Figure 10: The timescale t_s2 is characteristic of the time it takes s₂ to reach $s_2^{\lambda }$, and the timescale $t_{s_2}^{\ast }$ is the approximate time it takes s₂ to reach $s_2^{\lambda }$, respectively, in the auxiliary reaction mechanism (47)–(48).

The thick black curve is the numerically-integrated solution to the mass action equations (49a)–(49d). The leftmost dashed vertical line corresponds to t_s2, and the rightmost dashed vertical line corresponds to $t_{s_2}^{\ast }=-t_{s_2}\text{ln}{t}_{s_2}∕t_{c_2}$. The lower dotted horizontal line corresponds to the scaled characteristic value $\ell s_2^{\lambda }$, and the upper dashed/dotted vertical line corresponds to $s_2^{\lambda }$. The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=1000$, $e_2^0=100$, k₁ = 1, k₂ = 1, k₃ = 1, k₋₃ = 1, k₄ = 2 and k₋₁ = 1. Time has been mapped to the t_∞ scale: t_∞(t) = 1 – 1/ln[t + exp(1)], and s₂ has been numerically-scaled by its maximum value. For clarity, the mass action equations have been integrated from t = 0 to $t\approx t_{s_2}^{\ast }$.