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

When $D_u=D_v=D$, the linearized operator isFor a Fourier mode of wavenumber $k$, its two growth rates areBoth homogeneous growth rates have negative real part, and subtracting the nonnegative scalar $D{k}^2$ moves them farther into the left half-plane. Thus no spatial mode can become unstable. This is the impossibility of a two-species Turing instability at equal diffusivities.

Solved by gpt-5.6-sol high.