Front waves in the early RNA world: The Schlögl model and the logistic growth model

Front wave solutions of nonlinear reaction-diffusion models describing the spatio-temporal growth of RNA populations in the early RNA world are discussed. A two-variable model for RNA enzymes and enzyme complex molecules as well as single-variable models obtained via adiabatic elimination of the complex molecules are considered. In both models, the focus is on enzyme diffusion in one spatial dimension, assuming that the diffusion of complex molecules can be neglected. It is shown that one of the single-variable models corresponds to a Schlögl model of front propagation. In general, for the single-variable models it is found that front speed corresponds to the minimal speed of traveling fronts. In contrast, the two-variable model exhibits even slower front propagation. Front propagation might be an important factor in competitive evolutionary processes in the early RNA world.