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We study the linear instabilities and bifurcations in the Selkov model forglycolysis with diffusion. We show that this model has a zero wave-vector,finite frequency Hopf bifurcation to a growing oscillatory but spatiallyhomogeneous state and a saddle-node bifurcation to a growing inhomogeneousstate with a steady pattern with a finite wavevector. We further demonstratethat by tuning the relative diffusivity of the two concentrations, it ispossible to make both the instabilities to occur at the same point in theparameter space, leading to an unusual type of codimension-two bifurcation. Wethen show that in the vicinity of this bifurcation the initial conditionsdecide whether a spatially uniform oscillatory or a spatially periodic steadypattern emerges in the long time limit.

When Hopf meets saddle: bifurcations in the diffusive Selkov model for glycolysis