Grazing livestock move by Lévy walks: Implications for soil health and environment

Grazing livestock plays an important role in the context of food security, agricultural sustainability and climate change. Understanding how livestock move and interact with their environment may offer new insights on how grazing practices impact soil and ecosystem functions at spatial and temporal scales where knowledge is currently limited. We characterized daily and seasonal grazing patterns using Global Positioning System (GPS) data from two grazing strategies: conventionally- and rotationally-grazed pastures. Livestock movement was consistent with the so-called Lévy walks, and could thus be simulated with Lévy-walk based probability density functions. Our newly introduced "Moovement model" links grazing patterns with soil structure and related functions by coupling animal movement and soil structure dynamics models, allowing to predict spatially-explicit changes in key soil properties. Predicted post-grazing management-specific bulk densities were consistent with field measurements and confirmed that rotational grazing produced similar disturbance as conventional grazing despite hosting higher stock densities. Harnessing information on livestock movement and its impacts in soil structure within a modelling framework can help testing and optimizing grazing strategies for ameliorating their impact on soil health and environment.

Statistical modeling of mRNP transport in dendrites: A comparative analysis of β-actin and Arc mRNP dynamics

Localization of messenger RNA (mRNA) in dendrites is crucial for regulating gene expression during long-term memory formation. mRNA binds to RNA-binding proteins (RBPs) to form messenger ribonucleoprotein (mRNP) complexes that are transported by motor proteins along microtubules to their target synapses. However, the dynamics by which mRNPs find their target locations in the dendrite have not been well understood. Here, we investigated the motion of endogenous β-actin and Arc mRNPs in dissociated mouse hippocampal neurons using the MS2 and PP7 stem-loop systems, respectively. By evaluating the statistical properties of mRNP movement, we found that the aging Lévy walk model effectively describes both β-actin and Arc mRNP transport in proximal dendrites. A critical difference between β-actin and Arc mRNPs was the aging time, the time lag between transport initiation and measurement initiation. The longer mean aging time of β-actin mRNP (~100 s) compared with that of Arc mRNP (~30 s) reflects the longer half-life of constitutively expressed β-actin mRNP. Furthermore, our model also permitted us to estimate the ratio of newly generated and pre-existing β-actin mRNPs in the dendrites. This study offers a robust theoretical framework for mRNP transport, which provides insight into how mRNPs locate their targets in neurons.

Virtual prey with Lévy motion are preferentially attacked by predatory fish

Of widespread interest in animal behavior and ecology is how animals search their environment for resources, and whether these search strategies are optimal. However, movement also affects predation risk through effects on encounter rates, the conspicuousness of prey, and the success of attacks. Here, we use predatory fish attacking a simulation of virtual prey to test whether predation risk is associated with movement behavior. Despite often being demonstrated to be a more efficient strategy for finding resources such as food, we find that prey displaying Lévy motion are twice as likely to be targeted by predators than prey utilizing Brownian motion. This can be explained by the predators, at the moment of the attack, preferentially targeting prey that were moving with straighter trajectories rather than prey that were turning more. Our results emphasize that costs of predation risk need to be considered alongside the foraging benefits when comparing different movement strategies.

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.

Seed dispersal as a search strategy: dynamic and fragmented landscapes select for multi-scale movement strategies in plants

BACKGROUND

Plant dispersal is a critical factor driving ecological responses to global changes. Knowledge on the mechanisms of dispersal is rapidly advancing, but selective pressures responsible for the evolution of dispersal strategies remain elusive. Recent advances in animal movement ecology identified general strategies that may optimize efficiency in animal searches for food or habitat. Here we explore the potential for evolution of similar general movement strategies for plants.

METHODS

We propose that seed dispersal in plants can be viewed as a strategic search for suitable habitat, where the probability of finding such locations has been optimized through evolution of appropriate dispersal kernels. Using model simulations, we demonstrate how dispersal strategies can optimize key dispersal trade-offs between finding habitat, avoiding kin competition, and colonizing new patches. These trade-offs depend strongly on the landscape, resulting in a tight link between optimal dispersal strategy and spatiotemporal habitat distribution.

