Optimization of random search processes in the presence of an external bias

Abstract

We study the efficiency of random search processes based on L{\'e}vy 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 L\'evy 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 L{\'e}vy flights, due to which L{\'e}vy 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, DOI:10.1073/pnas.1320424111] we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favors L{\'e}vy flights over regular Brownian search. Contrary to the common belief that L{\'e}vy flights with a L{\'e}vy index $\alpha =1$ (i.e., Cauchy flights) are optimal for sparse targets, we find that the optimal value for $\alpha$ may range in the entire interval $(1,2)$ and include Brownian motion as the overall most efficient search strategy.