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.     

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.     

Measurement error causes scale-dependent threshold erosion of biological signals in animal movement data.

Recent advances in telemetry technology have created a wealth of tracking data available for many animal species moving over spatial scales from tens of meters to tens of thousands of kilometers. Increasingly, such data sets are being used for quantitative movement analyses aimed at extracting fundamental biological signals such as optimal searching behavior and scale-dependent foraging decisions. We show here that the location error inherent in various tracking technologies reduces the ability to detect patterns of behavior within movements. Our analyses endeavored to set out a series of initial ground rules for ecologists to help ensure that sampling noise is not misinterpreted as a real biological signal. We simulated animal movement tracks using specialized random walks known as Lévy flights at three spatial scales of investigation: 100-km, 10-km, and 1-km maximum daily step lengths. The locations generated in the simulations were then blurred using known error distributions associated with commonly applied tracking methods: the Global Positioning System (GPS), Argos polar-orbiting satellites, and light-level geolocation. Deviations from the idealized Lévy flight pattern were assessed for each track after incrementing levels of location error were applied at each spatial scale, with additional assessments of the effect of error on scale-dependent movement patterns measured using fractal mean dimension and first-passage time (FPT) analyses. The accuracy of parameter estimation (Lévy mu, fractal mean D, and variance in FPT) declined precipitously at threshold errors relative to each spatial scale. At 100-km maximum daily step lengths, error standard deviations of > or = 10 km seriously eroded the biological patterns evident in the simulated tracks, with analogous thresholds at the 10-km and 1-km scales (error SD > or = 1.3 km and 0.07 km, respectively). Temporal subsampling of the simulated tracks maintained some elements of the biological signals depending on error level and spatial scale. Failure to account for large errors relative to the scale of movement can produce substantial biases in the interpretation of movement patterns. This study provides researchers with a framework for understanding the limitations of their data and identifies how temporal subsampling can help to reduce the influence of spatial error on their conclusions.     

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.     

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.     

Distinguishing between Lévy walks and strong alternative models

Lévy walks are a widely used but contentious model of animal movement patterns. They are contentious because they have been wrongly ascribed to some animal species through use of incorrect statistical methods and because they have not been adequately compared against strong alternative models, such as composite correlated random walks. This lack of comparison has been partly because the strong alternative models do not have simple likelihood functions. Here I show that power-spectra and the distribution of the first significant digits (the leading non-zero digits) of the step lengths can distinguish between Lévy walks and composite correlated random walks. Using these diagnostic tools, I bolster previous claims that honey bees use a movement strategy that can be approximated by Lévy walks when searching for their hive or for a food source.     

Fishing sustainability via inclusion of man in predator-prey models: A case study in Lago Preto, Manacapuru, Amazonas

We modeled a fishery's system with two types of fishermen, commercial and subsistence fishermen, who exploit the fish stock at the Amazonian floodplain lakes. In the first model, we combined the Lotka-Volterra equations with Verhulst's Logistic model, by inserting hydrological cycle oscillations. The second model was based on the equations proposed by Berryman, which reflect the predator's functional response in relation to the prey's population behavior, taking into account the hydrological cycle. In both models, commercial fishermen and local direct consumers (called riverside dwellers - riverines - in the model), were considered the only predators acting upon fishing stocks. Primary data were collected in 48 riverside homes throughout 2006. The total number of interviewees corresponds to 69.6% of the universe of homes in the community defined as study area. The riverines were the predators that showed capacity to eliminate the opponent predators (commercial fishermen). The best scenery obtained regarding the number of prey, was the one that showed only commercial fishermen in the region. On the other hand, the simulations show that the coexistence is possible among predators, and between predators and their prey. The seasonal model with functional response provides a better response in relation to the system's current situation and to the established modeling conditions than the Lotka-Volterra seasonal model. The seasonal model with functional response also showed a better response pattern in all scenarios, with oscillations taking place more gradually, both for variations associated with the flooding pulse and for relations between predators and prey.     

