  This paper contributes an in-depth study of properties of continuous time
Markov chains (CTMCs) on non-negative integer lattices $\N_0^d$, with
particular interest in one-dimensional CTMCs with polynomial transitions rates.
Such stochastic processes are abundant in applications, in particular in
biology. We characterize the structure of the state space of general CTMCs on
$\N_0^d$ in terms of the set of jump vectors and their corresponding transition
rate functions. For CTMCs on $\N_0$ with polynomial transition rate functions,
we provide threshold criteria in terms of easily computable parameters for
various dynamical properties such as explosivity, recurrence, transience,
certain absorption, positive/null recurrence, implosivity, and existence and
non-existence of moments of hitting times. In particular, simple sufficient
conditions for exponential ergodicity of stationary distributions and
quasi-stationary distributions are obtained, and the few gap cases are
well-illustrated by examples. Subtle differences in conditions for different
dynamical properties are revealed in terms of examples. Finally, we apply our
results to stochastic reaction networks, an extended class of branching
processes, a general bursty single-cell stochastic gene expression model, and
population processes which are not birth-death processes.
