Journal of Mathematical Biology

, Volume 65, Issue 3, pp 553–580 | Cite as

Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution

  • Forrest W. Crawford
  • Marc A. SuchardEmail author


A birth–death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with n current particles, a new particle is born with instantaneous rate λ n and a particle dies with instantaneous rate μ n . Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth–death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics.


General birth–death process Continuous-time Markov chain Transition probabilities Population genetics Ecology Evolution 

Mathematics Subject Classification (2000)

60J27 92D15 92D20 92D40 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of BiomathematicsUniversity of California Los AngelesLos AngelesUSA
  2. 2.Department of BiostatisticsUniversity of California Los AngelesLos AngelesUSA
  3. 3.Department of Human GeneticsUniversity of California Los AngelesLos AngelesUSA

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