Citation

Anderson DF, Schnoerr D, Yuan C. J. Math. Biol. 2020; ePub(ePub): ePub.

Affiliation

University of Wisconsin-Madison, Wisconsin, USA.

Copyright

(Copyright © 2020, Holtzbrinck Springer Nature Publishing Group)

DOI

10.1007/s00285-020-01485-y

PMID

32211950

Abstract

It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).

Language: en

Keywords

Complex balancing; Deficiency; Poisson distribution; Reaction networks; Stochastic processes