Random Search with Memory in Patchy Media: Exploration-Exploitation Tradeoff

TL;DR

This paper introduces a coupled patch exploitation model that accounts for resource depletion and searcher movement, deriving optimal strategies for maximizing resource consumption in patchy environments.

Contribution

It presents a minimal model integrating resource depletion with search dynamics and analytically determines optimal exploration strategies in one and two dimensions.

Findings

Exploring multiple patches enhances resource intake.

Optimal strategies are robust across different resource distributions.

The model applies to both one-dimensional and two-dimensional environments.

Abstract

How to best exploit patchy resources? This long-standing question belongs to the extensively studied class of explore/exploit problems that arise in a wide range of situations, from animal foraging, to robotic exploration, and to human decision processes. Despite its broad relevance, the issue of optimal exploitation has previously only been tackled through two paradigmatic limiting models---patch-use and random search---that do not account for the interplay between searcher motion within patches and resource depletion. Here, we bridge this gap by introducing a minimal patch exploitation model that incorporates this coupling: the searcher depletes the resources along its random-walk trajectory within a patch and travels to a new patch after it takes $\mathcal{S}$ consecutive steps without finding resources. We compute the distribution of the amount of resources $F_t$ consumed by time $t$ for this non-Markovian random walker and show that exploring multiple patches is beneficial. In one dimension, we analytically derive the optimal strategy to maximize $F_t$. We show that this strategy is robust with respect to the distribution of resources within patches and the criterion for leaving a given patch. We also show that $F_t$ can be optimized in the ecologically-relevant case of two-dimensional patchy environments.