We review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager, and discuss its quantitative description using the theory of random walks. We illustrate Lévy flight foraging by comparison to Brownian random walks and discuss experimental observations of Lévy flights in biological foraging. We review recent findings suggesting that an inverse square probability density distribution P(ℓ)∼ℓ−2 of step lengths ℓ can lead to optimal searches. Finally, we survey the explanations put forth to account for these unexpected findings

We consider a Lévy flyer of order $\alpha$ that starts from a point $x_0$ on an interval $[O,L]$ with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of $\overrightarrow{\alpha }2,$ and the cases of $x_0\to 0$ and $x_0\to L.\mathrm{}$ For $\alpha >2,$ when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if $x_0$ is placed in the vicinity of absorbing boundaries, the average total length has a minimum at $\alpha =1,$ corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.