Spatially non-uniform perturbations

For a perturbation with a general non-zero n the eigenvalues (co12)n of the Jacobian matrix appropriate to eqns (10.64) and (10.65) are given by the roots of 

The nth component then grows or decays as the real part of co is positive or negative. 

Provided condition (10.70) is satisfied (i.e. provided the well-stirred system is unstable), this equation has a real solution for the wave number n for given values of /r and k. 

Of course the change in sign of the trace is only important if it leads to a change in the sign of the eigenvalues con. This will be the case, and will only be the case, provided the determinant det(J) remains positive all the time. However, det(J) also depends on n and itself becomes zero when 

This latter condition will have zero or two real solutions depending on the discriminant under the square root sign, and hence on k. If k is such that (tr(U))2 — 4det(U) is always negative, there will be no real roots to eqn (10.76). If, however, the discriminant can become positive, there can be two real roots for n. 

---

Thus, when the stirring stops, the uniform state remains a stationary solution of the system. Diffusion does not affect the existence of the uniform state, but it may influence its stability. In particular we are interested in determining whether this state can become unstable to spatially non-uniform perturbations. 

---