Mutation-rate threshold under changing environments with sharp-peak fitness function

We investigate the evolution of biological organisms under changing environments by using the parallel mutation-selection model for quasi-species with the sharp-peak fitness function. While in the static case with the sharp-peak fitness function f(ξ) = Aδ_ξ,N, a threshold μ_th³ -A appears as an upper bound on the mutation rate, above which no effective selection can occur, in the dynamic case where the position of the sharp peak oscillates between ξ = N and ξ = 0 with period T, the threshold on the mutation rate is found to be μ_th ^d = A/2, and the phase diagram for the mutation rate and the fitness phase space is modified according to the new threshold. We also calculate the stationary values for the maximum and the minimum populations of the master sequence.