Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters

Mathematical modelling of the directed movement of animals, microorganisms and cells is of great relevance in the fields of biology and medicine. Simple diffusive models of movement assume a random walk in the position, while more realistic models include the direction of movement by assuming a random walk in the velocity. These velocity jump processes, although more realistic, are much harder to analyse and an equation that describes the underlying spatial distribution only exists in one dimension. In this communication we set up a realistic reorientation model in two dimensions, where the mean turning angle is dependent on the previous direction of movement and bias is implicitly introduced in the probability distribution for the direction of movement. This model, and the associated reorientation parameters, is based on data from experiments on swimming microorganisms. Assuming a transport equation to describe the motion of a population of random walkers using a velocity jump process, together with this realistic reorientation model, we use a moment closure method to derive and solve a system of equations for the spatial statistics. These asymptotic equations are a very good match to simulated random walks for realistic parameter values.     

The home range fractal: From random walk to memory-dependent space use

We present theoretical developments of the multi-scaled random walk (MRW) model for cognitive map-influenced space use by animals. The extensions include a unified space–time scaling function, and further details with respect to statistical properties of the spatial distribution of a set of locations. Supported by numeric simulations we show how memory effects may open for a complex, multi-scaled and self-organized – i.e., intrinsically driven – habitat utilization pattern with fractal dimensional properties. These properties allow for testing for MRW compliance by using parameters from classic movement models like Brownian motion, correlated random walk and Levy walks as null models. In terms of applied ecology, empirical confirmation of memory-influenced space use by individuals will have consequences for interpretation and statistical analyses of habitat utilization. For example, under memory map influence, re-visits to intra-home range locations do not represent independent events. Further, the MRW formulation specifically implies home range movements over a continuum of process rates, spanning a range of spatio-temporal scales in parallel, in violation of the traditional low order Markovian (scale-specific) model architecture. The MRW approach requires an extension of classic Boltzmann–Gibbs statistical mechanics, which rests on the premise that spatio-temporal memory effects are averaged out beyond micro-scales. We suggest that the emergent coherence between spatial and temporal scaling from the MRW approach may open for a more realistic statistical mechanics theory for population processes under terms of memory-influenced space use by individuals.     

Extracellular Wire Tetrode Recording in Brain of Freely Walking Insects

Increasing interest in the role of brain activity in insect motor control requires that we be able to monitor neural activity while insects perform natural behavior. We previously developed a technique for implanting tetrode wires into the central complex of cockroach brains that allowed us to record activity from multiple neurons simultaneously while a tethered cockroach turned or altered walking speed. While a major advance, tethered preparations provide access to limited behaviors and often lack feedback processes that occur in freely moving animals. We now present a modified version of that technique that allows us to record from the central complex of freely moving cockroaches as they walk in an arena and deal with barriers by turning, climbing or tunneling. Coupled with high speed video and cluster cutting, we can now relate brain activity to various parameters of the movement of freely behaving insects.     

Inherent high correlation of individual motility enhances population dispersal in a heterotrophic, planktonic protist

Quantitative linkages between individual organism movements and the resulting population distributions are fundamental to understanding a wide range of ecological processes, including rates of reproduction, consumption, and mortality, as well as the spread of diseases and invasions. Typically, quantitative data are collected on either movement behaviors or population distributions, rarely both. This study combines empirical observations and model simulations to gain a mechanistic understanding and predictive ability of the linkages between both individual movement behaviors and population distributions of a single-celled planktonic herbivore. In the laboratory, microscopic 3D movements and macroscopic population distributions were simultaneously quantified in a 1L tank, using automated video- and image-analysis routines. The vertical velocity component of cell movements was extracted from the empirical data and used to motivate a series of correlated random walk models that predicted population distributions. Validation of the model predictions with empirical data was essential to distinguish amongst a number of theoretically plausible model formulations. All model predictions captured the essence of the population redistribution (mean upward drift) but only models assuming long correlation times (minute), captured the variance in population distribution. Models assuming correlation times of 8 minutes predicted the least deviation from the empirical observations. Autocorrelation analysis of the empirical data failed to identify a de-correlation time in the up to 30-second-long swimming trajectories. These minute-scale estimates are considerably greater than previous estimates of second-scale correlation times. Considerable cell-to-cell variation and behavioral heterogeneity were critical to these results. Strongly correlated random walkers were predicted to have significantly greater dispersal distances and more rapid encounters with remote targets (e.g. resource patches, predators) than weakly correlated random walkers. The tendency to disperse rapidly in the absence of aggregative stimuli has important ramifications for the ecology and biogeography of planktonic organisms that perform this kind of random walk.     

