Replace the discrete treatment of the Turing instability in a one-dimensional system discussed in the chapter with the corresponding continuum reaction-diffusion equations
$$\begin{array}{l}
\frac{\partial X}{\partial t}=D_{x} \frac{\partial^{2} X}{\partial x^{2}}+f(X, Y) \\
\frac{\partial Y}{\partial t}=D_{y} \frac{\partial^{2} Y}{\partial x^{2}}+g(X, Y)
\end{array}$$
for the morphogen concentrations $X$ and $Y$, which are both functions of position. Like for the case analyzed in the chapter, the functions $f(X, Y)$ and $g(\bar{X}, Y)$ reflect the chemical interactions between the morphogens. Also. we introduce the notation
$$\begin{array}{l}
a_{11}=\left.\frac{\partial f}{\partial X}\right|_{(h, k)}, \quad a_{12}=\left.\frac{\partial f}{\partial Y}\right|_{(h, k)} \\
a_{21}=\left.\frac{\partial g}{\partial X}\right|_{(h, k)}, \quad a_{22}=\left.\frac{\partial g}{\partial Y}\right|_{(h, k)}
\end{array}$$
where $h$ and $k$ are steady-state values of $X$ and $Y$. We are interested in the stability of the homogeneous steady state $(X, Y)=(h, k)$ with respect to a perturbation that is periodic in space, as an instability signals the spontaneous appearance of a spatially periodic pattern of morphogens.
(a) Linearize the reaction-diffusion equations in the vicinity of the fixed point $(X, Y)=(h, k) .$ Consider the time evolution of a periodic perturbation of the form $(8 X, 8 Y)=\left(\eta(t) e^{i q x}\right.$$\varepsilon(t) \mathrm{e}^{\mathrm{i} q x}$, in the vicinity of the fixed point $(h, k),$ following the same strategy that we used in the chapter. In particular, derive expressions for the eigenvalues of the rate matrix $\mathbf{A}$ analogous to the one defined in Equation $20.45,$ in terms of its determinant (DetA) and trace (Tr A).
(b) Show that a necessary condition for linear stability of
a fixed point in the absence of diffusion is that at least one of $a_{11}$ and $a_{22}$ is negative.
(c) Show that for a fixed point that is linearly stable in the absence of diffusion, a necessary condition for linear instability in the presence of diffusion is that exactly one of $a_{11}$ and $a_{22}$ is negative.
(d) Assuming that $a_{22}$ is negative, show that at a Turing bifurcation (that is, at the onset of instability: if we were to continuously adjust parameters such as reaction rate constants or diffusion constants until the fixed point just becomes unstable), the characteristic length of the emerging pattern from the homogeneous steady state (fixed point) scales like
$$\lambda^{-2} \sim \lambda_{x}^{-2}-\lambda_{y}^{-2}$$
where $\lambda_{X} \equiv \sqrt{D_{X} / a_{11}}$ and $\lambda_{y} \equiv \sqrt{D_{y} /\left(-a_{22}\right)}$
(e) Comment on the physical implications of these results. What is the wavelength of an oscillatory instability? (Problem courtesy of Justin Bois.)