Comparison of the numerical solutions of the mass action equations ( ) and (7c) with the numerical solution of the asymptotic approximation (29a)-(29b) suggests that the asymptotically-derived MM equations can be in qualitative agreement with the mass action equations (see Figs. -2(b)) over a spectrum of relative speeds. However, conditions for the validity of (29b) on timescales other than t s2 have not yet been established. This will be the central subject of the subsequent sections.