Search in Random Media with L\'evy Flights

TL;DR

This paper reviews mathematical models of search processes using Lévy flights, focusing on optimizing search success, energy expenditure, and searcher requirements in various applications like security and emergency management.

Contribution

It introduces a comprehensive mathematical framework for search strategies with Lévy flights, including probabilistic models, energy estimates, and numerical methods for non-homogeneous spaces.

Findings

Derived models for independent searchers with possible failures and time-outs.

Estimated energy and searcher requirements for successful searches within time B.

Developed an iterative numerical approach for non-homogeneous search spaces.

Abstract

We review some of our work regarding search which has been motivated by a variety of applications in engineering and technology, including traffic routing and security in communication networks, explosive mine detection and removal, tactical operations as well as emergency management. We develop the basic mathematical model representing N searchers that proceed independently, and which can be affected by destruction or loss, and time-outs. An approach based on Laplace transform is then developed for the case where the success of the search requires that k out of the N searchers be successful, and we estimate the amount of energy expended and the number of searchers that are required to conduct a successful search in time B, provided B is large. Finally, we present an iterative numerical solution approach that allows us to analyse the search time and the energy needed to find an object when the search space is non-homogeneous.