Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 13B / iii / Solution

The background has $\nabla a_0=0$, so the linear chemotaxis term is $-{\chi }_0{n}_0{\nabla }^2\xi$; derivatives of $\chi$ multiply products of disturbances and do not enter at first order. For a normal mode with ${\nabla }^2=-k^2$ and time dependence $e^{pt}$, put $q=k^2$. Thenand the characteristic polynomial isIts linear coefficient is positive for every $q\ge 0$. Thus instability occurs exactly when its constant term $Q(q)$ is negative: the roots are then real with opposite signs. The positive quadratic $Q$ is negative somewhere on $q>0$ precisely whenThereforeAt equality the neutral wavenumber is $q_c=\sqrt{A\gamma /D}$ and $p=0$. Above threshold the unstable band is $q_{-}<k^2<q_{+}$, where