Uniqueness of the fixed point for the quasi-species models under changing environments

We investigate the uniqueness of the stable fixed point of the quasi-species models: the Eigen model and the Parallel Mutation-Selection model. Under changing environments with a sharp-peak fitness function where the fitness peak oscillates periodically, we use the Perron-Frobenius theorem to show that the stable fixed point must be unique in those models. By using the symmetry, we also show that the stationary values for the maximum and the minimum population probabilities of the two peak sequences must be equal, respectively.