Bulletin of Mathematical Biology

, Volume 77, Issue 9, pp 1744–1767

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

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

DOI: 10.1007/s11538-015-0102-8

Cite this article as:
Anderson, D.F., Craciun, G., Gopalkrishnan, M. et al. Bull Math Biol (2015) 77: 1744. doi:10.1007/s11538-015-0102-8


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 

Supplementary material

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

Funding information

Funder NameGrant NumberFunding Note
National Science Foundation
  • DMS-1009275
National Science Foundation
  • DMS-1318832
Army Research Office
  • W911NF-14-1-0401
National Science Foundation
  • DMS-1412643
National Institutes of Health
  • R01GM086881
      Collstrups Fond
        Danish Research Council

          Copyright information

          © Society for Mathematical Biology 2015

          Authors and Affiliations

          • David F. Anderson
            • 1
          • 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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