  The risk-sensitive foraging theory formulated in terms of the (daily) energy
budget rule has been influential in behavioural ecology as well as other
disciplines. Predicting risk-aversion on positive budgets and risk-proneness on
negative budgets, however, the budget rule has recently been challenged both
empirically and theoretically. In this paper, we critically review these
challenges as well as the original derivation of the budget rule and propose a
`gradual' budget rule, which is normatively derived from a gradual nature of
risk sensitivity and encompasses the conventional budget rule as a special
case. The gradual budget rule shows that the conventional budget rule holds
when the expected reserve is close enough to a threshold for overnight
survival, selection pressure being significant. The gradual view also reveals
that the conventional budget rule does not need to hold when the expected
reserve is not close enough to the threshold, selection pressure being
insignificant. The proposed gradual budget rule better fits the empirical
findings including those that used to challenge the conventional budget rule.
