Detecting Lévy walks without turn designation

Many organisms have been reported to have movement patterns that are well approximated as Lévy walks. This is typically because distributions of straight line distances between consecutive significant turns in movement paths have heavy power law tails. This diagnostic tool has been called into question because there is currently no standard, unambiguous way to identify significant turns. Even if such a way could be found, statistical analyses based on significant turns cannot account for actual movements made between turns and as a consequence cannot distinguish between true Lévy walks and other fractal random walks such as Lévy modulated correlated random walks where organisms randomly meander rather than move in straight lines between consecutive reorientation events. Here, I show that structure functions (i.e. moments of net displacements made across fixed time intervals) can distinguish between different kinds of Lévy walks and between Lévy walks and random walks with a few scales such as composite correlated random walks and correlated random walks. Distinguishing between these processes will lead to a better understanding of how and why animals perform Lévy walks and help bridge the apparent divide between correlated random walks and Lévy walks. Structure functions do not require turn identification and instead take account of entire movement paths. Using this diagnostic tool, I bolster previous claims that honeybees use a movement strategy that can be approximated by Lévy walks when searching for their hive. I also show how structure functions can be used to establish the extent of self-similar behaviour in meandering Lévy walks.     

Overturning conclusions of Lévy flight movement patterns by fishing boats and foraging animals

A surprisingly diverse variety of foragers have previously been concluded to exhibit movement patterns known as Lévy flights, a special type of random walk. These foragers range in size from microzooplankton in experiments to fishermen in the Pacific Ocean and the North Sea. The Lévy flight conclusion implies that all the foragers have similar scale-free movement patterns that can be described by a single dimensionless parameter, the exponent micro of a power-law (Pareto) distribution. However, the previous conclusions have been made using methods that have since been shown to be problematic: inaccurate techniques were used to estimate micro, and the power-law distribution was usually assumed to hold without testing any alternative hypotheses. Therefore, I address the open question of whether the previous data still support the Lévy flight hypothesis, and thus determine whether Lévy flights really are so ubiquitous in ecology. I present a comprehensive reanalysis of 17 data sets from seven previous studies for which Lévy flight behavior had been concluded, covering marine, terrestrial, and experimental systems from four continents. I use the modern likelihood and Akaike weights approach to test whether simple alternative models are more supported by the data than Lévy flights. The previously estimated values of the power-law exponent micro do not match those calculated here using the accurate likelihood approach, and almost all of them lie outside of the likelihood-based 95% confidence intervals. Furthermore, the original power-law Lévy flight model is overwhelmingly rejected for 16 out of the 17 data sets when tested against three other simple models. For one data set, the data are consistent with coming from a bounded power-law distribution (a truncated Lévy flight). For three other data sets, an exponential distribution corresponding to a simple Poisson process is suitable. Thus, Lévy flight movement patterns are not the common phenomena that was once thought, and are not suitable for use as ecosystem indicators for fisheries management, as has been proposed.     

How many animals really do the Lévy walk?

Lévy walks (LW) are superdiffusive and scale-free random walks that have recently emerged as a new conceptual tool for modeling animal search paths. They have been claimed to be more efficient than the "classical" random walks, and they also seem able to account for the actual search patterns of various species. This suggests that many animals may move using a LW process. LW patterns look like the actual search patterns displayed by animals foraging in a patchy environment, where extensive and intensive searching modes alternate, and which can be generated by a mixture of classical random walks. In this context, even elementary composite Brownian walks are more efficient than LW but may be confounded with them because they present apparent move-length-heavy tail distributions and superdiffusivity. The move-length "survival" distribution (i.e., the cumulative number of moves greater than any given threshold) appears to be a better means to highlight a LW pattern. Even once such a pattern has been clearly identified, it remains to determine how it was actually generated, because a LW pattern is not necessarily produced by a LW process but may emerge from the way the animal interacted with the environment structure through more classical movement processes. In any case, emergent movement patterns should not be confused with the processes that gave rise to them.     

