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The L\'evy flight foraging hypothesis asserts that biological organisms haveevolved to employ (truncated) L\'evy flight searches due to such strategiesbeing more efficient than those based on Brownian motion. However, we providehere a concrete two-dimensional counterexample in which Brownian search is moreefficient. In fact, we show that the efficiency of L\'evy searches worsens thefarther the L\'evy flight tail index deviates from the Brownian limit. Ourcounterexample is based on the framework of the classic narrow capture problemin which a random search is performed for a small target within a confinedsearch domain. Our results are obtained via three avenues: Monte Carlosimulations of the discrete search processes, finite difference solutions and amatched asymptotic analysis of the elliptic (pseudo)-differential equations ofthe corresponding continuum limits.

A Counterexample to the L\'evy Flight Foraging Hypothesis in the Narrow Capture Framework