Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches

TL;DR

Problem addressed: establishing the optimality of inverse-square Lévy walks ($\\alpha=1$) for foraging with unrestricted revisits in dimensions $D\\ge 2$. Approach: analyze a Lévy walker confined to annuli or spherical shells with absorbing boundaries and derive analytical first-passage-time–based efficiency in the triple limit $l_c\\to a$, $L\\to\\infty$, $s\\to0$, linking to the 1D Riesz operator. Key findings: the scaled efficiency satisfies $\\eta_0\\sim \\delta^{-\\alpha/2}$ for $\\alpha<1$ and $\\eta_0\\sim \\delta^{-1+\\alpha/2}$ for $\\alpha>1$, with a sharp maximum at $\\alpha=1$, implying inverse-square steps minimize mean encounter time when revisits are allowed. Significance: provides the strongest formal support to LFH to date, extends 1D optimality to higher dimensions via a local flatness argument, and informs design principles for efficient search in sparse environments.

Abstract

The Lévy flight foraging hypothesis states that organisms must have evolved adaptations to exploit Lévy walk search strategies. Indeed, it is widely accepted that inverse square Lévy walks optimize the search efficiency in foraging with unrestricted revisits (also known as non-destructive foraging). However, a mathematically rigorous demonstration of this for dimensions $D \geq 2$ is still lacking. Here we study the very closely related problem of a Lévy walker inside annuli or spherical shells with absorbing boundaries. In the limit that corresponds to the foraging with unrestricted revisits, we show that inverse square Lévy walks optimize the search. This constitutes the strongest formal result to date supporting the optimality of inverse square Lévy walks search strategies.

Figures (6)

• Figure 1: The random search model is depicted in (a)--(b). (a) Always leaving from a position close to the previously found target, the walker follows the Lévy walk strategy (main text) looking for the next target. (b) Once the $n-1$--th target has been found, the $n$--th one can be either a revisit to the previous $n-1$--th target --- case ($A$) --- or the finding of any other than it --- case ($B$). (c) A random walker (of similar locomotion rules of the random search model) inside an annulus geometry. The circumferences ($A$) and ($B$) represent the inner and outer annuli of radii $a$ and $L$. $l_{c}$ marks the restart point each time the walker reaches one of the annuli borders. There is a proxy between ($A$) and ($B$) in (b) and ($A$) and ($B$) in (c), where being absorbing means finding a target.
• Figure 2: Efficiency $\eta_0$ as a function of the Lévy index $\alpha$ for different values of (a) Lévy scale parameter $\sigma$ ($\rho=5.6\times10^{-2}$ and $\delta=10^{-2}$), (b) the relative distance from the inner radius $\delta$ ($\sigma=10^{-2}$ and $\rho=5.6\times10^{-2}$), and (c) the effective density $\rho$ ($\sigma, \delta=10^{-2}$). The vertical line $\alpha=1$ is just guide for the eye.
• Figure 3: Efficiency $\eta_0$ as a function of the Lévy index for various values of $\sigma$, $\delta$, and $\rho$ for search inside a 2D annulus. As these parameters tend to zero, the optimal Lévy index goes to 1, corresponding to inverse square Lévy walks. The efficiency $\eta_0$ is so large for the case $\sigma=\delta=10^{-6}$ and $\rho=1.77\times10^{-3}$ (black curve) that the other curves appear to be zero on a linear scale. For large $\alpha > 1.2$ some points are not shown because the computational runtime can become extremely large. What is important to note, however, is the behavior near the peak.
• Figure 4: Efficiency $\eta_0$ as a function of the Lévy index for various values of $\sigma$, $\delta$ and $\rho$ for search inside a 3D spherical shell. As predicted theoretically, a maximum emerges near $\alpha=1$ as $\sigma, \delta, \rho\to 0$. Again, for $\alpha > 1.2$ the points are not shown because the computational runtime, even more critical in 3D.
• Figure 5: The spatial disposition and relevant distances of the searcher (dot) and nearest target (blue square). The searcher's detection radius is $a$, the initial distance from the searcher to the target is $l_c$, and $s$ is the scaling factor of the distribution of step lengths.