An activator-inhibitor system is described by the equations

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

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. Show that the system has a Turing instability when

$$d>\left(\frac{7}{2}+2 \sqrt{3}\right) a$$