Cubic autocatalytic reaction-diffusion equations: semi-analytical solutions

The Gray-Scott model of cubic-autocatalysis with linear decay is coupled with diffusion and considered in a one-dimensional reactor (a reaction-diffusion cell). The boundaries of the reactor are permeable, so diffusion occurs from external reservoirs that contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations. The ordinary differential equations are then analysed to obtain semi-analytical results for the reaction-diffusion cell. Steady-state concentration profiles and bifurcation diagrams are obtained both explicitly, for the one-term method, and as the solution to a pair of transcendental equations, for the two-term method. Singularity theory is used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. Also, in the semi-analytical model, a fifth bifurcation diagram occurs in an extremely small parameter region; its size being O(10(-13)). 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 in detail. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.