  We study a simplified scheme of $k$ coupled autocatalytic reactions,
previously introduced by Togashi and Kaneko. The role of stochastic
fluctuations is elucidated through the use of the van Kampen system-size
expansion and the results compared with direct stochastic simulations. Regular
temporal oscillations are predicted to occur for the concentration of the
various chemical constituents, with an enhanced amplitude resulting from a
resonance which is induced by the intrinsic graininess of the system. The
associated power spectra are determined and have a different form depending on
the number of chemical constituents, $k$. We make detailed comparisons in the
two cases $k=4$ and $k=8$. Agreement between the theoretical and numerical
results for the power spectrum are good in both cases. The resulting spectrum
is especially interesting in the $k=8$ system, since it has two peaks, which
the system-size expansion is still able to reproduce accurately.
