Turn designation, sampling rate and the misidentification of power laws in movement path data using maximum likelihood estimates

Many authors have claimed to observe animal movement paths that appear to be Lévy walks, i.e. a random walk where the distribution of move lengths follows an inverse power law. A Lévy walk is known to be the optimal search strategy of a particular class of random walks in certain environments; hence, it is important to distinguish correctly between Lévy walks and other types of random walks in observed animal movement paths. Evidence of a power law distribution in the step length distribution of observed animal movement paths is often used to classify a particular movement path as a Lévy walk. However, there is some doubt about the accuracy of early studies that apparently found Lévy walk behaviour. A recently accepted method to determine whether a movement path truly exhibits Lévy walk behaviour is based on an analysis of move lengths with a maximum likelihood estimate using Akaike weights. Here, we show that simulated (non-Lévy) random walks representing different types of animal movement behaviour (a composite correlated random walk; pooled data from a set of random walks with different levels of correlation and three-dimensional correlated random walks projected into one dimension) can all show apparent power law behaviour typical of Lévy walks when using the maximum likelihood estimation method. The probability of the movement path being identified as having a power law step distribution is related to both the sampling rate used by the observer and the way that ‘turns’ or ‘reorientations’ in the movement path are designated. However, identification is also dependent on the nature and properties of the simulated path, and there is currently no standard method of observation and analysis that is robust for all cases. Our results indicate that even apparently robust maximum likelihood methods can lead to a mismatch between pattern and process, as paths arising from non-Lévy walks exhibit Lévy-like patterns.     

Turn costs change the value of animal search paths

The tortuosity of the track taken by an animal searching for food profoundly affects search efficiency, which should be optimised to maximise net energy gain. Models examining this generally describe movement as a series of straight steps interspaced by turns, and implicitly assume no turn costs. We used both empirical‐ and modelling‐based approaches to show that the energetic costs for turns in both terrestrial and aerial locomotion are substantial, which calls into question the value of conventional movement models such as correlated random walk or Lévy walk for assessing optimum path types. We show how, because straight‐line travel is energetically most efficient, search strategies should favour constrained turn angles, with uninformed foragers continuing in straight lines unless the potential benefits of turning offset the cost.     

Effect of grazing on plant patterns in arid ecosystems of Patagonian Monte

Our objective was to assess the relationship between the spatial patterning of perennial grasses (total, grazed, and non‐grazed) and shrub patches in rangelands under different grazing pressures of the Patagonian Monte. We selected three grazed paddocks with the usual stocking rate for the area, where previous studies showed that a piosphere formation is common. At each paddock, we analysed the grain of heterogeneity at sites located at two distances from the single watering point (near, far), using high‐resolution aerial photographs. At these sites, we also assessed in the field the density, size, cover, and spatial patterning of grazed and non‐grazed perennial grasses and shrub patches. The grain of heterogeneity of shrub patches was coarser in sites near the watering point than in those distant from it, as a consequence of the increase in size of both, bare soil and shrub patches. Field sampling showed that a coarse grain of heterogeneity relative to fine‐grained sites resulted from changes in species composition, increased bare soil areas and reduced perennial grass cover. In coarse‐grained sites, lower perennial grass cover resulted from lower density and/or smaller size of grass bunches than in fine‐grained sites. We did not find significant differences among sites in the proportion of perennial grazed grasses. Since the density and cover of perennial grasses was higher in fine‐ than in coarse‐grained sites, we suggested that fine‐grained sites are more important as feeding stations than coarse‐grained sites. The consequences of this differential use could lead to degradation of fine‐grained sites and to higher homogeneity in spatial plant structure and floristic composition within paddocks with respect to the condition observed at present, increasing the size of the highly degraded zone within the piosphere. At the patch level, we found that at about one third of the sampled transects, both total and non‐grazed perennial grasses were spatially aggregated with shrub patches. However, in most transects grazed perennial grasses were indifferently distributed in relation with shrub patches, showing that grazers display high selectivity of foraging sites at macro level (i.e. high and low grazing pressure sites at the paddock level), but random occupancy of vegetation units (randomness in the distribution of grazed perennial grasses at the patch level). The intensity of the positive association between non‐grazed grasses and shrub patches was higher in fine‐grained than in coarse‐grained sites and may be attributed to higher protection against herbivores associated to denser shrub patches in fine‐ relative to coarse‐grained sites. We concluded that a feedback exists between the spatial distribution of species preferred by grazers and the spatial patterning of use of these species.     

