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

Collet, Pierre; Martı́nez, Servet; Méléard, Sylvie; Martı́n, Jaime San

doi:10.1007/s00440-010-0297-4

Cited by 30 publications

(33 citation statements)

References 23 publications

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“…Our main theorem is based on a general argument proved in [2]. We also show the extinction of the population when time increases.…”

Section: Introductionmentioning

confidence: 78%

“…In the case where d is bounded above by d * < ∞, we now choose C > D + d * and t = 1 and apply Theorem 4.2 of [2]. In the case where d is unbounded, we choose C > − log Q and apply Theorem 4.2 of [2].…”

Section: Existence Of Quasistationary Distributionsmentioning

confidence: 99%

“…In Section 3 of [2], it was shown that this extinction rate is an eigenvalue of the dual problem associated with the probability measure L † ν = −λν. In Section 3 of [2], it was shown that this extinction rate is an eigenvalue of the dual problem associated with the probability measure L † ν = −λν.…”

Section: Lemma 52 Let I Be An Open Interval Included Inmentioning

confidence: 99%

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Stochastic Models for a Chemostat and Long-Time Behavior

2013

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We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

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“…Our main theorem is based on a general argument proved in [2]. We also show the extinction of the population when time increases.…”

Section: Introductionmentioning

confidence: 78%

Section: Existence Of Quasistationary Distributionsmentioning

confidence: 99%

Section: Lemma 52 Let I Be An Open Interval Included Inmentioning

confidence: 99%

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Stochastic Models for a Chemostat and Long-Time Behavior

2013

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“…We show next how the method of the proof can be extended to treat several multi-dimensional examples in Section 4.1.2. One of these examples is infinite-dimensional (as in [10]) and assumes Brownian mutations in a continuous type space. Our last example is the neutron transport process in a bounded domain, absorbed at the boundary (Section 4.2).…”

Section: Introductionmentioning

confidence: 99%

Exponential convergence to quasi-stationary distribution and $$Q$$-process

Champagnat

Villemonais

2015

Probab. Theory Relat. Fields

139

271

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For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinitedimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.where · is the integer part function and · T V is the total variation norm. Conversely, if there is uniform exponential convergence for the total variation norm in (1.1), then Assumption (A) holds true.

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“…Furthermore, for each n P µ X 2n = y | τ > n = µ (y) , y ∈ {1, 2, .., N − 1} and µ is the (unique) quasi-stationary distribution of X n : As is well-known for Markov chains with finite state-space, the Yaglom limit coincides with the quasistationary distribution (qsd). See [16] and [3] for qsd examples in much more general contexts.…”

Section: Preliminary: Drift Reversal Of the Ehrenfest Urn Modelmentioning

confidence: 99%

Reversing the drift of the Ehrenfest urn model and three conditionings

Huillet¹

2009

J. Phys. A: Math. Theor.

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To cite this version:Thierry Huillet. Reversing the drift of the Ehrenfest urn model and three conditionings. J. Phys. A: Math. Theor., IOPscience, 2009, 42 (34) We consider a drift-reversed version of the celebrated Ehrenfest urn process with N balls. For this 'dual' process, the boundaries are assumed to be absorbing and so the killing times at the boundaries play a central role. Three natural conditionings on the fixation/extinction events pertaining to this model are investigated. Some spectral informations on the conditioned Markoff chains are obtained, allowing to draw precise new conclusions on their limiting behaviors.

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

Cited by 30 publications

References 23 publications

Stochastic Models for a Chemostat and Long-Time Behavior

Stochastic Models for a Chemostat and Long-Time Behavior

Exponential convergence to quasi-stationary distribution and $$Q$$-process

Reversing the drift of the Ehrenfest urn model and three conditionings

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