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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
Article

Abstract

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.

Keywords

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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References

  1. Abate J, Whitt W (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Syst 10: 5–87MathSciNetzbMATHCrossRefGoogle Scholar
  2. Abate J, Whitt W (1992) Numerical inversion of probability generating functions. Oper Res Lett 12: 245–251MathSciNetzbMATHCrossRefGoogle Scholar
  3. Abate J, Whitt W (1995) Numerical inversion of Laplace transforms of probability distributions. ORS J Comput 7(1): 36–43zbMATHCrossRefGoogle Scholar
  4. Abate J, Whitt W (1999) Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J Comput 11(4): 394–405MathSciNetzbMATHCrossRefGoogle Scholar
  5. Allee WC, Emerson AE, Park O (1949) Principles of animal ecology. Saunders, PhiladelphiaGoogle Scholar
  6. Bailey NTJ (1964) The Elements of stochastic processes with applications to the natural sciences. A Wiley publication in applied statistics. Wiley, New YorkGoogle Scholar
  7. Bankier JD, Leighton W (1942) Numerical continued fractions. Am J Math 64(1): 653–668MathSciNetzbMATHCrossRefGoogle Scholar
  8. Blanch G (1964) Numerical evaluation of continued fractions. SIAM Rev 6(4)Google Scholar
  9. Bordes G, Roehner B (1983) Application of stieltjes theory for s-fractions to birth and death processes. Adv Appl Probab 15(3): 507–530MathSciNetzbMATHCrossRefGoogle Scholar
  10. Craviotto C, Jones WB, Thron WJ (1993) A survey of truncation error analysis for Padé and continued fraction approximants. Acta Appl Math 33: 211–272MathSciNetzbMATHCrossRefGoogle Scholar
  11. Cuyt A, Petersen V, Verdonk B, Waadeland H, Jones W (2008) Handbook of continued fractions for special functions. Springer, BerlinzbMATHGoogle Scholar
  12. Dennis B (2002) Allee effects in stochastic populations. Oikos 96(3): 389–401MathSciNetCrossRefGoogle Scholar
  13. Donnelly P (1984) The transient behaviour of the Moran model in population genetics. Math Proc Cambridge 95(02): 349–358MathSciNetzbMATHCrossRefGoogle Scholar
  14. Feller W (1971) An introduction to probability theory and its applications. Wiley series in probability and mathematical statistics. Wiley, New YorkGoogle Scholar
  15. Flajolet P, Guillemin F (2000) The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv Appl Probab 32(3): 750–778MathSciNetzbMATHCrossRefGoogle Scholar
  16. Grassmann W (1977) Transient solutions in Markovian queues: an algorithm for finding them and determining their waiting-time distributions. Eur J Oper Res 1(6): 396–402MathSciNetzbMATHCrossRefGoogle Scholar
  17. Grassmann WK (1977) Transient solutions in Markovian queueing systems. Comput Oper Res 4(1): 47–53CrossRefGoogle Scholar
  18. Guillemin F, Pinchon D (1999) Excursions of birth and death processes, orthogonal polynomials, and continued fractions. J Appl Probab 36(3): 752–770MathSciNetzbMATHCrossRefGoogle Scholar
  19. Ismail MEH, Letessier J, Valent G (1988) Linear birth and death models and associated Laguerre and Meixner polynomials. J Approx Theory 55(3): 337–348MathSciNetzbMATHCrossRefGoogle Scholar
  20. Karlin S, McGregor J (1957) The classification of birth and death processes. Trans Am Math Soc 86(2): 366–400MathSciNetzbMATHCrossRefGoogle Scholar
  21. Karlin S, McGregor J (1957) The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans Am Math Soc 85(2): 589–646MathSciNetCrossRefGoogle Scholar
  22. Karlin S, McGregor J (1958) Linear growth, birth and death processes. J Math Mech 7(4): 643–662MathSciNetzbMATHGoogle Scholar
  23. Karlin S, McGregor J (1958) Many server queueing processes with Poisson input and exponential service times. Pacific J Math 8(1): 87–118MathSciNetzbMATHGoogle Scholar
  24. Karlin S, McGregor J (1962) On a genetics model of Moran. Math Proc Cambridge 58(02): 299–311MathSciNetCrossRefGoogle Scholar
  25. Kendall DG (1948) On the generalized “birth-and-death” process. Ann Math Stat 19(1): 1–15MathSciNetCrossRefGoogle Scholar