PRIMO/PRIMONA: A coarse‐grained model for proteins and nucleic acids that preserves near‐atomistic accuracy

The new coarse graining model PRIMO/PRIMONA for proteins and nucleic acids is proposed. This model combines one to several heavy atoms into coarse‐grained sites that are chosen to allow an analytical, high‐resolution reconstruction of all‐atom models based on molecular bonding geometry constraints. The accuracy of proposed reconstruction method in terms of structure and energetics is tested and compared with other popular reconstruction methods for a variety of protein and nucleic acid test sets. Proteins 2010. © 2009 Wiley‐Liss, Inc.     

Large‐scale comparison of protein essential dynamics from molecular dynamics simulations and coarse‐grained normal mode analyses

A large‐scale comparison of essential dynamics (ED) modes from molecular dynamic simulations and normal modes from coarse‐grained normal mode methods (CGNM) was performed on a dataset of 335 proteins. As CGNM methods, the elastic network model (ENM) and the rigid cluster normal mode analysis (RCNMA) were used. Low‐frequency normal modes from ENM correlate very well with ED modes in terms of directions of motions and relative amplitudes of motions. Notably, a similar performance was found if normal modes from RCNMA were used, despite a higher level of coarse graining. On average, the space spanned by the first quarter of ENM modes describes 84% of the space spanned by the five ED modes. Furthermore, no prominent differences for ED and CGNM modes among different protein structure classes (CATH classification) were found. This demonstrates the general potential of CGNM approaches for describing intrinsic motions of proteins with little computational cost. For selected cases, CGNM modes were found to be more robust among proteins that have the same topology or are of the same homologous superfamily than ED modes. In view of recent evidence regarding evolutionary conservation of vibrational dynamics, this suggests that ED modes, in some cases, might not be representative of the underlying dynamics that are characteristic of a whole family, probably due to insufficient sampling of some of the family members by MD. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.     

Historical data reveal power‐law dispersal patterns of invasive aquatic species

Understanding how invasive species spread is of particular concern in the current era of globalisation and rapid environmental change. The occurrence of super‐diffusive movements within the context of Lévy flights has been discussed with respect to particle physics, human movements, microzooplankton, disease spread in global epidemiology and animal foraging behaviour. Super‐diffusive movements provide a theoretical explanation for the rapid spread of organisms and disease, but their applicability to empirical data on the historic spread of organisms has rarely been tested. This study focuses on the role of long‐distance dispersal in the invasion dynamics of aquatic invasive species across three contrasting areas and spatial scales: open ocean (north‐east Atlantic), enclosed sea (Mediterranean) and an island environment (Ireland). Study species included five freshwater plant species, Azolla filiculoides, Elodea canadensis, Lagarosiphon major, Elodea nuttallii and Lemna minuta; and ten species of marine algae, Asparagopsis armata, Antithamnionella elegans, Antithamnionella ternifolia, Codium fragile, Colpomenia peregrina, Caulerpa taxifolia, Dasysiphonia sp., Sargassum muticum, Undaria pinnatifida and Womersleyella setacea. A simulation model is constructed to show the validity of using historical data to reconstruct dispersal kernels. Lévy movement patterns similar to those previously observed in humans and wild animals are evident in the re‐constructed dispersal pattern of invasive aquatic species. Such patterns may be widespread among invasive species and could be exacerbated by further development of trade networks, human travel and environmental change. These findings have implications for our ability to predict and manage future invasions, and improve our understanding of the potential for spread of organisms including infectious diseases, plant pests and genetically modified organisms.     

