We address the general question of what is the best strategy to employ when searching for targets of unpredictable sizes. Motivated by biological scenarios, we focus on generalized random walk strategies, in which step-lengths follow a given distribution, while their direction is randomly uniform. To capture common biological scenarios in which motion degrades detection, we further restrict attention to intermittent settings, which means that the searcher can detect a target only while being "immobile", that is , in-between moving steps. We prove that in a finite two-dimensional torus, an intermittent L\'evy walk whose step-lengths are distributed according to an ] inverse-square power-law is a near-optimal strategy for finding ] sparse targets of any size and shape. Specifically , in a square torus of area  n, and assuming that the detection radius is normalized to 1, the strategy finds any connected set of diameter D in O(n/D) expected time, whereas \Omega(n/D) is an unconditional lower bound on the expected time, that holds even when assuming that the shape and size of the target are known. Furthermore, this particular L\'evy process stands in stark contrast to many other basic intermittent processes, including all other L\'evy walks, which we prove to be extremely inefficient for wide ranges of target scales. Our results thus provide strong theoretical support for the optimality and robustness of intermittent L\'evy walks under general conditions.
 L\'evy walks are random walks processes whose step length follows a long-tailed power law distribution. Due to their abundance as movement patterns of URLanisms, significant theoretical efforts have been devoted to identify the foraging circumstances that would make such patterns advantageous Viswanathan et al. Nature, 1999]. Despite numerous attempts, there is currently no conclusive analytical evidence that L\'evy flights are preferable strategies in higher dimensions than one. Here we show that the optimality of inverse-square L\'evy walks in two-dimensions becomes striking when targets are sparse and unpredictable in size, and when detection is weak. Specifically, we prove that under the intermittent model, in which detection is not possible while moving ballistically B\'enichou et al. Reviews of Modern Physics, 2011], this strategy optimally finds sparse targets of any size and shape. That is , in a square torus of area ~ n, and assuming that the detection radius is normalized to 1, the strategy finds any connected set of diameter D in O(n/D) expected time, whereas \Omega(n/D) is an unconditional lower bound on the expected time, that holds even when assuming that the shape and size of the target are known. Furthermore, this particular L\'evy process stands in stark contrast to many other basic intermittent processes, including all other L\'evy walks, whose competitive ratio is shown to be polynomial in n, for wide ranges of target scales. Our results thus provide strong theoretical support for the optimality and robustness of intermittent L\'evy walks under general conditions.
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We address the general question of what is the best strategy to employ when searching for targets of unpredictable sizes. Motivated by biological scenarios, we focus on generalized random walk strategies, in which step-lengths follow a given distribution, while their direction is randomly uniform. To capture common biological scenarios in which motion degrades detection, we further restrict attention to intermittent settings, which means that the searcher can detect a target only while being "immobile", that is , in-between moving steps. We prove that in a finite two-dimensional torus, an intermittent
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walk whose step-lengths are distributed according to an
walks are random walks processes whose step length follows a long-tailed power law distribution. Due to their abundance as movement patterns of URLanisms, significant theoretical efforts have been devoted to identify the foraging circumstances that would make such patterns advantageous
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Viswanathan et al. Nature, 1999
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. Despite numerous attempts, there is currently no conclusive analytical evidence that L\'evy flights are preferable strategies in higher dimensions than one. Here we show that the optimality of
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power-law is a near-optimal strategy for finding
L\'evy walks in two-dimensions becomes striking when targets are sparse and unpredictable in size, and when detection is weak. Specifically, we prove that under the intermittent model, in which detection is not possible while moving ballistically
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B\'enichou et al. Reviews of Modern Physics, 2011
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, this strategy optimally finds
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Specifically
That is
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~
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which we prove to be extremely inefficient
whose competitive ratio is shown to be polynomial in n,