Mathematical Models of Evolution for Replicator Systems: Fitness Landscape Adaptation

Two classical approaches to replicator systems are considered: quasispecies and hypercycle models. We expand the adaptive landscape metaphor by S. Wright and Fisher’s fundamental theorem of natural selection, combining it with Kimura’s maximal principals to the case of dynamical fitness landscape. We assume that the parameters of the replicator system depend continuously on the evolutionary time, which distinguish from the internal system dynamics time. We suppose that evolutionary time is much slower that ordinary time of the system. Each step of the process of the fitness landscape adaptation place occurs in a steady-state. This process is equivalent to maximization the mean fitness of the system. From mathematical point of view, it is reduced to a series mathematical programming problems or to a first eigenvalue maximization problem. New properties of the adapted systems are discussed.