Role of Depletion on the Dynamics of a Diffusing Forager

TL;DR

This paper analyzes the survival dynamics of a forager depleting resources in various dimensions, deriving explicit results in one dimension and approximations for higher dimensions, revealing how resource depletion affects forager lifespan and movement.

Contribution

It provides the first explicit solutions for the starving random walk in one dimension and develops a novel circular depletion zone approximation for two dimensions.

Findings

Exact average lifetime in 1D

Distribution of visited sites at starvation

Lower bounds for 2D scaling behavior

Abstract

We study the dynamics of a starving random walk in general spatial dimension $d$. This model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources. The forager depletes its environment by consuming resources and dies if it wanders too long without finding food. In the exactly-solvable case of one dimension, we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation. We also give a heuristic derivation for the averages of these two quantities. We tackle the complex but ecologically-relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone. Within this framework, we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation. We also determine the asymptotic distribution of the number of distinct sites visited at starvation. Finally, we solve the case of high spatial dimensions within a mean-field approach.