Figure 13: The validity of $t_{s_2}^{\chi }$ and $t_{s_2}^{\chi ,\ast }$ in the auxiliary reaction mechanism (47)–(48) when ϵ ≪ 1.

The thick black curve is the numerically-integrated solution to the mass action equations (49a)–(49d), and the unfilled circles mark the inner solution given by (103). The leftmost dashed vertical line corresponds to $t_{s_2}^{\chi }$, and the rightmost dashed vertical line corresponds to $t_{s_2}^{\chi ,\ast }=-t_{s_2}^{\chi }\text{ln}{t}_{s_2}^{\chi }∕t_{s_1}$. The lower dotted horizontal line corresponds to $y=\ell s_2^{\max }$, and the upper dotted horizontal line corresponds to $y=s_2^{\max }$. The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=1000$, $e_2^0=1$, k₁ = 1, k₂ = 1, k₃ = 1, k₋₃ = 1, k₄ = 100 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. Note that the mass action equations have only been integrated from t = 0 to $t\approx t_{s_2}^{\chi ,\ast }$ for clarity.