Socially informed random walks: incorporating group dynamics into models of population spread and growth

Simple correlated random walk (CRW) models are rarely sufficient to describe movement of animals over more than the shortest time scales. However, CRW approaches can be used to model more complex animal movement trajectories by assuming individuals move in one of several different behavioural or movement states, each characterized by a different CRW. The spatial and social context an individual experiences may influence the proportion of time spent in different movement states, with subsequent effects on its spatial distribution, survival and fecundity. While methods to study habitat influences on animal movement have been previously developed, social influences have been largely neglected. Here, we fit a 'socially informed' movement model to data from a population of over 100 elk (Cervus canadensis) reintroduced into a new environment, radio-collared and subsequently tracked over a 4-year period. The analysis shows how elk move further when they are solitary than when they are grouped and incur a higher rate of mortality the further they move away from the release area. We use the model to show how the spatial distribution and growth rate of the population depend on the balance of fission and fusion processes governing the group structure of the population. The results are briefly discussed with respect to the design of species reintroduction programmes.     

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.     

The limitation of species range: A consequence of searching along resource gradients

Ecological modelers have long puzzled over the spatial distribution of species. The random walk or diffusive approach to dispersal has yielded important results for biology and mathematics, yet it has been inadequate in explaining all phenomenological features. Ranges can terminate non-smoothly absent a complementary shift in the characteristics of the environment. Also unexplained is the absence of a species from nearby areas of adequate, or even abundant, resources. In this paper, I show how local searching behavior–keyed to a density-dependent fitness–can limit the speed and extent of a species’ spread. In contrast to standard diffusive processes, pseudo-rational movement facilitates the clustering of populations. It also can be used to estimate the speed of an expanding population range, explain expansion stall, and provides a mechanism by which a population can colonize seemingly removed regions — biogeographic islands in a continental framework. Finally, I discuss the effect of resource degradation and different resource impact/utilization curves on the model.     

Stochastic modelling of animal movement

Modern animal movement modelling derives from two traditions. Lagrangian models, based on random walk behaviour, are useful for multi-step trajectories of single animals. Continuous Eulerian models describe expected behaviour, averaged over stochastic realizations, and are usefully applied to ensembles of individuals. We illustrate three modern research arenas. (i) Models of home-range formation describe the process of an animal ‘settling down’, accomplished by including one or more focal points that attract the animal''s movements. (ii) Memory-based models are used to predict how accumulated experience translates into biased movement choices, employing reinforced random walk behaviour, with previous visitation increasing or decreasing the probability of repetition. (iii) Lévy movement involves a step-length distribution that is over-dispersed, relative to standard probability distributions, and adaptive in exploring new environments or searching for rare targets. Each of these modelling arenas implies more detail in the movement pattern than general models of movement can accommodate, but realistic empiric evaluation of their predictions requires dense locational data, both in time and space, only available with modern GPS telemetry.     

Behavioral states help translate dispersal movements into spatial distribution patterns of floaters

Within the field of spatial ecology, it is important to study animal movements in order to better understand population dynamics. Dispersal is a nonlinear process through which different behavioral mechanisms could affect movement patterns. One of the most common approaches to analyzing the trajectories of organisms within patches is to use random-walk models to describe movement features. These models express individual movements within a specific area in terms of random-walk parameters in an effort to relate movement patterns to the distributions of organisms in space. However, only using the movement trajectories of individuals to predict the spatial spread of animal populations may not fit the complex distribution of individuals across heterogeneous environments. When we empirically tested the results from a random-walk model (a residence index) used to predict the spatial equilibrium distribution of individuals, we found that the index severely underestimated the spatial spread of dispersing individuals. We believe this is because random-walk models only account for the effects of environmental conditions on individual movements, completely overlooking the crucial influence of behavior changes over time. In the future, both aspects should be accounted for when predicting general rules of (meta)population abundance, distribution, and dynamics from patterns of animal movements.     

