Assessing Lévy walks as models of animal foraging

Random walks with spatial and temporal resets can explain individual and colony-level searching patterns in ants

Central place foragers, such as many ants, exploit the environment around their nest. The extent of their foraging range is a function of individual movement, but how the movement patterns of large numbers of foragers result in an emergent colony foraging range remains unclear. Here, we introduce a random walk model with stochastic resetting to depict the movements of searching ants. Stochastic resetting refers to spatially resetting at random times the position of agents to a given location, here the nest of searching ants. We investigate the effect of a range of resetting mechanisms and compare the macroscopic predictions of our model to laboratory and field data. We find that all returning mechanisms very robustly ensure that scouts exploring the surroundings of a nest will be exponentially distributed with distance from the nest. We also find that a decreasing probability for searching ants to return to their nest is compatible with empirical data, resulting in scouts going further away from the nest as the number of foraging trips increases. Our findings highlight the importance of resetting random walk models for depicting the movements of central place foragers and nurture novel questions regarding the searching behaviour of ants.

Caterpillar movement mediates spatially local interactions and determines the relationship between population density and contact

BACKGROUND

While interactions in nature are inherently local, ecological models often assume homogeneity across space, allowing for generalization across systems and greater mathematical tractability. Density-dependent disease models are a prominent example of models that assume homogeneous interactions, leading to the prediction that disease transmission will scale linearly with population density. In this study, we examined how the scale of larval butterfly movement interacts with the resource landscape to influence the relationship between larval contact and population density in the Baltimore checkerspot (Euphydryas phaeton). Our study was inspired by the recent discovery of a viral pathogen that is transmitted horizontally among Baltimore checkerspot larvae.

METHODS

We used multi-year larvae location data across six Baltimore checkerspot populations in the eastern U.S. to test whether larval nests are spatially clustered. We then integrated these spatial data with larval movement data in different resource contexts to investigate whether heterogeneity in spatially local interactions alters the assumed linear relationship between larval nest density and contact. We used Correlated Random Walk (CRW) models and field observations of larval movement behavior to construct Probability Distribution Functions (PDFs) of larval dispersal, and calculated the overlap in these PDFs to estimate conspecific contact within each population.

RESULTS

We found that all populations exhibited significant spatial clustering in their habitat use. Subsequent larval movement rates were influenced by encounters with host plants and larval age, and under many movement scenarios, the scale of predicted larval movement was not sufficient to allow for the "homogeneous mixing" assumed in density dependent disease models. Therefore, relationships between population density and larval contact were typically non-linear. We also found that observed use of available habitat patches led to significantly greater contact than would occur if habitat use were spatially random.

CONCLUSIONS

These findings strongly suggest that incorporating larval movement and spatial variation in larval interactions is critical to modeling disease outcomes in E. phaeton. Epidemiological models that assume a linear relationship between population density and larval contact have the potential to underestimate transmission rates, especially in small populations that are already vulnerable to extinction.

Smart active particles learn and transcend bacterial foraging strategies

Throughout evolution, bacteria and other microorganisms have learned efficient foraging strategies that exploit characteristic properties of their unknown environment. While much research has been devoted to the exploration of statistical models describing the dynamics of foraging bacteria and other (micro-) organisms, little is known, regarding the question of how good the learned strategies actually are. This knowledge gap is largely caused by the absence of methods allowing to systematically develop alternative foraging strategies to compare with. In the present work, we use deep reinforcement learning to show that a smart run-and-tumble agent, which strives to find nutrients for its survival, learns motion patterns that are remarkably similar to the trajectories of chemotactic bacteria. Strikingly, despite this similarity, we also find interesting differences between the learned tumble rate distribution and the one that is commonly assumed for the run and tumble model. We find that these differences equip the agent with significant advantages regarding its foraging and survival capabilities. Our results uncover a generic route to use deep reinforcement learning for discovering search and collection strategies that exploit characteristic but initially unknown features of the environment. These results can be used, e.g., to program future microswimmers, nanorobots, and smart active particles for tasks like searching for cancer cells, micro-waste collection, or environmental remediation.

