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Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model

A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.