Classification and threshold dynamics of stochastic reaction networks

Abstract

Stochastic reaction networks (SRNs) provide models of many realworld networks. Examples include networks in epidemiology, pharmacology, genetics, ecology, chemistry, and social sciences. Here, we model stochastic reaction networks by continuous time Markov chains (CTMCs) and derive new results on the decomposition of the ambient space N d 0 (with d ≥ 1 the number of species) into communicating classes. In particular, we propose to study (minimal) core networks of an SRN, and show that these characterize the decomposition of the ambient space.Special attention is given to one-dimensional mass-action SRNs (1-d stoichiometric subspace). In terms of (up to) four parameters, we provide sharp checkable criteria for various dynamical properties (including explosivity, recurrence, ergodicity, and the tail asymptotics of stationary or quasi-stationary distributions) of SRNs in the sense of their underlying CTMCs. As a result, we prove that all 1-d endotactic networks are non-explosive, and positive recurrent with an ergodic stationary distribution with Conley-Maxwell-Poisson (CMP)-like tail, provided they are essential. In particular, we prove the recently proposed positive recurrence conjecture in one dimension: Weakly reversible mass-action SRNs with 1-d stoichiometric subspaces are positive recurrent. The proofs of the main results rely on our recent work on CTMCs with polynomial transition rate functions.

On the sum of chemical reactions

“…Hence, having the same cl(R) for two RNs is in general stronger than having the same reachability sets for all initial states (in the latter case, the RNs are said to be structurally identical [45]).…”

Absolute concentration robustness in networks with low-dimensional stoichiometric subspace

“…Remark 2.5. One-dimensional networks have been analyzed recently in terms of multistationarity and multistability [26] and in the stochastic setting [28,29].…”