A chemical system with concentrations $u(x, t), v(x, t)$ obeys the coupled reactiondiffusion equations

$$\begin{aligned} &\frac{d u}{d t}=r u+u^{2}-u v+\kappa_{1} \frac{d^{2} u}{d x^{2}} \\ &\frac{d v}{d t}=s\left(u^{2}-v\right)+\kappa_{2} \frac{d^{2} v}{d x^{2}} \end{aligned}$$

where $r, s, \kappa_{1}, \kappa_{2}$ are constants with $s, \kappa_{1}, \kappa_{2}$ positive.

(a) Find conditions on $r, s$ such that there is a steady homogeneous solution $u=u_{0}$, $v=u_{0}^{2}$ which is stable to spatially homogeneous perturbations.

(b) Investigate the stability of this homogeneous solution to disturbances proportional to $\exp (i k x)$. Assuming that a solution satisfying the conditions of part (a) exists, find the region of parameter space in which the solution is stable to space-dependent disturbances, and show in particular that one boundary of this region for fixed $s$ is given by

$$d \equiv \sqrt{\frac{\kappa_{2}}{\kappa_{1}}}=\sqrt{2 s}+\frac{1}{u_{0}} \sqrt{s\left(2 u_{0}^{2}-u_{0}\right)}$$

Sketch the various regions of existence and stability of steady, spatially homogeneous solutions in the $\left(d, u_{0}\right)$ plane for the case $s=2$.

(c) Show that the critical wavenumber $k=k_{c}$ for the onset of the instability satisfies the relation

$$k_{c}^{2}=\frac{1}{\sqrt{\kappa_{1} \kappa_{2}}}\left[\frac{s(d-\sqrt{2 s})}{d(2 \sqrt{2 s}-d)}\right] .$$

Explain carefully what happens when $d<\sqrt{2 s}$ and when $d>2 \sqrt{2 s}$.