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

The biology and ecology of the basking shark: A review

Here we review the literature on the basking shark (Cetorhinus maximus, Gunnerus, 1765), well known as the second largest extant shark (and fish) species globally. Previous reviews were published by Kunzlik in 1988 and Sims in 2008, but in the last 15 years modern electronic and DNA sequencing technologies have resulted in considerable advances in our knowledge of the species' behaviour and ecology. Basking sharks are planktivores and under appropriate conditions spend prolonged periods at the ocean surface feeding on copepod prey that primarily make up their diet, the behaviour that gave rise to their common name. In general, they are migratory and move into higher latitude waters during the summer months, when loose surface-feeding aggregations may form at favoured sites, the best known of which at present occur at hotspots on the west coasts of Britain and Ireland. The species is found circumglobally in temperate waters, but they are also now known on occasion to migrate at depth between northern and southern hemispheres, as well as across oceans within the northern hemisphere. In the past basking shark were more abundant across much of their range, but, consequent on targeted fisheries and in some places intentional eradication, became everywhere scarce, with recent population recovery in the north-east Atlantic being the result of protective measures initiated in the 1990s. Despite their charismatic nature, some of their most fundamental biological processes including copulation, gestation and birth remain largely unknown, due to their migratory and often deep-water lifestyle. In contrast, the deployment of small-scale archival and satellite tags has revealed the details of both broadscale migratory movements and horizontal and vertical foraging behaviours. Recent genetic studies support evidence suggesting a degree of site fidelity in relation to seasonal feeding grounds, which likely explains why in the past local populations have collapsed following periods of intensive fishing. Other recent research using aerial drones and towed cameras has revealed within loose feeding aggregations elements of social behaviour that may have a courtship function as well as enhance feeding efficiency.

Examination of head versus body heading may help clarify the extent to which animal movement pathways are structured by environmental cues?

Understanding the processes that determine how animals allocate time to space is a major challenge, although it is acknowledged that summed animal movement pathways over time must define space-time use. The critical question is then, what processes structure these pathways? Following the idea that turns within pathways might be based on environmentally determined decisions, we equipped Arabian oryx with head- and body-mounted tags to determine how they orientated their heads - which we posit is indicative of them assessing the environment - in relation to their movement paths, to investigate the role of environment scanning in path tortuosity. After simulating predators to verify that oryx look directly at objects of interest, we recorded that, during routine movement, > 60% of all turns in the animals' paths, before being executed, were preceded by a change in head heading that was not immediately mirrored by the body heading: The path turn angle (as indicated by the body heading) correlated with a prior change in head heading (with head heading being mirrored by subsequent turns in the path) twenty-one times more than when path turns occurred due to the animals adopting a body heading that went in the opposite direction to the change in head heading. Although we could not determine what the objects of interest were, and therefore the proposed reasons for turning, we suggest that this reflects the use of cephalic senses to detect advantageous environmental features (e.g. food) or to detect detrimental features (e.g. predators). The results of our pilot study suggest how turns might emerge in animal pathways and we propose that examination of points of inflection in highly resolved animal paths could represent decisions in landscapes and their examination could enhance our understanding of how animal pathways are structured.

Lévy movements and a slowly decaying memory allow efficient collective learning in groups of interacting foragers

Many animal species benefit from spatial learning to adapt their foraging movements to the distribution of resources. Learning involves the collection, storage and retrieval of information, and depends on both the random search strategies employed and the memory capacities of the individual. For animals living in social groups, spatial learning can be further enhanced by information transfer among group members. However, how individual behavior affects the emergence of collective states of learning is still poorly understood. Here, with the help of a spatially explicit agent-based model where individuals transfer information to their peers, we analyze the effects on the use of resources of varying memory capacities in combination with different exploration strategies, such as ordinary random walks and Lévy flights. We find that individual Lévy displacements associated with a slow memory decay lead to a very rapid collective response, a high group cohesion and to an optimal exploitation of the best resource patches in static but complex environments, even when the interaction rate among individuals is low.

