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Asymmetric Lévy 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 Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. 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 Lévy flights with β≤0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α→1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices α close to unity.