Nonnegativity, reducibility, and semistability of mass action kinetics

Mass action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. We provide a general construction of the kinetic equation from reaction laws based upon the formulation given by Erdi et al. (1988). We consider the nonnegativity of the solutions to the kinetic equation, the reducibility of the mass action kinetics, and the stability of the equilibria of the kinetic equation. To do this, we apply Lyapunov methods to the kinetic equation and obtain results that guarantee semistability, that is, convergence to an equilibrium that depends upon initial concentrations. This notion was previously developed by the authors (1995, 1999), which extends the linear semistability theory to nonlinear systems. Finally, we revisit the “zero deficiency” result given by Feinberg (1995), which provides rate-independent conditions guaranteeing stability