An activator-inhibitor system is described by the equations

$$\begin{aligned} &\frac{\partial u}{\partial t}=u(c+u-v)+\frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial t}=v(a u-b v)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

where $a, b, c, d>0$.

Find and sketch the range of $a, b$ for which the spatially homogeneous system has a stable stationary solution with $u>0$ and $v>0$.

Considering spatial perturbations of the form $\cos (k x)$ about the solution found above, find conditions for the system to be unstable. Sketch this region in the $(a, b)$-plane for fixed $d$ (for a value of $d$ such that the region is non-empty).

Show that $k_{c}$, the critical wavenumber at the onset of the instability, is given by

$$k_{c}=\sqrt{\frac{2 a c}{d-a}}$$