Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks

David F Anderson; Gheorghe Craciun; Manoj Gopalkrishnan; Carsten Wiuf

doi:10.1007/s11538-015-0102-8

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks

David F Anderson, Gheorghe Craciun, Manoj Gopalkrishnan, Carsten Wiuf

Institut for Matematiske Fag

26 Citationer (Scopus)

Abstract

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.

Originalsprog	Engelsk
Tidsskrift	Bulletin of Mathematical Biology
Vol/bind	77
Udgave nummer	9
Sider (fra-til)	1744-1767
Antal sider	24
ISSN	0092-8240
DOI	https://doi.org/10.1007/s11538-015-0102-8
Status	Udgivet - 1 sep. 2015

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10.1007/s11538-015-0102-8

Citationsformater

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AU - Craciun, Gheorghe

AU - Gopalkrishnan, Manoj

AU - Wiuf, Carsten

PY - 2015/9/1

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N2 - We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.

AB - We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.

KW - Kinetics

KW - Markov Chains

KW - Mathematical Concepts

KW - Metabolic Networks and Pathways

KW - Models, Biological

KW - Population Dynamics

KW - Stochastic Processes

KW - Journal Article

KW - Research Support, N.I.H., Extramural

KW - Research Support, Non-U.S. Gov't

KW - Research Support, U.S. Gov't, Non-P.H.S.

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DO - 10.1007/s11538-015-0102-8

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SP - 1744

EP - 1767

JO - Bulletin of Mathematical Biology

JF - Bulletin of Mathematical Biology

IS - 9

ER -

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks

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