Exact phase diagram of a quasispecies model with a mutation rate modifier

We consider an infinite asexual population with a mutator allele which can elevate mutation rates. With probability f, a transition from nonmutator to mutator state occurs but the reverse transition is forbidden. We find that at f=0, the population is in the state with minimum mutation rate, and at f=fc, a phase transition occurs between a mixed phase with both nonmutators and mutators and a pure mutator phase. We calculate the critical probability fc and the total mutator fraction Q in the mixed phase exactly. Our predictions for Q are in agreement with those seen in microbial populations in static environments.