Search reliability and search efficiency of combined L\'evy-Brownian motion: long relocations mingled with thorough local exploration

TL;DR

This paper models a combined Brownian and Lévy flight search process using a fractional Fokker-Planck equation, revealing how this hybrid dynamics affects the probability and efficiency of locating a target.

Contribution

It introduces an analytical and numerical framework for understanding search reliability and efficiency in combined Brownian-Lévy dynamics, highlighting the impact of Lévy flight parameters.

Findings

Superposition of Brownian motion and Lévy flights with α<1 results in transient motion with finite hitting probability.

Exact dependence of search reliability and efficiency on initial distance and Lévy exponent α is characterized.

The model demonstrates how long relocations and local exploration balance in search processes.

Abstract

A combined dynamics consisting of Brownian motion and L\'evy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and L\'evy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and L\'evy flights with stable exponent $\alpha<1$, by itself implying zero probability of hitting a point on a line, lead to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent $\alpha$ of the L\'evy flight component.