Optimally Frugal Foraging

We introduce the \emph{frugal foraging} model in which a forager performs a discrete-time random walk on a lattice, where each site initially contains $\mathcal{S}$ food units. The forager metabolizes one unit of food at each step and starves to death when it last ate $\mathcal{S}$ steps in the past. Whenever the forager decides to eat, it consumes all food at its current site and this site remains empty (no food replenishment). The crucial property of the forager is that it is \emph{frugal} and eats only when encountering food within at most $k$ steps of starvation. We compute the average lifetime analytically as a function of frugality threshold and show that there exists an optimal strategy, namely, a frugality threshold $k^*$ that maximizes the forager lifetime.