An optimized protocol for large‐scale in situ sampling and analysis of volatile organic compounds

Chemical ecology is an ever‐expanding field with a growing interest in population‐ and community‐level studies. Many such studies are hindered due to lack of an efficient and accelerated protocol for large‐scale sampling and analysis of chemical compounds. Here, we present an optimized protocol for such large‐scale study of volatiles. A large‐scale in situ study to understand role of semiochemicals in variation in mating success of lekking blackbuck was conducted. Suitable methods for sampling and statistical analysis were identified by testing and comparing the efficiencies of available techniques to reduce analysis time while retaining sensitivity and comprehensiveness. Solid‐phase extraction using polydimethylsiloxane, analysis using a semiautomated detection of retention time and base peak, and statistical analysis using random forest algorithm were identified as the most efficient methods for large‐scale in situ sampling and analysis of volatiles. The protocol for large‐scale volatile analysis can facilitate evolutionary and metaecological studies of volatiles in situ from all types of biological samples. The protocol has potential for wider application with the analysis and interpretation methods being suitable for all kinds of semiochemicals, including nonvolatile chemicals.     

Cross‐scale variation in species richness–environment associations

Aim The scale dependence of many ecological patterns and processes implies that general inference is reliant on obtaining scale‐response curves over a large range of grains. Although environmental correlates of richness have been widely studied, comparisons among groups have usually been applied at single grains. Moreover, the relevance of environment–richness associations to fine‐grain assemblages has remained surprisingly unclear. We present a first global cross‐scale assessment of environment–richness associations for birds, mammals and amphibians from 2000 km down to c. 20 km. Location World‐wide. Methods We performed an extensive survey of the literature for well‐sampled terrestrial vertebrate inventories over clearly defined small extents. Coarser grain richness was estimated from the intersection of extent‐of‐occurrence range maps with concentric equal‐distance circles around fine‐grain assemblage location centroids. General linear and simultaneous autoregressive models were used to relate richness at the different grains to environmental correlates. Results The ability of environmental variables to explain species richness decreases markedly toward finer grains and is lowest for fine‐grained assemblages. A prominent transition in importance occurs between productivity and temperature at increased grains, which is consistent with the role of energy affecting regional, but not local, richness. Variation in fine‐grained predictability across groups is associated with their purported grain of space use, i.e. highest for amphibians and narrow‐ranged and small‐bodied species. Main conclusions We extend the global documentation of environment–richness associations to fine‐grained assemblages. The relationship between fine‐grained predictability of a group and its ecological characteristics lends empirical support to the idea that variation in species fine‐grained space use may scale up to explain coarse‐grained diversity patterns. Our study exposes a dramatic and taxonomically variable scale dependence of environment–richness associations and suggests that environmental correlates of richness may hold limited information at the level of communities.     

Behavioral intermittence, Lévy patterns, and randomness in animal movement

The recent debate on both the existence and the cause of fractal (Lévy) patterns in animal movement resonates with much deeper and richer problems in movement ecology: (1) establishing mechanistic links between animal behavior and statistical patterns of movement, and (2) understanding what is the role of randomness (stochasticity) in animal motion. Here, the idea of behavioral intermittence is shown to be crucial to establish mechanistic connections between the behavior of organisms and the statistical properties they generate when moving. Attention is drawn to the fact that some random walk modeling procedures can impair the identification of intermittent biological mechanisms which could govern major statistical properties of movement. This fact, together with some misconceptions and prejudices regarding the role of randomness in animal motion may explain why stochastic processes have been disregarded as a potential source of adaptation in animal movement. In the near future, the advances in biotelemetry together with a more explicit consideration of behavioral intermittence, and the development of novel random walk approaches, could help us to set up the bases for a landscape-level behavioral ecology.     

Self-similar mitochondrial DNA

We show that repeated sequences, like palindromes (local repetitions) and homologies between two different nucleotide sequences (motifs along the genome), compose a self-similar (fractal) pattern in mitochondrial DNA. This self-similarity comes from the looplike structures distributed along the genome. The looplike structures generate scaling laws in a pseudorandom DNA walk constructed from the sequence, called a Lévy flight. We measure the scaling laws from the generalized fractal dimension and singularity spectrum for mitochondrial DNA walks for 35 different species. In particular, we report characteristic loop distributions for mammal mitochondrial genomes.     