Displaced honey bees perform optimal scale-free search flights

Honey bees (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, but the nature and efficiency of their searching strategy have hitherto been unknown. We used harmonic radar to record the flight paths of honey bees that were searching for their hives. Our subsequent analysis of these paths revealed that they can be represented by a series of straight line segments that have a scale-free, Lévy distribution with an inverse-square-law tail. We show that these results, combined with the "no preferred direction" characteristic of the segments, demonstrate that the bees were flying an optimal search pattern. Lévy movements have already been identified in a number of other animals. Our results are the best reported example where the movements are mostly attributable to the adoption of an optimal, scale-free searching strategy.     

Underwater and above-water search patterns of an Arctic seabird: reduced searching at small spatiotemporal scales

How predators vary search patterns in response to prey predictability is poorly known. For example, marine invertebrates may be predictable but of low energy value, while fish may be of higher energy value but unpredictable at large (pelagic schools) or small (solitary benthics) spatial scales. We investigated the search patterns of the thick-billed murre (Uria lomvia), an Arctic seabird feeding on invertebrates, pelagic fish, or benthic fish. Foraging ranges at the Coats Island colony are generally smaller (<240 min per trip) than at larger colonies, and many birds specialize in foraging tactics and diet. Underwater search times for benthic fish were higher than for pelagic fish or invertebrates while above-water search times for pelagic fish were higher than for benthic fish or invertebrates. There were few stops during trips. Total trip time, flying time, number of flights, and number of dives were intercorrelated and increased with prey energy content, suggesting that longer trips involved fewer prey encounters due to selection of higher-quality, but rarer, prey items. Flight times were not Lévy-distributed and seabirds may have used area-restricted searches. The high degree of specialization, apparent absence of information center effects, and reduced above-water searching times may be linked to the relatively small colony size and the resulting short commuting distances to feeding areas, leading to greater prey predictability. We concluded that prey predictability over various scales affected predator search patterns.     

Individual-based modelling of fishermen search behaviour with neural networks and reinforcement learning

A model to mimic the search behaviour of fishermen is built with two neural networks to cope with two separate decision-making processes in fishing activities. One neural network deals with decisions to stay or move to new fishing grounds and the other is constructed for the purpose of finding prey within the fishing areas. Some similarities with the behaviour of real fishermen are found: concentrated local search once a prey has been located to increase the probability of remaining near a prey patch and the straightforward movement to other fishing grounds. The artificial fisherman prefers areas near the port when conditions in different fishing grounds are similar or when there is high uncertainty in its world. In the latter case a reluctance to navigate to other areas is observed. The artificial fisherman selects areas with higher concentration of prey, even if they are far from the port of departure, unless a high uncertainty is related to the fishing ground. Connected areas are preferred and followed in orderly fashion if a higher catch is expected. The observed behaviour of the artificial fisherman in uncertain scenarios can be described as a risk-averse attitude. The approach seems appropriate for an individual-based modelling of fishery systems, focusing on the learning and adaptive characteristics of fishermen and on interactions that take place at a fine scale.     

Reduction of the impact produced by sea lions on the fisheries in Mexico

Activities of fishermen in the Bay of La Paz, B. C. S. are focused to satisfy the local demand of fish and shellfish by using approximately 300 small outboard crafts equipped with gillnets.Sea lions in this Bay attack the fishes captured damaging both product and gear. We did experimental gillnet throws to determine the frequency and preferences of sea lions in fishing areas. 52 experimental gillnet throws with time averages of 2 hr were performed, rending an average of 30 kg of fish captured and less than 10% of damages to the net.. Traditional fishermen in this Bay usually left the net the whole night, (approximately 7.50 hr, obtaining an average of 50 kg of captured fish, but the damages to the nets is in average of 40%. The cost-benefit balance comparing our alternative fishing method, which includes the use of the gillnets during the afternoon, watching for sea lions and retiring the nets at their arrival, it is more sustainable and profitable than the traditional fishing method currently used by the local fishermen. This paper suggests how to minimize the harmful effects of the sea lions on the fishermen productivity and gear, maximizing the production and reducing the damage. Our alternative method is applicable to other regions where this harmful interaction is taking place. We conclude that the coexistence of sea lions-fisheries is feasible, by applying the simple measures that we propose.     

