Search efficiency of discrete fractional Brownian motion in a random distribution of targets

TL;DR

This paper investigates the efficiency of fractional Brownian motion as a search strategy for randomly distributed targets, revealing that optimal search depends on various parameters and scenarios, and proposing a comprehensive classification framework.

Contribution

It introduces a detailed analysis of fractional Brownian motion for search tasks, highlighting its complex behavior and developing a unifying framework that includes Le9vy walks as a special case.

Findings

Search efficiency varies with parameters and scenarios.

Different motion modes optimize success depending on conditions.

A classification framework unifies search strategies including Le9vy walks.

Abstract

Efficiency of search for randomly distributed targets is a prominent problem in many branches of the sciences. For the stochastic process of L\'evy walks, a specific range of optimal efficiencies was suggested under variation of search intrinsic and extrinsic environmental parameters. In this article, we study fractional Brownian motion as a search process, which under parameter variation generates all three basic types of diffusion, from sub- to normal to superdiffusion. In contrast to L\'evy walks, fractional Brownian motion defines a Gaussian stochastic process with power law memory yielding anti-persistent, respectively persistent motion. Computer simulations of search by time-discrete fractional Brownian motion in a uniformly random distribution of targets show that maximising search efficiencies sensitively depends on the definition of efficiency, the variation of both intrinsic and extrinsic parameters, the perception of targets, the type of targets, whether to detect only one or many of them, and the choice of boundary conditions. In our simulations we find that different search scenarios favour different modes of motion for optimising search success, defying a universality across all search situations. Some of our numerical results are explained by a simple analytical model. Having demonstrated that search by fractional Brownian motion is a truly complex process, we propose an over-arching conceptual framework based on classifying different search scenarios. This approach incorporates search optimisation by L\'evy walks as a special case.