Three-Dimensional Multi-Agent Foraging Strategy Based on Local Interaction

This paper considers a multi-agent foraging problem, where multiple autonomous agents find resources (called pucks) in a bounded workspace and carry the found resources to a designated location, called the base. This article considers the case where autonomous agents move in unknown 3-D workspace with many obstacles. This article describes 3-D multi-agent foraging based on local interaction, which does not rely on global localization of an agent. This paper proposes a 3-D foraging strategy which has the following two steps. The first step is to detect all pucks inside the 3-D cluttered unknown workspace, such that every puck in the workspace is detected in a provably complete manner. The next step is to generate a path from the base to every puck, followed by collecting every puck to the base. Since an agent cannot use global localization, each agent depends on local interaction to bring every puck to the base. In this article, every agent on a path to a puck is used for guiding an agent to reach the puck and to bring the puck to the base. To the best of our knowledge, this article is novel in letting multiple agents perform foraging and puck carrying in 3-D cluttered unknown workspace, while not relying on global localization of an agent. In addition, the proposed search strategy is provably complete in detecting all pucks in the 3-D cluttered bounded workspace. MATLAB simulations demonstrate the outperformance of the proposed multi-agent foraging strategy in 3-D cluttered workspace.

Linking neural circuits to the mechanics of animal behavior in Drosophila larval locomotion

The motions that make up animal behavior arise from the interplay between neural circuits and the mechanical parts of the body. Therefore, in order to comprehend the operational mechanisms governing behavior, it is essential to examine not only the underlying neural network but also the mechanical characteristics of the animal's body. The locomotor system of fly larvae serves as an ideal model for pursuing this integrative approach. By virtue of diverse investigation methods encompassing connectomics analysis and quantification of locomotion kinematics, research on larval locomotion has shed light on the underlying mechanisms of animal behavior. These studies have elucidated the roles of interneurons in coordinating muscle activities within and between segments, as well as the neural circuits responsible for exploration. This review aims to provide an overview of recent research on the neuromechanics of animal locomotion in fly larvae. We also briefly review interspecific diversity in fly larval locomotion and explore the latest advancements in soft robots inspired by larval locomotion. The integrative analysis of animal behavior using fly larvae could establish a practical framework for scrutinizing the behavior of other animal species.

The potential underlying mechanisms during learning flights

Hymenopterans, such as bees and wasps, have long fascinated researchers with their sinuous movements at novel locations. These movements, such as loops, arcs, or zigzags, serve to help insects learn their surroundings at important locations. They also allow the insects to explore and orient themselves in their environment. After they gained experience with their environment, the insects fly along optimized paths guided by several guidance strategies, such as path integration, local homing, and route-following, forming a navigational toolkit. Whereas the experienced insects combine these strategies efficiently, the naive insects need to learn about their surroundings and tune the navigational toolkit. We will see that the structure of the movements performed during the learning flights leverages the robustness of certain strategies within a given scale to tune other strategies which are more efficient at a larger scale. Thus, an insect can explore its environment incrementally without risking not finding back essential locations.

Optimal foraging strategies for mutually avoiding competitors

Many animals are known to exhibit foraging patterns where the distances they travel in a given direction are drawn from a heavy-tailed Lévy distribution. Previous studies have shown that, under sparse and random resource conditions, solitary non-destructive (with regenerating resources) foragers perform a maximally efficient search with Lévy exponent μ equal to 2, while for destructive foragers, efficiency decreases with μ monotonically and there is no optimal μ. However, in nature, there also exist situations where multiple foragers, displaying avoidance behavior, interact with each other competitively. To understand the effects of such competition, we develop a stochastic agent-based simulation that models competitive foraging among mutually avoiding individuals by incorporating an avoidance zone, or territory, of a certain size around each forager which is not accessible for foraging by other competitors. For non-destructive foraging, our results show that with increasing size of the territory and number of agents the optimal Lévy exponent is still approximately 2 while the overall efficiency of the search decreases. At low values of the Lévy exponent, however, increasing territory size actually increases efficiency. For destructive foraging, we show that certain kinds of avoidance can lead to qualitatively different behavior from solitary foraging, such as the existence of an optimal search with 1<μ<2. Finally, we show that the variance among the efficiencies of the agents increases with increasing Lévy exponent for both solitary and competing foragers, suggesting that reducing variance might be a selective pressure for foragers adopting lower values of μ. Taken together, our results suggest that, for multiple foragers, mutual avoidance and efficiency variance among individuals can lead to optimal Lévy searches with exponents different from those for solitary foragers.

Using pseudo-absence models to test for environmental selection in marine movement ecology: the importance of sample size and selection strength

BACKGROUND

Understanding the selection of environmental conditions by animals requires knowledge of where they are, but also of where they could have been. Presence data can be accurately estimated by direct sampling, sightings, or through electronic tag deployments. However, absence data are harder to determine because absences are challenging to measure in an uncontrolled setting. To address this problem, ecologists have developed different methods for generating pseudo-absence data relying on theoretical movement models. These null models represent the movement of environmentally naive individuals, creating a set of locations that animals could have been if they were not exhibiting environmental selection.

