Distribution of the quasispecies for a Galton-Watson process on the sharp peak landscape

We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe − 1) a k! ∑