An activator-inhibitor system is described by the equations

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

where $a, b, d>0$.

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 $(d, b)$ plane for fixed $a \in(0,1)$.

Find $k_{c}$, the critical wavenumber at the onset of the instability, in terms of $a$ and $b$.