Consider the two-variable reaction-diffusion system

$$\begin{gathered} \frac{\partial u}{\partial t}=a-u+u^{2} v+\nabla^{2} u \\ \frac{\partial v}{\partial t}=b-u^{2} v+d \nabla^{2} v \end{gathered}$$

where $a, b$ and $d$ are positive constants.

Show that there is one possible spatially homogeneous steady state with $u>0$ and $v>0$ and show that it is stable to small-amplitude spatially homogeneous disturbances provided that $\gamma<\beta$, where

$$\gamma=\frac{b-a}{b+a} \quad \text { and } \quad \beta=(a+b)^{2}$$

Now assuming that the condition $\gamma<\beta$ is satisfied, investigate the stability of the homogeneous steady state to spatially varying perturbations by considering the timedependence of disturbances whose spatial form is such that $\nabla^{2} u=-k^{2} u$ and $\nabla^{2} v=-k^{2} v$, with $k$ constant. Show that such disturbances vary as $e^{p t}$, where $p$ is one of the roots of

$$p^{2}+\left(\beta-\gamma+d k^{2}+k^{2}\right) p+d k^{4}+(\beta-d \gamma) k^{2}+\beta$$

By comparison with the stability condition for the homogeneous case above, give a simple argument as to why the system must be stable if $d=1$.

Show that the boundary between stability and instability (as some combination of $\beta, \gamma$ and $d$ is varied) must correspond to $p=0$.

Deduce that $d \gamma>\beta$ is a necessary condition for instability and, furthermore, that instability will occur for some $k$ if

$$d>\frac{\beta}{\gamma}\left\{1+\frac{2}{\gamma}+2 \sqrt{\frac{1}{\gamma}+\frac{1}{\gamma^{2}}}\right\} .$$

Deduce that the value of $k^{2}$ at which instability occurs as the stability boundary is crossed is given by

$$k^{2}=\sqrt{\frac{\beta}{d}}$$