Quasi-stationary distributions and population processes
By S. M'el'eard, Denis Villemonais. 2011
265 citations recorded by the source corpus.
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.
Subfields: Modeling and Simulation, Public Health, Environmental and Occupational Health, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/eb403564dbcab1ff3e71a505ee68904e854e495c
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Citations in the corpus
- General criteria for the study of quasi-stationarity (2017, background): “In many general studies of quasi-stationary distributions [68, 15], one usually assumes that Px (τ∂ <∞) = 1 for all x ∈ E (so that the conditioning becomes singular in the limit of large time) and Px (n < τ∂) > 0 for all n > 0 and all x ∈ E so”
- General approximation method for the distribution of Markov processes conditioned not to be killed (2011, background): “iting distribution, i.e. limt→+∞ Pµ0 (ZT ∈ · | T<τ∂), is the subject of several studies and is closely related (in fact equivalent) to the concept of quasistationary distribution (see for instance [19]). Theorem 1 implies that the quasi-limiting distribution, if it exists, is equal to limt→+∞ lim”
- Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes (2017, background): “First, it implies the existence of a so-called mortality/extinction plateau: there exists λ0 > 0 limit of the extinction rate of the population (see [20]).”