Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths

It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step.     

Fractal analysis of narwhal space use patterns

Quantifying animal movement in response to a spatially and temporally heterogeneous environment is critical to understanding the structural and functional landscape influences on population viability. Generalities of landscape structure can easily be extended to the marine environment, as marine predators inhabit a patchy, dynamic system, which influences animal choice and behavior. An innovative use of the fractal measure of complexity, indexing the linearity of movement paths over replicate temporal scales, was applied to satellite tracking data collected from narwhals (Monodon monoceros) (n = 20) in West Greenland and the eastern Canadian high Arctic. Daily movements of individuals were obtained using polar orbiting satellites via the ARGOS data location and collection system. Geographic positions were filtered to obtain a daily good quality position for each whale. The length of total pathway was measured over seven different temporal length scales (step lengths), ranging from one day to one week, and a seasonal mean was calculated. Fractal dimension (D) was significantly different between seasons, highest during summer (D = 1.61, SE 0.04) and winter (D = 1.69, SE 0.06) when whales made convoluted movements in focal areas. Fractal dimension was lowest during fall (D = 1.34, SE 0.03) when whales were migrating south ahead of the forming sea ice. There were no significant effects of size category or sex on fractal dimension by season. The greater linearity of movement during the migration period suggests individuals do not intensively forage on patchy resources until they arrive at summer or winter sites. The highly convoluted movements observed during summer and winter suggest foraging or searching efforts in localized areas. Significant differences between the fractal dimensions on two separate wintering grounds in Baffin Bay suggest differential movement patterns in response to the dynamics of sea ice.     

How to reliably estimate the tortuosity of an animal's path: straightness, sinuosity, or fractal dimension?

The tortuosity of an animal's path is a key parameter in orientation and searching behaviours. The tortuosity of an oriented path is inversely related to the efficiency of the orientation mechanism involved, the best mechanism being assumed to allow the animal to reach its goal along a straight line movement. The tortuosity of a random search path controls the local searching intensity, allowing the animal to adjust its search effort to the local profitability of the environment. This paper shows that (1) the efficiency of an oriented path can be reliably estimated by a straightness index computed as the ratio between the distance from the starting point to the goal and the path length travelled to reach the goal, but such a simple index, ranging between 0 and 1, cannot be applied to random search paths; (2) the tortuosity of a random search path, ranging between straight line movement and Brownian motion, can be reliably estimated by a sinuosity index which combines the mean cosine of changes of direction with the mean step length; and (3) in the current state of the art, the fractal analysis of animals' paths, which may appear as an alternative and promising way to measure the tortuosity of a random search path as a fractal dimension ranging between 1 (straight line movement) and 2 (Brownian motion), is only liable to generate artifactual results. This paper also provides some help for distinguishing between oriented and random search paths, and depicts a general, comprehensive framework for analysing individual animals' paths in a two-dimensional space.     

Using animal movement paths to measure response to spatial scale

Animals live in an environment that is patchy and hierarchical. I present a method of detecting the scales at which animals perceive their world. The hierarchical nature of habitat causes movement path structure to vary with spatial scale, and the patchy nature of habitat causes movement path structure to vary throughout space. These responses can be measured by a combination of path tortuousity (measured with fractal dimension) versus spatial scale, the variation in tortuousity of small path segments along the movement path, and the correlation between tortuousities of adjacent path segments. These statistics were tested using simulated animal movements. When movement paths contained no spatial heterogeneity, then fractal D and variance continuously increased with scale, and correlation was zero at all scales. When movement paths contained spatial heterogeneity, then fractal D sometimes showed a discontinuity at transitions between domains of scale, variation showed peaks at transitions, and correlations showed a statistically significant positive value at scales smaller than patch size, decreasing to below zero at scales greater than patch size. I illustrated these techniques with movement paths from deer mice and red-backed voles. These new analyses should help understand how animals perceive and react to their landscape structure at various spatial scales, and to answer questions about how habitat structure affects animal movement patterns.     

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.     

