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Quasi-stationary distributions for structured birth and death processes with mutations
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  • Published: 05 May 2010

Quasi-stationary distributions for structured birth and death processes with mutations

  • Pierre Collet1,
  • Servet Martínez2,
  • Sylvie Méléard3 &
  • …
  • Jaime San Martín2 

Probability Theory and Related Fields volume 151, pages 191–231 (2011)Cite this article

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Abstract

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.

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Authors and Affiliations

  1. CNRS Physique Théorique, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Pierre Collet

  2. Departamento Ingeniería Matemática, Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile

    Servet Martínez & Jaime San Martín

  3. Ecole Polytechnique, CMAP, 91128, Palaiseau Cedex, France

    Sylvie Méléard

Authors
  1. Pierre Collet
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  2. Servet Martínez
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  3. Sylvie Méléard
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  4. Jaime San Martín
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Corresponding author

Correspondence to Pierre Collet.

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Collet, P., Martínez, S., Méléard, S. et al. Quasi-stationary distributions for structured birth and death processes with mutations. Probab. Theory Relat. Fields 151, 191–231 (2011). https://doi.org/10.1007/s00440-010-0297-4

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  • Received: 17 March 2009

  • Revised: 01 April 2010

  • Published: 05 May 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0297-4

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Keywords

  • Quasi-stationary distribution
  • Birth–death process
  • Population dynamics
  • Measured valued Markov processes

Mathematics Subject Classification (2000)

  • Primary 92D25
  • Secondary 60K35
  • 60J70
  • 60J80
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