In this paper we discuss steady-state solutions of the system of reaction-diffusion equations known as the Sel'kov model. This model has been the subject of much discussion; in particular, analytical and numerical results have been discussed by Lopez-Gomez et al (1992, IMA J. Num. Anal. 12, 405-28). We show that a simple analysis of the bifurcation function associated with the system can explain many of the numerical observations, such as the formation and development of loops of nontrivial solutions, in a simpler and more complete manner than the analysis of Lopez-Gomez et al. This allows for a clearer understanding of the qualitative behaviour of the set of nontrivial solutions and hence of the bifurcation diagram.