Probability Theory and Related Fields

, Volume 164, Issue 1–2, pp 285–332 | Cite as

Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

  • J.-R. ChazottesEmail author
  • P. Collet
  • S. Méléard


We study a general class of birth-and-death processes with state space \({\mathbb {N}}\) that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a ‘carrying capacity’ \(K\). When \(K\) is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of \(K\), for large \(K\). We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when \(K\) is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process.

Mathematics Subject Classification

Primary 92D25 Secondary 60J27 60J28 60J80 47A75 92D40 



The third author benefited from the support of the “Chaire Modélisation Mathématique et Biodiversité” funded by Veolia Environnement, the Ecole polytechnique and the Muséum national d’Histoire naturelle. The authors thank the referees for their careful reading and comments.


  1. 1.
    Allen, L.J.S.: An Introduction to Stochastic Processes with Applications to Biology. CRC Press, New York (2011)zbMATHGoogle Scholar
  2. 2.
    Barbour, A.D., Pollett, P.K.: Total variation approximation for quasi-stationary distributions. J. Appl. Probab. 47, 934–946 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bansaye, V., Méléard, S., Richard, M.: How do birth and death processes come down from infinity?, preprint (2013). arXiv:1310.7402 [math.PR]
  4. 4.
    Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., San Martín, J.: Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37(5), 1926–1969 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Champagnat, N., Villemonais, D.: Exponential convergence to quasi-stationary distribution and Q-process, preprint (2014). arXiv:1404.1349v1 [math.PR]
  6. 6.
    Cloez, B., Thai, M.N.: Quantitative results for the Fleming-Viot particle system in discrete space, preprint (2014). arXiv:1312.2444v2 [math.PR]
  7. 7.
    Collet, P., Martínez, S.: Quasi-Stationary Distributions. Probability and its Applications. Springer, New York (2013)CrossRefGoogle Scholar
  8. 8.
    Diaconis, P., Miclo, L.: On quantitative convergence to quasi-stationarity, preprint (2014). arXiv:1406.1805v1 [math.PR]
  9. 9.
    Doering, C., Sargsyan, K., Sander, L.: Extinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation. Multiscale Model. Simul. 3(2), 283–299 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    van Doorn, E.: Quasi-stationary distributions and convergence to quasi-stationarity for birth-death processes. Adv. Appl. Probab. 23, 683–700 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fedoryuk, M.: Asymptotic Analysis. Linear Ordinary Differential Equations. Springer, Berlin (1993)zbMATHGoogle Scholar
  12. 12.
    Karlin, S., McGregor, J.L.: The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Am. Math. Soc. 86, 489–546 (1957)CrossRefGoogle Scholar
  13. 13.
    Karlin, S., Taylor, H.M.: An Introduction to Stochastic Modeling, 3rd edn. Academic Press, New York (1998)zbMATHGoogle Scholar
  14. 14.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, New York (1966)CrossRefGoogle Scholar
  15. 15.
    Kessler, D., Shnerb, N.: Extinction rates for fluctuation-induced metastabilities: a real-space WKB approach. J. Stat. Phys. 127(5), 861–886 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49–58Google Scholar
  17. 17.
    Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Méléard, S., Villemonais, D.: Quasi-stationary distributions and population processes. Probab. Surv. 9, 340–410 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nåsell, I.: Extinction and quasi-stationarity in the stochastic logistic SIS model. Lecture Notes in Mathematics, Mathematical Biosciences Subseries, vol. 2022. Springer, New York (2011)zbMATHGoogle Scholar
  20. 20.
    Ovaskainen, O., Meerson, B.: Stochastic models of population extinction. Trends Ecol. Evol. 25, 643–652 (2010)CrossRefGoogle Scholar
  21. 21.
    Sagitov, S., Shahmerdenova, A.: Extinction times for a birth-death process with weak competition. Lith. Math. J. 53, 220–234 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sotomayor, J.: Inversion of smooth mappings. Z. Angew. Math. Phys. 41(2), 306–310 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Fundamental Principles of Mathematical Sciences, vol. 293. Springer, Berlin (1991)CrossRefGoogle Scholar
  24. 24.
    Yosida, K.: Functional analysis. Reprint of the sixth (1980) edition. Classics in Mathematics. Springer, Berlin (1995)Google Scholar

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Authors and Affiliations

  1. 1.Centre de Physique Théorique, CNRS UMR 7644Ecole polytechniquePalaiseau CedexFrance
  2. 2.Centre de Mathématiques Appliquées, CNRS UMR 7641Ecole polytechniquePalaiseau CedexFrance

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