Cubic autocatalysis with Michaelis–Menten kinetics: semi-analytical solutions for the reaction–diffusion cell

Abstract

Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of transcendental equations. Singularity theory is then used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limit-cycle is also considered. Comparison with numerical solutions of the governing partial differential equations shows that the semi-analytical solutions are very accurate.