Markovian Switching of Mutation Rates in Evolutionary Network Dynamics

Abstract

The replicator-mutator dynamic was originally derived to model the evolution of language, and since the model was derived in such a general manner, it has been applied to the dynamics of social behavior and decision making in multi-agent networks. For the two type population, a bifurcation point of the mutation rate is derived, displaying different long-run behaviors above and below this point. The long-run behavior would naturally be subjected to noise from the environment, however, to date there does not exist a model that dynamically accounts for the effects of the environment. To account for the environmental impacts on the evolution of the populace, mutation rates above and below this bifurcation point are switched according to a continuous-time Markov chain. The long-run behaviors of this model are derived, showing a counterintuitive result that the majority of initial conditions will favor the dominated type.