Optimally frugal foraging

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