Stability of Surface Mechanisms with Three Species and Mass-Action Kinetics

The linear stability problem for surface mechanisms with free sites and two adsorbed species is investigated under the assumptions of mass action (Langmuir) surface kinetics, fast mass transport to and from the surface, and a conservation condition. The results also apply to enzyme kinetics for systems with a single enzyme occurring in the free form and two combined forms, and with fast mass transport of the substrates and products. Mechanisms are classified according to their stability and the presence or absence of complex eigenvalues, and specific reactions with numerical values of the rate constants and surface concentrations are given to illustrate the results. Some mechanisms, e.g., proportionate reactions, are shown to be stable for all values of the rate constants and stoichiometric coefficients. The two most common types of mechanisms, namely sequential mechanisms and the simple Langmuir–Hinshelwood mechanism (one adsorbate per site), are always stable. The possibility of complex eigenvalues arises for sequential mechanisms (providing a counterexample to a condition for real eigenvalues given previously in the literature). More general Langmuir–Hinshelwood mechanisms can be unstable (e.g., those in which one adsorbate occupies two sites). Some results are generalized to mechanisms with three or more adsorbed species, and global stability is investigated using monotone dynamical systems theory.