An optimal search theory, the so-called Lévy-flight foraging hypothesis 1 , predicts that predators should adopt search strategies known as Lévy flights where prey is sparse and distributed unpredictably, but that Brownian movement is sufficiently efficient for locating abundant prey [2][3][4] . Empirical studies have generated controversy because the accuracy of statistical methods that have been used to identify Lévy behaviour has recently been questioned 5,6 . Consequently, whether foragers exhibit Lévy flights in the wild remains unclear. Crucially, moreover, it has not been tested whether observed movement patterns across natural landscapes having different expected resource distributions conform to the theory's central predictions. Here we use maximum-likelihood methods to test for Lévy patterns in relation to environmental gradients in the largest animal movement data set assembled for this purpose. Strong support was found for Lévy search patterns across 14 species of open-ocean predatory fish (sharks, tuna, billfish and ocean sunfish), with some individuals switching between Lévy and Brownian movement as they traversed different habitat types. We tested the spatial occurrence of these two principal patterns and found Lévy behaviour to be associated with less productive waters (sparser prey) and Brownian movements to be associated with productive shelf or convergence-front habitats (abundant prey). These results are consistent with the Lévy-flight foraging hypothesis 1,7 , supporting the contention 8,9 that organism search strategies naturally evolved in such a way that they exploit optimal Lévy patterns.Lévy flights are a special class of random walk with movement displacements (steps) drawn from a probability distribution with a power-law tail (the so-called Pareto-Lévy distribution) 1,10 , and give rise to stochastic processes closely linked to fractal geometry and anomalous diffusion phenomena 7,11 . Lévy flights describe a movement pattern characterized by many small steps connected by longer relocations, with this pattern having scale invariance under projection, such that the probability density function, P(l j ), has a power-law tail in the long-distance regime: P(l j ) < l j 2m , where l j is the flight length (step length of move j), and m, 1 , m # 3, is the power-law exponent. Lévy flights comprise instantaneous steps and hence involve infinite velocities, whereas a Lévy walk 10 refers to a finitevelocity walk such that displacement is determined after a time t, reflecting a dynamical process such as movement 1,10,11 . Lévy flights and walks are theorized to be the most efficient movement pattern for locating patchy prey in low concentrations on spatial scales beyond a searcher's sensory range, with an optimal search having a power-law exponent of m < 2 (refs 4, 13). It is proposed that organisms have therefore naturally evolved search patterns that can be modelled as optimal Lévy flights 1,7,13 .However, burgeoning empirical support for this hypothesis recently foundered following studies sug...