Let $m \ge 1$, let $\Omega \subseteq \mathbb{R}^m$ be a bounded open set with $C^1$ boundary, and let $\nu$ denote the outward unit normal to $\partial\Omega$. Let $V \subseteq \mathbb{R}^2$ be open, let $f, g: V \to \mathbb{R}$ be $C^1$, and let $D_1, D_2 > 0$ be constants. Consider the reaction-diffusion system for $u, v: \Omega \times (0, \infty) \to \mathbb{R}$: \begin{align*} \partial_t u &= f(u, v) + D_1\,\Delta u, \\ \partial_t v &= g(u, v) + D_2\,\Delta v, \end{align*} on $\Omega \times (0, \infty)$, with homogeneous Neumann boundary conditions \begin{align*} \frac{\partial u}{\partial \nu} &= 0, \quad \frac{\partial v}{\partial \nu} = 0 \quad \text{on } \partial\Omega \times (0, \infty). \end{align*} Let $(u^*, v^*) \in V$ satisfy $f(u^*, v^*) = 0$ and $g(u^*, v^*) = 0$, so that the spatially homogeneous function $(u, v) \equiv (u^*, v^*)$ is a steady state. Define the Jacobian matrix $J \in \mathbb{R}^{2 \times 2}$ and the diffusion matrix $D \in \mathbb{R}^{2 \times 2}$ by \begin{align*}

J &:= \begin{pmatrix} f_u(u^*, v^*) & f_v(u^*, v^*) \\ g_u(u^*, v^*) & g_v(u^*, v^*) \end{pmatrix}, \quad D := \begin{pmatrix} D_1 & 0 \\ 0 & D_2 \end{pmatrix}, \end{align*} where $f_u := \partial f/\partial u$, etc. Suppose the following four conditions hold: (i) $\operatorname{tr}(J) < 0$. (ii) $\det(J) > 0$. (iii) $D_2\,J_{11} + D_1\,J_{22} > 0$, where $J_{11} := f_u(u^*, v^*)$ and $J_{22} := g_v(u^*, v^*)$. (iv) $(D_2\,J_{11} + D_1\,J_{22})^2 > 4\,D_1\,D_2\,\det(J)$. Then the steady state $(u^*, v^*)$ is linearly unstable to spatially inhomogeneous perturbations. More precisely, let $\mu$ be an eigenvalue of $-\Delta$ on $\Omega$ with homogeneous Neumann boundary conditions (i.e., $-\Delta\varphi = \mu\varphi$ in $\Omega$, $\partial_\nu\varphi = 0$ on $\partial\Omega$) with eigenfunction $\varphi$. If $\mu \in (k_-^2, k_+^2)$, then there exist $\lambda > 0$ and $\zeta \in \mathbb{R}^2 \setminus \{0\}$ such that $(U, V)^\top := \zeta\,e^{\lambda t}\,\varphi(x)$ solves the linearised system \begin{align*} \partial_t \begin{pmatrix} U \\ V \end{pmatrix} &= J \begin{pmatrix} U \\ V \end{pmatrix} + D\,\Delta \begin{pmatrix} U \\ V \end{pmatrix}.

\end{align*} Here \begin{align*} k_\pm^2 &:= \frac{(D_2\,J_{11} + D_1\,J_{22}) \pm \sqrt{(D_2\,J_{11} + D_1\,J_{22})^2 - 4\,D_1\,D_2\,\det(J)}}{2\,D_1\,D_2}. \end{align*} Moreover, conditions (i)–(iii) together imply that $J_{11}$ and $J_{22}$ have opposite signs and $D_1 \neq D_2$.