replication that leads to a subexponential or a parabolic growth. This could be observed during the stages preceding a catalytic replication that are necessary for the formation of hypercycles. The coexistence of various non-enzymatically replicating sequences could help to maintain a sufficient diversity of RNA modules used later to build molecules with catalytic functions. From the mathematical point of view, it is possible to find conditions required for cooperation of several hypercycles. However, in reality, the cooperation of hypercycles would be extremely difficult, because it requires the existence of a complicated multi-step biochemical mechanism or an incorporation of more than two types of molecules. Both conditions seem very improbable; therefore, the existence of coupled hypercycles is assumed impossible in practice. Evolution of a hypercycle ensues from the creation of new components by the mutation of its internal species. Mutations can be incorporated into the hypercycle, enlarging it if, and only if, two requirements are satisfied. First, a new information carrier Inew created by the mutation must be better recognized by one of the hypercycle's members Ii than the chain Ii+1 that was previously recognized by it. Secondly, the new member Inew of the cycle has to better catalyse the formation of the polynucleotide Ii+1 that was previously catalysed by the product of its predecessor Ii. In theory, it is possible to incorporate into the hypercycle mutations that do not satisfy the second condition. They would form parasitic branches that use the system for their own replication but do not contribute to the system as a whole. However, it was noticed that such mutants do not pose a threat to the hypercycle, because other constituents of the hypercycle grow nonlinearly, which prevents the parasitic branches from growing. Evolutionary dynamics: a mathematical model According to the definition of a hypercycle, it is a nonlinear, dynamic system, and, in the simplest case, it can be assumed that it grows at a rate determined by a system of quadratic differential equations. Then, the competition between evolving hypercycles can be modelled using the differential equation: where Here, Cl is the total concentration of all polynucleotide chains belonging to a hypercycle Hl, C is the total concentration of polynucleotide chains belonging to all hypercycles, ql is the rate of growth, and φ is a dilution flux that guarantees that the total concentration is constant. According to the above model, in the initial phase, when several hypercycles