A mathematical model concerning the origin and the growth of self-replicating polymers is developed. The polymers compete for activated monomers which can be carried into the system under constant or periodic fluxes. Changes in their concentrations are determined by the processes of: spontaneous generation of the shortest polymers, ligation with free monomers at the end of their chains, template-instructed synthesis for polymers above a length threshold, and decomposition. The rate equations for polymer and monomer concentrations are based on the Guldberg-Waage law. In this first analysis of the model, we are supposing that primordial competition occurred among short self-replicating homopolymers. The dynamics of this population is investigated by computer simulations. Our purpose is to determine which changes in the rate parameters and what kind of monomer flux favors replication of the longer polymers. Within the domain of the investigated values, we have observed that the average length of polymers increases more efficiently with enhancements on the flux intensity or on the parameter related to the instability of the double strands that appear as intermediate states following self-duplications, or yet by reducing the rate constant related to the polymer degradation reaction. In the case of periodic monomer fluxes, we have noted that the polymer population is much more sensitive to changes on the flux intensity than on the flux periodicity.