Bulletin of Mathematical Biology

, Volume 72, Issue 8, pp 1947–1970

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks


    • Department of MathematicsUniversity of Wisconsin
  • Gheorghe Craciun
    • Department of MathematicsUniversity of Wisconsin
  • Thomas G. Kurtz
    • Department of MathematicsUniversity of Wisconsin
Original Article

DOI: 10.1007/s11538-010-9517-4

Cite this article as:
Anderson, D.F., Craciun, G. & Kurtz, T.G. Bull. Math. Biol. (2010) 72: 1947. doi:10.1007/s11538-010-9517-4


We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.


Product-form stationary distributionsDeficiency zero

Copyright information

© Society for Mathematical Biology 2010