Figure 9: The timescale t_c2 is characteristic of the time it takes c₂ to reach $c_2^{\max }$, and the timescale $t_{c_2}^{\ast }$ is the approximate time it takes c₂ to reach $c_2^{\max }$, respectively, in the auxiliary reaction mechanism (47)–(48).

The thick black curve is the numerically-integrated solutions to the mass action equations (49a)–(49d). The leftmost dashed vertical line corresponds to t_c2, and the rightmost dashed vertical line corresponds to $t_{c_2}^{\ast }=-t_{c_2}\text{ln}{t}_{c_2}∕t_{s_1}$. The lower dotted horizontal line corresponds to the scaled characteristic value $\ell c_2^{\max }$, and the upper dotted horizontal line corresponds to $c_2^{\max }$. 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 c₂ has been numerically scaled by its maximum value. Note that the mass action equations have only been integrated from t = 0 to $t\approx t_{c_2}^{\ast }$ for clarity.