Stochastic stability and time-dependent mutations

This paper considers stochastic stability analysis in evolutionary models with time-dependent mutations. It takes a class of time-homogeneous Markov models where the transition probabilities are approximately polynomial functions of the mutation parameter and allows the mutation parameter to decline to zero over time. The main result shows that as long as the mutation parameter converges to zero slowly enough and its variation is finite, the resulting time-inhomogeneous model has a limiting distribution regardless of the details of the mutation process. Moreover, a bound on the required rate of decline is explicitly expressed as a function of the minimum coradius of the limit sets and the transition probabilities within the minimum coradius set.