Figure 17: The trajectory follows the s₂-nullcline in the phase–plane of the auxiliary reaction mechanism (47)-(48) when $t_{s_2}^{\chi }∕t_{s_1}\ll 1$.

The thick black curve is the numerically-integrated solution to the mass action equations (49a)–(49d), the dashed/dotted red curve is the c₂-nullcline and the dashed/dotted blue curve is a snapshot of s₂-nullcline when t ≈ 1.1 · t_s1. The black dot is the corresponding snapshot of the numerical solution to (49a)–(49d). In this simulation, $t_{s_2}^{\chi }∕t_{s_1}\approx 0.001<{\mu }_2\approx 0.1$; consequently, the trajectory follows the s₂-nullcline but fails to closely follow x* (see Movie 1 in Supplementary Materials). The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=100$, $e_2^0=100$, k₁ = 1, k₂ = 1, k₃ = 1, k₋₃ = 1, k₄ = 0.1 and k₋₁ = 1.