METHODS

Here, we use four different kinds of null animal movement models-Brownian motion, Lévy walks, Correlated random walks, and Joint correlated random walks to test the ability and power of each of these null movement models to serve as appropriate animal absence models. We use Kolmogorov-Smirnov tests to detect environmental selection using two data sets, one of simulated animal tracks biased towards warmer sea surface temperatures, and one of 57 observed blue shark tracks of unknown sea surface temperature selection.

RESULTS

The four different types of movement models showed minimal difference in the ability to serve as appropriate null models for environmental selection studies. Selection strength and sample size were more important in detecting true environmental selection. We show that this method can suffer from high false positive rates, especially in the case where animals are not selecting for specific environments. We provide estimates of test accuracy at different sample sizes and selection strengths to avoid false positives when using this method.

CONCLUSION

We show how movement models can be used to generate pseudo-absences and test for habitat selection in marine organisms. While this approach efficiently detects environmental selection in marine organisms, it cannot detect the underlying mechanisms driving this selection.

Emergence of spatially structured populations by area-concentrated search

The idea that populations are spatially structured has become a very powerful concept in ecology, raising interest in many research areas. However, despite dispersal being a core component of the concept, it typically does not consider the movement behavior underlying any dispersal. Using individual-based simulations in continuous space, we explored the emergence of a spatially structured population in landscapes with spatially heterogeneous resource distribution and with organisms following simple area-concentrated search (ACS); individuals do not, however, perceive or respond to any habitat attributes per se but only to their foraging success. We investigated the effects of different resource clustering pattern in landscapes (single large cluster vs. many small clusters) and different resource density on the spatial structure of populations and movement between resource clusters of individuals. As results, we found that foraging success increased with increasing resource density and decreasing number of resource clusters. In a wide parameter space, the system exhibited attributes of a spatially structured populations with individuals concentrated in areas of high resource density, searching within areas of resources, and "dispersing" in straight line between resource patches. "Emigration" was more likely from patches that were small or of low quality (low resource density), but we observed an interaction effect between these two parameters. With the ACS implemented, individuals tended to move deeper into a resource cluster in scenarios with moderate resource density than in scenarios with high resource density. "Looping" from patches was more likely if patches were large and of high quality. Our simulations demonstrate that spatial structure in populations may emerge if critical resources are heterogeneously distributed and if individuals follow simple movement rules (such as ACS). Neither the perception of habitat nor an explicit decision to emigrate from a patch on the side of acting individuals is necessary for the emergence of such spatial structure.

Intrinsic and environmental factors modulating autonomous robotic search under high uncertainty

Autonomous robotic search problems deal with different levels of uncertainty. When uncertainty is low, deterministic strategies employing available knowledge result in most effective searches. However, there are domains where uncertainty is always high since information about robot location, environment boundaries or precise reference points is unattainable, e.g., in cave, deep ocean, planetary exploration, or upon sensor or communications impairment. Furthermore, latency regarding when search targets move, appear or disappear add to uncertainty sources. Here we study intrinsic and environmental factors that affect low-informed robotic search based on diffusive Brownian, naive ballistic, and superdiffusive strategies (Lévy walks), and in particular, the effectiveness of their random exploration. Representative strategies were evaluated considering both intrinsic (motion drift, energy or memory limitations) and extrinsic factors (obstacles and search boundaries). Our results point towards minimum-knowledge based modulation approaches that can adjust distinct spatial and temporal aspects of random exploration to lead to effective autonomous search under uncertainty.

Mechanistic modelling of COVID-19 and the impact of lockdowns on a short-time scale

BACKGROUND

To mitigate the spread of the COVID-19 coronavirus, some countries have adopted more stringent non-pharmaceutical interventions in contrast to those widely used. In addition to standard practices such as enforcing curfews, social distancing, and closure of non-essential service industries, other non-conventional policies also have been implemented, such as the total lockdown of fragmented regions, which are composed of sparsely and highly populated areas.

METHODS

In this paper, we model the movement of a host population using a mechanistic approach based on random walks, which are either diffusive or super-diffusive. Infections are realised through a contact process, whereby a susceptible host is infected if in close spatial proximity of the infectious host with an assigned transmission probability. Our focus is on a short-time scale (∼ 3 days), which is the average time lag time before an infected individual becomes infectious.

