In recent years many researchers have become interested in the existence and stability of periodic wavetrains for various reaction-diffusion systems [1,2,4,6,8,9]. These systems are thought to be adequate models of various chemical and biological phenomena [1,9]. It is generally accepted that certain large amplitude long waves which arise from stable limit cycle kinetics are, themselves, stable. On the other hand, stable small amplitude waves bifurcating from a uniform rest state have not been demonstrated. Several authors [6,8] constructed bifurcating waves that were later shown to be unstable. More recently, Cohen et al. [2] have used a multiscale perturbation scheme to demonstrate the instability of wavetrains for very general systems. This result led them to conclude that all small amplitude waves are unstable.