  How to best exploit patchy resources? This long-standing question belongs to
the extensively studied class of explore/exploit problems that arise in a wide
range of situations, from animal foraging, to robotic exploration, and to human
decision processes. Despite its broad relevance, the issue of optimal
exploitation has previously only been tackled through two paradigmatic limiting
models---patch-use and random search---that do not account for the interplay
between searcher motion within patches and resource depletion. Here, we bridge
this gap by introducing a minimal patch exploitation model that incorporates
this coupling: the searcher depletes the resources along its random-walk
trajectory within a patch and travels to a new patch after it takes
$\mathcal{S}$ consecutive steps without finding resources. We compute the
distribution of the amount of resources $F_t$ consumed by time $t$ for this
non-Markovian random walker and show that exploring multiple patches is
beneficial. In one dimension, we analytically derive the optimal strategy to
maximize $F_t$. We show that this strategy is robust with respect to the
distribution of resources within patches and the criterion for leaving a given
patch. We also show that $F_t$ can be optimized in the ecologically-relevant
case of two-dimensional patchy environments.
