inventory systems; in the economic sciences they describe the development of the number of enterprises in an area and manpower fluctuations, and in physics they model flows of radioactive, cosmic, and other particles.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup> In biology the state counts individuals in a population of organisms, the application that gave the processes their names.<sup>[1](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)</sup>

The class is also flexible enough to reproduce arbitrary probability distributions on the counting numbers. Faddy (1997) showed that a pure birth process with rates \( \lambda_k = \lambda (\gamma + k)^{c} \) and zero death rates has transition probabilities matching any discrete distribution on the non-negative integers; the exponent c = 0 recovers a Poisson process with rate \( \lambda \), c > 0 produces overdispersion relative to Poisson, and c < 0 produces underdispersion.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC6014701/)</sup>

## Relation to birth-death processes

Adding death rates \( \mu_k \), so that from state k the process can also step down to k − 1, yields the general birth-death process, a continuous-time Markov chain in which transitions from state i go only to i + 1 or i − 1.<sup>[12](https://www.sciencedirect.com/topics/mathematics/continuous-time-markov-chain)</sup> The pure birth process models early stages of population growth but not growth in which members die or are removed, which is why the two-rate generalization was introduced.<sup>[3](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)</sup> In phylogenetics the distinction matters directly: a pure birth (Yule) model assumes no extinction, while birth-death models of diversification estimate speciation and extinction rates separately, and the choice between them affects what can be inferred from reconstructed trees.<sup>[8](https://www.phyloeco.bio.ens.psl.eu/pdf/Morlon2014ELE.pdf)</sup>

## References

1. [Beichelt, F. Stochastic Processes, Chapter 5: Continuous-Time Markov Processes (Section 5.6, Birth- and Death Processes), Yale University course reading](https://www.stat.yale.edu/~jtc5/251/readings/Stochastic%20Processes%20Beichelt.pdf)
2. [NPTEL course notes: Birth-Death Processes, Pure Birth and Pure Death Processes](https://archive.nptel.ac.in/content/storage2/courses/117103017/module2/lec2/3.html)
3. [Introduction and Review of Probability / Continuous Time Markov Processes (University of Vienna lecture notes, Chapter 6)](https://www.mat.univie.ac.at/~bruin/SP_US.pdf)
4. [Continuous-Time Birth-Death Chains, Statistics LibreTexts (Siegrist)](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/16%3A_Markov_Processes/16.21%3A_Continuous-Time_Birth-Death_Chains)
5. [Chen, G. Math 263 lecture notes: Continuous-Time Markov Chains, San José State University](https://www.sjsu.edu/faculty/guangliang.chen/Math263/lec9ContinuousTimeMC.pdf)
6. [Stability and Explosions in Continuous Time Markov Chains (course text, Benemérita Universidad Autónoma de Puebla)](https://www.fcfm.buap.mx/jzacarias/cursos/procesos/libros/book3pe.pdf)
7. [Boucherie, R. J. Lecture notes, Markov chains chapter, University of Twente](https://www.utwente.nl/en/eemcs/sor/boucherie/education/mqsn/mqsn-lnmb-chap-1.pdf)
8. [Morlon, H. (2014). Phylogenetic approaches for studying diversification. Ecology Letters](https://www.phyloeco.bio.ens.psl.eu/pdf/Morlon2014ELE.pdf)
9. [Pure Birth Process: an overview, ScienceDirect Topics (Pinsky & Karlin, An Introduction to Stochastic Modeling)](https://www.sciencedirect.com/topics/mathematics/pure-birth-process)
10. [Pure-birth process / Yule process chapter, Princeton University Press (Stochastic Modeling chapters PDF)](https://assets.press.princeton.edu/chapters/s13_8458.pdf)
11. [Crawford, F. W., Minin, V. N., Suchard, M. A. (2018). Computational methods for birth-death processes. Statistics in Medicine](https://pmc.ncbi.nlm.nih.gov/articles/PMC6014701/)
12. [Continuous Time Markov Chain: an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/continuous-time-markov-chain)
13. [mdpi.com](https://www.mdpi.com/2227-7390/8/12/2207)
14. [Applied Probability](https://arnavs1ngh.github.io/notes/docs/Year3/ap/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Markov chains and processes › Continuous-time Markov processes › Birth–death processes*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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