An activator-inhibitor system for $u(x, t)$ and $v(x, t)$ is described by the equations

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

where $a, D>0$.

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

For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ to the equilibrium solution found above. Give a condition on $D$ in terms of $a$ for the system to have a Turing instability (a spatial instability).