P matrix properties, injectivity, and stability in chemical reaction systems

In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are P matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behaviour such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the P matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process, and characterise conditions which ensure the P matrix property survives the negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature

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