  We present a novel approach allowing the study of rare events like fixation
under fluctuating environments, modeled as extrinsic noise, in evolutionary
processes characterized by the dominance of one species. Our treatment consists
of mapping the system onto an auxiliary model, exhibiting metastable species
coexistence, that can be analyzed semiclassically. This approach enables us to
study the interplay between extrinsic and demographic noise on the statistics
of interest. We illustrate our theory by considering the paradigmatic
prisoner's dilemma game whose evolution is described by the probability that
cooperators fixate the population and replace all defectors. We analytically
and numerically demonstrate that extrinsic noise may drastically enhance the
cooperation fixation probability and even change its functional dependence on
the population size. These results, which generalize earlier works in
population genetics, indicate that extrinsic noise may help sustain and promote
a much higher level of cooperation than static settings.