Steepness of thermal gradient is essential to obtain a unified view of thermotaxis in C. elegans

One of the adaptive behaviors of animals in their environment is thermotaxis, by which they migrate toward a preferred temperature. This sensorimotor integration is accomplished by choosing one of two behaviors depending on the surrounding temperature, namely thermophilic or cryophilic movement. Caenorhabditis elegans exhibits thermotaxis and its migration behavior has been analyzed experimentally at both the population and individual levels. However, some experimental data are inconsistent especially for thermophilic movement, which is expected to be observed in lower than favorable temperatures. There are no experimental analyzes that find thermophilic tendencies in the individual behavior of worms, despite multiple reports supporting thermophilic movement of the population. Although theoretical methods have been used to study thermotaxis of C. elegans, no mathematical model provides a consistent explanation for this discrepancy. Here we develop a simple biased random walk model, which describes population behavior, but which is based on the results of individual assays. Our model can integrate all previous experiments without any contradiction. We regenerate all the population patterns reported in past studies and give a consistent explanation for the conflicting results. Our results suggest that thermophilic movement is observed, even in individual movements, when the thermal gradient is sufficiently slight. On the contrary, thermophilic movement disappears when the thermal gradient is too steep. The thermal gradient is thus essential for a comprehensive understanding of the experimental studies of thermotaxis in C. elegans. Our model provides insight into an integrative understanding of the neural activity and thermotactic behavior in C. elegans.     

High-content behavioral analysis of Caenorhabditis elegans in precise spatiotemporal chemical environments

To quantitatively understand chemosensory behaviors, it is desirable to present many animals with repeatable, well-defined chemical stimuli. To that end, we describe a microfluidic system to analyze Caenorhabditis elegans behavior in defined temporal and spatial stimulus patterns. A 2 cm × 2 cm structured arena allowed C. elegans to perform crawling locomotion in a controlled liquid environment. We characterized behavioral responses to attractive odors with three stimulus patterns: temporal pulses, spatial stripes and a linear concentration gradient, all delivered in the fluid phase to eliminate variability associated with air-fluid transitions. Different stimulus configurations preferentially revealed turning dynamics in a biased random walk, directed orientation into an odor stripe and speed regulation by odor. We identified both expected and unexpected responses in wild-type worms and sensory mutants by quantifying dozens of behavioral parameters. The devices are inexpensive, easy to fabricate, reusable and suitable for delivering any liquid-borne stimulus.     

Scaling marine fish movement behavior from individuals to populations

Understanding how, where, and when animals move is a central problem in marine ecology and conservation. Key to improving our knowledge about what drives animal movement is the rising deployment of telemetry devices on a range of free‐roaming species. An increasingly popular way of gaining meaningful inference from an animal's recorded movements is the application of hidden Markov models (HMMs), which allow for the identification of latent behavioral states in the movement paths of individuals. However, the use of HMMs to explore the population‐level consequences of movement is often limited by model complexity and insufficient sample sizes. Here, we introduce an alternative approach to current practices and provide evidence of how the inclusion of prior information in model structure can simplify the application of HMMs to multiple animal movement paths with two clear benefits: (a) consistent state allocation and (b) increases in effective sample size. To demonstrate the utility of our approach, we apply HMMs and adapted HMMs to over 100 multivariate movement paths consisting of conditionally dependent daily horizontal and vertical movements in two species of demersal fish: Atlantic cod (Gadus morhua; n = 46) and European plaice (Pleuronectes platessa; n = 61). We identify latent states corresponding to two main underlying behaviors: resident and migrating. As our analysis considers a relatively large sample size and states are allocated consistently, we use collective model output to investigate state‐dependent spatiotemporal trends at the individual and population levels. In particular, we show how both species shift their movement behaviors on a seasonal basis and demonstrate population space use patterns that are consistent with previous individual‐level studies. Tagging studies are increasingly being used to inform stock assessment models, spatial management strategies, and monitoring of marine fish populations. Our approach provides a promising way of adding value to tagging studies because inferences about movement behavior can be gained from a larger proportion of datasets, making tagging studies more relevant to management and more cost‐effective.     

