Depletion-Controlled Starvation of a Diffusing Forager

TL;DR

This paper models a forager's starvation dynamics on a lattice, revealing how the average lifetime depends on the spatial dimension and the forager's starvation threshold, with distinct behaviors in different dimensions.

Contribution

It introduces a lattice random walk model with depletion-controlled starvation, analyzing how spatial dimension affects the forager's lifetime and exploration patterns.

Findings

In 1D, average lifetime scales linearly with S.

In dimensions greater than 2, lifetime grows exponentially with S.

In 2D, lifetime scales as S to the power of approximately 2.

Abstract

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.