# Dynamical properties of Discrete Reaction Networks

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## Abstract

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of *Discrete Reaction Networks* (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: *irreducibility*, i.e., the ability to reach any discrete state from any other state; and *recurrence*, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.

## Mathematics Subject Classification

37N25 92C42 92D25 92C45 80A30## Notes

### Acknowledgments

We thank Andrei Caldararu for helpful comments and suggestions. The work of LP was supported by the Swiss SystemsX.ch project. The work of GC was supported by NIH grant R01GM086881. The work of HK was supported by the Swiss National Science Foundation, grant no. PP00P2 128503.

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