Depletion-Controlled Starvation of a Diffusing Forager

We study the starvation of a lattice random walker in which each site initially contains one food unit and the walker can travel $\mathcal{S}$ steps without food before starving. When the walker encounters food, the food is completely eaten, and the walker can again travel $\mathcal{S}$ steps without food before starving. When the walker hits an empty site, the time until the walker starves decreases by 1. In spatial dimension $d=1$, the average lifetime of the walker $<\tau>\propto \mathcal{S}$, while for $d > 2$, $<\tau>\simeq\exp(\mathcal{S}^\omega)$, with $\omega\to 1$ as $d\to\infty$. In the marginal case of $d=2$, $<\tau>\propto \mathcal{S}^z$, with $z\approx 2$. Long-lived walks explore a highly ramified region so they always remains close to sources of food and the distribution of distinct sites visited does not obey single-parameter scaling.