Department of Theoretical Physics, University of Geneva, 1211 Geneva 4, Switzerland.
Department of Biochemistry, University of Geneva, 1211 Geneva 4, Switzerland.
Phys Rev Lett. 2020 Feb 28;124(8):080601. doi: 10.1103/PhysRevLett.124.080601.
The Lévy hypothesis states that inverse square Lévy 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 Lévy walks to explore space because of their optimal efficiency. Here we provide analytically the dependence on target density of the encounter rate of Lévy walks for any space dimension d; in particular, this scaling is shown to be independent of the Lévy exponent α for the biologically relevant case d≥2, which proves that the founding result of the Lévy hypothesis is incorrect. As a consequence, we show that optimizing the encounter rate with respect to α is irrelevant: it does not change the scaling with density and can lead virtually to any optimal value of α depending on system dependent modeling choices. The conclusion that observed inverse square Lévy patterns are the result of a common selection process based purely on the kinetics of the search behavior is therefore unfounded.
莱维假设指出，平方反比莱维漫步是最优的搜索策略，因为它最大限度地提高了与稀疏、随机分布、可补充目标的相遇率。它为解释从分子马达到寻找资源的动物等各种尺度的大量实验数据提供了理论依据，提出了许多生物体之所以会进行莱维漫步来探索空间，是因为它们的效率最优。在这里，我们分析了任何空间维度 d 下莱维漫步的目标密度对相遇率的依赖性；特别是，对于生物学上相关的情况 d≥2，这种标度与莱维指数α无关，这证明了莱维假设的基本结果是不正确的。因此，我们表明，根据α来优化相遇率是无关紧要的：它不会改变与密度的标度，并且实际上可以根据系统相关的建模选择导致任何最优的α值。因此，观察到的平方反比莱维模式是基于搜索行为动力学的纯粹共同选择过程的结果的结论是没有根据的。
