  This paper is concerned with the dynamical properties of deterministically
modeled chemical reaction systems. Specifically, this paper provides a proof of
the Global Attractor Conjecture in the setting where the underlying reaction
diagram consists of a single linkage class, or connected component. The
conjecture dates back to the early 1970s and is the most well known and
important open problem in the field of chemical reaction network theory. The
resolution of the conjecture has important biological and mathematical
implications in both the deterministic and stochastic settings. One of our main
analytical tools, which is introduced here, will be a method for partitioning
the relevant monomials of the dynamical system along sequences of trajectory
points into classes with comparable growths. We will use this method to
conclude that if a trajectory converges to the boundary, then a whole family of
Lyapunov functions decrease along the trajectory. This will allow us to
overcome the fact that the usual Lyapunov functions of chemical reaction
network theory are bounded on the boundary of the positive orthant, which has
been the technical sticking point to a proof of the Global Attractor Conjecture
in the past.
