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 Probability Surveys > Vol. 9 (2012) open journal systems 

Quasi-stationary distributions and population processes

Sylvie Méléard, CMAP, École Polytechnique
Denis Villemonais, INRIA Nancy

This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.

AMS 2000 subject classifications: Primary 92D25, 60J70, 60J80, 65C50.

Keywords: Population dynamics, quasi-stationarity, Yaglom limit, birth and death process, logistic Feller diffusion, Fleming-Viot particle system.

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Méléard, Sylvie, Villemonais, Denis, Quasi-stationary distributions and population processes, Probability Surveys, 9, (2012), 340-410 (electronic). DOI: 10.1214/11-PS191.


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