  We study fluctuation effects in a two species reaction-diffusion system, with
three competing reactions $A+A\rightarrow\emptyset$, $B+B\rightarrow\emptyset$,
and $A+B\rightarrow\emptyset$. Asymptotic density decay rates are calculated
for $d\leq 2$ using two separate methods - the Smoluchowski approximation, and
also field theoretic/renormalisation group (RG) techniques. Both approaches
predict power law decays, with exponents which asymptotically depend only on
the ratio of diffusion constants, and not on the reaction rates. Furthermore,
we find that, for $d<2$, the Smoluchowski approximation and the RG improved
tree level give identical exponents. However, whereas the Smoluchowski approach
cannot easily be improved, we show that the RG provides a systematic method for
incorporating additional fluctuation effects. We demonstrate this advantage by
evaluating one loop corrections for the exponents in $d<2$, and find good
agreement with simulations and exact results.
