Graphically balanced equilibria and stationary measures of reaction networks

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models of dynamics of the reaction network. Correspondingly, in the stochastic setting, when modeled as a continuous time Markov chain, these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun, and Kurtz [Bull Math. Biol., 72 (2010), pp. 1947-1970] identified stationary distributions of a complex balanced network; later, Cappelletti and Wiuf [SIAM J. Appl. Math., 76 (2016), pp. 411-432] developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure, reaction vector balanced measure, and cycle balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. Furthermore, in the spirit of an earlier work by Joshi [Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), pp. 1077-1105] we give sufficient conditions under which detailed balance of the stationary distribution of Markov chain models implies the existence of positive detailed balance equilibria for the related deterministic reaction network model. Finally, we provide a complete map of the implications between balancing properties of deterministic and corresponding stochastic reaction systems, such as complex balance, reaction balance, reaction vector balance, and cycle balance.