Title: Tier structure of strongly endotactic reaction networks,
Abstract: 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 used when the
counts are in high abundance. 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.
We provide an analytical characterization of strongly endotactic networks in
terms of tier structures. By doing so, we shed light on the connection between
the two points of view, and 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).