Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / A / 1 / v / Solution

For $\mu >0$, the positive equilibrium isSet $S=S_{\ast }+\epsilon S_1$ and $N=N_{\ast }+\epsilon N_1$ and retain first-order terms. The linear system isDifferentiating its first row and using the second givesIn particular the trace is positive and determinant is $4{\mu }^{7/6}>0$. The steady drainage state is linearly unstable for every positive supply, whether its instability is oscillatory or nonoscillatory.