Consider the reaction-diffusion system

$$\begin{aligned} &\frac{\partial u}{\partial \tau}=\beta_{u}\left(\frac{u^{2}}{v}-u\right)+D_{u} \frac{\partial^{2} u}{\partial x^{2}} \\ &\frac{\partial v}{\partial \tau}=\beta_{v}\left(u^{2}-v\right)+D_{v} \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

for an activator $u$ and inhibitor $v$, where $\beta_{u}$ and $\beta_{v}$ are degradation rate constants and $D_{u}$ and $D_{v}$ are diffusion rate constants.

(a) Find a suitably scaled time $t$ and length $s$ such that

$$\begin{aligned} \frac{\partial u}{\partial t} &=\frac{u^{2}}{v}-u+\frac{\partial^{2} u}{\partial s^{2}} \\ \frac{1}{Q} \frac{\partial v}{\partial t} &=u^{2}-v+P \frac{\partial^{2} v}{\partial s^{2}} \end{aligned}$$

and find expressions for $P$ and $Q$.

(b) Show that the Jacobian matrix for small spatially homogenous deviations from a nonzero steady state of $(*)$ is

$$J=\left(\begin{array}{cc} 1 & -1 \\ 2 Q & -Q \end{array}\right)$$

and find the values of $Q$ for which the steady state is stable.

[Hint: The eigenvalues of a $2 \times 2$ real matrix both have positive real parts iff the matrix has a positive trace and determinant.]

(c) Derive linearised ordinary differential equations for the amplitudes $\hat{u}(t)$ and $\hat{v}(t)$ of small spatially inhomogeneous deviations from a steady state of $(*)$ that are proportional to $\cos (s / L)$, where $L$ is a constant.

(d) Assuming that the system is stable to homogeneous perturbations, derive the condition for inhomogeneous instability. Interpret this condition in terms of how far activator and inhibitor molecules diffuse on average before they are degraded.

(e) Calculate the lengthscale $L_{\text {crit }}$ of disturbances that are expected to be observed when the condition for inhomogeneous instability is just satisfied. What are the dominant mechanisms for stabilising disturbances on lengthscales (i) much less than and (ii) much greater than $L_{\text {crit }}$ ?