On the Temporal Characteristics of a Model for the Zhabotinskii-Belousov Reaction H. G. OTHMER Department of Mathematics, Rutgers Unioersily, New Brunswick, New Jersey Communicated by R. Aris ABSTRACT The number and stability of stationary and periodic solutions in a semi- phenomenological reaction mechanism that qualitatively models the Zhabotinskii- Belousov reaction are studied as functions of parameters in the model. Both multiple stationary solutions and multiple periodic solutions can exist simultaneously. Periodic solutions bifurcate in one of three ways: at zero amplitude as predicted from the Hopf theorem; pairwise at finite amplitude, the so-called “hard” bifurcation; or by coalescense with separatrix loops. Detailed computations for several cases reveal the dependence of the period and amplitude of the periodic solutions on the parameters. The results show that the model exhibits all the qualitative features that might be expected in intracellular reactions and so can serve as a model system for theoretical studies of pattern formation in developing systems. 1. INTRODUCTION There have been numerous theoretical studies of chemically reacting systems, both with and without transport, in the sixty-odd years since Lotka postulated a model that predicts sustained temporal oscillations [ 19, 35, 38, 41, 44, 471. The qualitative dynamics of closed systems are well understood in that the number of equilibrium points possible and the convergence to these equilibrium points are known for various classes of systems. For instance, for both adiabatic and isothermal homogeneous, ideal, closed systems there is a unique equilibrium point (31, and convergence to the equilibrium point is ultimately monotonic [23, 32, 481. Open systems, by contrast, show a wider variety of behavior and few general results are MATHEMATICAL BIOSCIENCES 24, 205-238 (1975) 0 American Elsevier Publishing Company, Inc., 1975 205