The effect of stirring on the autocatalytic bromate-bromide-catalyst reaction has been studied at various initial concentrations of bromate, [BrO3-](0), bromide, [Br-](0), and catalyst, [C](0), in a batch closed reactor. As catalysts, ferroin and Ce-III were used. An enhancement of the stirring rate was found to lead to a longer induction period T-ind of the autocatalytic reaction. A hypothesis has been substantiated that the magnitude of the stirring effect, expressed as T-ind(max)/T-ind(min), is determined by the rate of [Br-] approaching the critical concentration [Br-](cr) and by the amplitude of asymmetrical nonequilibrium fluctuations. When this rate decreases in response to larger [Br-](0) or smaller [BrO3-](0), as well as when the fluctuation amplitude grows in response to larger [C](0), the magnitude of T-ind(max)/T-ind(min) increases. The role of nuclei has been discussed. The stirring effect in the oscillatory Belousov-Zhabotinsky reaction and in the autocatalytic bromate-bromide-catalyst reaction was shown to have the same origin.