Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks
- 387 Downloads
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.
KeywordsComplex 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.
- Feinberg M (1979) Lectures on chemical reaction networks. Delivered at the Mathematics Research Center, Univ. Wisc.-Madison. http://crnt.engineering.osu.edu/LecturesOnReactionNetworks
- Gardiner CW (1985) Handbook of stochastic methods, 2nd edn. Spinger, BerlinGoogle Scholar
- Gradshteyn IS, Ryzhik IM (2007) Tables of integrals, series, and products, 7th edn. Academic Press, LondonGoogle Scholar
- Gunawardena J (2003) Chemical reaction network theory for in-silico biologists. http://vcp.med.harvard.edu/papers/crnt.pdf
- Gupta A, Khammash M (2013) Determining the long-term behavior of cell populations: a new procedure for detecting ergodicity in large stochastic reaction networks. arXiv:1312.2879
- Kurtz TG (1977/78) Strong approximation theorems for density dependent Markov chains. Stoch Proc Appl 6:223–240Google Scholar
- Kurtz TG (1981) Approximation of population processes. In: CBMS-NSF regular conference series in applied mathematics, vol. 36. SIAMGoogle Scholar
- Perko L (2000) Differential equations and dynamical systems, 3rd edn. Springer, BerlinGoogle Scholar