Counterexample to the Lévy flight foraging hypothesis in the narrow capture framework

The Lévy flight foraging hypothesis asserts that biological organisms have evolved to employ (truncated) Lévy flight searches due to such strategies being more efficient than those based on Brownian motion. However, we provide here a concrete two-dimensional counterexample in which Brownian search is more efficient. In fact, we show that the efficiency of Lévy searches worsens the farther the Lévy flight tail index deviates from the Brownian limit. Our counterexample is based on the framework of the classic narrow capture problem in which a random search is performed for a small target within a confined search domain. Our results are obtained via three avenues: Monte Carlo simulations of the discrete search processes, finite-difference solutions, and a matched asymptotic analysis of the elliptic (pseudo)differential equations of the corresponding continuum limits. Asymptotic analysis of the Lévy search yields an expression for the average search time accurate to O(1), providing insights into how the latter is impacted by various features of the target and search domain.