  Stochastic reaction networks, which are usually modeled as continuous-time
Markov chains on $\mathbb Z^d_{\ge 0}$, and simulated via a version of the
"Gillespie algorithm," have proven to be a useful tool for the understanding of
processes, chemical and otherwise, in homogeneous environments. There are
multiple avenues for generalizing away from the assumption that the environment
is homogeneous, with the proper modeling choice dependent upon the context of
the problem being considered. One such generalization was recently introduced
in (Duso and Zechner, PNAS, 2020), where the proposed model includes a varying
number of interacting compartments, or cells, each of which contains an
evolving copy of the stochastic reaction system. The novelty of the model is
that these compartments also interact via the merging of two compartments
(including their contents), the splitting of one compartment into two, and the
appearance and destruction of compartments. In this paper we begin a systematic
exploration of the mathematical properties of this model. We (i) obtain
basic/foundational results pertaining to explosivity, transience, recurrence,
and positive recurrence of the model, (ii) explore a number of examples
demonstrating some possible non-intuitive behaviors of the model, and (iii)
identify the limiting distribution of the model in a special case that
generalizes three formulas from an example in (Duso and Zechner, PNAS, 2020).
