Paper Info
Title
The Dynamics of Weakly Reversible Population Processes near Facets
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.
Year
DOI
Venue
2010
10.1137/090764098
SIAM JOURNAL ON APPLIED MATHEMATICS
Keywords
Field
DocType
persistence,global stability,dynamical systems,population processes,chemical reaction systems,mass action kinetics,deficiency,complex-balancing,detailed-balancing,polyhedron
Attractor,Population,Combinatorics,Population process,Mathematical analysis,Dynamical systems theory,Invariant (mathematics),Invariant manifold,Strongly connected component,Manifold,Mathematics
Journal
Volume
Issue
ISSN
70
6
0036-1399
Citations 
PageRank 
References 
12
1.71
4
Authors
2
Name
Order
Citations
PageRank
David F. Anderson17412.25
Anne Shiu28714.47