  We study a minimal model for the growth of a phenotypically heterogeneous
population of cells subject to a fluctuating environment in which they can
replicate (by exploiting available resources) and modify their phenotype within
a given landscape (thereby exploring novel configurations). The model displays
an exploration-exploitation trade-off whose specifics depend on the statistics
of the environment. Most notably, the phenotypic distribution corresponding to
maximum population fitness (i.e. growth rate) requires a non-zero exploration
rate when the magnitude of environmental fluctuations changes randomly over
time, while a purely exploitative strategy turns out to be optimal in two-state
environments, independently of the statistics of switching times. We obtain
analytical insight into the limiting cases of very fast and very slow
exploration rates by directly linking population growth to the features of the
environment.
