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.
Highlights
Finding randomly located objects is a challenge for every human being, be it the search for mushrooms [1], for lost keys [2], or for food [3]
We studied the efficiency of search generated by fractional Brownian motion (fBm) in a random field of targets
In more general terms we explored the sensitivity of search succes on the specific setting and the parameters defining the search process, the environment and the interaction of both with each other
Summary
Finding randomly located objects is a challenge for every human being, be it the search for mushrooms [1], for lost keys [2], or for food [3]. An important paradigm for this modeling was put forward by Karl Pearson, who suggested at the beginning of the last century that organisms may migrate according to simple random walks [11] characterized by Gaussian position distributions in a suitable scaling limit This paradigm was challenged two decades ago by the experimental observation [12] and a corresponding theory [13] that wandering albatrosses searching for food performed flights. According to non-Gaussian step length distributions [14] In this case, the mean square displacement (MSD) of an ensemble of moving agents may not grow linearly in time like for Gaussian spreading generated by random walks or Brownian motion. We hope that our work will set the scene for further studies to understand biological foraging on the basis of stochastic theory
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