We present a rational scheme for modeling natural convection-driven polymerase chain reaction (PCR), where many copies of a DNA template are made by cycling between hot and cold regions via a circulatory, buoyancy-driven flow. This process is described here in the framework of multiple-species formulation, using evolution equations which govern the concentrations of the various DNA species in the carrying solution. In the intermediate asymptotic limit, where a stationary amplification rate is achieved, these equations provide an eigenvalue problem for computing the exponential amplification rate of double-stranded DNA. The scheme is demonstrated using a simplified model of a Rayleigh-Bénard cell. In contrast to what may have been anticipated, diffusion tends to enhance the growth rate. The present model, intended to be used as a template for more device-specific analyses, provides a starting point for understanding the effects of the competing mechanisms (reaction, convection and diffusion) upon the amplification efficiency.