Dynamics of Small Autocatalytic Reaction Network IV:
Inhomogeneous Replicator Equations

Peter F. Stadler

The inhomogeneous replicator equation is derived as the continuous time model for parallel first and second order autocatalytic replication of macromolecules in a flow reactor based on mass action kinetics. It is shown that the total concentration of replicating material determines the relative importance of the first order and the second order mechanism. A complete description of the dynamics of the first order model and some special features of the inhomogeneous replicator equation are presented. A minimal prebiotic scenario with the potentiality to develop cooperation is derived from the inhomogeneous replicator equation. In this model cooperation can emerge when the total concentration of replication material exceeds a certain threshold. Below this value, a single species is selected; which one is determined by the rate constants of the first order reaction alone. Above this threshold the second order process becomes important and may lead to cooperative behaviour such as hypercycles.
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