Ultimate failure of the Lévy Foraging Hypothesis: Two-scale searching strategies outperform scale-free ones even when prey are scarce and cryptic

Pre-asymptotic analysis of Lévy flights

We study the properties of Lévy flights with index 0<α<2 at elapsed times smaller than those required for reaching the diffusive limit, and we focus on the bulk of the walkers' distribution rather than on its tails. On the basis of the analogs of the Kramers-Moyal expansion and of the Pawula theorem, we show that, for any α≤2/3, the bulk of the walkers' distribution occurs at wave-numbers greater than (2/α)1/(2α)≥1, and it remains non-self-similar for a time-scale longer than the Markovian time-lag of at least one order of magnitude. This result highlights the fact that for Lévy flights, the Markovianity time-lag is not the only time-scale of the process and indeed another and longer time-scale controls the transition to the familiar power-law regime in the final diffusive limit. The magnitude of this further time-scale is independent of the index α and may compromise the reliability of applications of Lévy flights to real world cases related with recurrence and transience as optimal searching, animal foraging, and site fidelity.

Geometrical multiscale tortuosity of desert ant walking trajectories

Desert ants stand out as some of the most intriguing insect navigators, having captured the attention of scientists for decades. This includes the structure of walking trajectories during goal approach and search behaviour for the nest and familiar feeding sites. In the present study, we analysed such trajectories with regard to changes in walking direction. The directional change of the ants was quantified, i.e. an angle θ between trajectory increments of a given arclength λ was computed. This was done for different length scales λ, according to our goal of analysing desert ant path characteristics with respect to length scale. First, varying λ through more than two orders of magnitude demonstrated Brownian motion characteristics typical of the random walk component of search behaviour. Unexpectedly, this random walk component was also present in - supposedly rather linear - approach trajectories. Second, there were small but notable deviations from a uniform angle distribution that is characteristic of random walks. This was true for specific search situations, mostly close to the (virtual) goal position. And third, experience with a feeder position resulted in straighter approaches and more focused searches, which was also true for nest searches, albeit to a lesser extent. Taken together, these results both verify and extend previous studies on desert ant path characteristics. Of particular interest are the ubiquitous Brownian motion signatures and specific deviations thereof close to the goal position, indicative of unexpectedly structured search behaviour.

Intermittent Search, Not Strict Lévy Flight, Evolves under Relaxed Foraging Distribution Constraints

AbstractThe survival of an animal depends on its success as a forager, and understanding the adaptations that result in successful foraging strategies is an enduring endeavour of behavioral ecology. Random walks are one of the primary mathematical descriptions of foraging behavior. Power law distributions are often used to model random walks, as they can characterize a wide range of behaviors, including Lévy walks. Empirical evidence indicates the prevalence and efficiency of Lévy walks as a foraging strategy, and theoretical work suggests an evolutionary origin. However, previous evolutionary models have assumed a priori that move lengths are drawn from a power law or other families of distributions. Here, we remove this restriction with a model that allows for the evolution of any distribution. Instead of Lévy walks, our model unfailingly results in the evolution of intermittent search, a random walk composed of two disjoint modes-frequent localized walks and infrequent extensive moves-that consistently outcompeted Lévy walks. We also demonstrate that foraging using intermittent search may resemble a Lévy walk because of interactions with the resources within an environment. These extrinsically generated Lévy-like walks belie an underlying behavior and may explain the prevalence of Lévy walks reported in the literature.

Learning and Animal Movement

Integrating diverse concepts from animal behavior, movement ecology, and machine learning, we develop an overview of the ecology of learning and animal movement. Learning-based movement is clearly relevant to ecological problems, but the subject is rooted firmly in psychology, including a distinct terminology. We contrast this psychological origin of learning with the task-oriented perspective on learning that has emerged from the field of machine learning. We review conceptual frameworks that characterize the role of learning in movement, discuss emerging trends, and summarize recent developments in the analysis of movement data. We also discuss the relative advantages of different modeling approaches for exploring the learning-movement interface. We explore in depth how individual and social modalities of learning can matter to the ecology of animal movement, and highlight how diverse kinds of field studies, ranging from translocation efforts to manipulative experiments, can provide critical insight into the learning process in animal movement.

