Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 13B / ii / Solution

The positive homogeneous equilibrium is $n=n_0$, $C=\alpha n_0/\beta$. For spatially constant perturbations $u,v$, linearization gives $\dot{u}=-\gamma n_{0}u$ and $\dot{v}=\alpha u-\beta v$. The triangular matrix has eigenvalues $-\gamma n_0$ and $-\beta$, both negative, proving asymptotic stability to homogeneous perturbations. The other homogeneous equilibrium, zero population and zero chemical, has positive population eigenvalue $\gamma n_0$ and is not the desired stable state.