RNA viruses offer a unique opportunity for the study of evolution at the molecular level. Recent experiments involving clonal populations of RNA viruses have shown that competition among virus strains of approximately equal relative fitness can result in the eventual competitive exclusion of one of the species. As competition proceeds in time, both the winners and the losers exhibited absolute gains in fitness, consistent with the "Red Queen" hypothesis of evolution. Further experiments involving closely related evolving quasispecies revealed a highly predictable nonlinear behavior suggesting a deterministic component in the underlying quasispecies dynamics. This is apparently in contradiction with the standard view of RNA virus evolution as a highly unpredictable process. In this paper we present a simple model which allows previous hypothesis to be tested and provides an interpretation for the observed experimental results.