We investigate the small-scale properties of a stochastic cubic-autocatalytic reaction-diffusion (CARD) model using renormalization techniques. We renormalize noise-induced ultraviolet divergences and obtain β functions for the decay rate and coupling at one loop. Assuming colored (power-law) noise, our results show that the behavior of both decay rate and coupling with scale depends crucially on the noise exponent. Interpreting the CARD model as a proxy for a (very simple) living system, our results suggest that power-law correlations in environmental fluctuations can both decrease or increase the growth of structures at smaller scales.