An isothermal biochemical in vitro amplification system with two trans-cooperatively coupled amplifying DNA molecules was investigated as a simple reaction-diffusion system. In a model of the recently developed experimental system, no reaction mechanism higher than second-order occurs, Numerical simulations in two spatial dimensions show a variety of complex spatiotemporal patterns which arise in response to finite amplitude perturbations. In a certain domain of the kinetic parameters, the system shows self-replicating spots. These spots are supposed to stabilize the cooperative amplification in evolving systems against emerging parasites. The results are of high relevance for experimental studies of evolutionary effects of the system in open two-dimensional reaction-diffusion reactors. For experimental spatially resolved evolution studies one- and two-dimensional microstructured flow reactors were designed, which enable the continuous influx of reactants, homogeneous flow through the active reaction layer and the removal of reaction products. First experimental results show homogeneous as well as inhomogeneous amplification of the biochemical system in microstructured reactors.