Geometric singular perturbation theory for stochastic differential equations

Berglund, Nils; Gentz, Barbara

doi:10.1016/s0022-0396(03)00020-2

Cited by 87 publications

(97 citation statements)

References 22 publications

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“…In terms of the mathematical approach, dynamical systems evolving in a fast/slow time framework can be analyzed using singular perturbation analysis (e.g., Wasow, 1965;Fenichel, 1979;Berglund, 1998;Berglund and Gentz, 2003). Two cases are of main interest in problems with the state variables evolving in di¤erent time scales, the case of an adiabatic system and the case of a fast/slow system.…”

mentioning

confidence: 99%

Time Scale Externalities and the Management of Renewable Resources

Vardas

Xepapadeas

2015

SSRN Journal

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The evolution of renewable resources is characterized in many cases by di¤erent time scales where some state variables such as biomass, may evolve relatively faster than other state variables such as carrying capacity. Ignoring this time scale separation means that a slowly changing variable is treated as constant over time. Management rules designed without accounting for time scale separation will result in ine¢ ciencies in resource management. We call this ine¢ ciency time scale externality and we analyze renewable resource harvesting when carrying capacity evolves slowly, either in response to exogenous forcing or in response to emissions generated by the industrial sector of the economy. We study cooperative and non-cooperative solutions under time scale separation. Using singular perturbation reduction methods (Fenichel 1979), we examine the role of di¤erent time scales in environmental management and the potential errors in optimal regulation when time scale separation is ignored.

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confidence: 99%

Time Scale Externalities and the Management of Renewable Resources

Vardas

Xepapadeas

2015

SSRN Journal

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“…In this direction, and restricting to the discrete time case, we will consider the aggregation of density-dependent branching models [14,38] The aggregation of systems in the context of stochastic time continuous models has been studied only in the case of stochastic differential equations (SDEs). [28] explores the (very restrictive) conditions under which it is possible to carry out the perfect aggregation of SDEs and [9] carries out the study of the approximate reduction of SDEs in the case that the random noise is small enough. It would be interesting to extend those results to deal with the case in which the noise intensity is not restricted.…”

Section: Discussionmentioning

confidence: 99%

Approximate Aggregation Methods in Discrete Time Stochastic Population Models

Sanz

Alonso

2010

Math. Model. Nat. Phenom.

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Abstract. Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler "aggregated system" governed by a few variables. Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales in the system under consideration. In this work we deal with aggregation techniques in stochastic discrete time models and their application to the study of multiregional models, i.e., of models for an age structured population distributed amongst different spatial patches and in which migration between the patches is usually fast with respect to the demography (reproduction-survival) in each patch. Stochasticity in population models can be of two kinds: environmental and demographic. We review the formulation and the main properties of the dynamics of the different models for populations evolving in discrete time and subjected to the effects of environmental and demographic stochasticity. Then we present different stochastic multiregional models with two time scales in which migration is fast with respect to demography and we review the main relationships between the dynamics of the original complex system and the aggregated simpler one. Finally, and within the context of models with environmental stochasticity in which the environmental variation is Markovian, we make use these techniques to analyze qualitatively the behavior of two multiregional models in which the original complex system is intractable. In particular we study conditions under which the population goes extinct or grows exponentially.

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“…Note that if the deterministic solution y 0 t of the deterministic reduced equationẏ 0 = g(x(y 0 , ε), y 0 ) leaves the interval of existence I in a time of order 1, then sample paths of the stochastic equation are likely to leave I in a comparable time, cf. (Berglund and Gentz, 2003, Theorem 2.6) and (Berglund and Gentz, 2006, Section 5.1.4).…”

Section: Stochastic Casementioning

confidence: 96%

“…Theorem 1.2.4. (Berglund and Gentz 2003) Assume the initial condition lies on the invariant curve, that is, x 0 =x(y 0 , ε) for some y 0 ∈ I. Then there exist constants h 0 , c, L > 0 such that for all h h 0 ,…”

Section: Stochastic Casementioning

confidence: 99%