Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

Abstract

We study the efficiency of random search processes based on Lévy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the bias of the searcher based on prior experience. While Lévy flights turn out to be efficient search processes when relative to the starting point the target is upstream, in the downstream scenario regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Lévy flights, due to which Lévy flights typically overshoot a point or small interval. Extending our recent work on biased LF search [V. V. Palyulin, A. V. Chechkin, and R. Metzler, Proc. Natl. Acad. Sci. USA, 111, 2931 (2014).] we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favors Lévy flights over regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α = 1 (i.e., Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and include Brownian motion as the overall most efficient search strategy.

Comparison of pure and combined search strategies for single and multiple targets

“…The arrival time density ℘ fa (t) can be determined from the (fractional) Fokker-Planck equation of the search process, equipped with a sink term [38,39,42]. For search in one dimension by LFs without a bias the search reliability is unity if α > 1 and zero otherwise [38,39,42], which is consistent with previous results [109]. For search in the presence of a bias (water stream for marine searchers, winds for airborne foragers, etc.)…”

“…For search in the presence of a bias (water stream for marine searchers, winds for airborne foragers, etc.) the search reliability can vary between zero and unity [38,39], which is true also for Brownian motion [40]: when the bias pushes a searcher away from the target the search reliability is exponentially suppressed by a Boltzmann-like factor [40]. A search reliability of unity does not necessarily imply recurrence of the motion.…”

“…For a single target and saltatory motion we argued that the efficiency should be defined from Eq. (4) with proper averaging [38,39]. In our continuous time model the number of steps is naturally substituted by the time of the process.…”

“…This means that the search needs to be efficient. However, on top of this it often must also be reliable in that a given target is not missed out but found with sufficiently high probability [38,39].…”

“…On the other hand, landing on a single target after a relocation process has been completed can be formulated as a first arrival problem [38,39,42]. This distinction relates to the situation described above, when a searcher does not perceive targets while moving but only if it comes at rest exactly on the target, or sufficiently close to it (see Fig.…”