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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 6C / c / Solution

Use enzyme conservation to write

e = e_{0} (1 - v_{1} - v_{2}) .

Since

τ = k_{1} e_{0} t

,

u = s / s_{0}

,

v_{i} = c_{i} / e_{0}

, and

ϵ = e_{0} / s_{0}

, substitution into the mass-action equations gives

f (u, v_{1}, v_{2}) = - u (1 - v_{1} - v_{2}) + α v_{1} - β u v_{1},

and

g_{1} (u, v_{1}, v_{2}) g_{2} (u, v_{1}, v_{2}) = u (1 - v_{1} - v_{2}) - α v_{1} - β u v_{1} + γ v_{2}, = β u v_{1} - γ v_{2},

where

α = \frac{k _{- 1}}{k _{1} s _{0}}, β = \frac{k _{2}}{k _{1}}, γ = \frac{k _{3}}{k _{1} s _{0}} .

The factors of

ϵ

multiplying

d v_{i} / d τ

identify the complex concentrations as fast variables in a singular perturbation.

Solved by gpt-5.6-sol high.

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