Path tortuosity of Eurasian badgers (<Emphasis Type="Italic">Meles meles</Emphasis>) in a heterogeneous Mediterranean landscape

Movement is the process by which individual organisms are displaced over time to eat, reproduce and defend resources. Fractal analysis is a technique used to study animal movement that measures spatial complexity of path tortuosity; here, we apply it to characterize the movement patterns of the Eurasian badger (Meles meles) in a Mediterranean landscape. We calculated path tortuosity overall and seasonally, and for individuals of different sexes and social groups. The influence of variables related to badgers’ resources (food, shelter, water), human infra-structures and weather conditions were analysed with respect to the tortuosity of each badger’s path. A total of 55 search paths from six badgers were considered for this study. Although badgers generally displayed convoluted movement, there were two exceptions: (a) males overall and (b) all badgers in summer; for both, movements had a lower fractal value, i.e. were less tortuous. The convoluted movement pattern generally observed is probably adapted to the clumped distribution of food in the study area. Nevertheless, our results suggest that the use of dens and latrines were the principal determinants of tortuosity of badgers’ paths while foraging.     

Fractals and the analysis of growth paths

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns, a single numberD, the fractal dimension, to each curve.D=log(n)/[log(n)+log(d/L)], where:n is the number of line segments,L is the total length of the line segments, andd is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves,D=1; at the other end of the spectrum, for random walk curves,D→2. Standard statistics are done on the logarithms of the fractal dimension [log(D)]. With this measure, trails of biological movement, such as the growth paths of the cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.     

Effects of intrinsic and extrinsic factors on the diet of Bathyraja macloviana,a benthophagous skate

The effects of intrinsic (sex, maturity stage and body size) and extrinsic (depth and region) factors on the diet of Bathyraja macloviana, in the south‐west Atlantic Ocean, were evaluated using a multiple‐hypothesis modelling approach. Bathyraja macloviana fed mainly on polychaetes followed by amphipods, isopods and decapods. Effects of intrinsic and extrinsic factors on diet composition of this species were found. The consumption of polychaetes had a humped relationship with total length (L_T), and isopods and decapods increased with increasing L_T. Immature individuals preyed on amphipods more heavily than mature individuals. Furthermore, region and depth had an important effect on the consumption of isopods, decapods and amphipods. Such ontogenetic changes and spatial patterns may provide insights into understanding the regulatory mechanisms of marine communities.     

Effect of Colony, Patch Distance, And Trajectory Sense on Movement Complexity in Foraging Ants

Animals display foraging trajectories when exploiting food patches. These displacements have been studied, particularly in organisms that forage from a central place. The complexity of a movement path may be analyzed by means of the fractal dimension, an index that estimates the tortuosity of a trajectory. In field experiments we studied the effect of trajectory sense, distance to the resource patch and colony on the movement complexity in a common ant of central Chile. We found that these factors and their interactions significantly affected the complexity of movement paths. We discuss whether mortality risk determines more linear trajectories when the resource patch is distant from the nest, and whether the information acquired from the environment by workers determines less complex return trips.     

Texture of locomotor path: a replicable characterization of a complex behavioral phenotype

A database of mouse locomotor path in spatial tests can be used to search in silico for behavioral measures that better discriminate between genotypes and are more replicable across laboratories. In this study, software for the exploration of exploration (SEE) was used to search a large database for a novel behavioral measure that would characterize complex movement paths. The database included mouse open-field behavior assessed in 3 laboratories, 7 inbred strains, several pharmacological treatments and hundreds of animals. The new behavioral measure, "path texture", was characterized using the local curvature of the path (the change of direction per unit distance, in degrees/cm) across several spatial scales, starting from scales smaller than the animal's body length and up to the scale of the arena size. Path texture analysis differs from fractal dimension analysis in that it does not assume self-similarity across scales. Path texture was found to discriminate inbred strains with relatively high broad-sense heritability (43%-71%) and high replicability across laboratories. Even genotypes that had similar path curvatures in some scales usually differed in other scales, and self-similarity across scales was not displayed by all genotypes. Amphetamine decreased the path curvature of C57BL/6 mice in small and medium scales, while having no effect on DBA/2J mice. Diazepam dose-dependently decreased the curvature of C57BL/6 mice across all scales, while 2 anxiogenic drugs, FG-7142 and pentylenetetrazole, increased it. Path texture thus has high potential for behavioral phenotyping and the study of drug effects in the mouse.     

