 Solution

If $D_1=D_2=D$, the linearized matrix for wavenumber $k$ isIts eigenvalues are those of the homogeneous Jacobian shifted left by $D{k}^2$, so a stable homogeneous equilibrium remains stable for every spatial mode. Thus equal diffusivities cannot produce a Turing instability. In part (b), equal diffusivities mean $\lambda =1$; because homogeneous stability requires $\beta >1$, the necessary condition $1-\lambda \beta >0$ fails, consistently ruling out instability.