Population extinction in a time-modulated environment

Assaf, Michael; Kamenev, Alex; Meerson, Baruch

doi:10.1103/physreve.78.041123

Cited by 59 publications

(116 citation statements)

References 26 publications

(83 reference statements)

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“…Without the perturbation Q 1 , the above flow equation could thus be solved by extending the previous ansatz (38) to two species. To accommodate Q 1 as well, it proves useful to generalize that ansatz to…”

Section: Methods For the Analysis Of The Generating Function's Flow Ementioning

confidence: 99%

“…A HamiltonJacobi equation can be solved by the method of characteristics [368], with the characteristic curves q(s) and x(s) obeying Hamilton's equations 2 ) to the "passive state" (0, 0) constitutes the "optimal path to extinction" (see below) [35,38,369].…”

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

“…The value of this factor has been determined in [357,365,366] and reads, in terms of our rate coefficients, A = 2 π/n ∞ . More information on the above procedure can be found in [35,38]. The method has also been applied to classic epidemiological models [369] and to a variant of the Verhulst logistic model [370] (exact results on this model with added immigration have been obtained in [371] using the generating function technique).…”

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

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Master equations and the theory of stochastic path integrals

2017

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Abstract. This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of masters equations most often rely on lownoise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a "generating functional", which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a "forward" and a "backward" path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit. arXiv:1609.02849v2 [cond-mat...

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Section: Methods For the Analysis Of The Generating Function's Flow Ementioning

confidence: 99%

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

Section: Wkb Approximations and Related Approachesmentioning

confidence: 99%

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Master equations and the theory of stochastic path integrals

2017

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“…This has already been done in explicitly time dependent chemical systems for the simpler extinction trajectories [30]. Herein we have shown that for exponentially long times the sort of Hamiltonian chaos we have described is possible for the simple reaction (31) by virtue of chemical noise.…”

Section: Chemical Chaosmentioning

confidence: 85%

“…We have also studied the optimal paths to extinction in a plankton population dynamics model. Although extinction phenomena have been considered numerous times within this framework [13,14,30], the approach was based on the Hamiltonian dynamics on the stationary H = 0 manifold. The Hamiltonian dynamical system in our case was however degenerated and all the extinction trajectories fell on the H = 0 manifolds.…”

Section: Discussionmentioning

confidence: 99%

Chemical Oscillations out of Chemical Noise

Escudero¹,

Rivera²,

Torres³

2011

SIAM J. Appl. Dyn. Syst.

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The dynamics of one species chemical kinetics is studied. Chemical reactions are modelled by means of continuous time Markov processes whose probability distribution obeys a suitable master equation. A large deviation theory is formally introduced, which allows developing a Hamiltonian dynamical system able to describe the system dynamics. Using this technique we are able to show that the intrinsic fluctuations, originated in the discrete character of the reagents, may sustain oscillations and chaotic trajectories which are impossible when these fluctuations are disregarded. An important point is that oscillations and chaos appear in systems whose mean-field dynamics has too low a dimensionality for showing such a behavior. In this sense these phenomena are purely induced by noise, which does not limit 1 arXiv:1004.2722v2 [cond-mat.stat-mech] 2 Oct 2011 itself to shifting a bifurcation threshold. On the other hand, they are large deviations of a short transient nature which typically only appear after long waiting times. We also discuss the implications of our results in understanding extinction events in population dynamics models expressed by means of stoichiometric relations.

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The WKB Method: A User-Guide

Ashcroft

2016

The Statistical Physics of Fixation and Equilibration in Individual-Based Models

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Population extinction in a time-modulated environment

Cited by 59 publications

References 26 publications

Master equations and the theory of stochastic path integrals

Master equations and the theory of stochastic path integrals

Chemical Oscillations out of Chemical Noise

The WKB Method: A User-Guide

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