Levy Foraging in a Dynamic Environment – Extending the Levy Search

Vincenzo Fioriti,Stefano Chiesa,Fabio Fratichini,Claudio Moriconi

doi:10.5772/60414

Abstract

A common task for robots is the patrolling of an unknown area with inadequate information about target locations. Under these circumstances it has been suggested that animal foraging could provide an optimal or at least sub-optimal search methodology, namely the Levy flight search. Although still in debate, it seems that predators somehow follow this search pattern when foraging, because it avoids being trapped in a local search if the food is beyond the sensory range. A Levy flight is a particular case of the random walk. Its displacements on a 2-D surface are drawn from the Pareto-Levy probability distribution, characterized by power law tails. The Levy flight search has many applications in optical material, ladars, optics, large database search, earthquake data analysis, location of DNA sites, human mobility, stock return analysis, online auctions, astronomy, ecology and biology. Almost all studies and simulations concerning the Levy flight foraging examine static or slowly moving (with respect to the forager) uniformly distributed resources. Moreover, in recent works a small swarm of underwater autonomous vehicles has been used to test the standard Levy search in the underwater environment, with good results. In this paper we extend the classical Levy foraging framework taking into consideration a moving target allocated on a 2-D surface according to a radial probability distribution and comparing its performance with the random walk search. The metric used in the numerical simulations is the detection rate. Simulations include the sensor resolution, intended as the maximum detection distance of the forager from the target. Furthermore, contrarily to the usual Levy foraging framework, we use only one target. Results show that Levy flight outperforms the random walk if the sensor detection radius is not too small or too large. We also find the Levy flight in the velocity of the center of mass model of a fish school according the Kuramoto equation, a famous model of synchronization phenomena. Finally, a discussion about the controversy concerning the innate or evolutionary origin of the Levy foraging is given.

Highlights

In this paper, we extend the standard Levy search and show that the center of mass of a biological or of a robot swarm is able to move following a Levy statistics if coordinated by the Kuramoto equation
The more the search is reduced to a local search, the less the Levy flight (LF) is an optimal strategy
The Levy flight search has been suggested as the animal foraging strategy, because large areas are covered with a minimum energy expenditure

Summary

Introduction

We extend the standard Levy search and show that the center of mass of a biological or of a robot swarm is able to move following a Levy statistics if coordinated by the Kuramoto equation. A random walk with steps (spatial displacements) from a Pareto-Levy distribution [1] is called a Levy flight (LF). It is a set of many small dis‐ placements with just a few very large "jumps". The Levy search scenario consists of a number of non-mobile targets uniformly distributed on a surface We extend these scenarios to a single mobile target initially positioned according to non-uniform probability distributions, with a given sensor resolution, testing the performance of the Levy flight search and of the random walk. We show in numeric simulations how the Kuramoto model spontaneously generates the Levy statistics, we can ascertain that a biological school is led to a Levy search. An innate LF would imply a build-in structure, maybe a hardware device, that in the case of a swarm could be as simple as an oscillating electronic circuit

The Levy Distribution

Radial Probability and Moving Target

Simulation Results

Innate or Evolutionary

Conclusions