 Solution

ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2009/b/6/d/i/solution

Distinguish physical time

t

from scaled time

T = t / τ

. Put

P = P_{0} + P_{s} p, P_{s} = \frac{M}{G P _{0}}, \dot{V} = \frac{M}{P _{0}} v, v (T) = [p (T - 1)]_{+} .

Then

K (P_{s} / τ) p^{'} = M - [P_{0} + P_{s} p] (M / P_{0}) v

p^{'} = α [1 - (1 + μ p) v], α = \frac{τ G P _{0}}{K}, μ = \frac{M}{G P _{0}^{2}}, p_{1} = p (T - 1) .

A steady state must be positive, since otherwise ventilation vanishes while production remains positive. It solves

μ p_{*}^{2} + p_{*} - 1 = 0

, so

p_{*} = \frac{2}{1 + 1 + 4 μ} = 1 - μ + 2 μ^{2} + O (μ^{3}) .

The value

1/ (1 + μ) = 1 - μ + μ^{2} + O (μ^{3})

printed in the question agrees only through first order. Thus it is a small-

μ

approximation with error of order

μ^{2}

, not an exact equilibrium for nonzero

μ

. The other quadratic root is negative and inadmissible for this steady ventilation balance.

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