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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
Article

Abstract

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 

Notes

Acknowledgments

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

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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