Campos D., Bartumeus F., Raposo E.P., Méndez V. (2015) First-passage times in multiscale random walks: The impact of movement scales on search efficiency. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 92: 0-0.EnllaçDoi: 10.1103/PhysRevE.92.052702
An efficient searcher needs to balance properly the trade-off between the exploration of new spatial areas and the exploitation of nearby resources, an idea which is at the core of scale-free Lévy search strategies. Here we study multiscale random walks as an approximation to the scale-free case and derive the exact expressions for their mean-first-passage times in a one-dimensional finite domain. This allows us to provide a complete analytical description of the dynamics driving the situation in which both nearby and faraway targets are available to the searcher, so the exploration-exploitation trade-off does not have a trivial solution. For this situation, we prove that the combination of only two movement scales is able to outperform both ballistic and Lévy strategies. This two-scale strategy involves an optimal discrimination between the nearby and faraway targets which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters. So, this optimization necessarily requires some prior information (albeit crude) about target distances or distributions. Furthermore, we found that the incorporation of additional (three, four, ...) movement scales and its adjustment to target distances does not improve further the search efficiency. This allows us to claim that optimal random search strategies arise through the informed combination of only two walk scales (related to the exploitative and the explorative scales, respectively), expanding on the well-known result that optimal strategies in strictly uninformed scenarios are achieved through Lévy paths (or, equivalently, through a hierarchical combination of multiple scales). © 2015 American Physical Society.
Kolzsch A., Alzate A., Bartumeus F., De Jager M., Weerman E.J., Hengeveld G.M., Naguib M., Nolet B.A., Van De Koppel J. (2015) Experimental evidence for inherent lévy search behaviour in foraging animals. Proceedings of the Royal Society B: Biological Sciences. 282: 0-0.EnllaçDoi: 10.1098/rspb.2015.0424
Recently, Lévy walks have been put forward as a new paradigm for animal search and many cases have been made for its presence in nature. However, it remains debated whether Lévy walks are an inherent behavioural strategy or emerge from the animal reacting to its habitat. Here, we demonstrate signatures of Lévy behaviour in the search movement of mud snails (Hydrobia ulvae) based on a novel, direct assessment of movement properties in an experimental set-up using different food distributions. Our experimental data uncovered clusters of small movement steps alternating with long moves independent of food encounter and landscape complexity. Moreover, size distributions of these clusters followed truncated power laws. These two findings are characteristic signatures of mechanisms underlying inherent Lévy-like movement. Thus, our study provides clear experimental evidence that such multi-scale movement is an inherent behaviour rather than resulting from the animal interacting with its environment. © 2015 The Author(s) Published by the Royal Society.
Tromer R.M., Barbosa M.B., Bartumeus F., Catalan J., Da Luz M.G.E., Raposo E.P., Viswanathan G.M. (2015) Inferring Lévy walks from curved trajectories: A rescaling method. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 92: 0-0.EnllaçDoi: 10.1103/PhysRevE.92.022147
An important problem in the study of anomalous diffusion and transport concerns the proper analysis of trajectory data. The analysis and inference of Lévy walk patterns from empirical or simulated trajectories of particles in two and three-dimensional spaces (2D and 3D) is much more difficult than in 1D because path curvature is nonexistent in 1D but quite common in higher dimensions. Recently, a new method for detecting Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The key new idea is to exploit the fact that the 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. Here, we ask whether or not this projection method is powerful enough to cleanly distinguish 2D Lévy walk with added curvature from a simple Markovian correlated random walk. We study the especially challenging case in which both 2D walks have exactly identical probability density functions (pdf) of step sizes as well as of turning angles between successive steps. Our approach extends the original projection method by introducing a rescaling of the projected data. Upon projection and coarse-graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, no fat tail appears when a (Markovian) correlated random walk is analyzed in this way. We show that this procedure works extremely well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and in air and oceans (3D). © 2015 American Physical Society.
