 Solution

ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2009/a/1/v/solution

For

μ > 0

, the positive equilibrium is

Q_{*} = m_{*} = μ, S_{*} = μ^{3/4}, N_{*} = μ^{1/12} .

S = S_{*} + ε S_{1}

and

N = N_{*} + ε N_{1}

\frac{d}{d t} (S_{1} N_{1}) = (μ^{1/4} /3 4 μ^{1/4} /3 - 3 μ^{11/12} 0) (S_{1} N_{1}) .

Differentiating its first row and using the second gives

S_{1}^{''} = \frac{1}{3} μ^{1/4} S_{1}^{'} - 4 μ^{7/6} S_{1} .

In particular the trace is positive and determinant is

4 μ^{7/6} > 0

. The steady drainage state is linearly unstable for every positive supply, whether its instability is oscillatory or nonoscillatory.

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