A random walk model that accounts for space occupation and movements of a large herbivore

Animal movement has been identified as a key feature in understanding animal behavior, distribution and habitat use and foraging strategies among others. Large datasets of invididual locations often remain unused or used only in part due to the lack of practical models that can directly infer the desired features from raw GPS locations and the complexity of existing approaches. Some of them being disputed for their lack of biological justifications in their design. We propose a simple model of individual movement with explicit parameters, based on a two-dimensional biased and correlated random walk with three forces related to advection (correlation), attraction (bias) and immobility of the animal. These forces can be directly estimated using individual data. We demonstrate the approach by using GPS data of 5 red deer with a high frequency sampling. The results show that a simple random walk template can account for the spatial complexity of wild animals. The practical design of the model is also verified for detecting spatial feature abnormalities and for providing estimates of density and abundance of wild animals. Integrating even more additional features of animal movement, such as individuals’ interactions or environmental repellents, could help to better understand the spatial behavior of wild animals.

Spatially and temporally distributed data foraging decisions in disciplinary field science

How do scientists generate and weight candidate queries for hypothesis testing, and how does learning from observations or experimental data impact query selection? Field sciences offer a compelling context to ask these questions because query selection and adaptation involves consideration of the spatiotemporal arrangement of data, and therefore closely parallels classic search and foraging behavior. Here we conduct a novel simulated data foraging study-and a complementary real-world case study-to determine how spatiotemporal data collection decisions are made in field sciences, and how search is adapted in response to in-situ data. Expert geoscientists evaluated a hypothesis by collecting environmental data using a mobile robot. At any point, participants were able to stop the robot and change their search strategy or make a conclusion about the hypothesis. We identified spatiotemporal reasoning heuristics, to which scientists strongly anchored, displaying limited adaptation to new data. We analyzed two key decision factors: variable-space coverage, and fitting error to the hypothesis. We found that, despite varied search strategies, the majority of scientists made a conclusion as the fitting error converged. Scientists who made premature conclusions, due to insufficient variable-space coverage or before the fitting error stabilized, were more prone to incorrect conclusions. We found that novice undergraduates used the same heuristics as expert geoscientists in a simplified version of the scenario. We believe the findings from this study could be used to improve field science training in data foraging, and aid in the development of technologies to support data collection decisions.

Foraging behavior in visual search: A review of theoretical and mathematical models in humans and animals

Visual search (VS) is a fundamental task in daily life widely studied for over half a century. A variant of the classic paradigm-searching one target among distractors-requires the observer to look for several (undetermined) instances of a target (so-called foraging) or several targets that may appear an undefined number of times (recently named as hybrid foraging). In these searches, besides looking for targets, the observer must decide how much time is needed to exploit the area, and when to quit the search to eventually explore new search options. In fact, visual foraging is a very common search task in the real world, probably involving additional cognitive functions than typical VS. It has been widely studied in natural animal environments, for which several mathematical models have been proposed, and just recently applied to humans: Lévy processes, composite and area-restricted search models, marginal value theorem, and Bayesian learning (among others). We conducted a systematic search in the literature to understand those mathematical models and study its applicability in human visual foraging. The review suggests that these models might be the first step, but they seem to be limited to fully comprehend foraging in visual search. There are essential variables involving human visual foraging still to be established and understood. Indeed, a jointly theoretical interpretation based on the different models reviewed could better account for its understanding. In addition, some other relevant variables, such as certain individual differences or time perception might be crucial to understanding visual foraging in humans.

