Lévy Walks Suboptimal under Predation Risk

<div><p>A key challenge in movement ecology is to understand how animals move in nature. Previous studies have predicted that animals should perform a special class of random walks, called Lévy walk, to obtain more targets. However, some empirical studies did not support this hypothesis, and the relationship between search strategy and ecological factors is still unclear. We focused on ecological factors, such as predation risk, and analyzed whether Lévy walk may not be favored. It was remarkable that the ecological factors often altered an optimal search strategy from Lévy walk to Brownian walk, depending on the speed of the predator’s movement, density of predators, etc. This occurred because higher target encounter rates simultaneously led searchers to higher predation risks. Our findings indicate that animals may not perform Lévy walks often, and we suggest that it is crucial to consider the ecological context for evaluating the search strategy performed by animals in the field.</p></div

The relative fitness of a Lévy searcher with life-cycle type I.

<p>The strategy of predators is (A) sit-and-wait (<i>v</i>_p = 0); (B) slow Lévy walker (<i>v</i>_p/<i>v</i>_s = 0.2); (C) middle Lévy walker (<i>v</i>_p/<i>v</i>_s = 1); and (D) fast Lévy walker (<i>v</i>_p/<i>v</i>_s = 5). The horizontal axis represents the number of predators introduced, and the vertical axis represents the Lévy index <i>μ</i> of the searcher. The total search time is 10⁷ for sit-and-wait, slow, and middle predator conditions and 5×10⁷ for fast predator conditions.</p

The relationship between relative encounter rates of targets and predators.

<p>Data in (A–C) and (D–F) correspond to life-cycle types I and II, respectively. (A and D) The lower encounter rate condition, <i>k</i>_BW = 0.1 (e.g., lower predator density or slower predators). (B and E) The middle encounter rate condition, <i>k</i>_BW = 1. (C and F) The higher encounter rate condition, <i>k</i>_BW = 10 (e.g., higher predator density or faster predators). The colors represent the fitness of LW (or any search strategies) compared to BW. The result shows that the life-cycle type and the three parameters, encounter rate to targets, that to predators and strength of predation pressure critically determine the fitness. When the strength of predation pressure is high (C and F), the LW obtains lower fitness even if the encounter rate with targets is high.</p

The relative fitness of a Lévy searcher with life-cycle type II.

<p>The strategy of predators is (A) sit-and-wait (<i>v</i>_p = 0); (B) slow Lévy walker (<i>v</i>_p/<i>v</i>_s = 0.2); (C) middle Lévy walker (<i>v</i>_p/<i>v</i>_s = 1); and (D) fast Lévy walker (<i>v</i>_p/<i>v</i>_s = 5). The horizontal axis represents the number of predators introduced, and the vertical axis represents the Lévy index <i>μ</i> of the searcher. The total search time is 10⁷ for sit-and-wait, slow, and middle predator conditions and 5×10⁷ for fast predator conditions.</p

The relative mean search time changes depending on the density and velocity of predators.

<p>The strategy of predators is (A) sit-and-wait; (<i>v</i>_p = 0); (B) slow Lévy walker (<i>v</i>_p/<i>v</i>_s = 0.2); (C) middle Lévy walker (<i>v</i>_p/<i>v</i>_s = 1); and (D) fast Lévy walker (<i>v</i>_p/<i>v</i>_s = 5). The horizontal axis represents the number of predators introduced, and the vertical axis represents the Lévy index <i>μ</i> of the searcher. The total search time is 10⁷ for sit-and-wait, slow, and middle predator conditions and 5×10⁷ for fast predator conditions.</p

Encounter rates with targets and predators in our simulation setting.

<p>(A) The relative encounter rate with targets increases at an intermediate <i>μ</i>. (B) The mean encounter number of a BW searcher (<i>k</i>_BW) for <i>T</i>_max. As the number or velocity of predators increases, the encounter number increases. (C) The relative encounter rate with predators increases when the movement of a searcher approaches a straight line (i.e., smaller <i>μ</i>). However, the faster the movement of predators, the lower the rate of increase. Similar results for <i>N</i>_p = 30 (left) and <i>N</i>_p = 120 (right) indicate that the relative encounter rate does not change depending on predator densities.</p

Movie S3 from Optimizing mating encounters by sexually dimorphic movements

Change over time in distance between a male and a female performing Lévy walk with different power-law exponents μ under one-dimensional boundless conditions. Red bars indicate the proportions of pairs that have already encountered one another

Movie S2 from Optimizing mating encounters by sexually dimorphic movements

Change over time in distance between a male and a female performing Lévy walk with the same power-law exponent μ under one-dimensional boundless conditions. Red bars indicate the proportions of pairs that have already encountered one another

Movie S1 from Optimizing mating encounters by sexually dimorphic movements

Change over time in locations of individuals performing Lévy walk with various power-law exponents (μ = 1.1, 1.5, 2.0, 2.5, 3.0)

Supplementary Materials from Optimizing mating encounters by sexually dimorphic movements

Figs S1-S7 and Table S