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Background  The Lévy flight foraging hypothesis predicts a transition from scale-free Lévy walk (LW) to scale-specific Brownian motion (BM) as an animal moves from resource-poor towards resource-rich environment. However, the LW-BM continuum implies a premise of memory-less search, which contradicts the cognitive capacity of vertebrates.    Results  We describe methods to test if apparent support for LW-BM transitions may rather be a statistical artifact from movement under varying intensity of site fidelity. A higher frequency of returns to previously visited patches (stronger site fidelity) may erroneously be interpreted as a switch from LW towards BM. Simulations of scale-free, memory-enhanced space use illustrate how the ratio between return events and scale-free exploratory movement translates to varying strength of site fidelity. An expanded analysis of GPS data of 18 female red deer,  Cervus elaphus,  strengthens previous empirical support of memory-enhanced and scale-free space use in a northern forest ecosystem.    Conclusion  A statistical mechanical model architecture that describes foraging under environment-dependent variation of site fidelity may allow for higher realism of optimal search models and movement ecology in general, in particular for vertebrates with high cognitive capacity.    Electronic supplementary material  The online version of this article (doi:10.1186/2051-3933-1-9) contains supplementary material, which is available to authorized users.

The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion