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 S steps without food before starving. When the walker encounters food, it is completely eaten, and the walker can again travel 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 ⟨τ⟩∝S, while for d>2, ⟨τ⟩≃exp(S^{ω}), with ω→1 as d→∞; the latter behavior suggests that the upper critical dimension is infinite. In the marginal case of d=2, ⟨τ⟩∝S^{z}, with z≈2. Long-lived walks explore a highly ramified region so they always remain close to sources of food and the distribution of distinct sites visited does not obey single-parameter scaling.