An activator-inhibitor system is described by the equations

$$\begin{aligned} \frac{\partial u}{\partial t} &=\frac{a u}{v}-u^{2}+d_{1} \frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=v^{2}-\frac{v}{u^{2}}+d_{2} \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

where $a, d_{1}, d_{2}>0$.

Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$. Determine when the equilibrium is a stable focus, and sketch the phase diagram for this case (restricting attention to $u>0$ and $v>0)$.

For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ of the solution found above. Briefly explain why the system will be stable to spatial perturbations with very small or very large $k$. Find conditions for the system to be unstable to a spatial perturbation (for some range of $k$ which need not be given). Sketch the region satisfying these conditions in the $\left(a, d_{1} / d_{2}\right)$ plane.

Find $k_{c}$, the critical wavenumber at the onset of instability, in terms of $a$ and $d_{1}$.