Self-organisation in streams: the relationship between movement behaviour and body condition in a headwater salamander

1. When movement behaviour is correlated with traits affecting fitness, it may affect population performance directly, independent of extrinsic habitat conditions. 2. In a previous 3‐year, capture–recapture study, upstream movement by the salamander Gyrinophilus porphyriticus compensated for low reproduction in the upper 500 m of Merrill Brook, a first‐order stream in northern New Hampshire (U.S.A.). This immigrant subsidy resulted from excess reproduction in the downstream section (i.e. the lower 500 m of stream length) and from consistently upstream‐biased movement by salamanders. Reproduction in the two stream sections was positively correlated with mean body condition. Using 6 years of capture–recapture data, this study examines whether the movement behaviour of G. porphyriticus is related to body condition and thereby directly influences mean body condition and reproduction in the two sections of Merrill Brook. 3. Upstream‐biased movement and greater mean body condition in the downstream section were consistent across 6 years of data collection. In Merrill Brook and four other streams, however, individuals with high body condition were more likely to move upstream and low‐condition individuals were more likely to move downstream. Movement direction was unrelated to the size, sex and initial location of individuals. Body condition was positively related to growth rate, further supporting its link to reproductive potential, and positively autocorrelated through time in individuals that moved. 4. Results of this 6‐year study suggest that the movement behaviour of G. porphyriticus partially compensated for environmental factors differentiating mean body condition and reproduction along Merrill Brook and illustrate the potential for this form of self‐organisation to occur in linear habitats such as streams and rivers.     

Optimization of two-joint arm movements: a model technique or a result of natural selection?

The fossil record of early hominids suggests that their Arm length, and presumably stature and weight, had a tendency to increase. Using the minimum jerk principle and a related formulation of averaged specific power, ASP, with regard to selected two-joint Arm movements, the current paper explores relationships between ASP, hand trajectory length (or Arm length, or body mass) and mean movement speed, deriving relationships which indicate that ASP is proportional to cubic mean movement speed, but inversely proportional to hand trajectory length (or Arm length, or 1/3 power of body mass). Accordingly, an `ecological niche’ is modeled in a three-parameter space. Either ASP maximization for fixed movement time, or ASP minimization for fixed mean movement speed, taken as selective optimization criterion, allows the increasing of human Arm length during evolution, regardless of the arm-to-forearm length ratio.     

Population size and growth rate, sex ratio and behaviour in the ant isopod, Platyarthrus hoffmannseggi

Female-biased sex ratios are often associated with small, isolated populations. These conditions are exhibited in populations of Platyarthrus hoflmannseggi (Brandt, 1833), a small, blind woodlouse which lives almost solely in ant nests. This study was undertaken to determine how sex ratio varies with population size in P. hoffmunnseggi and how both of these factors affect population growth rate. To accomplish this, a total of 2155 of these isopods were collected from 20 nests of the ant LusiusJaour (Fabricius, 1781). The majority of the isopod populations were female-biased. The behaviour and survivorship of the isopods in their own and foreign antcolonies were compared and suggest that the free movement of P. hoffmannseggi individuals among ant colonies may be extremely limited. These studies also show how isopods are nutritionally linked to their hosts. Measurement of the woodlice revealed two distinct size/age classes in each population and allowed the change in sex ratio from one generation to the next to be estimated. Female-biased populations produce in future generations relatively more females and fewer males than populations of a similar size with a sex ratio of unity. There was no correlation between the degree of female bias and the rate of population growth. These findings are reviewed in the light of recent theoretical sex ratio models and the possible control of the isopods sex allocation by unusual chromosomal mechanisms and intracellular parasites.     

Olfactory orientation responses of the eucalyptus woodborer, Phoracantha semipunctata, to host plant in a wind tunnel

The eucalyptus woodborer, Phoracantha semipunctata Fabricius (Coleoptera: Cerambycidae), attacks mainly species of Eucalyptus (Myrtaceae). This study investigated walking and flight behaviour of P. semipunctata males and females exposed to an odour plume originating from a log of E. globulus placed vertically in the upwind end of a wind tunnel. In control experiments, beetles were exposed to a PVC drainpipe in the same position as the log, providing a visual stimulus without host‐tree odour. No statistical differences were found between behavioural responses of either sex when exposed to the log or PVC pipe. No beetles landed on the PVC pipe, whereas 49% of the beetles exposed to host‐tree odour plume landed on the log. Beetles aged over 24 days after emergence from the host tree were more responsive than beetles aged 20–24 days, and accounted vor 86% of the beetles that landed on the log. While walking, host‐tree odour affected the behaviour of the beetles that landed on the log as follows: upwind movement and path linearity increased, whereas turning rate, stopping frequency, mean stopping time and time to take‐off flight decreased. During flight, host‐tree odour affected the behaviour of the beetles that landed on the log as follows: increased upwind flight, turning rate, flight time, flight distance, and decreased flight speed. For beetles that never lost contact with the odour plume, flight progressed upwind with narrow zigzags, and showed higher directedness upwind, path linearity, faster flight speed and lower turning rate than for beetles that lost contact with the odour plume. After loosing contact with the plume, beetles tended to decrease their upwind progression, exhibiting a sharp turn or quick counterturns followed by crosswind or downwind excursions. This led to regaining contact with the odour plume and resumed upwind progression at higher speed provided they flew within the boundaries of the plume. The results showed that host‐tree odour affects both walking and flight behaviour of P. semipunctata beetles, inducing a more directed upwind movement and landing on the visual stimulus of a tree trunk.     

