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.
Davidson, F. A., & Rynne, B. P. (1996). Local and global behaviour of steady-state solutions of the Sel'kov model. IMA Journal of Applied Mathematics, 56(2), 145-155. https://doi.org/10.1093/imamat/56.2.145