Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 14C / a / Solution

Letbe the reaction Jacobian at the homogeneous equilibrium. Stability without diffusion givesFor a spatial Fourier mode with $-{\nabla }^2$ eigenvalue $k^2$, the linearized matrix isIts trace is even more negative than $\mathrm{tr}M$, so instability occurs exactly when its determinant becomes negative. Put $z=D_v{k}^2$ and $d=D_u/D_v$. ThenThis upward-opening quadratic is negative for some $z>0$ exactly when its minimum occurs at positive $z$ and lies below zero:The second strict inequality implies the first when written with a positive square root, so the two-species diffusion-driven instability criterion becomes