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     

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.     

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.     

A General Model for Analyzing Data from Mark-recapture Experiments with an Application to the Pacific Halibut

A general model is developed to examine the patterns of the regional movement of tagged and released fish from mark-recapture experiments. It is a stochastic model that incorporates fishing mortality, natural mortality, fish movement, tag-shedding, and different rates of reporting. A likelihood function is constructed for estimating its parameters. We used this model to analyze data on the Pacific halibut from mark-recapture experiments conducted by the International Pacific Halibut Commission (IPHC), with a total of 36,058 releases from 1982 to 1986 and 5,826 recoveries from 1982 to 2000. We estimated their rates of movement among IPHC management areas, along with their instantaneous rates of natural and fishing mortalities. Our analysis revealed that fish movement was not significant among areas, with a resident probability of > 0.92. This lends support to the IPHC catch-at-age stock assessment model (which has no built-in movement components). The estimated instantaneous rate of natural mortality (0.198 year⁻¹) lies between that assumed in all IPHC stock assessments before 1998 (0.20 year⁻¹) and that from 1999 onwards (0.15 year⁻¹). The estimates of the instantaneous rates of fishing mortality were consistent with those from the IPHC stock assessment model. Received: April 2003 / Revised: May 2005     

Fishery resource management in Chilika lagoon: a study on coastal conservation in the Eastern Coast of India

A significant proportion of the fishing population comprises small-scale fishermen and many studies illustrate that these people are exploited by middlemen in the process of fish marketing combined with money lending. The negative dependency gives rise to poverty and triggers indiscriminate fish catch that threatens fishery resources depletion. This article explores the root causes of failures in resource-led development from the viewpoint of coastal resource conservation. The study presents a case study of Chilika lagoon, India and focuses on the interaction between small-scale fishermen and middlemen. The findings reveal that most of the small-scale fishermen have been exploited by specific middlemen and the underlying causes of the present fish marketing structure stem from (i) indebtedness and (ii) the unstable situation because of perpetual conflicts over fishery resources among the fishers across Chilika lagoon. Based on these observations, this article presents some recommendations on fishery resource conservation from the perspective of a fish marketing structure.

Managing a protected marine area for the conservation of critically endangered vaquita (Phocoena sinus Norris, 1958) in the Upper Gulf of California

We analysed fisheries trends in the northern region of the Gulf of California, within the Biosphere Reserve of the Upper Gulf of California and Colorado Delta River and the Vaquita Refuge Area, and suggest measures to protect the vaquita, Phocoena sinus. We compiled and analysed catch reports of artisanal fishermen in the three fishing communities of the Upper Gulf of California (San Felipe in the State of Baja California, and Golfo de Santa Clara and Puerto Peñasco in the State of Sonora) from 1995 to 2007. This information was categorised with respect to geographic information systems, and all fishing sites within two marine protected areas in the region were identified. In addition, from a survey based on direct interviews with artisanal fishermen in each of the three ports, we identified that 23% of fishermen will continue fishing despite on-going fishing buy-out programmes in the region. We suggest several specific courses of action to decrease the fishing impact on this critically endangered cetacean. However, given the critical situation of this critically endangered species, it is very uncertain whether enforcing a no-take zone within the biosphere reserve and the Vaquita Refuge Area, or even a wider fishing moratorium, will be enough to save this endangered species from extinction.     

