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SIAM J. APPL. MATH. c 2011 Society for Industrial and Applied Mathematics Vol. 71, No. 4, pp. 14871508
 

Summary: SIAM J. APPL. MATH. c 2011 Society for Industrial and Applied Mathematics
Vol. 71, No. 4, pp. 14871508
A PROOF OF THE GLOBAL ATTRACTOR CONJECTURE IN THE
SINGLE LINKAGE CLASS CASE
DAVID F. ANDERSON
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. 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.
Key words. persistence, global stability, population processes, chemical reaction systems, mass-
action kinetics, deficiency, complex-balancing, detailed-balancing

  

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

 

Collections: Mathematics