  26. Kingman JFC (1982) The coalescent. Stat Proc Appl 13(3): 235–248MathSciNetzbMATHCrossRefGoogle Scholar
  27. Kingman JFC (1982) On the genealogy of large populations. J Appl Probab 19: 27–43MathSciNetCrossRefGoogle Scholar
  28. Klar B, Parthasarathy PR, Henze N (2010) Zipf and Lerch limit of birth and death processes. Probab Eng Inform Sc 24(01): 129–144MathSciNetzbMATHCrossRefGoogle Scholar
  29. Krone SM, Neuhauser C (1997) Ancestral processes with selection. Theor Popul Biol 51: 210–237zbMATHCrossRefGoogle Scholar
  30. Lentz WJ (1976) Generating Bessel functions in Mie scattering calculations using continued fractions. Appl Opt 15(3): 668–671CrossRefGoogle Scholar
  31. Levin D (1973) Development of non-linear transformations for improving convergence of sequences. Int J Comput Math 3(B): 371–388zbMATHGoogle Scholar
  32. Lorentzen L, Waadeland H (1992) Continued fractions with applications. North-Holland, AmsterdamzbMATHGoogle Scholar
  33. Mederer M (2003) Transient solutions of Markov processes and generalized continued fractions. IMA J Appl Math 68(1): 99–118MathSciNetzbMATHCrossRefGoogle Scholar
  34. Mohanty S, Montazer-Haghighi A, Trueblood R (1993) On the transient behavior of a finite birth–death process with an application. Comput Oper Res 20(3): 239–248MathSciNetzbMATHCrossRefGoogle Scholar
  35. Moran PAP (1958) Random processes in genetics. Math Proc Cambridge 54(01): 60–71zbMATHCrossRefGoogle Scholar
  36. Murphy JA, O’Donohoe MR (1975) Some properties of continued fractions with applications in Markov processes. IMA J Appl Math 16(1): 57–71MathSciNetzbMATHCrossRefGoogle Scholar
  37. Novozhilov AS, Karev GP, Koonin EV (2006) Biological applications of the theory of birth-and-death processes. Brief Bioinform 7(1): 70–85CrossRefGoogle Scholar
  38. Numerical Recipes Software (2007) Derivation of the Levin transformation. Numerical recipes webnote No 6. http://www.nr.com/webnotes?6
  39. Parthasarathy PR, Sudhesh R (2006) Exact transient solution of a state-dependent birth–death process. J Appl Math Stoch Anal 82(6): 1–16CrossRefGoogle Scholar
  40. Parthasarathy PR, Sudhesh R (2006) A formula for the coefficients of orthogonal polynomials from the three-term recurrence relations. Appl Math Lett 19(10): 1083–1089MathSciNetzbMATHCrossRefGoogle Scholar
  41. Press WH (2007) Numerical recipes: the art of scientific computing. Cambridge University Press, New YorkzbMATHGoogle Scholar
  42. Renshaw E (1993) Modelling biological populations in space and time. Cambridge studies in mathematical biology. Cambridge University Press, CambridgeGoogle Scholar
  43. Rosenlund SI (1978) Transition probabilities for a truncated birth–death process. Scand J Stat 5(2): 119–122MathSciNetzbMATHGoogle Scholar
  44. Sharma OP, Dass J (1988) Multi-server Markovian queue with finite waiting space. Sankhy Ser B 50(3): 428–431MathSciNetzbMATHGoogle Scholar
  45. Tan WY, Piantadosi S (1991) On stochastic growth processes with application to stochastic logistic growth. Stat Sinica 1: 527–540MathSciNetzbMATHGoogle Scholar
  46. Taylor H, Karlin S (1998) An introduction to stochastic modeling. Academic Press, San DiegozbMATHGoogle Scholar
  47. Thompson IJ, Barnett AR (1986) Coulomb and Bessel functions of complex arguments and order. J Comput Phys 64: 490–509MathSciNetzbMATHCrossRefGoogle Scholar
  48. Thorne J, Kishino H, Felsenstein J (1991) An evolutionary model for maximum likelihood alignment of DNA sequences. J Mol Evol 33(2): 114–124CrossRefGoogle Scholar
  49. Wall HS (1948) Analytic theory of continued fractions. University Series in Higher Mathematics, D. Van Nostrand Company Inc., New YorkzbMATHGoogle Scholar
  50. Wallis J (1695) Opera Mathematica, vol 1. Oxoniae e Theatro Shedoniano, reprinted by Georg Olms Verlag, Hildeshein, New York, 1972Google Scholar
  51. Yule GU (1925) A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis. FRS Philos T R Soc Lon B 213: 21–87CrossRefGoogle Scholar

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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