  Under constant selection, each trait has a fixed fitness, and small mutation
rates allow populations to efficiently exploit the optimal trait. Therefore it
is reasonable to expect mutation rates will evolve downwards. However, we find
this need not be the case, examining several models of mutation. While upwards
evolution of mutation rate has been found with frequency or time dependent
fitness, we demonstrate its possibility in a much simpler context. This work
uses adaptive dynamics to study the evolution of mutation rate, and the
replicator-mutator equation to model trait evolution. Our approach differs from
previous studies by considering a wide variety of methods to represent
mutation. We use a finite string approach inspired by genetics, as well as a
model of local mutation on a discretization of the unit intervals, handling
mutation beyond the endpoints in three ways. The main contribution of this work
is a demonstration that the evolution of mutation rate can be significantly
more complicated than what is usually expected in relatively simple models.
