  Stochastic reaction networks (SRNs) provide models of many real-world
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 pay special attention to
one-dimensional mass-action SRNs (1-d stoichiometric subspace). We classify all
states of the underlying CTMC of 1-d SRNs. 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 the state space of the
associated CTMCs consists of closed communicating classes. 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.
