We introduce a periodic extension of the Kingman model [11] for the balance between selection and mutation in large populations. In its original form, the model describes a population’s fitness distribution by a probability measure on the unit interval evolving through a simple discrete-time dynamical system, in which selection operates via size-biasing, and the mutation distribution remains constant along time. We allow the mutation environment to vary periodically over time and prove the convergence of the fitness distribution along subsequences; crucially, we derive an explicit criterion, phrased in term of the Perron eigenvalue of a characteristic matrix, to determine whether an atom emerges at the largest fitness in the limit, a phenomenon called condensation. Our results provide new insights on the role of periodic mutation effects in population Darwinian evolution.
