 Solution

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

Since the steady state is strictly positive, the positive-part function is differentiable in a neighbourhood of it. Write

p = p_{*} + q

. Retaining first-order terms gives

q^{'} (T) = - αμ p_{*} q (T) - α (1 + μ p_{*}) q (T - 1) .

Substitute the exponential mode

q = e^{λ T}

to obtain

λ = - β - γ e^{- λ}, β = αμ p_{*}, γ = α (1 + μ p_{*}) = α / p_{*} .

Here the last equality uses the exact steady-state equation. For small

μ

β = αμ + O (α μ^{2})

and

γ = α (1 + μ) + O (α μ^{2})

. Linearizing at the exact root avoids carrying a spurious constant residual from the printed approximate equilibrium.

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