Applications of Very High‐Resolution Imagery in the Study and Conservation of Large Predators in the Southern Ocean

The Southern Ocean is one of the most rapidly changing ecosystems on the planet due to the effects of climate change and commercial fishing for ecologically important krill and fish. Because sea ice loss is expected to be accompanied by declines in krill and fish predators, decoupling the effects of climate and anthropogenic changes on these predator populations is crucial for ecosystem‐based management of the Southern Ocean. We reviewed research published from 2007 to 2014 that incorporated very high‐resolution satellite imagery to assess distribution, abundance, and effects of climate and other anthropogenic changes on populations of predators in polar regions. Very high‐resolution imagery has been used to study 7 species of polar animals in 13 papers, many of which provide methods through which further research can be conducted. Use of very high‐resolution imagery in the Southern Ocean can provide a broader understanding of climate and anthropogenic forces on populations and inform management and conservation recommendations. We recommend that conservation biologists continue to integrate high‐resolution remote sensing into broad‐scale biodiversity and population studies in remote areas, where it can provide much needed detail. Aplicaciones de Imágenes de Muy Alta Resolución en el Estudio y Conservación de Grandes Depredadores en el Océano Antártico     

Modeling the Effects of Fishing and Implications for the Design of Marine Protected Areas: Juvenile Fish Responses to Variations in Seafloor Habitat

Abstract: A number of recent studies have linked post-settlement survivorship of Atlantic cod (  Gadus morhua ) with the complexity of the seafloor to which fish settle. Survivorship is greater in habitats of higher complexity (e.g., pebble-cobble substratum with emergent epifauna> pebble-cobble> sand), where cover provides shelter from predators. Fishing with mobile gear such as bottom trawls and dredges reduces the complexity of seafloor habitats. We used a dynamic model to (1) link patterns in habitat-mediated survivorship of post-settlement juvenile cod with spatial variations in habitat complexity, (2) simulate habitat change based on fishing activities, and (3) determine the role of marine protected areas in enhancing recruitment success. Density-dependent natural mortality was specified as three alternative functional response curves to assess the influence of different predator foraging strategies on juvenile survivorship during the first 12 months of demersal existence. We applied the model to a theoretical patch of hard-bottom substrata and to a case study based on seafloor habitat distributions at Stellwagen Bank National Marine Sanctuary (Gulf of Maine, Northwest Atlantic). Our results demonstrate that patterns in the shape of response surfaces that show the relationship between juvenile cod survivorship and density as well as movement rate were similar regardless of functional response type, that juvenile cod movement rates and post-settlement density were critical for predicting the effects of marine protected-area size on survivorship, and that habitat change caused by fishing has significant negative effects on juvenile cod survivorship and use of marine protected areas can ameliorate such effects.     

Comanagement Practices Enhance Fisheries in Marine Protected Areas

Abstract:  Fishing activities worldwide have dramatically affected marine fish stocks and ecosystems. Marine protected areas (MPAs) with no-take zones may enhance fisheries, but empirical evidence of this is scant. We conducted a 4-year survey of fish catches around and within an MPA that was previously fully closed to fishing and then partially reopened under regulated comanaged fishing. In collaboration with the fishers and the MPA authority, we set the fishing effort and selected the gear to limit fishing impact on key fish predators, juvenile fish stage, and benthic communities and habitats. Within an adaptive comanagement framework, fishers agreed to reduce fishing effort if symptoms of overfishing were detected. We analyzed the temporal trends of catch per unit of effort (CPUE) of the whole species assemblages and CPUE of the four most valuable and frequent species observed inside the opened buffer zone and outside the MPA investigated. After the comanaged opening, CPUE first declined and then stabilized at levels more than twice that of catches obtained outside the MPA. Our results suggest that working closely with fishers can result in greater fisheries catches. Partial protection of coastal areas together with adaptive comanagement involving fishers, scientists, and managers can effectively achieve conservation and fishery management goals and benefit fishing communities and alleviate overfishing.