Local and Global Stability of Equilibria for a Class of Chemical Reaction Networks

A class of chemical reaction networks is described with the property that each positive equilibrium is locally asymptotically stable relative to its stoichiometry class, an invariant subspace on which it lies. The reaction systems treated are characterized primarily by the existence of a certain factorization of their stoichiometric matrix and strong connectedness of an associated graph. Only very mild assumptions are made about the rates of reactions, and, in particular, mass action kinetics are not assumed. In many cases, local asymptotic stability can be extended to global asymptotic stability of each positive equilibrium relative to its stoichiometry class. The results are proved via the construction of Lyapunov functions whose existence follows from the fact that the reaction networks define monotone dynamical systems with increasing integrals.

  • 1.  P. Érdi and  J. Tóth , Mathematical Models of Chemical Reactions , Nonlinear Sci. Theory Appl. , Princeton University Press , Princeton, NJ , 1989 . Google Scholar

  • 2.  M. Feinberg , Complex balancing in general kinetic systems , Arch. Ration. Mech. Anal. , 49 ( 1972 ), pp. 187 -- 194 . CrossrefISIGoogle Scholar

  • 3.  F. Horn and  R. Jackson , General mass action kinetics , Arch. Ration. Mech. Anal. , 47 ( 1972 ), pp. 81 -- 116 . CrossrefISIGoogle Scholar

  • 4.  M. W. Hirsch and  H. Smith , Monotone dynamical systems , in Handbook of Differential Equations: Ordinary Differential Equations , Vol. II , Elsevier B. V. , Amsterdam , 2005 , pp. 239 -- 357 . Google Scholar

  • 5.  H. Smith and  Monotone Dynamical Systems : An Introduction to the Theory of Competitive and Cooperative Systems , AMS , Providence, RI , 1995 . Google Scholar

  • 6.  A. I. Volpert V. A. Volpert and  V. A. Volpert , Travelling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. 140 , AMS , Providence, RI , 1994 . Google Scholar

  • 7.  H. Kunze and  D. Siegel , Monotonicity properties of chemical reactions with a single initial bimolecular step , J. Math. Chem. , 31 ( 2002 ), pp. 339 -- 344 . CrossrefISIGoogle Scholar

  • 8.  M. Mincheva and  D. Siegel , Comparison principles for reaction-diffusion systems with respect to proper polyhedral cones , Dynam. Contin. Discrete Impuls. Systems , 10 ( 2003 ), pp. 477 -- 490 . ISIGoogle Scholar

  • 9.  P. De Leenheer D. Angeli and  E. D. Sontag , Monotone chemical reaction networks , J. Math. Chem. , 41 ( 2007 ), pp. 295 -- 314 . CrossrefISIGoogle Scholar

  • 10.  M. Banaji , Monotonicity in chemical reaction systems , Dyn. Syst. , 24 ( 2009 ), pp. 1 -- 30 . CrossrefISIGoogle Scholar

  • 11.  D. Angeli P. De Leenheer and  E. D. Sontag , Graph-theoretic characterizations of monotonicity of chemical reaction networks in reaction coordinates , J. Math. Biol. , 61 ( 2010 ), pp. 581 -- 616 . CrossrefISIGoogle Scholar

  • 12.  D. Angeli and  E. D. Sontag , Translation invariant monotone systems and a global convergence result for enzymatic futile cycles , Nonlinear Anal. , 9 ( 2008 ), pp. 128 -- 140 . CrossrefISIGoogle Scholar

  • 13.  M. Banaji and  J. Mierczyński , Global convergence in systems of differential equations arising from chemical reaction networks , J. Differential Equations , 254 ( 2013 ), pp. 1359 -- 1374 . CrossrefISIGoogle Scholar

  • 14.  J. Mierczyński , Strictly cooperative systems with a first integral , SIAM J. Math. Anal. , 18 ( 1987 ), pp. 642 -- 646 . LinkISIGoogle Scholar

  • 15.  M. Banaji and  D. Angeli , Convergence in strongly monotone systems with an increasing first integral , SIAM J. Math. Anal. , 42 ( 2010 ), pp. 334 -- 353 . LinkISIGoogle Scholar

  • 16.  M. Banaji D. Angeli and  Addendum “Convergence in strongly monotone systems with an increasing first integral ,'' SIAM J. Math. Anal. , 44 ( 2012 ), pp. 536 -- 537 . LinkISIGoogle Scholar

  • 17.  M. Banaji and  G. Craciun , Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements , Commun. Math. Sci. , 7 ( 2009 ), pp. 867 -- 900 . CrossrefISIGoogle Scholar

  • 18.  M. Feinberg , Chemical reaction network structure and the stability of complex isothermal reactors I. The deficiency zero and deficiency one theorems , Chem. Eng. Sci. , 42 ( 1987 ), pp. 2229 -- 2268 . CrossrefISIGoogle Scholar

  • 19.  G. Craciun F. Nazarov and  C. Pantea , Persistence and permanence of mass-action and power-law dynamical systems , SIAM J. Appl. Math. , 73 ( 2013 ), pp. 305 -- 329 . LinkISIGoogle Scholar

  • 20.  C. Pantea , On the persistence and global stability of mass-action systems , SIAM J. Math. Anal. , 44 ( 2012 ), pp. 1636 -- 1673 . LinkISIGoogle Scholar

  • 21.  D. A. Fell , Understanding the Control of Metabolism , Portland Press , London , 1997 . Google Scholar

  • 22.  A. Berman and  R. Plemmons , Nonnegative Matrices in the Mathematical Sciences , Academic Press , New York , 1979 . Google Scholar

  • 23.  M. Banaji and  A. Burbanks , A graph-theoretic condition for irreducibility of a set of cone preserving matrices , Linear Algebra Appl. , 438 ( 2013 ), pp. 4103 -- 4113 . CrossrefISIGoogle Scholar

  • 24.  E. H. Spanier, Algebraic Topology, Springer, New York, 1981.Google Scholar

  • 25.  M. Banaji P. Donnell and  S. Baigent , \em$P$ matrix properties, injectivity, and stability in chemical reaction systems , SIAM J. Appl. Math. , 67 ( 2007 ), pp. 1523 -- 1547 . LinkISIGoogle Scholar

  • 26.  D. Angeli P. De Leenheer and  E. D. Sontag , A Petri net approach to the study of persistence in chemical reaction networks , Math. Biosci. , 210 ( 2007 ), pp. 598 -- 618 . CrossrefISIGoogle Scholar

  • 27.  A. Shiu and  B. Sturmfels , Siphons in chemical reaction networks , Bull. Math. Biol. , 72 ( 2010 ), pp. 1448 -- 1463 . CrossrefISIGoogle Scholar