Stability and Uniqueness of the Equilibrium Point in Chemical Reaction Systems

Abstract

It has recently been established that equilibria are stable in the large for coupled chemical reaction systems of arbitrary complexity and arbitrary mixtures of orders, so long as the rates of change of the amounts of the species present are derivable from mass-action rate equations. The proof, employing Liapunov's direct method, is entirely kinetic. It is now shown that there can never be a continuum of equilibrium points accessible to a reaction system. An argument based on stability in the large is used to show that if there were a second equilibrium point accessible to a reaction system, there would have to be a continuum of equilibrium points connecting the two. Consequently, there cannot be more than one equilibrium point for any given reaction system. A proof that there must be at least one equilibrium point has also been given, based on Brouwer's fixed-point theorem. Thus, each reaction system has one, and only one, equilibrium point.