Fig. The validity in the transition region, where t ≈ ts 1 , depends on how well the timescale s 1 serves as a depletion timescale for the s 1 mass action equation (dashed and dotted black curve). The numerical solution of inner solution (38a) is the dashed red curve, and provides a good approximation to the numerical solution of the mass action equation (7a)-(7d) for s 2 (solid black curve) when t < ts 1 and the outer solution (38b) (dashed blue curve) when δ S 1. The asymptotic solution matches better in panel (b) since ts 1 yields a more precise approximation to the depletion timescale of the non-observable reaction. Time has been mapped to the t∞ scale: t∞(t) = 1 -1/ ln(t + e).