Tier structure of strongly endotactic reaction networks

Reaction networks are mainly used to model the time-evolution of molecules of interacting chemical species. Stochastic models are typically used when the counts of the molecules are low, whereas deterministic models are often used when the counts are in high abundance. The mathematical study of reaction networks has increased dramatically over the last two decades as these models are now routinely used to investigate cellular behavior. In 2011, the notion of “tiers” was introduced to study the long time behavior of deterministically modeled reaction networks that are weakly reversible and have a single linkage class. This “tier” based argument was analytical in nature. Later, in 2014, the notion of a strongly endotactic network was introduced in order to generalize the previous results from weakly reversible networks with a single linkage class to this wider family of networks. The point of view of this later work was more geometric and algebraic in nature. The notion of strongly endotactic networks was later used in 2018 to prove a large deviation principle for a class of stochastically modeled reaction networks. In the current paper we provide an analytical characterization of strongly endotactic networks in terms of tier structures. By doing so, we not only shed light on the connection between the two points of view, but also make available a new proof technique for the study of strongly endotactic networks. We show the power of this new technique in two distinct ways. First, we demonstrate how the main previous results related to strongly endotactic networks, both for the deterministic and stochastic modeling choices, can be quickly obtained from our characterization. Second, we demonstrate how new results can be obtained by proving that a sub-class of strongly endotactic networks, when modeled stochastically, is positive recurrent. Finally, and similarly to recent independent work by Agazzi and Mattingly, we provide an example which closes a conjecture in the negative by showing that stochastically modeled strongly endotactic networks can be transient (and even explosive).