Asymmetric Levy Flights Are More Efficient in Random Search

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Levy flights by leveraging the Fokker-Planck equation with a delta-sink and an asymmetric space-fractional derivative operator with stable index alpha and asymmetry (skewness) parameter beta. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Levy flights with beta <= 0 (with a rightward bias) for short initial distances, while for beta>0 (with a leftward bias) Levy flights with alpha -> 1 are more efficient. When increasing the initial distance of the searcher to the target, Levy flight search (except for alpha=1 with beta=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Levy search compared to symmetric Levy flights at both short and long distances, and the effect is more pronounced for stable indices alpha close to unity.

Automated summary

The study focused on the dynamics and efficiency of asymmetric Levy flights in a one-dimensional random search model, using the Fokker-Planck equation with delta-sink and an asymmetric space-fractional derivative operator. Exact analytical results were found for the probability density of first-arrival times and search efficiency, and the behavior was analyzed for short and long times. Asymmetry in jumps resulted in higher efficiency of Levy search compared to symmetric Levy flights, with more pronounced effects for stable indices alpha close to unity.