Environmental fluctuation between generations can lead to mixed optima
l strategies (so-called ''bet-hedging''). Due to mathematical intracta
bility, however, optimality modelling has up to now largely ignored th
e effects of such fluctuations. The cases that have been considered so
far are relatively simple. We show that the optimal strategy can be c
alculated explicitly for the case of non-structured populations. an en
vironmental parameter which varies over generations according to an er
godic process and certain types of payoff functions (specifying the re
lationship between an individual's trait value, the environment, and i
ts expected contribution to the populations size in the next generatio
n). Our results also lead to numerical solutions for other cases. Situ
ations with and without information about the environment are consider
ed. In both cases, there is a minimum environmental variability above
which mixed strategies are optimal. Examination of the differences bet
ween long-term reproductive success of constrained purr strategies and
optimal mixed strategies indicates that there is a high selection pre
ssure for mixed strategies to evolve unless individuals can acquire hi
ghly accurate information about their environment. We show that the so
lution is robust in the sense that it has near to optimal long-term re
productive success under small perturbations of the payoff function an
d/or the distribution of the environmental parameter. (C) 1995 Academi
c Press, Inc.