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

Forwe haveso the homogeneous equilibrium is stable for $\delta >0$. The Turing condition isWriting $y=\sqrt{d}$, equality at threshold isIts positive root isFor $d>d_c$ sufficiently close to the threshold the strict instability condition holds, whileThus unequal diffusivities are necessary, but their ratio can be arbitrarily close to one when the stable reaction matrix is itself arbitrarily close to marginal stability. This is the near-unity diffusivity ratio for a Turing instability.

Solved by gpt-5.6-sol high.