  By a refinement of the technique used by Johnson and Zumbrun to show
stability under localized perturbations, we show that spectral stability
implies nonlinear modulational stability of periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. The main new ingredient
is a detailed analysis of linear behavior under modulational data $\bar
u'(x)h_0(x)$, where $\bar u$ is the background profile and $h_0$ is the initial
modulation