Improving ranking of models for protein complexes with side chain modeling and atomic potentials

An atomically detailed potential for docking pairs of proteins is derived using mathematical programming. A refinement algorithm that builds atomically detailed models of the complex and combines coarse grained and atomic scoring is introduced. The refinement step consists of remodeling the interface side chains of the top scoring decoys from rigid docking followed by a short energy minimization. The refined models are then re‐ranked using a combination of coarse grained and atomic potentials. The docking algorithm including the refinement and re‐ranking, is compared favorably to other leading docking packages like ZDOCK, Cluspro, and PATCHDOCK, on the ZLAB 3.0 Benchmark and a test set of 30 novel complexes. A detailed analysis shows that coarse grained potentials perform better than atomic potentials for realistic unbound docking (where the exact structures of the individual bound proteins are unknown), probably because atomic potentials are more sensitive to local errors. Nevertheless, the atomic potential captures a different signal from the residue potential and as a result a combination of the two scores provides a significantly better prediction than each of the approaches alone. Proteins 2013. © 2012 Wiley Periodicals, Inc.     

Genome‐scale models of metabolism and gene expression extend and refine growth phenotype prediction

Growth is a fundamental process of life. Growth requirements are well‐characterized experimentally for many microbes; however, we lack a unified model for cellular growth. Such a model must be predictive of events at the molecular scale and capable of explaining the high‐level behavior of the cell as a whole. Here, we construct an ME‐Model for Escherichia coli—a genome‐scale model that seamlessly integrates metabolic and gene product expression pathways. The model computes ～80% of the functional proteome (by mass), which is used by the cell to support growth under a given condition. Metabolism and gene expression are interdependent processes that affect and constrain each other. We formalize these constraints and apply the principle of growth optimization to enable the accurate prediction of multi‐scale phenotypes, ranging from coarse‐grained (growth rate, nutrient uptake, by‐product secretion) to fine‐grained (metabolic fluxes, gene expression levels). Our results unify many existing principles developed to describe bacterial growth.     

Individual analyses of Lévy walk in semi-free ranging Tonkean macaques (Macaca tonkeana)

Animals adapt their movement patterns to their environment in order to maximize their efficiency when searching for food. The Lévy walk and the Brownian walk are two types of random movement found in different species. Studies have shown that these random movements can switch from a Brownian to a Lévy walk according to the size distribution of food patches. However no study to date has analysed how characteristics such as sex, age, dominance or body mass affect the movement patterns of an individual. In this study we used the maximum likelihood method to examine the nature of the distribution of step lengths and waiting times and assessed how these distributions are influenced by the age and the sex of group members in a semi free-ranging group of ten Tonkean macaques. Individuals highly differed in their activity budget and in their movement patterns. We found an effect of age and sex of individuals on the power distribution of their step lengths and of their waiting times. The males and old individuals displayed a higher proportion of longer trajectories than females and young ones. As regards waiting times, females and old individuals displayed higher rates of long stationary periods than males and young individuals. These movement patterns resembling random walks can probably be explained by the animals moving from one location to other known locations. The power distribution of step lengths might be due to a power distribution of food patches in the enclosure while the power distribution of waiting times might be due to the power distribution of the patch sizes.     

Identifying climatic niche shifts using coarse‐grained occurrence data: a test with non‐native freshwater fish

Aim We tested whether coarse‐grained occurrence data can be used to detect climatic niche shifts between native and non‐native ranges for a set of widely introduced freshwater fishes. Location World‐wide. Methods We used a global database of freshwater fish occurrences at the river basin scale to identify native and non‐native ranges for 18 of the most widely introduced fish species. We also examined climatic conditions within each river basin using fine‐grained climate data. We combined this information to test whether climatic niche shifts have occurred between native and non‐native ranges. We defined climatic niche shifts as instances where the ranges of a climatic variable within native and non‐native basins exhibit zero overlap. Results We detected at least one climatic niche shift for each of the 18 studied species. However, we did not detect common patterns in the thermal preference or biogeographic origin of the non‐native fish, hence suggesting a species‐specific response. Main conclusions Coarse‐grained occurrence data can be used to detect climatic niche shifts. They also enable the identification of the species experiencing niche shifts, although the mechanisms responsible for these shifts (e.g. local adaptation, dispersal limitation or physiological constraints) have yet to be determined. Furthermore, the coarse‐grained approach, which highlights regions where climatic niche shifts have occurred, can be used to select specific river basins for more detailed, fine‐grained studies.     

