The Dynamics of Weakly Reversible Population Processes near Facets

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 thi...

[1]  Eduardo D. Sontag.  Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction , 2001, IEEE Trans. Autom. Control..

[2]  T. McKeithan,et al.  Kinetic proofreading in T-cell receptor signal transduction. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Martin Feinberg,et al.  Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity , 1989 .

[4]  Eduardo Sontag,et al.  A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks , 2007 .

[5]  G. Ziegler.  Lectures on Polytopes , 1994 .

[6]  F. Horn.  Necessary and sufficient conditions for complex balancing in chemical kinetics , 1972 .

[7]  Gilles Gnacadja.  Univalent positive polynomial maps and the equilibrium state of chemical networks of reversible binding reactions , 2009, Adv. Appl. Math..

[8]  M. Feinberg.  Complex balancing in general kinetic systems , 1972 .

[9]  J. Bauer,et al.  Chemical reaction network theory for in-silico biologists , 2003 .

[10]  L. Pachter,et al.  Algebraic Statistics for Computational Biology: Preface , 2005 .

[11]  D. Siegel,et al.  Global stability of complex balanced mechanisms , 2000 .

[12]  Gilles Gnacadja,et al.  Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor. , 2007, Journal of theoretical biology.

[13]  Bernd Sturmfels,et al.  Siphons in Chemical Reaction Networks , 2009, Bulletin of mathematical biology.

[14]  A. I. Vol'pert,et al.  Analysis in classes of discontinuous functions and equations of mathematical physics , 1985 .

[15]  T. Kurtz.  The Relationship between Stochastic and Deterministic Models for Chemical Reactions , 1972 .

[16]  M. Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems , 1987 .

[17]  Eduardo Sontag,et al.  A Petri net approach to the study of persistence in chemical reaction networks. , 2006, Mathematical biosciences.

[18]  T. Kurtz.  Approximation of Population Processes , 1987 .

[19]  David F. Anderson,et al.  Global Asymptotic Stability for a Class of Nonlinear Chemical Equations , 2007, SIAM J. Appl. Math..

[20]  R. Jackson,et al.  General mass action kinetics , 1972 .

[21]  J. Hedrick,et al.  Observer design for a class of nonlinear systems , 1994 .

[22]  Alicia Dickenstein,et al.  Toric dynamical systems , 2007, J. Symb. Comput..