Bartumeus F., Raposo E.P., Viswanathan G.M., Da Luz M.G.E. (2014) Stochastic optimal foraging: Tuning intensive and extensive dynamics in random searches. PLoS ONE. 9: 0-0.EnllaçDoi: 10.1371/journal.pone.0106373
Recent theoretical developments had laid down the proper mathematical means to understand how the structural complexity of search patterns may improve foraging efficiency. Under information-deprived scenarios and specific landscape configurations, Lévy walks and flights are known to lead to high search efficiencies. Based on a one-dimensional comparative analysis we show a mechanism by which, at random, a searcher can optimize the encounter with close and distant targets. The mechanism consists of combining an optimal diffusivity (optimally enhanced diffusion) with a minimal diffusion constant. In such a way the search dynamics adequately balances the tension between finding close and distant targets, while, at the same time, shifts the optimal balance towards relatively larger close-to-distant target encounter ratios. We find that introducing a multiscale set of reorientations ensures both a thorough local space exploration without oversampling and a fast spreading dynamics at the large scale. Lévy reorientation patterns account for these properties but other reorientation strategies providing similar statistical signatures can mimic or achieve comparable efficiencies. Hence, the present work unveils general mechanisms underlying efficient random search, beyond the Lévy model. Our results suggest that animals could tune key statistical movement properties (e.g. enhanced diffusivity, minimal diffusion constant) to cope with the very general problem of balancing out intensive and extensive random searching. We believe that theoretical developments to mechanistically understand stochastic search strategies, such as the one here proposed, are crucial to develop an empirically verifiable and comprehensive animal foraging theory. © 2014 Bartumeus et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Campos D., Bartumeus F., Mendez V., Espadaler X. (2014) Reorientation patterns in central-place foraging: Internal clocks and klinokinesis. Journal of the Royal Society Interface. 11: 0-0.EnllaçDoi: 10.1098/rsif.2013.0859
We study central-place foraging patterns of Aphaenogaster senilis ants at a population level by video framing individual ant trajectories in a circular arena with a nest connected to its centre. The ants naturally leave and enter the nest and forage generating non-trivial movement patterns around the nest. Our data analysis indicated that the trajectories observed can be classified into two strategies: the risk-averse strategy, which involves wandering around the nest without departing far from it and the riskprone strategy, which involves long exploration paths with periodic returns to the central region, nearby the nest. We found that both risk-prone and risk-averse strategies exhibit qualitatively the same reorientation patterns, with the time between consecutive reorientations covering a wide range of scales, and fitting a stretched exponential function. Nevertheless, differences in the temporal scales and the time variability of such reorientation events differ, together with other aspects of motion, such as average speed and turns. Our results give experimental evidence that the internal mechanisms driving reorientations in ants tend to favour frequently long relocations, as theory predicts for efficient exploration in patchy landscapes, but ants engaged in central-place foraging can modulate such behaviour to control distances from the nest. Previous works on the species support the idea that risk-prone and risk-averse strategies may reflect actual differences between individuals age and experience; these factors (age and experience) should be then relevant in modulating the internal reorientation clocks. To support the validity of our findings, we develop a random-walk model combining stretched exponential reorientation clocks with klinokinesis that fits the time length and the travelled distance distributions of the observed trajectories. © 2013 The Author(s) Published by the Royal Society. All rights reserved.
Salvador L.C.M., Bartumeus F., Levin S.A., Ryu W.S. (2014) Mechanistic analysis of the search behaviour of Caenorhabditis elegans. Journal of the Royal Society Interface. 11: 0-0.EnllaçDoi: 10.1098/rsif.2013.1092
A central question in movement research is how animals use information and movement to promote encounter success. Current random search theory identifies reorientation patterns as key to the compromise between optimizing encounters for both nearby and faraway targets, but how the balance between intrinsic motor programmes and previous environmental experience determines the occurrence of these reorientation behaviours remains unknown. We used high-resolution tracking and imaging data to describe the complete motor behaviour of Caenorhabditis elegans when placed in a novel environment (one in which food is absent). Movement in C. elegans is structured around different reorientation behaviours, and we measured how these contributed to changing search strategies as worms became familiar with their new environment. This behavioural transition shows that different reorientation behaviours are governed by two processes: (i) an environmentally informed 'extrinsic' strategy that is influenced by recent experience and that controls for area-restricted search behaviour, and (ii) a time-independent, 'intrinsic' strategy that reduces spatial oversampling and improves random encounter success. Our results show how movement strategies arise from a balance between intrinsic and extrinsic mechanisms, that search behaviour in C. elegans is initially determined by expectations developed from previous environmental experiences, and which reorientation behaviours are modified as information is acquired from new environments. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
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