We analyzed the movement of fishing vessels during fishing trips in order to understand how fishermen behave in space while searching for fish. For that purpose we used hourly geo-referenced positions of vessels, provided by a satellite vessel monitoring system, for the entire industrial fleet (809 vessels) of the world's largest single species fishery (Peruvian anchovy, Engraulis ringens) from December 1999 to March 2003. Observed trajectories of fishing vessels are well modeled by Lévy random walks, suggesting that fishermen use a stochastic search strategy which conforms to the same search statistics as non-human predators. We show that human skills (technology, communication, or others) do not result in the fishermen's spatial behavior being fundamentally different from that of animal predators. With respect to probability of prey encounter, our results suggest that fishermen, on average, evolved an optimal movement pattern (mu = 2.00) among the family of Lévy random walks. This Lagrangian approach opens several perspectives in terms of operational management of the pelagic fish stock.