RESULTS

We find that the level of infection remains approximately constant with an increase in population diffusion, and also in the case of faster population dispersal (super-diffusion). Moreover, we demonstrate how the efficacy of imposing a lockdown depends heavily on how susceptible and infectious individuals are distributed over space.

CONCLUSION

Our results indicate that on a short-time scale, the type of movement behaviour does not play an important role in rising infection levels. Also, lock-down restrictions are ineffective if the population distribution is homogeneous. However, in the case of a heterogeneous population, lockdowns are effective if a large proportion of infectious carriers are distributed in sparsely populated sub-regions.

Three-dimensional random walk models of individual animal movement and their application to trap counts modelling

BACKGROUND

Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps.

METHODS

We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we focus on the geometrical aspects of the 3D trap and thus consider three types of shape; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency.

RESULTS

We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion.

CONCLUSION

Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.

Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes

Lévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks.

A mechanistic theory of personality-dependent movement behaviour based on dynamic energy budgets

Consistent between-individual differences in movement are widely recognised across taxa. In addition, foraging plasticity at the within-individual level suggests a behavioural dependency on the internal energy demand. Because behaviour co-varies with fast-slow life history (LH) strategies in an adaptive context, as theoretically predicted by the pace-of-life syndrome hypothesis, mass/energy fluxes should link behaviour and its plasticity with physiology at both between- and within-individual levels. However, a mechanistic framework driving these links in a fluctuating ecological context is lacking. Focusing on home range behaviour, we propose a novel behavioural-bioenergetics theoretical model to address such complexities at the individual level based on energy balance. We propose explicit mechanistic links between behaviour, physiology/metabolism and LH by merging two well-founded theories, the movement ecology paradigm and the dynamic energetic budget theory. Overall, our behavioural-bioenergetics model integrates the mechanisms explaining how (1) behavioural between- and within-individual variabilities connect with internal state variable dynamics, (2) physiology and behaviour are explicitly interconnected by mass/energy fluxes, and (3) different LHs may arise from both behavioural and physiological variabilities in a given ecological context. Our novel theoretical model reveals encouraging opportunities for empiricists and theoreticians to delve into the eco-evolutionary processes that favour or hinder the development of between-individual differences in behaviour and the evolution of personality-dependent movement syndromes.

Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long "flights". Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for "reprogramming" the Lévy walks into either diffusive or ballistic movements.

The prevalence of olfactory- versus visual-signal encounter by searching bumblebees

While the phrase 'foraging bumblebee' brings to mind a bumbling bee flying flower to flower in a sunny meadow, foraging is a complicated series of behaviors such as: locating a floral patch; selecting a flower-type; learning handling skills for pollen and nectar extraction; determining when to move-on from a patch; learning within-patch paths (traplining); and learning efficient hive-to-patch routes (spatial navigation). Thus the term 'forager' encompasses multiple distinct behaviors that rely on different sensory modalities. Despite a robust literature on bumblebee foraging behavior, few studies are directly relevant to sensory-guided search; i.e. how workers locate novel patches. The first step in answering this question is to determine what sensory information is available to searching bumblebees. This manuscript presents a computational model that elucidates the relative frequency of visual and olfactory cues that are available to workers searching for floral resources under a range of ecologically relevant scenarios. Model results indicate that odor is the most common sensory cue encountered during search flights. When the likelihood of odor-plume contact is higher, odor-encounter is ubiquitous. While integrative (visual + olfactory) cues are common when foragers are searching for larger flowers (e.g. Echinacea), they become rare when foragers are searching for small flowers (e.g. Penstemon). Visual cues are only encountered in isolation when foragers are seeking large flowers with a low odor-plume contact probability. These results indicate that despite the multisensory nature of floral signals, different modalities may be encountered in isolation during search-behavior, as opposed to the reliably multimodal signals encountered during patch-exploitation or nectar/ pollen acquisition.

