  In the framework of the AdS/CFT correspondence, we find a neutral complex
scalar field dynamics in a $2+1$ dimensional black hole background which can
provide a scheme for studying the pattern formation process in $1+1$
dimensional reaction-diffusion systems. The patterns include plane wave, defect
turbulence, phase turbulence, spatio-temporal intermittency where defect chaos
coexists with stable plane wave, and coherent structures. A phase diagram is
obtained by studying the linear instability of the plane wave solutions to
determine the onset of the holographic version of the BFN instability. Near the
critical temperature the holographic model is dual to the one-dimensional
complex Ginzburg-Landau equation (CGLE), which has been studied extensively in
reaction-diffusion systems. While at low temperature the holographic theory is
different from CGLE.