RESULTS

Our findings reveal that multi-scale seed dispersal strategies that combine a broad range of dispersal scales, including Lévy-like dispersal, are optimal across a wide range of dynamic and patchy landscapes. At the extremes, static and patchy landscapes select for dispersal strategies dominated by short distances, while uniform and highly unpredictable landscapes both select for dispersal strategies dominated by long distances.

CONCLUSIONS

By viewing plant seed dispersal as a strategic search for suitable habitat, we provide a reference framework for the analysis of plant dispersal data. Consideration of the entire dispersal kernel, including distances across the full range of scales, is key. This reference framework helps identify plant species' dispersal strategies, the evolutionary forces determining these strategies and their ecological consequences, such as a potential mismatch between plant dispersal strategy and altered spatiotemporal habitat dynamics due to land use change. Our perspective opens up directions for future studies, including exploration of composite search behaviour and 'informed searches' in plant species with directed dispersal.

Lévy Walk in Swarm Models Based on Bayesian and Inverse Bayesian Inference

While swarming behavior is regarded as a critical phenomenon in phase transition and frequently shows the properties of a critical state such as Lévy walk, a general mechanism to explain the critical property in swarming behavior has not yet been found. Here, we address this problem with a simple swarm model, the Self-Propelled Particle (SPP) model, and propose a way to explain this critical behavior by introducing agents making decisions via the data-hypothesis interaction in Bayesian inference, namely, Bayesian and inverse Bayesian inference (BIB). We compare three SPP models, namely, the simple SPP, the SPP with Bayesian-only inference (BO) and the SPP with BIB models. We show that only the BIB model entails coexisting tornado, splash and translation behaviors, and the Lévy walk pattern.

Tumble Suppression Is a Conserved Feature of Swarming Motility

Bacteria within a swarm move characteristically in packs, displaying an intricate swirling motion in which hundreds of dynamic rafts continuously form and dissociate as the swarm colonizes an increasing expanse of territory. The demonstrated property of E. coli to reduce its tumble bias and hence increase its run duration during swarming is expected to maintain and promote side-by-side alignment and cohesion within the bacterial packs. In this study, we observed a similar low tumble bias in five different bacterial species, both Gram positive and Gram negative, each inhabiting a unique habitat and posing unique problems to our health. The unanimous display of an altered run-tumble bias in swarms of all species examined in this investigation suggests that this behavioral adaptation is crucial for swarming.

Many bacteria use flagellum-driven motility to swarm or move collectively over a surface terrain. Bacterial adaptations for swarming can include cell elongation, hyperflagellation, recruitment of special stator proteins, and surfactant secretion, among others. We recently demonstrated another swarming adaptation in Escherichia coli, wherein the chemotaxis pathway is remodeled to decrease tumble bias (increase run durations), with running speeds increased as well. We show here that the modification of motility parameters during swarming is not unique to E. coli but is shared by a diverse group of bacteria we examined—Proteus mirabilis, Serratia marcescens, Salmonella enterica, Bacillus subtilis, and Pseudomonas aeruginosa—suggesting that increasing run durations and speeds are a cornerstone of swarming.

Resource distribution and internal factors interact to govern movement of a freshwater snail

Movement enables mobile organisms to respond to local environmental conditions and is driven by a combination of external and internal factors operating at multiple scales. Here, we explored how resource distribution interacted with the internal state of organisms to drive patterns of movement. Specifically, we tracked snail movements on experimental landscapes where resource (algal biofilm) distribution varied from 0 to 100% coverage and quantified how that movement changed over a 24 h period. Resource distribution strongly affected snail movement. Trajectories were tortuous (i.e. Brownian-like) within resource patches but straighter (i.e. Lévy) in resource-free (bare) patches. The average snail speed was slower in resource patches, where snails spent most of their time. Different patterns of movement between resource and bare patches explained movement at larger spatial scales; movement was ballistic-like Lévy in resource-free landscapes, Lévy in landscapes with intermediate resource coverage and approximated Brownian in landscapes covered in resources. Our temporal analysis revealed that movement patterns changed predictably for snails that satiated their hunger and then performed other behaviours. These changes in movement patterns through time were similar across all treatments that contained resources. Thus, external and internal factors interacted to shape the inherently flexible movement of these snails.

