Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

Barendregt, Nicholas W.; Thomas, Peter J.

doi:10.1007/s00285-022-01859-4

Cited by 7 publications

(2 citation statements)

References 65 publications

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“…Examples include underdamped linear mass-spring system immersed in a heat bath [31], subthreshold oscillations in nerve cells sustained by channel noise [32], models of EEG oscillations and intermittent cortical network activity [33][34][35][36], and oscillations in predator-prey systems sustained by demographic (finite-population) noise [2]. Demographic fluctuations can also sustain oscillations in systems with rock-paper-scissors interactions by yet another mechanism: noisy heteroclinic cycle dynamics [37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

A Universal Description of Stochastic Oscillators

Pérez-Cervera¹,

Gutkin²,

Thomas³

et al. 2023

Preprint

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Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here we introduce a nonlinear transformation of stochastic oscillators to a new complex-valued function Q * 1 (x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function Q * 1 (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but non-vanishing) eigenvalue λ1 = µ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width µ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

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Section: Introductionmentioning

confidence: 99%

A Universal Description of Stochastic Oscillators

Pérez-Cervera¹,

Gutkin²,

Thomas³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Examples include underdamped linear mass-spring system immersed in a heat bath ( 31 ), subthreshold oscillations in nerve cells sustained by channel noise ( 32 ), models of EEG oscillations and intermittent cortical network activity ( 33 – 36 ), and oscillations in predator-prey systems sustained by demographic (finite-population) noise ( 2 ). Demographic fluctuations can also sustain oscillations in systems with rock–paper–scissors interactions by yet another mechanism: noisy heteroclinic cycle dynamics ( 37 – 40 ).…”

mentioning

confidence: 99%

A universal description of stochastic oscillators

Pérez-Cervera

Gutkin

Thomas

et al. 2023

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

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Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function Q 1 * ( x ) that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function Q 1 * ( x ) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ 1 = μ 1 + iω 1 . The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω 1 and half-width μ 1 ; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω 1 ; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

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Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Peixe,

Rodrigues

2023

J Nonlinear Sci

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Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

Cited by 7 publications

References 65 publications

A Universal Description of Stochastic Oscillators

A Universal Description of Stochastic Oscillators

A universal description of stochastic oscillators

Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

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