Nahua mushroom gatherers use area-restricted search strategies that conform to marginal value theorem predictions

We develop a method of analysis for testing the marginal value theorem (MVT) in natural settings that does not require an independent definition or mapping of patches. We draw on recent theoretical work on area-restricted search (ARS) that links turning-angle and step-size changes to geographically localized encounter-rates. These models allow us to estimate "giving-up times" using encounter-annotated GPS tracking data. Applied to a case study of Nahua mushroom foragers, these models identify distinct forms of intrapatch and interpatch search behavior, with intrapatch search transitioning to interpatch search after a predictable interval of time since the last encounter with a harvested mushroom. Our empirical estimate of giving-up time coincides with the theoretically optimal giving-up time derived under the MVT in the same environment. The MVT is currently underused in studies of human foraging and settlement patterns, due in large part to the difficulty of identifying discrete resource patches and quantifying their characteristics. Our methods mitigate the need to make such discrete maps of patches and thus have the potential to broaden the scope for empirical evaluations of the MVT and related theory in humans. Beyond studies of naturalistic foraging in humans and other animals, our approach has implications for optimization of search behavior in a range of applied fields where search dynamics must be adapted to shifting patterns of environmental heterogeneity affecting prey density and patchiness.

Evidence for encounter-conditional, area-restricted search in a preliminary study of Colombian blowgun hunters

Active search for prey is energetically costly, so understanding how foragers optimize search has been central to foraging theory. Some theoretical work has suggested that foragers of randomly distributed prey should search using Lévy flights, while work on area-restricted and intermittent search strategies has demonstrated that foragers can use the information provided by prey encounters to more effectively adapt search direction and velocity. Previous empirical comparisons of these search modes have tended to rely on distribution-level analyses, due to the difficulty of collecting event-level data on encounters linked to the GPS tracks of foragers. Here we use a preliminary event-level data-set (18.7 hours of encounter-annotated focal follows over 6 trips) to show that two Colombian blowgun hunters use adaptive encounter-conditional heuristics, not non-conditional Lévy flights, when searching for prey. Using a theoretically derived Bayesian model, we estimate changes in turning-angle and search velocity as a function of encounters with prey at lagged time-steps, and find that: 1) hunters increase average turning-angle in response to encounters, producing a more tortuous search of patches of higher prey density, but adopt more efficient uni-directional, inter-patch movement after failing to encounter prey over a sufficient period of time; and, 2) hunters reduce search velocity in response to encounters, causing them to spend more of their search time in patches with demonstrably higher prey density. These results illustrate the importance of using event-level data to contrast encounter-conditional, area-restricted search and Lévy flights in explaining the search behavior of humans and other organisms.

A general model of forager search: Adaptive encounter-conditional heuristics outperform Lévy flights in the search for patchily distributed prey

A theoretical and applied literature has suggested that foragers search using Lévy flights, since Lévy flights can maximize the efficiency of search in the absence of information on the location of randomly distributed prey. Foragers, however, often have available to them at least some information about the distribution of prey, gained either through evolved mechanisms, experience and memory, or social transmission of information. As such, we might expect selection for heuristics that make use of such information to further improve the efficiency of random search. Here we present a general model of random search behavior that includes as special cases: area-restricted search, correlated random walks, Brownian search, and Lévy flights. This generative model allows foragers to adjust search parameters based on encounter-conditional and other heuristics. Using a simulation model, we demonstrate the efficiency gains of these search heuristics, and illustrate the resulting differences in the distributions of step-size and heading angle change they imply, relative to Lévy flights. We conclude by presenting a statistical model that can be fit to empirical data and a set of testable, quantitative predictions that contrast our model of adaptive search with the Lévy flight foraging hypothesis.