Lévy walk process in self-organization of pedestrian crowds

Similar to other animal groups, human crowds exhibit various collective patterns that emerge from self-organization. Recent studies have emphasized that individuals anticipate their neighbours' motions to seek their paths in dynamical pedestrian flow. This path-seeking behaviour results in deviation of pedestrians from their desired directions (i.e. the direct path to their destination). However, the strategies that individuals adopt for the behaviour and how the deviation of individual movements impact the emergent organization are poorly understood. We here show that the path-seeking behaviour is performed through a scale-free movement strategy called a Lévy walk, which might facilitate transition to the group-level behaviour. In an experiment of lane formation, a striking example of self-organized patterning in human crowds, we observed how flows of oppositely moving pedestrians spontaneously separate into several unidirectional lanes. We found that before (but not after) lane formation, pedestrians deviate from the desired direction by Lévy walk process, which is considered optimal when searching unpredictably distributed resources. Pedestrians balance a trade-off between seeking their direct paths and reaching their goals as quickly as possible; they may achieve their optimal paths through Lévy walk process, facilitating the emergent lane formation.

Escherichia coli Remodels the Chemotaxis Pathway for Swarming

The fundamental motile behavior of E. coli is a random walk, where straight “runs” are punctuated by “tumbles.” This behavior, conferred by the chemotaxis signaling system, is used to track chemical gradients in liquid. Our study results show that when migrating collectively on surfaces, E. coli modifies its chemosensory physiology to decrease its tumble bias (and hence to increase run durations) by post-transcriptional changes that alter the levels of a key signaling protein. We speculate that the low tumble bias may contribute to the observed Lévy walk (LW) trajectories within the swarm, where run durations have a power law distribution. In animals, LW patterns are hypothesized to maximize searches in unpredictable environments. Swarming bacteria face several challenges while moving collectively over a surface—maintaining cohesion, overcoming constraints imposed by a physical substrate, searching for nutrients as a group, and surviving lethal levels of antimicrobials. The altered chemosensory behavior that we describe in this report may help with these challenges.

Many flagellated bacteria “swarm” over a solid surface as a dense consortium. In different bacteria, swarming is facilitated by several alterations such as those corresponding to increased flagellum numbers, special stator proteins, or secreted surfactants. We report here a change in the chemosensory physiology of swarming Escherichia coli which alters its normal “run tumble” bias. E. coli bacteria taken from a swarm exhibit more highly extended runs (low tumble bias) and higher speeds than E. coli bacteria swimming individually in a liquid medium. The stability of the signaling protein CheZ is higher in swarmers, consistent with the observed elevation of CheZ levels and with the low tumble bias. We show that the tumble bias displayed by wild-type swarmers is the optimal bias for maximizing swarm expansion. In assays performed in liquid, swarm cells have reduced chemotactic performance. This behavior is specific to swarming, is not specific to growth on surfaces, and persists for a generation. Therefore, the chemotaxis signaling pathway is reprogrammed for swarming.

Optimizing mating encounters by sexually dimorphic movements

All organisms with sexual reproduction undergo a process of mating, which essentially involves the encounter of two individuals belonging to different sexes. During mate search, both sexes should mutually optimize their encounters, thus raising a question of how they achieve this. Here, we show that a population with sexually dimorphic movement patterns achieves the highest individual mating success under a limited lifespan. Extensive simulations found and analytical approximations corroborated the existence of conditions under which sexual dimorphism in the movement patterns (i.e. how diffusively they move) is advantageous over sexual monomorphism. Mutual searchers with limited lifespans need to balance the speed and accuracy of finding their mates, and dimorphic movements can solve this trade-off. We further demonstrate that the sexual dimorphism can evolve from an initial sexually monomorphic population. Our results emphasize the importance of considering mutual optimization in problems of random search.