Immune-inspired search strategies for robot swarms

Detection of targets distributed randomly in space is a task common to both robotic and biological systems. Lévy search has previously been used to characterize T cell search in the immune system. We use a robot swarm to evaluate the effectiveness of a Lévy search strategy and map the relationship between search parameters and target configurations. We show that the fractal dimension of the Lévy search which optimizes search efficiency depends strongly on the distribution of targets but only weakly on the number of agents involved in search. Lévy search can therefore be tuned to the target configuration while also being scalable. Implementing search behaviors observed in T cells in a robot swarm provides an effective, adaptable, and scalable swarm robotic search strategy. Additionally, the adaptability and scalability of Lévy search may explain why Lévy-like movement has been observed in T cells in multiple immunological contexts.

Rationalizing spatial exploration patterns of wild animals and humans through a temporal discounting framework

Understanding the exploration patterns of foragers in the wild provides fundamental insight into animal behavior. Recent experimental evidence has demonstrated that path lengths (distances between consecutive turns) taken by foragers are well fitted by a power law distribution. Numerous theoretical contributions have posited that "Lévy random walks"-which can produce power law path length distributions-are optimal for memoryless agents searching a sparse reward landscape. It is unclear, however, whether such a strategy is efficient for cognitively complex agents, from wild animals to humans. Here, we developed a model to explain the emergence of apparent power law path length distributions in animals that can learn about their environments. In our model, the agent's goal during search is to build an internal model of the distribution of rewards in space that takes into account the cost of time to reach distant locations (i.e., temporally discounting rewards). For an agent with such a goal, we find that an optimal model of exploration in fact produces hyperbolic path lengths, which are well approximated by power laws. We then provide support for our model by showing that humans in a laboratory spatial exploration task search space systematically and modify their search patterns under a cost of time. In addition, we find that path length distributions in a large dataset obtained from free-ranging marine vertebrates are well described by our hyperbolic model. Thus, we provide a general theoretical framework for understanding spatial exploration patterns of cognitively complex foragers.

Optimal search strategies of run-and-tumble walks

The run-and-tumble walk, consisting of randomly reoriented ballistic excursions, models phenomena ranging from gas kinetics to bacteria motility. We evaluate the mean time required for this walk to find a fixed target within a two- or three-dimensional spherical confinement. We find that the mean search time admits a minimum as a function of the mean run duration for various types of boundary conditions and run duration distributions (exponential, power-law, deterministic). Our result stands in sharp contrast to the pure ballistic motion, which is predicted to be the optimal search strategy in the case of Poisson-distributed targets.

Robustness of movement models: can models bridge the gap between temporal scales of data sets and behavioural processes?

Discrete-time random walks and their extensions are common tools for analyzing animal movement data. In these analyses, resolution of temporal discretization is a critical feature. Ideally, a model both mirrors the relevant temporal scale of the biological process of interest and matches the data sampling rate. Challenges arise when resolution of data is too coarse due to technological constraints, or when we wish to extrapolate results or compare results obtained from data with different resolutions. Drawing loosely on the concept of robustness in statistics, we propose a rigorous mathematical framework for studying movement models' robustness against changes in temporal resolution. In this framework, we define varying levels of robustness as formal model properties, focusing on random walk models with spatially-explicit component. With the new framework, we can investigate whether models can validly be applied to data across varying temporal resolutions and how we can account for these different resolutions in statistical inference results. We apply the new framework to movement-based resource selection models, demonstrating both analytical and numerical calculations, as well as a Monte Carlo simulation approach. While exact robustness is rare, the concept of approximate robustness provides a promising new direction for analyzing movement models.

A framework for analyzing the robustness of movement models to variable step discretization

When sampling animal movement paths, the frequency at which location measurements are attempted is a critical feature for data analysis. Important quantities derived from raw data, e.g. travel distance or sinuosity, can differ largely based on the temporal resolution of the data. Likewise, when movement models are fitted to data, parameter estimates have been demonstrated to vary with sampling rate. Thus, biological statements derived from such analyses can only be made with respect to the resolution of the underlying data, limiting extrapolation of results and comparison between studies. To address this problem, we investigate whether there are models that are robust against changes in temporal resolution. First, we propose a mathematically rigorous framework, in which we formally define robustness as a model property. We then use the framework for a thorough assessment of a range of basic random walk models, in which we also show how robustness relates to other probabilistic concepts. While we found robustness to be a strong condition met by few models only, we suggest a new method to extend models so as to make them robust. Our framework provides a new systematic, mathematically founded approach to the question if, and how, sampling rate of movement paths affects statistical inference.

