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SIAM J. APPL. MATH. c 2010 Society for Industrial and Applied Mathematics Vol. 70, No. 6, pp. 18401858
 

Summary: SIAM J. APPL. MATH. c 2010 Society for Industrial and Applied Mathematics
Vol. 70, No. 6, pp. 18401858
THE DYNAMICS OF WEAKLY REVERSIBLE POPULATION
PROCESSES NEAR FACETS
DAVID F. ANDERSON AND ANNE SHIU
Abstract. This paper concerns the dynamical behavior of weakly reversible, deterministically
modeled population processes near the facets (codimension-one faces) of their invariant manifolds and
proves that the facets of such systems are "repelling." It has been conjectured that any population
process whose network graph is weakly reversible (has strongly connected components) is persistent.
We prove this conjecture to be true for the subclass of weakly reversible systems for which only facets
of the invariant manifold are associated with semilocking sets, or siphons. An important application
of this work pertains to chemical reaction systems that are complex-balancing. For these systems it
is known that within the interior of each invariant manifold there is a unique equilibrium. The global
attractor conjecture states that each of these equilibria is globally asymptotically stable relative to
the interior of the invariant manifold in which it lies. Our results pertaining to weakly reversible
systems imply that this conjecture holds for all complex-balancing systems whose boundary equilibria
lie in the relative interior of the boundary facets. As a corollary, we show that the global attractor
conjecture holds for those systems for which the associated invariant manifolds are two-dimensional.
Key words. persistence, global stability, dynamical systems, population processes, chemical
reaction systems, mass action kinetics, deficiency, complex-balancing, detailed-balancing, polyhedron

  

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

 

Collections: Mathematics