Original Article

Bulletin of Mathematical Biology

, Volume 72, Issue 8, pp 1947-1970

First online:

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

  • David F. AndersonAffiliated withDepartment of Mathematics, University of Wisconsin Email author 
  • , Gheorghe CraciunAffiliated withDepartment of Mathematics, University of Wisconsin
  • , Thomas G. KurtzAffiliated withDepartment of Mathematics, University of Wisconsin

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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 distributions Deficiency zero