The brusselator model of oscillatory reactions. Relationships between two-variable and four-variable models with rigorous application of mass conservation and detailed balance

Abstract

The classical Brusselator [graphic omitted] is known to display oscillatory behaviour in the species X and Y when reverse reactions are neglected and the concentrations of A and B are kept constant. Its validity as a model for any imaginable oscillatory chemical system has been queried on the grounds that all real chemical reactions must be to some extent reversible, and it has been alleged that the reversible Brusselator oscillates only if the two principles of detailed balance and conservation of matter are violated. We have studied mathematically the reversible Brusselator in circumstances where all the requirements of detailed balance are satisfied, and the consumption of A and B is not ignored; the system modelled is closed and matter is strictly conserved. Oscillatory behaviour is possible at least for some combinations of reaction parameters (rate constants, initial concentrations etc.) and may be modelled self-consistently. When they occur, oscillations are finite in number and they last a finite time. After they stop they are always followed by a monotonic decay to equilibrium. Oscillatory behaviour in the irreversible system is obtainable as the limiting case of the reversible system. The irreversible Brusselator is not the simplest scheme of its kind to show any of the above behaviour and it is not without other difficulties. It shows infinities in intermediate concentrations and in oscillatory amplitudes and it loses oscillations in a simple open system (c.s.t.r.). Nevertheless, it deserves its place with the Lotka–Volterra scheme as part of the history of the subject.