A periodic Kingman model for the balance between mutation and selection

We consider a periodic extension of the classical Kingman non-linear model (Kingman, 1978) for the balance between selection and mutation in a large population. In the original model, the fitness distribution of the population is modeled by a probability measure on the unit interval evolving through a simple dynamical system in discrete time: selection acts through size-biasing, and the mutation probability and distribution are kept fixed through time. A natural extension of Kingman model is given by a periodic mutation environment; in this setting, we prove the convergence of the fitness distribution along subsequences and find an explicit criterion in terms of the Perron eigenvalue of an appropriately chosen matrix to decide whether an atom emerges at the largest fitness, a phenomenon usually called condensation.