  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.
