  Reaction networks are mathematical models of interacting chemical species
that are primarily used in biochemistry. There are two modeling regimes that
are typically used, one of which is deterministic and one that is stochastic.
In particular, the deterministic model consists of an autonomous system of
differential equations, whereas the stochastic system is a continuous time
Markov chain. Connections between the two modeling regimes have been studied
since the seminal paper by Kurtz (1972), where the deterministic model is shown
to be a limit of a properly rescaled stochastic model over compact time
intervals. Further, more recent studies have connected the long-term behaviors
of the two models when the reaction network satisfies certain graphical
properties, such as weak reversibility and a deficiency of zero.
  These connections have led some to conjecture a link between the long-term
behavior of the two models exists, in some sense. In particular, one is tempted
to believe that positive recurrence of all states for the stochastic model
implies the existence of positive equilibria in the deterministic setting, and
that boundary equilibria of the deterministic model imply the occurrence of an
extinction event in the stochastic setting. We prove in this paper that these
implications do not hold in general, even if restricting the analysis to
networks that are bimolecular and that conserve the total mass. In particular,
we disprove the implications in the special case of models that have absolute
concentration robustness, thus answering in the negative a conjecture stated in
the literature in 2014.
