Stationary distribution of population size in

Peters, Craig S.; Mangel, Marc; Costantino, R. F.

doi:10.1016/s0092-8240(89)80104-1

Cited by 8 publications

(10 citation statements)

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“…In contrast to the FP approximation, the Wentzel-Kramers-Brillouin (WKB) approximation [55], first employed in the context of the master equation in Refs. [56][57][58][59], is ideally suited for rare event statistics in large populations, as it accurately predicts quantities such as the MTE and the most likely pathway to extinction [36,[60][61][62][63] (Box 3). Both the FP and WKB approximations replace the master equation with an analytically tractable equation, but in the case of the WKB approach, the approximation is mathematically controlled.…”

Section: Methods For Estimating Time To Extinctionmentioning

confidence: 99%

Stochastic models of population extinction

Ovaskainen

Meerson

2010

Trends in Ecology & Evolution

384

432

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Theoretical ecologists have long sought to understand how the persistence of populations depends on biotic and abiotic factors. Classical work showed that demographic stochasticity causes the mean time to extinction to increase exponentially with population size, whereas variation in environmental conditions can lead to a power-law scaling. Recent work has focused especially on the influence of the autocorrelation structure ('color') of environmental noise. In theoretical physics, there is a burst of research activity in analyzing large fluctuations in stochastic population dynamics. This research provides powerful tools for determining extinction times and characterizing the pathway to extinction. It yields, therefore, sharp insights into extinction processes and has great potential for further applications in theoretical biology.

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Section: Methods For Estimating Time To Extinctionmentioning

confidence: 99%

Stochastic models of population extinction

Ovaskainen

Meerson

2010

Trends in Ecology & Evolution

384

432

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“…It decreases because of cannibalism on pre-adult stages. But note that B(O) = b 0 b 1 > 0; this represents "recruitment" into the adult stage from other stages oflife history, even when there are no adults present (see Peters et al 1989 for further interpretation). We could identify the "carrying capacity" as the positive steady state ns satisfying B(ns) = D(ns).…”

Section: Introductionmentioning

confidence: 99%

“…In summary, fact 2 shows that (thanks to the development of powerful microcomputers and high-level computer languages) the computation of the mean and variance of extinction times can now be done in just a few lines of computer code (although it does take some serious, biological work to get the matrix M, as in the description of the flour beetle dynamics above; Peters et al 1989). Thus, although one must still use a numerical method, it is not a Monte Carlo simulation, which may require thousands of runs to obtain reasonable estimates of the statistics on persistence time (cf.…”

Section: Introductionmentioning

confidence: 99%

Four Facts Every Conservation Biologists Should Know about Persistence

Mangel¹,

Tier²

1994

Ecology

204

124

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“…We calculate the probability P n (t) by means of the contour integral with a saddle point approximation. The saddle point condition (53) combined with the solution of the approximated classical equation of motion (44) gives the dependence…”

Section: Binary and Single Annihilationmentioning

confidence: 99%

“…The WKB approximation is applied to the evolution equation of the generating function. For related studies on the WKB approximation see [45][46][47][48][49][50][51][52][53] and for reviews see [54][55][56]. For stochastic population models the WKB approximation allows one also to calculate the mean extinction times and probabilities [57][58][59][60] and for switching rates in multistep reactions see [61].…”

Section: Introductionmentioning

confidence: 99%

WKB-type-of approximation for rare event statistics in reacting systems

Mühlbacher,

Guhr

2019

Preprint

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We calculate the probabilities to find systems of reacting particles in states which largely deviate from typical behavior. The rare event statistics is obtained from the master equation which describes the dynamics of the probability distribution of the particle number. We transform the master equation by means of a generating function into a time-dependent "Schrödinger equation". Its solution is provided by a separation ansatz and an approximation for the stationary part which is of Wentzel-Kramers-Brillouin (WKB) type employing a small parameter. The solutions of the "classical" equations of motions and a saddle point approximation yield the proper generating function. Our approach extends a method put forward by Elgart and Kamenev. We calculate the rare event statistics for systems where the dynamics cannot be entirely analyzed in an analytical manner. We consider different examples.

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Stationary distribution of population size in

Cited by 8 publications

References 0 publications

Stochastic models of population extinction

Stochastic models of population extinction

Four Facts Every Conservation Biologists Should Know about Persistence

WKB-type-of approximation for rare event statistics in reacting systems

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