Inverse Square Levy Walks are not Optimal Search Strategies for d ≥ 2

The Levy hypothesis states that inverse square Levy walks are optimal search strategies because they maximize the encounter rate with sparse, randomly distributed, replenishable targets. It has served as a theoretical basis to interpret a wealth of experimental data at various scales, from molecular motors to animals looking for resources, putting forward the conclusion that many living organisms perform Levy walks to explore space because of their optimal efficiency. Here we provide analytically the dependence on target density of the encounter rate of Levy walks for any space dimension d; in particular, this scaling is shown to be independent of the Levy exponent alpha for the biologically relevant case d >= 2, which proves that the founding result of the Levy hypothesis is incorrect. As a consequence, we show that optimizing the encounter rate with respect to alpha is irrelevant: it does not change the scaling with density and can lead virtually to any optimal value of alpha depending on system dependent modeling choices. The conclusion that observed inverse square Levy patterns are the result of a common selection process based purely on the kinetics of the search behavior is therefore unfounded.