We perform a rate equation analysis of the dynamics of the exciton Mott transition (EMT) assuming a detailed balance between excitons and unbound electron-hole (e-h) pairs. Using the Saha equation and adopting an empirical expression for the band-gap renormalization effect caused by unbound e-h pairs, we show that the ionization ratio of excitons exhibits bistability as a function of the total e-h pair density at low temperatures. We demonstrate that an incubation time emerges in the dynamics of the EMT from the oversaturated exciton gas phase on the verge of the bistable region. The incubation time shows slowing down behavior when the pair density approaches saddle-node bifurcation of the hysteresis curve of the exciton ionization ratio.