  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.