Honeybees use a Lévy flight search strategy and odour-mediated anemotaxis to relocate food sources

The availability of food resources changes over time and space, and foraging animals are constantly faced with choices about how to respond when a resource becomes depleted. We hypothesise that flying insects like bees discover new food sources using an optimal Lévy flight searching strategy and odour-mediated anemotaxis, as well as visual cues. To study these searching patterns, foraging honeybees were trained to a scented feeder which was then removed. Two new unrewarding feeders, or ‘targets’, were then positioned up- and downwind of the original location of the training feeder. The subsequent flight patterns of the bees were recorded over several hundred metres using harmonic radar. We show that the flight patterns constitute an optimal Lévy flight searching strategy for the location of the training feeder, a strategy that is also optimal for the location of alternative food sources when patchily distributed. Scented targets that were positioned upwind of the original training feeder were investigated most with the numbers of investigations declining with increasing distance from the original feeder. Scented targets in downwind locations were rarely investigated and unscented targets were largely ignored, despite having the same visual appearance as the rewarding training feeder.     

Are Lévy flight patterns derived from the Weber–Fechner law in distance estimation?

Honeybees (Apis mellifera) are regularly faced with the task of navigating back to their hives from remote food sources. They have evolved several methods to do this, including compass-directed “vector” flights and the use of landmarks. If these hive-centered mechanisms are disrupted, bees revert to searching for the hive, using an optimal Lévy flight searching strategy. The same strategy is adopted when a food source at a known location ceases to be available. Here, we show that the programming for this Lévy strategy does not need to be very sophisticated or clever on the bee’s part, as Lévy flight patterns can be derived from the Weber–Fechner law in a bee’s odometer. Odometry errors of a different kind occur in desert ants (Cataglyphis spp., Melophorus bagoti). The searching behaviors of these ants are very similar in overall structure to that of honeybees but do not display any Lévy flight characteristics. We suggest that errors in the estimation of distance can be implicitly involved in shaping the structure of systematic search behavior and should not be regarded as merely deficiencies in the odometer.     

Lévy flight patterns are predicted to be an emergent property of a bumblebees’ foraging strategy

Bumblebees forage uninterrupted for long periods of time because they are not distracted by sex or territorial defense and have few predators. This has led to a long running debate about whether bumblebees forage optimally. This debate has been enriched by the possibility that bumblebees foraging within clover patches have flight patterns that can be approximated by Lévy flights. Such flight patterns optimise the success of random searches. Bumblebees foraging within a flower patch tend to approach the nearest flower but then often depart without landing or probing it if it has been visited previously; unvisited flowers are not rejected in this manner. Here, this foraging behaviour has been replicated in numerical simulations. Lévy flight patterns are found to be an inconsequential emergent property of a bumblebees’ foraging behaviour. Lévy flights are predicted to emerge when bees reject at least 99% of previously visited flowers. A foraging bumblebee can certainly empty a clover flower head of nectar in one visit, but lower rates of rejection are observed for many other flowers. These findings suggest that Lévy flight patterns in foraging bumblebees are rare and specific to a few flower species and that if they exist, then they are not part of an innate, evolved optimal searching strategy.     

Comparisons in subcellular and biochemical behaviors of cadmium between Iow-Cd and high-Cd accumulation cultivars of pakchoi （Brassica chinensis L.）

Subcellular distributions and chemical forms of cadmium （Cd） in the leaves, stems and roots were investigated in low-Cd accumulation cultivars and high-Cd accumulation cultivars ofpakchoi （Brassica chinensis L.）. Root cell wall played a key role in limiting soil Cd from entering the protoplast, especially in the low-Cd cultivars. The high-Cd cultivars had significantly higher leaf and stem Cd concentrations than the low-Cd cultivars in cell wall fraction, chloroplast/trophoplast fraction, organelle fraction and soluble fraction. In low-Cd cultivars, which were more sensitive and thus had greater physiological needs of Cd detoxification than high-Cd cultivars, leaf vacuole sequestrated higher proportions of Cd. Cd in the form of pectate/protein complexes （extracted by 1 tool. L~ NaC1） played a decisive role in Cd translocation from root to shoot, which might be one of the mechanisms that led to the differences in shoot Cd accumulation between the two types of cultivars. Furthermore, the formation of Cd- phosphate complexes （extracted by 2% HAc） was also involved in Cd detoxification within the roots of pakchoi under high Cd stress, suggesting that the mechanisms of Cd detoxification might be different between low- and high-Cd cultivars.     

