Replicator equations and the principle of minimal production of information

Many complex systems in mathematical biology and other areas can be described by the replicator equation. We show that solutions of a wide class of replicator equations minimize the KL-divergence of the initial and current distributions under time-dependent constraints, which in their turn, can be computed explicitly at every instant due to the system dynamics. Therefore, the Kullback principle of minimum discrimination information, as well as the maximum entropy principle, for systems governed by the replicator equations can be derived from the system dynamics rather than postulated. Applications to the Malthusian inhomogeneous models, global demography, and the Eigen quasispecies equation are given.