Stable system + Stabilizing effect = Instability. WTF?

"The chemical basis for Morphogenesis" is one of the lesser known contributions of Turing but still a seminal paper in the field of mathematical biology. He tried to explain pattern formation in Morphogenesis( The process by which an organism in earlier stage of life generates shapes) using a simple model of a reaction-diffusion equation. He used this formalism to explain stripes on a tiger skin, spots in a leopards skin, pattern on a fish scales, digit formation in limbs etc (Check figure below). 

One of the key mathematical/ physical insights that this paper gives rise to, is the idea of Turing instability.  I came across the idea of Turing instability very recently. Frankly I had read Turing's morphogenesis paper some time back but the counter-intuitiveness of the Turing instability was something that I glossed over. In my defense, Turing explained the instability as a long paragraph rather than writing it down in equations, which made it very off-putting for me. 

The instability can occur when you take two linear stable hyperbolic systems and you couple two of them using diffusion, which in itself stabilizing. This counter intuitively gives rise to an instability. 

Turing explains this using a simple case of two chemicals X,Y which are created by the rate 5X-6Y+1 and 6X-7Y+1 respectively. We look at the stable fixed points of the equation given below which are (X*,Y*) = (1,1). The stability can be analyzed by the matrix M =[ 5,-6;6,-7], which has Eigenvalues -1,-1; therefore the fixed point is hyperbolically stable. 

Let's now say that there are two cells A,B which both have chemical X,Y each. Let's mark it as (X1,Y1) and (X2,Y2) respectively. Let's add the effect of diffusion between these two cells where X diffuses at a rate of Dx = 0.5 per unit conc. difference and Dy = 4. 5 for  Y per unit conc. difference. After introducing the diffusion coefficient we have a modified equation which is given below. 

We can now look at the  stability of this 4-D dynamical system. Using the 4x4 deformation gradient matrix. The eigenvalues are (-1,-1,-14,2). Therefore we can see a positive eigenvalue  which implies that the system is unstable.

The formal definition from a mathematical perspective is dX/dt = AX is a stable dynamic and you add diffusion to it by the form dX/dt = BX. So if we have dX/dt = (A+B)X, the system could be unstable based on some criterion which I am not mentioning here. (X is phase-space coordinates and A,B are matrices).

This would seem super obvious from a dynamical system point of view but from a physics perspective it is very surprising on how diffusion(stabilizer) + stable system gives rise to an instability.

You can write down this principal in general to reaction diffusion equations with chemical concentration being fields. f(U,V) and g(U,V) are reaction terms of the chemicals U and V respectively. The conditions given below if satisfied generates Turing instability.