Cannibalism as an interacting phenotype: precannibalistic aggression is influenced by social partners in the endangered Socorro Isopod (Thermosphaeroma thermophilum)

Models for the evolution of cannibalism highlight the importance of asymmetries between individuals in initiating cannibalistic attacks. Studies may include measures of body size but typically group individuals into size/age classes or compare populations. Such broad comparisons may obscure the details of interactions that ultimately determine how socially contingent characteristics evolve. We propose that understanding cannibalism is facilitated by using an interacting phenotypes perspective that includes the influences of the phenotype of a social partner on the behaviour of a focal individual and focuses on variation in individual pairwise interactions. We investigated how relative body size, a composite trait between a focal individual and its social partner, and the sex of the partners influenced precannibalistic aggression in the endangered Socorro isopod, Thermosphaeroma thermophilum. We also investigated whether differences in mating interest among males and females influenced cannibalism in mixed sex pairs. We studied these questions in three populations that differ markedly in range of body size and opportunities for interactions among individuals. We found that relative body size influences the probability of and latency to attack. We observed differences in the likelihood of and latency to attack based on both an individual's sex and the sex of its partner but found no evidence of sexual conflict. The instigation of precannibalistic aggression in these isopods is therefore a property of both an individual and its social partner. Our results suggest that interacting phenotype models would be improved by incorporating a new conditional ψ, which describes the strength of a social partner's influence on focal behaviour.     

Combining animal movements and behavioural data to detect behavioural states

Animal movement paths show variation in space caused by qualitative shifts in behaviours. I present a method that (1) uses both movement path data and ancillary sensor data to detect natural breakpoints in animal behaviour and (2) groups these segments into different behavioural states. The method can also combine analyses of different path segments or paths from different individuals. It does not assume any underlying movement mechanism. I give an example with simulated data. I also show the effects of random variation, # of states and # of segments on this method. I present a case study of a fisher movement path spanning 8 days, which shows four distinct behavioural states divided into 28 path segments when only turning angles and speed were considered. When accelerometer data were added, the analysis shows seven distinct behavioural states divided into 41 path segments.     

Effects of macrophyte heterogeneity and food availability on structural parameters of the macroinvertebrate community in a Pampean stream

Environmental heterogeneity in natural ecosystems influences several parameters at the population and community levels. In freshwater ecosystems, habitat heterogeneity can be provided by macrophyte species with different structural shapes. Previous studies suggest that aquatic plants with more complex architectures will support higher number, biomass, and taxon richness of macroinvertebrates than plants with simpler shape. We investigated the influence of macrophyte structural heterogeneity (quantified by fractal dimension) and food availability (represented by epiphytic biomass) on several parameters (number of individuals, biomass, body size distribution, taxon richness, and diversity) of the macroinvertebrate community in a Pampean stream. Four submerged macrophyte species (Egeria densa, Elodea ernstae, Ceratophyllum demersum, and Stuckenia striata) and associated macroinvertebrates were sampled in late spring, summer, and autumn. Plants were photographed and fractal dimension was estimated from the images by the box-counting method. Fractal dimension was independent of plant surface area per unit of macrophyte biomass and differed significantly among species. Mean fractal dimension varied between 1.29 and 1.62, and increased following the sequence E. densa → S. striata → E. ernstae → C. demersum. Macrophyte species with higher fractal dimension supported a greater abundance of macroinvertebrates, especially those of small body size (500–1,000 μm); but fractal dimension was unrelated to macroinvertebrate biomass, richness, and diversity. However, overall animal biomass was significantly associated to the epiphytic abundance. Consequently, macrophyte heterogeneity influences macroinvertebrate density and body size distribution, while animal biomass depends on epiphytic food resources provided by plants.     

