Contributions to the Theory of Mass A ction Kinetics III. Some Dynamic Properties of Second Order Mass Action Kinetics

Mass action kinetics whose elementary reactions are of at most second order provide the simplest models of nonlinear chemical systems. Despite their simplicity, however, only few7 investigations deal with second order mass action kinetics in a system­ atic and general setting. In [1] Aris showed the connection between these kinetics and certain nonassociative algebras. Jenks [2, 3] studied general multidimensional homogeneous differential equa­ tions w'hich also describe mass action kinetics. In [4] Horn developed a graph theoretical method for second order kinetics w hich allows to determine the stability properties of a kinetics by inspecting a graph. In [5] it could be demonstrated that open and closed mass action kinetics can be considered within a common m athematical framewrork facilitat­ ing the investigation of the dynamic behavior of nonlinear kinetic systems. The convenience of this mathematical setting will become clear in the following discussion of some qualitative dynamic properties of mass action kinetics. The aim of this note is to investigate the dynamic properties of mass action kinetics, viz. location, number, and stability pattern of their critical points. First, conditions for the occurrence of boundary critical points and their stability properties will be discussed. Then it will be demonstrated that a wide class of mass action kinetics, namely weakly reversible ones (defined belowr), exhibit only unstable boundary critical points. These mass action kinetics will be analyzed further by using well-known algebro-geometrical and topological theorems (Bezout theorem, Poincare-Hopf theorem).