Immigration-extinction dynamics of stochastic populations

Meerson, Baruch; Ovaskainen, Otso

doi:10.1103/physreve.88.012124

“…Therefore, the approximation is better for larger value of a/b − r. The graph of 1 − p 0 (t) obtained from the quasi-steady approximation, Eq. (24), is also shown in Figures 4 and 5, where we indeed see that the leakage is faster than that of the exact solution at K/b = ∞, but captures the exact behavior much better at a/b − r = 9 where the time-scales are more separated, compared to a/b − r = 4.…”

Section: The Time-scale Separation and The Rate Of Leakagesupporting

confidence: 59%

Quantitative Analysis of a Transient Dynamics of a Gene Regulatory Network

Lee

¹

,

Lee

²

2018

Preprint

0

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In a stochastic process, noise often modifies the picture offered by the mean field dynamics.In particular, when there is an absorbing state, the noise erases a stable fixed point of the mean field equation from the stationary distribution, and turns it into a transient peak. We make a quantitative analysis of this effect for a simple genetic regulatory network with positive feedback, * jul@ssu.ac.kr 1 This is also called the transcriptional leakage, but we will refrain from using this terminology because we will be using the word leakage in quite the opposite sense. 2 The concentration is defined as x ≡ m/m, where m is the number of proteins andm is a large number chosen to be of the order of average number of proteins [31]. The ratesr,k 1 , andã of the rate equation, and the corresponding quantities r, k, and a in the master equation, are related byr = r/m,k 1 = k 1 /m, andã = a/m. See appendix E for details.

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“…This is because m = 0 is not an absorbing state any more. Similar situation is encountered in population dynamics, where influx of immigration plays the role of baseline production in the current model [24].…”

Section: Effect Of the Baseline Productionmentioning

confidence: 58%

Quantitative analysis of a transient dynamics of a gene regulatory network

Lee

¹

,

Lee

²

2018

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In a stochastic process, noise often modifies the picture offered by the mean field dynamics.In particular, when there is an absorbing state, the noise erases a stable fixed point of the mean field equation from the stationary distribution, and turns it into a transient peak. We make a quantitative analysis of this effect for a simple genetic regulatory network with positive feedback, where the proteins become extinct in the presence of stochastic noise, contrary to the prediction of the deterministic rate equation that the protein number converges to a non-zero value. We show that the transient peak appears near the stable fixed point of the rate equation, and the extinction time diverges exponentially as the stochastic noise approaches zero. We also show how the baseline production from the inactive gene ameliorates the effect of the stochastic noise, and interpret the opposite effects of the noise and the baseline production in terms of the position shift of the unstable fixed point. The order of magnitude estimates using biological parameters suggest that for a real gene regulatory network, the stochastic noise is sufficiently small so that not only is the extinction time much larger than biologically relevant time-scales, but also the effect of the baseline production dominates over that of the stochastic noise, leading to the protection from the catastrophic rare event of protein extinction. * jul@ssu.ac.kr 1 This is also called the transcriptional leakage, but we will refrain from using this terminology because we will be using the word leakage in quite the opposite sense. 2 The concentration is defined as x ≡ m/m, where m is the number of proteins andm is a large number chosen to be of the order of average number of proteins [31]. The ratesr,k 1 , andã of the rate equation, and the corresponding quantities r, k, and a in the master equation, are related byr = r/m,k 1 = k 1 /m, andã = a/m. See appendix E for details.

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“…For a i = +A > 0, the demographic noise ξ, leads to extinction in a finite time that diverges as σ ξ → 0 [4,6]. The external noise η reduces the most probable value of the population size, that becomes very close to zero when σ η > 2A/b [31].…”

Section: Stochastic Casementioning

confidence: 99%

“…These fragments, also known as patches, are not completely isolated as they are coupled due to movements of individuals in space. For modeling purposes, as a first step one can adopt a single patch viewpoint, taking into account the impact of the surrounding population in an effective manner [4][5][6][7]. As a further step beyond the single patch level, one can resort to a spatially explicit model.…”

mentioning

confidence: 99%

Metapopulation dynamics in a complex ecological landscape

Colombo

¹

,

Anteneodo

²

2015

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We propose a general model to study the interplay between spatial dispersal and environment spatiotemporal fluctuations in metapopulation dynamics. An ecological landscape of favorable patches is generated like a Lévy dust, which allows to build a range of patterns, from dispersed to clustered ones. Locally, the dynamics is driven by a canonical model for the evolution of the population density, consisting of a logistic expression plus multiplicative noises. Spatial coupling is introduced by means of two spreading mechanisms: diffusive dispersion and selective migration driven by patch suitability. We focus on the long-time population size as a function of habitat configurations, environment fluctuations and coupling schemes. We obtain the conditions, that the spatial distribution of favorable patches and the coupling mechanisms must fulfill, to grant population survival. The fundamental phenomenon that we observe is the positive feedback between environment fluctuations and spatial spread preventing extinction.

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Immigration-extinction dynamics of stochastic populations

Cited by 18 publications

References 23 publications

Quantitative Analysis of a Transient Dynamics of a Gene Regulatory Network

Quantitative Analysis of a Transient Dynamics of a Gene Regulatory Network

Quantitative analysis of a transient dynamics of a gene regulatory network

Metapopulation dynamics in a complex ecological landscape

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