Negative or positive density‐dependence in movements depends on climatic seasons: The case of a Neotropical marsupial

One of the major challenges in animal ecology is to understand the factors and processes driving movement behaviour. Although density may influence movement patterns, the occurrence and nature of density‐dependence in animal movements are still unclear, particularly whether it may vary among populations of a species, or across time within a population. Here, we evaluate the occurrence and nature of density‐dependence in the movements of a Neotropical marsupial, the Grey four‐eyed opossum Philander frenatus (Didelphidae, Didelphimorphia). We quantified fine‐scale path tortuosity of individuals inhabiting continuous forest areas and forest fragments, in different climatic seasons (humid vs. super‐humid). We also determined the relative importance of population size compared to sex and body mass on movements, using a model‐selection approach. In forest fragments, path tortuosity increased with population size in the super‐humid season, but decreased in the humid season. In the continuous forest, path tortuosity was affected only by sex and body mass, being slightly higher in males and negatively related to body mass. The occurrence of density‐dependence on movements only in forest fragments is likely to reflect the higher overall density of P. frenatus in small forest fragments. The variation in the nature of density‐dependence between climatic seasons is likely to reflect a trade off between foraging over large areas (humid season, low resource availability) versus avoiding agonistic encounters (super‐humid season, high resource availability). Our results show that (i) density‐dependence in movements may be context‐dependent occurring only in areas of relatively high overall population density; and (ii) density may affect movements in different ways at different climatic seasons.     

Feeding habits of the Magellan skate: effects of sex, maturity stage, and body size on diet

The aim of this study was to evaluate the effects of sex, maturity stage, and body size on the diet of the Magellan skate, Bathyraja magellanica, in the Southwest Atlantic off Argentina, by examining stomach contents using a multiple hypothesis modeling approach. Relationships between the number of prey and sex, maturity stage, and total length (TL) were assessed by built generalized linear models (GLM). Furthermore, we tested whether there was a threshold size at which B. magellanica started or quit consuming a given prey. The overall diet of B. magellanica was mainly consisted of teleosts, followed by amphipods, isopods, and decapods. Ontogenetic diet shifts were independent of sex and maturity stage. However, discrete shifts in diet with TL were found, with individuals larger than 554 and 623 mm TL ceasing to consume amphipods and isopods, respectively. The consumption of teleosts progressively increased with increasing predator size. Likewise, ontogenetic shifts in foraging behavior were also observed with smaller individuals showing specialization on amphipods with larger specimens consuming teleosts. These results confirm that ontogenetic shifts in diet of B. magellanica are more a function of predator size rather than any other life-history traits. We propose that these food shifts are probably related to morphological limitations and abilities associated with feeding habits of skate, so when specimens of B. magellanica reach an optimum body size, they may have access to higher quality trophic resources. Our results suggest that evaluating the importance of life-history stages on the feeding habits of a species is essential for understanding how that species exploits food resources, which, in turn, is an important factor in developing a suitable plan of marine ecosystem conservation.     

The effects of overwintering and habitat type on body condition and locomotion of the wolf spider Pardosa alacris

Overwintering in temperate regions is a prominent mortality risk for invertebrates and may affect their behaviour and body condition. Pardosa alacris is a common ground dwelling spider in central European native and plantation forests, and habitat type and prey availability may play important roles in their overwintering. The effect of overwintering on body condition and behaviour of spiders in semi natural and exotic habitats is relatively unknown. Here we assess the effects of winter on spiders from native poplar and exotic pine plantations. The locomotory behaviour of P. alacris (distance covered and speed) was assessed by tracking their movement in a white circular plastic arena. We assessed body condition, body size, and total fat content. Forest type and sex had significant effects on body length. Fat content was significantly higher in the spring than in autumn, and spiders covered larger distances and were faster in autumn than in spring. Fat content had a significant negative effect on average speed. Spiders in native forests were smaller but grew more during the winter than in exotic plantations, possibly due to higher prey availability in native forests. Visually-hunting predators may significantly affect spiders. Fat spiders with better body condition moved less, and were thus less detectable by predators. However the low movement rate may result in a low rate of encountering prey items, thus lowering feeding efficiency.