Bulletin of Mathematical Biology

, Volume 77, Issue 9, pp 1744–1767 | Cite as

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

  • David F. AndersonEmail author
  • Gheorghe Craciun
  • Manoj Gopalkrishnan
  • Carsten Wiuf
Original Article


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.


Complex balanced Dynamical system Birth-death process Continuous time Markov chain Long-term dynamics 



We thank the American Institute of Mathematics for hosting a workshop at which this research was initiated. Anderson was supported by NSF Grants DMS-1009275 and DMS-1318832 and Army Research Office Grant W911NF-14-1-0401. Craciun was supported by NSF Grant DMS1412643 and NIH Grant R01GM086881. Wiuf was supported by the Lundbeck Foundation (Denmark), the Carlsberg Foundation (Denmark), Collstrups Fond (Denmark), and the Danish Research Council. Part of this work was carried out while Wiuf visited the Isaac Newton Institute in 2014.

Supplementary material

11538_2015_102_MOESM1_ESM.pdf (178 kb)
Supplementary material 1 (pdf 177 KB)


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Copyright information

© Society for Mathematical Biology 2015

Authors and Affiliations

  • David F. Anderson
    • 1
    Email author
  • Gheorghe Craciun
    • 1
    • 2
  • Manoj Gopalkrishnan
    • 3
  • Carsten Wiuf
    • 4
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of Biomolecular ChemistryUniversity of Wisconsin-MadisonMadisonUSA
  3. 3.School of Technology and Computer ScienceTata Institute of Fundamental ResearchMumbaiIndia
  4. 4.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

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