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Reaction networks are mathematical models of interacting chemical speciesthat are primarily used in biochemistry. There are two modeling regimes thatare typically used, one of which is deterministic and one that is stochastic.In particular, the deterministic model consists of an autonomous system ofdifferential equations, whereas the stochastic system is a continuous timeMarkov chain. Connections between the two modeling regimes have been studiedsince the seminal paper by Kurtz (1972), where the deterministic model is shownto be a limit of a properly rescaled stochastic model over compact timeintervals. Further, more recent studies have connected the long-term behaviorsof the two models when the reaction network satisfies certain graphicalproperties, such as weak reversibility and a deficiency of zero.  These connections have led some to conjecture a link between the long-termbehavior of the two models exists, in some sense. In particular, one is temptedto believe that positive recurrence of all states for the stochastic modelimplies the existence of positive equilibria in the deterministic setting, andthat boundary equilibria of the deterministic model imply the occurrence of anextinction event in the stochastic setting. We prove in this paper that theseimplications do not hold in general, even if restricting the analysis tonetworks that are bimolecular and that conserve the total mass. In particular,we disprove the implications in the special case of models that have absoluteconcentration robustness, thus answering in the negative a conjecture stated inthe literature in 2014.

Discrepancies between extinction events and boundary equilibria in reaction networks