  Biological populations are subject to fluctuating environmental conditions.
Different adaptive strategies can allow them to cope with these fluctuations:
specialization to one particular environmental condition, adoption of a
generalist phenotype that compromise between conditions, or population-wise
diversification (bet-hedging). Which strategy provides the largest selective
advantage in the long run depends on the range of accessible phenotypes and the
statistics of the environmental fluctuations. Here, we analyze this problem in
a simple mathematical model of population growth. First, we review and extend a
graphical method to identify the nature of the optimal strategy when the
environmental fluctuations are uncorrelated. Temporal correlations in
environmental fluctuations open up new strategies that rely on memory but are
mathematically challenging to study: we present here new analytical results to
address this challenge. We illustrate our general approach by analyzing optimal
adaptive strategies in the presence of trade-offs that constrain the range of
accessible phenotypes. Our results extend several previous studies and have
applications to a variety of biological phenomena, from antibiotic resistance
in bacteria to immune responses in vertebrates.