The evolutionary origins of Lévy walk foraging

We study through a reaction-diffusion algorithm the influence of landscape diversity on the efficiency of search dynamics. Remarkably, the identical optimal search strategy arises in a wide variety of environments, provided the target density is sparse and the searcher’s information is restricted to its close vicinity. Our results strongly impact the current debate on the emergentist vs. evolutionary origins of animal foraging. The inherent character of the optimal solution (i.e., independent on the landscape for the broad scenarios assumed here) suggests an interpretation favoring the evolutionary view, as originally implied by the Lévy flight foraging hypothesis. The latter states that, under conditions of scarcity of information and sparse resources, some organisms must have evolved to exploit optimal strategies characterized by heavy-tailed truncated power-law distributions of move lengths. These results strongly suggest that Lévy strategies—and hence the selection pressure for the relevant adaptations—are robust with respect to large changes in habitat. In contrast, the usual emergentist explanation seems not able to explain how very similar Lévy walks can emerge from all the distinct non-Lévy foraging strategies that are needed for the observed large variety of specific environments. We also report that deviations from Lévy can take place in plentiful ecosystems, where locomotion truncation is very frequent due to high encounter rates. So, in this case normal diffusion strategies—performing as effectively as the optimal one—can naturally emerge from Lévy. Our results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

How organisms improve the search for food, mates, etc., is a key factor to their survival. Mathematically, the best strategy to look for randomly distributed re-visitable resources—under scarce information and sparse conditions—results from Lévy distributions of move lengths (the probability of taking a step ℓ is proportional to 1/ℓ²). Today it is well established that many animal species in different habitats do perform Lévy foraging. This fact has raised a heated debate, viz., the emergent versus evolutionary hypotheses. For the former, a Lévy foraging is an emergent property, a consequence of searcher-environment interactions: certain landscapes induce Lévy patterns, but others not. In this view, the optimal strategy depends on the particular habitat. The evolutionary explanation, in contrast, is that Lévy foraging strategies are adaptations that evolved via natural selection. In this article, through simulations we exhaustively analyze the influence of distinct environments on the foraging efficiency. We find that the optimal procedure is the same in all situations, provided density is low and landscape information is scarce. So, the best search strategy is remarkably independent of details. These results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

Ungulates rely less on visual cues, but more on adapting movement behaviour, when searching for forage

Finding suitable forage patches in a heterogeneous landscape, where patches change dynamically both spatially and temporally could be challenging to large herbivores, especially if they have no a priori knowledge of the location of the patches. We tested whether three large grazing herbivores with a variety of different traits improve their efficiency when foraging at a heterogeneous habitat patch scale by using visual cues to gain a priori knowledge about potential higher value foraging patches. For each species (zebra (Equus burchelli), red hartebeest (Alcelaphus buselaphus subspecies camaa) and eland (Tragelaphus oryx)), we used step lengths and directionality of movement to infer whether they were using visual cues to find suitable forage patches at a habitat patch scale. Step lengths were significantly longer for all species when moving to non-visible patches than to visible patches, but all movements showed little directionality. Of the three species, zebra movements were the most directional. Red hartebeest had the shortest step lengths and zebra the longest. We conclude that these large grazing herbivores may not exclusively use visual cues when foraging at a habitat patch scale, but would rather adapt their movement behaviour, mainly step length, to the heterogeneity of the specific landscape.

Big data analyses reveal patterns and drivers of the movements of southern elephant seals

The growing number of large databases of animal tracking provides an opportunity for analyses of movement patterns at the scales of populations and even species. We used analytical approaches, developed to cope with “big data”, that require no ‘a priori’ assumptions about the behaviour of the target agents, to analyse a pooled tracking dataset of 272 elephant seals (Mirounga leonina) in the Southern Ocean, that was comprised of >500,000 location estimates collected over more than a decade. Our analyses showed that the displacements of these seals were described by a truncated power law distribution across several spatial and temporal scales, with a clear signature of directed movement. This pattern was evident when analysing the aggregated tracks despite a wide diversity of individual trajectories. We also identified marine provinces that described the migratory and foraging habitats of these seals. Our analysis provides evidence for the presence of intrinsic drivers of movement, such as memory, that cannot be detected using common models of movement behaviour. These results highlight the potential for “big data” techniques to provide new insights into movement behaviour when applied to large datasets of animal tracking.