Figure 16: The lag time in the auxiliary reaction mechanism (47)–(48) when t_c1 ≪ t_c2 = t_s2 ≪ t_s1.

This is a close-up of Figure 15 near x*. 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 the stationary s₂-nullcline. The solid black circle marks the trajectory when $t=t_{s_2}^{\ast }=-t_{s_2}\text{ln}{t}_{s_2}∕t_{s_1}$. Notice that Tikhonov’s Theorem still provides a reasonable estimate of the lag time, which is synonymous with the matching timescale corresponding to either s₂ or c₂. The constants (without units) used in the numerical simulation are: $e_1^0=1$, $s_1^0=1000$, $e_2^0=10$, k₁ = 1, k₂ = 1, k₃ = 10, k₋₃ = 1, k₄ = 100 and k₋₁ = 1.