Mouth dimorphism in scale‐eating cichlid fish from Lake Tanganyika advances individual fitness

Random asymmetry, that is the coexistence of left‐ and right‐sided (or ‐handed) individuals within a population, is a particular case of natural variation; what triggers and maintains such dimorphisms remains unknown in most cases. Here, we report a field‐based cage experiment in the scale‐eating Tanganyikan cichlid Perissodus microlepis, which occurs in two morphs in nature: left‐skewed and right‐skewed individuals with respect to mouth orientation. Using underwater cages stocked with scale‐eaters and natural prey fish, we first confirm that, under semi‐natural conditions, left‐skewed scale‐eaters preferentially attack the right flank of their prey, whereas right‐skewed individuals feed predominantly from the left side. We then demonstrate that scale‐eaters have a higher probability for successful attacks when kept in dimorphic experimental populations (left‐ and right‐skewed morphs together) as compared to monomorphic populations (left‐ or right‐skewed morphs), most likely because prey fishes fail to accustom to strikes from both sides. The significantly increased probability for attacks appears to be the selective agent responsible for the evolution and maintenance of mouth dimorphism in P. microlepis, lending further support to the hypothesis that negative frequency‐dependent selection is the stabilizing force balancing the mouth dimorphism at quasi‐equal ratios in scale‐eating cichlids.     

Folding and stability of helical bundle proteins from coarse‐grained models

We develop a coarse‐grained model where solvent is considered implicitly, electrostatics are included as short‐range interactions, and side‐chains are coarse‐grained to a single bead. The model depends on three main parameters: hydrophobic, electrostatic, and side‐chain hydrogen bond strength. The parameters are determined by considering three level of approximations and characterizing the folding for three selected proteins (training set). Nine additional proteins (containing up to 126 residues) as well as mutated versions (test set) are folded with the given parameters. In all folding simulations, the initial state is a random coil configuration. Besides the native state, some proteins fold into an additional state differing in the topology (structure of the helical bundle). We discuss the stability of the native states, and compare the dynamics of our model to all atom molecular dynamics simulations as well as some general properties on the interactions governing folding dynamics. Proteins 2013; 81:1200–1211. © 2013 Wiley Periodicals, Inc.     

Of scales and stationarity in animal movements

With recent technological advances in tracking devices, movements of numerous animal species can be recorded with a high resolution over large spatial and temporal ranges. This opens promising perspectives for understanding how an animal perceives and reacts to the multi‐scale structure of its environment. Yet, conceptual issues such as confusion between movement scales and searching modes prevent us from properly inferring the movement processes at different scales. Here, I propose to build on stationarity (i.e. stability of statistical parameters) to develop a consistent theoretical framework in which animal movements are modelled as a generic composite multi‐scale multi‐mode random walk model. This framework makes it possible to highlight scales that are relevant to the studied animal, the nature of the behavioural processes that operate at each of these different scales, and the way in which the processes involved at any given scale can interact with those operating at smaller or larger scales. This explicitly scale‐focused approach should help properly analyse actual movements by relating, for each scale and each mode, the values of the main model parameters (speed, short‐ and long‐term persistences, degree of stochasticity) to the animal's needs and skills and its response to its environment at multiple scales.     

Understanding movement data and movement processes: current and emerging directions

Animal movement has been the focus on much theoretical and empirical work in ecology over the last 25 years. By studying the causes and consequences of individual movement, ecologists have gained greater insight into the behavior of individuals and the spatial dynamics of populations at increasingly higher levels of organization. In particular, ecologists have focused on the interaction between individuals and their environment in an effort to understand future impacts from habitat loss and climate change. Tools to examine this interaction have included: fractal analysis, first passage time, Lévy flights, multi‐behavioral analysis, hidden markov models, and state‐space models. Concurrent with the development of movement models has been an increase in the sophistication and availability of hierarchical bayesian models. In this review we bring these two threads together by using hierarchical structures as a framework for reviewing individual models. We synthesize emerging themes in movement ecology, and propose a new hierarchical model for animal movement that builds on these emerging themes. This model moves away from traditional random walks, and instead focuses inference on how moving animals with complex behavior interact with their landscape and make choices about its suitability.