  Foraging is a fundamental behavior as animals' search for food is crucial for
their survival. Patch leaving is a canonical foraging behavior, but classic
theoretical conceptions of patch leaving decisions lack some key naturalistic
details. Optimal foraging theory provides general rules for when an animal
should leave a patch, but does not provide mechanistic insights about how those
rules change with the structure of the environment. Such a mechanistic
framework would aid in designing quantitative experiments to unravel behavioral
and neural underpinnings of foraging. To address these shortcomings, we develop
a normative theory of patch foraging decisions. Using a Bayesian approach, we
treat patch leaving behavior as a statistical inference problem. We derive the
animals' optimal decision strategies in both non-depleting and depleting
environments. A majority of these cases can be analyzed explicitly using
methods from stochastic processes. Our behavioral predictions are expressed in
terms of the optimal patch residence time and the decision rule by which an
animal departs a patch. We also extend our theory to a hierarchical model in
which the forager learns the environmental food resource distribution. The
quantitative framework we develop will therefore help experimenters move from
analyzing trial based behavior to continuous behavior without the loss of
quantitative rigor. Our theoretical framework both extends optimal foraging
theory and motivates a variety of behavioral and neuroscientific experiments
investigating patch foraging behavior.
