In this paper we study the steady-state solutions of a reaction-diffusion model, the Selkov scheme for glycolysis, under homogeneous Dirichlet boundary conditions. Near to thermodynamic equilibrium, the structure and stability of solutions are fully described. A bifurcation analysis is carried out, using the size of the region in which the reaction takes place and one diffusion coefficient as main bifurcation parameters. The analysis helps us to understand the nature of the bifurcation points, and determines the shapes and stability of the bifurcating manifolds in the neighbourhood of the constant state. Local convergence of spectral methods is shown, and some global pictures are calculated using path-following techniques. The framework we use can be applied to a wide variety of reaction-diffusion systems. © 1992 Oxford University Press.