Experimental evidence for inherent Lévy search behaviour in foraging animals

Recently, Lévy walks have been put forward as a new paradigm for animal search and many cases have been made for its presence in nature. However, it remains debated whether Lévy walks are an inherent behavioural strategy or emerge from the animal reacting to its habitat. Here, we demonstrate signatures of Lévy behaviour in the search movement of mud snails (Hydrobia ulvae) based on a novel, direct assessment of movement properties in an experimental set-up using different food distributions. Our experimental data uncovered clusters of small movement steps alternating with long moves independent of food encounter and landscape complexity. Moreover, size distributions of these clusters followed truncated power laws. These two findings are characteristic signatures of mechanisms underlying inherent Lévy-like movement. Thus, our study provides clear experimental evidence that such multi-scale movement is an inherent behaviour rather than resulting from the animal interacting with its environment.

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.

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.

Capture of fixation by rotational flow; a deterministic hypothesis regarding scaling and stochasticity in fixational eye movements

Visual scan paths exhibit complex, stochastic dynamics. Even during visual fixation, the eye is in constant motion. Fixational drift and tremor are thought to reflect fluctuations in the persistent neural activity of neural integrators in the oculomotor brainstem, which integrate sequences of transient saccadic velocity signals into a short term memory of eye position. Despite intensive research and much progress, the precise mechanisms by which oculomotor posture is maintained remain elusive. Drift exhibits a stochastic statistical profile which has been modeled using random walk formalisms. Tremor is widely dismissed as noise. Here we focus on the dynamical profile of fixational tremor, and argue that tremor may be a signal which usefully reflects the workings of oculomotor postural control. We identify signatures reminiscent of a certain flavor of transient neurodynamics; toric traveling waves which rotate around a central phase singularity. Spiral waves play an organizational role in dynamical systems at many scales throughout nature, though their potential functional role in brain activity remains a matter of educated speculation. Spiral waves have a repertoire of functionally interesting dynamical properties, including persistence, which suggest that they could in theory contribute to persistent neural activity in the oculomotor postural control system. Whilst speculative, the singularity hypothesis of oculomotor postural control implies testable predictions, and could provide the beginnings of an integrated dynamical framework for eye movements across scales.

How superdiffusion gets arrested: ecological encounters explain shift from Lévy to Brownian movement

Ecological theory uses Brownian motion as a default template for describing ecological movement, despite limited mechanistic underpinning. The generality of Brownian motion has recently been challenged by empirical studies that highlight alternative movement patterns of animals, especially when foraging in resource-poor environments. Yet, empirical studies reveal animals moving in a Brownian fashion when resources are abundant. We demonstrate that Einstein's original theory of collision-induced Brownian motion in physics provides a parsimonious, mechanistic explanation for these observations. Here, Brownian motion results from frequent encounters between organisms in dense environments. In density-controlled experiments, movement patterns of mussels shifted from Lévy towards Brownian motion with increasing density. When the analysis was restricted to moves not truncated by encounters, this shift did not occur. Using a theoretical argument, we explain that any movement pattern approximates Brownian motion at high-resource densities, provided that movement is interrupted upon encounters. Hence, the observed shift to Brownian motion does not indicate a density-dependent change in movement strategy but rather results from frequent collisions. Our results emphasize the need for a more mechanistic use of Brownian motion in ecology, highlighting that especially in rich environments, Brownian motion emerges from ecological interactions, rather than being a default movement pattern.

Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models

1. Increased inference regarding underlying behavioural mechanisms of animal movement can be achieved by comparing GPS data with statistical mechanical movement models such as random walk and Lévy walk with known underlying behaviour and statistical properties. 2. GPS data are typically collected with ≥ 1 h intervals not exactly tracking every mechanistic step along the movement path, so a statistical mechanical model approach rather than a mechanistic approach is appropriate. However, comparisons require a coherent framework involving both scaling and memory aspects of the underlying process. Thus, simulation models have recently been extended to include memory-guided returns to previously visited patches, that is, site fidelity. 3. We define four main classes of movement, differing in incorporation of memory and scaling (based on respective intervals of the statistical fractal dimension D and presence/absence of site fidelity). Using three statistical protocols to estimate D and site fidelity, we compare these main movement classes with patterns observed in GPS data from 52 females of red deer (Cervus elaphus). 4. The results show best compliance with a scale-free and memory-enhanced kind of space use; that is, a power law distribution of step lengths, a fractal distribution of the spatial scatter of fixes and site fidelity. 5. Our study thus demonstrates how inference regarding memory effects and a hierarchical pattern of space use can be derived from analysis of GPS data.