How cockroaches exploit tactile boundaries to find new shelters

Animals such as cockroaches depend on exploration of unknown environments, and their strategies may inspire robotic approaches. We have previously shown that cockroach behavior, with respect to shelters and the walls of an otherwise empty arena, can be captured with a stochastic state-based algorithm. We call this algorithm RAMBLER, randomized algorithm mimicking biased lone exploration in roaches. In this work, we verified and extended this model by adding a barrier in the previously used arena and conducted more cockroach experiments. In two arena configurations, our simulated model's path length distribution was similar to the experimental distribution (mean experimental path length 3.4 and 3.2 m, mean simulated path length 3.9 and 3.3 m). By analyzing cockroach behavior before, along, and at the end of the barrier, we have generalized RAMBLER to address arbitrarily complex 2D mazes. For biology, this is an abstract behavioral model of a decision-making process in the cockroach brain. For robotics, this is a strategy that may improve exploration for goals, especially in unpredictable environments with non-convex obstacles. Generally, cockroach behavior seems to recommend variability in the absence of planning, and following paths defined by walls.

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

The "Lévy Foraging Hypothesis" promotes Lévy walk (LW) as the best strategy to forage for patchily but unpredictably located prey. This strategy mixes extensive and intensive searching phases in a mostly cue-free way through strange, scale-free kinetics. It is however less efficient than a cue-driven two-scale Composite Brownian walk (CBW) when the resources encountered are systematically detected. Nevertheless, it could be assumed that the intrinsic capacity of LW to trigger cue-free intensive searching at random locations might be advantageous when resources are not only scarcely encountered but also so cryptic that the probability to detect those encountered during movement is low. Surprisingly, this situation, which should be quite common in natural environments, has almost never been studied. Only a few studies have considered "saltatory" foragers, which are fully "blind" while moving and thus detect prey only during scanning pauses, but none of them compared the efficiency of LW vs. CBW in this context or in less extreme contexts where the detection probability during movement is not null but very low. In a study based on computer simulations, we filled the bridge between the concepts of "pure continuous" and "pure saltatory" foraging by considering that the probability to detect resources encountered while moving may range from 0 to 1. We showed that regularly stopping to scan the environment can indeed improve efficiency, but only at very low detection probabilities. Furthermore, the LW is then systematically outperformed by a mixed cue-driven/internally-driven CBW. It is thus more likely that evolution tends to favour strategies that rely on environmental feedbacks rather than on strange kinetics.

Mathematical and computational approaches to epidemic modeling: a comprehensive review

Mathematical and computational approaches are important tools for understanding epidemic spread patterns and evaluating policies of disease control. In recent years, epidemiology has become increasingly integrated with mathematics, sociology, management science, complexity science, and computer science. The cross of multiple disciplines has caused rapid development of mathematical and computational approaches to epidemic modeling. In this article, we carry out a comprehensive review of epidemic models to provide an insight into the literature of epidemic modeling and simulation. We introduce major epidemic models in three directions, including mathematical models, complex network models, and agent-based models. We discuss the principles, applications, advantages, and limitations of these models. Meanwhile, we also propose some future research directions in epidemic modeling.

Optimal Lévy-flight foraging in a finite landscape

We present a simple model to study Lévy-flight foraging with a power-law step-size distribution [P(l) ∞ l-μ] in a finite landscape with countable targets. We find that different optimal foraging strategies characterized by a wide range of power-law exponent μopt, from ballistic motion (μopt → 1) to Lévy flight (1 < μopt < 3) to Brownian motion (μopt ≥ 3), may arise in adaptation to the interplay between the termination of foraging, which is regulated by the number of foraging steps, and the environmental context of the landscape, namely the landscape size and number of targets. We further demonstrate that stochastic returning can be another significant factor that affects the foraging efficiency and optimality of foraging strategy. Our study provides a new perspective on Lévy-flight foraging, opens new avenues for investigating the interaction between foraging dynamics and the environment and offers a realistic framework for analysing animal movement patterns from empirical data.

