Consider the activator-inhibitor system in the fast-inhibitor limit

$$\begin{gathered} u_{t}=D u_{x x}-u(u-r)(u-1)-\rho(v-u) \\ 0=v_{x x}-(v-u) \end{gathered}$$

where $D$ is small, $0<r<1$ and $0<\rho<1$.

Examine the linear stability of the state $u=v=0$ using perturbations of the form $\exp (i k x+\sigma t)$. Sketch the growth-rate $\sigma$ as a function of the wavenumber $k$. Find the growth-rate of the most unstable wave, and so determine the boundary in the $r$ - $\rho$ parameter plane which separates stable and unstable modes.

Show that the system is unchanged under the transformation $u \rightarrow 1-u, v \rightarrow 1-v$ and $r \rightarrow 1-r$. Hence write down the equation for the boundary between stable and unstable modes of the state $u=v=1$.