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

In foraging for food, biological organisms explore terrain in search of irregularly dispersed sources. Mathematically, such foraging processes can be modelled using stochastic processes, and in the simplest paradigm – Gaussian spreading generated by random walks or Brownian motion – the mean square displacement of an ensemble of moving agents grows linearly in time. More generally, a variety of other processes lead to so-called anomalous diffusion in which the mean square displacement grows in a different way, with〈x2(t)〉∼ tα and α not equal to one, where x(t) is the position of an agent in space at time t. Here α >1 corresponds to superdiffusion, α <1 to subdiffusion and α= 1 to normal diffusion.

Much recent research has explored the mathematical properties of superdiffusive Lévy walks – characterised  by power-law step length distributions – as well as normal diffusive intermittent motion and correlated random walks. Yet, as LML External Fellow Rainer Klages and colleagues explore in a recent paper, an over-arching framework for stochastic search is still missing. In particular, little is known about the efficiency of foraging through one paradigmatic process, fractional Brownian motion (fBm). In the article, the authors explore such processes systematically using extensive computer simulations and analytical arguments. In particular, they compute the efficiency of a searcher moving by fBm in a two-dimensional array of random distributed targets, and consider two distinct measures of search efficiency. The first employs an ensemble average of moving fBm particles for finding one target only, and the second considers the time-ensemble average of searchers for finding many targets along their whole trajectories.

One intriguing and somewhat counterintuitive finding is that subdiffusion can sometimes enhance search efficiency. Although subdiffusion is effectively slower than normal diffusion, it can lead to a more careful search of particular regions, resulting in a better sampling.

But the main lesson of the study, the authors suggest, is that fBm processes do not reveal universal behaviour in connection with search efficiency. Rather, different mechanisms determine search success in distinct scenarios. For example, boundary effects play a crucial role for search in some cases, but a minimal role in others. Overall, they conclude, search is a highly complex process which depends sensitively on the interplay between many fine details which reach far beyond superficial universalities.

The paper is available at https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.023169

 

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