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A PROOF OF THE GLOBAL ATTRACTOR CONJECTURE IN THE SINGLE LINKAGE CLASS CASE
 

Summary: A PROOF OF THE GLOBAL ATTRACTOR CONJECTURE IN THE
SINGLE LINKAGE CLASS CASE
DAVID F. ANDERSON1
Abstract. 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. This will allow us to conclude that if a trajectory converges to the boundary,
then a whole family of Lyapunov functions decrease along the trajectory, a fact that will allow us to
overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity
as trajectories converge to the boundary, which has been the technical sticking point to a proof of
the Global Attractor Conjecture over the years.
Key words. persistence, global stability, population processes, chemical reaction systems, mass-
action kinetics, deficiency, complex-balancing, detailed-balancing
AMS subject classifications. 37C10, 80A30, 92C40, 92D25
1. Introduction. This paper is concerned with the qualitative behavior of de-

  

Source: Anderson, David F. - Department of Mathematics, University of Wisconsin at Madison

 

Collections: Mathematics