Lévy walk patterns in the foraging movements of spider monkeys (<Emphasis Type="Italic">Ateles geoffroyi</Emphasis>)

Scale invariant patterns have been found in different biological systems, in many cases resembling what physicists have found in other, nonbiological systems. Here we describe the foraging patterns of free-ranging spider monkeys (Ateles geoffroyi) in the forest of the Yucatan Peninsula, Mexico and find that these patterns closely resemble what physicists know as Lévy walks. First, the length of a trajectorys constituent steps, or continuous moves in the same direction, is best described by a power-law distribution in which the frequency of ever larger steps decreases as a negative power function of their length. The rate of this decrease is very close to that predicted by a previous analytical Lévy walk model to be an optimal strategy to search for scarce resources distributed at random. Second, the frequency distribution of the duration of stops or waiting times also approximates to a power-law function. Finally, the mean square displacement during the monkeys first foraging trip increases more rapidly than would be expected from a random walk with constant step length, but within the range predicted for Lévy walks. In view of these results, we analyze the different exponents characterizing the trajectories described by females and males, and by monkeys on their own and when part of a subgroup. We discuss the origin of these patterns and their implications for the foraging ecology of spider monkeys.Communicated by D. Watts     

Population genetics of the fissiparous holothurians Stichopus chloronotus and Holothuria atra (Aspidochirotida): a comparison between the Torres Strait and La Réunion

Collections of about 50 individuals from each of five populations of the fissiparous holothurian species Stichopus chloronotus and four populations of Holothuria atra were made in 1999. These populations were located in the Torres Strait (western Pacific) and La Réunion (western Indian Ocean). Allozyme electrophoretic surveys of five (S. chloronotus) and six (H. atra) loci were conducted to compare patterns of asexual reproduction and to investigate connectivity between regions separated by large geographic distances. Deviations from genotype frequencies expected under Hardy-Weinberg equilibrium, mostly heterozygote excesses, were observed in all populations of both species. The maximum contribution of sexual reproduction (calculated as the maximum number of sexually produced individuals: sample size=N*/N_i) was similar for all S. chloronotus (58-64%) and H. atra (76-92%) populations, and on the same level as previously reported for midshelf reefs of the Great Barrier Reef. The higher values in the latter species indicated greater contributions of asexual reproduction to S. chloronotus populations. Variability was strongly reduced in S. chloronotus populations at La Réunion, with only one locus being variable in that population. When the dataset was reduced to one representative per multi-locus genotype per population to reduce the effect of asexual reproduction on calculations on gene flow, F_ST values were not significantly different from zero, suggesting high gene flow between these regions. However UPGMA cluster analyses using Rogers' genetic distance, roughly clustered populations by region. In the case of H. atra, pooled populations within each region were significantly different from those of the other region. Thus, although some restrictions in gene flow and greater genetic distances between the regions may exist, those differences are distinctly less than those reported in previous studies on echinoderms over similar geographic scales. Despite the importance of asexual reproduction for the maintenance of local population size, this study also confirmed that the potential for widespread dispersal mediated by sexually produced larvae is large.     

Modelling of coastal vulnerability in the stretch between the beaches of Porto de Mós and Falésia, Algarve (Portugal)

Coastal systems are characterized by high geophysical and biophysical sensitivity as being their massive occupation a serious issue concerning their self-regulation. Worldwide coastal areas are exposed to problems such as coastal erosion, degradation and destruction of marine habitats, pollution and rising sea level. Thus, it is crucial to design models of coastal vulnerability assessment to ensure a better management of coastal areas. This study characterizes the vulnerability of the coastal stretch between the beaches of Porto de Mós and Falésia in the Algarve, corresponding to 52?km of the south coast of Portugal. The expansion of urban areas was modelled using a Cellular Automata (CA) based approach. Results show that 65% of the coastal stretch has high and very high vulnerability caused by both physical and human factors. Results further indicate that urban growth may interfere with the natural evolution of the coastal geomorphology. The scenario of urban expansion for the year 2015 highlights the need to develop effective urban planning processes to ensure a correct balance between the geophysical resilience of coastal systems and the promotion of the coastal sector as a strategic asset for the regional and national economy.