  The notion of information pervades informal descriptions of biological
systems, but formal treatments face the problem of defining a quantitative
measure of information rooted in a concept of fitness, which is itself an
elusive notion. Here, we present a model of population dynamics where this
problem is amenable to a mathematical analysis. In the limit where any
information about future environmental variations is common to the members of
the population, our model is equivalent to known models of financial
investment. In this case, the population can be interpreted as a portfolio of
financial assets and previous analyses have shown that a key quantity of
Shannon's communication theory, the mutual information, sets a fundamental
limit on the value of information. We show that this bound can be violated when
accounting for features that are irrelevant in finance but inherent to
biological systems, such as the stochasticity present at the individual level.
This leads us to generalize the measures of uncertainty and information usually
encountered in information theory.
