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

## TL;DR

This paper presents a specific counterexample within the narrow capture framework showing that Brownian motion can outperform Le9vy flights in search efficiency, challenging the widely held hypothesis.

## Contribution

It provides the first concrete two-dimensional counterexample demonstrating that Le9vy flights are not always the most efficient search strategy.

## Key findings

- Brownian search can be more efficient than Le9vy flights in certain scenarios
- The efficiency of Le9vy flights worsens as the tail index deviates from Brownian motion
- Multiple analytical and simulation methods confirm the counterexample

## Abstract

The L\'evy flight foraging hypothesis asserts that biological organisms have evolved to employ (truncated) L\'evy 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 L\'evy searches worsens the farther the L\'evy 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.