Impossibility of a two-species Turing instability at equal diffusivities
= Impossibility of a two-species Turing instability at equal diffusivities
When $D_u=D_v=D$, every spatial Fourier mode replaces the reaction Jacobian $J$ by $J-Dk^2I$. Each eigenvalue is shifted left by $Dk^2$, so diffusion cannot destabilize a stable homogeneous equilibrium.