Zigzag Generalized Lévy Walk: the In Vivo Search Strategy of Immunocytes

Immune responses are based on the coordinated searching behaviors of immunocytes that are aimed at tracking down specific targets. The search efficiency of immunocytes significantly affects the speed of initiation and development of immune responses. Previous studies have shown that not only the intermittent walk but also the zigzag turning preference of immunocytes contributes to the search efficiency. However, among existing models describing immunocytes' search strategy, none has captured both features. Here we propose a zigzag generalized Lévy walk model to describe the search strategy of immunocytes more accurately and comprehensively by considering both the intermittent and the zigzag-turning walk features. Based on the analysis of the searching behaviors of typical immune cell types, dendritic cells and leukocytes, in their native physiological environment, we demonstrate that the model can describe the in vivo search strategy of immunocytes well. Furthermore, by analyzing the search efficiency, we find that this type of search strategy enables immunocytes to capture rare targets in approximately half the time than the previously proposed generalized Lévy walk. This study sheds new light on the fundamental mechanisms that drive the efficient initiation and development of immune responses and in turn may lead to the development of novel therapeutic approaches for diseases ranging from infection to cancer.

Levy Foraging in a Dynamic Environment – Extending the Levy Search

A common task for robots is the patrolling of an unknown area with inadequate information about target locations. Under these circumstances it has been suggested that animal foraging could provide an optimal or at least sub-optimal search methodology, namely the Levy flight search. Although still in debate, it seems that predators somehow follow this search pattern when foraging, because it avoids being trapped in a local search if the food is beyond the sensory range. A Levy flight is a particular case of the random walk. Its displacements on a 2-D surface are drawn from the Pareto-Levy probability distribution, characterized by power law tails. The Levy flight search has many applications in optical material, ladars, optics, large database search, earthquake data analysis, location of DNA sites, human mobility, stock return analysis, online auctions, astronomy, ecology and biology. Almost all studies and simulations concerning the Levy flight foraging examine static or slowly moving (with respect to the forager) uniformly distributed resources. Moreover, in recent works a small swarm of underwater autonomous vehicles has been used to test the standard Levy search in the underwater environment, with good results. In this paper we extend the classical Levy foraging framework taking into consideration a moving target allocated on a 2-D surface according to a radial probability distribution and comparing its performance with the random walk search. The metric used in the numerical simulations is the detection rate. Simulations include the sensor resolution, intended as the maximum detection distance of the forager from the target. Furthermore, contrarily to the usual Levy foraging framework, we use only one target. Results show that Levy flight outperforms the random walk if the sensor detection radius is not too small or too large. We also find the Levy flight in the velocity of the center of mass model of a fish school according the Kuramoto equation, a famous model of synchronization phenomena. Finally, a discussion about the controversy concerning the innate or evolutionary origin of the Levy foraging is given.

Extending Lévy search theory from one to higher dimensions: Lévy walking favours the blind

A diverse range of organisms, including T cells, E. coli, honeybees, sharks, turtles, bony fish, jellyfish, wandering albatrosses and even human hunter-gatherers have movement patterns that can be approximated by Lévy walks (LW; sometimes called Lévy flights in the biological and ecological literature). These observations lend support to the 'Lévy flight foraging hypothesis' which asserts that natural selection should have led to adaptations for Lévy flight foraging, because Lévy flights can optimize search efficiencies. The hypothesis stems from a rigorous theory of one-dimensional searching and from simulation data for two-dimensional searching. The potential effectiveness of three-dimensional Lévy searches has not been examined but is central to a proper understanding of marine predators and T cells which have provided the most compelling empirical evidence for LW. Here I extend Lévy search theory from one to three dimensions. The new theory predicts that three-dimensional Lévy searching can be advantageous but only when targets are large compared with the perceptual range of the searchers, i.e. only when foragers are effectively blind and need to come into contact with a target to establish its presence. This may explain why effective blindness is a common factor among three-dimensional Lévy walkers.

Effect of resource spatial correlation and hunter-fisher-gatherer mobility on social cooperation in Tierra del Fuego

This article presents an agent-based model designed to explore the development of cooperation in hunter-fisher-gatherer societies that face a dilemma of sharing an unpredictable resource that is randomly distributed in space. The model is a stylised abstraction of the Yamana society, which inhabited the channels and islands of the southernmost part of Tierra del Fuego (Argentina-Chile). According to ethnographic sources, the Yamana developed cooperative behaviour supported by an indirect reciprocity mechanism: whenever someone found an extraordinary confluence of resources, such as a beached whale, they would use smoke signals to announce their find, bringing people together to share food and exchange different types of social capital. The model provides insight on how the spatial concentration of beachings and agents' movements in the space can influence cooperation. We conclude that the emergence of informal and dynamic communities that operate as a vigilance network preserves cooperation and makes defection very costly.