Effects of neck and circumoesophageal connective lesions on posture and locomotion in the cockroach

Few studies in arthropods have documented to what extent local control centers in the thorax can support locomotion in absence of inputs from head ganglia. Posture, walking, and leg motor activity was examined in cockroaches with lesions of neck or circumoesophageal connectives. Early in recovery, cockroaches with neck lesions had hyper-extended postures and did not walk. After recovery, posture was less hyper-extended and animals initiated slow leg movements for multiple cycles. Neck lesioned individuals showed an increase in walking after injection of either octopamine or pilocarpine. The phase of leg movement between segments was reduced in neck lesioned cockroaches from that seen in intact animals, while phases in the same segment remained constant. Neither octopamine nor pilocarpine initiated changes in coordination between segments in neck lesioned individuals. Animals with lesions of the circumoesophageal connectives had postures similar to intact individuals but walked in a tripod gait for extended periods of time. Changes in activity of slow tibial extensor and coxal depressor motor neurons and concomitant changes in leg joint angles were present after the lesions. This suggests that thoracic circuits are sufficient to produce leg movements but coordinated walking with normal motor patterns requires descending input from head ganglia.Electronic Supplementary Material Supplementary material is available for this article at     

Exploitation of home range and spatial distribution of resources in German cockroaches (Dictyoptera: Blattellidae)

We analyzed the expression of edge following behavior in German cockroaches, Blattella germanica (L.). Structure of the environment and spatial distribution of resources modified both expression of locomotory behavior of cockroaches and its distribution between the edge and the central area of an experimental arena. When they are in a familiar environment, German cockroaches are definitely not edge followers and they exploit similarly the different parts of the accessible surface of their home range. Cockroaches placed in a new environment tended to follow edges more than undisturbed insects. This modification could be assimilated to a response to stressful conditions.     

Nonrandom spatial distribution by mammalian cells in culture

Quadrat analysis was used to investigate the spatial distribution of seven mammalian cell lines in culture. The number of cells in replicate unit areas of the culture was determined, and the variance to mean ratio used as an index of random and nonrandom spatial distribution. Only mouse SV3T3 cells distributed themselves randomly throughout the entire culture growth cycle. The remaining six lines all assumed a nonrandom distribution at some point in their growth cycles. Mouse L929 cells displayed avoidance behavior, and spaced themselves at regular intervals in a uniform spatial distribution. The five remaining lines (mouse S180, rat C6, hamster CHO, canine MDCK, and human BeWo) formed multicellular clusters, and were distributed aggregatively rather than randomly. Random walk migration can account for the random distribution of SV3T3 cells. Random walk combined with contact inhibition of movement provides a satisfactory explanation for the uniform distribution of L929 cells. Random walk and contact inhibition are incompatible with cell clustering, however. Thus other mechanisms of motility or adhesiveness must contribute to cell clustering. It is possible that random walk and contact inhibition may be less common components of cell movement than generally assumed.     

A biomechanical hypothesis explaining upstream movements by the freshwater snail Elimia

1. Many taxa of freshwater invertebrates show active upstream movements, particularly the snails. Hypotheses explaining this behaviour invoke the search for food or space, compensation for drift, avoidance of predation and hydrodynamic effects. The pervasiveness of upstream movements among remote lineages of snails (two subclasses, three orders, 10 families), however, suggests that snails may move upstream for mechanical rather than adaptive reasons.
2. It is proposed that upstream movements by snails are a function of torque on the snail's foot generated by hydrodynamic drag on the shell. When subject to high broadside drag-forces on their shells, snails are able to reduce torque and stabilize orientation only by directing their anterior aspect upstream.
3. Movements of the freshwater pleurocerid snail Elimia were studied by following marked free-ranging individuals in six streams in Alabama, USA (four species, eight populations).
4. Populations showed either no net movement (two streams) or significant upstream movements ranging to a mean of ≈40 m over a 3-month period (four streams). Movement patterns were stream specific rather than species or population specific. Within populations showing significant upstream movements, snails with shell lengths ≤10 mm showed little net movement. Larger snails showed movements from 0 to >200 m upstream.
5. A torque-constrained random walk model was used to perform a post hoc test of the hypothesis that upstream movements were a function of torque on the snail's foot generated by hydrodynamic drag on the shell. The model predicted upstream and size-dependent movement patterns that approximated those observed for snails in the field.     

Automatic detection of single fluorophores in live cells

Recent developments in light microscopy enable individual fluorophores to be observed in aqueous conditions. Biological molecules, labeled with a single fluorophore, can be localized as isolated spots of light when viewed by optical microscopy. Total internal reflection fluorescence microscopy greatly reduces background fluorescence and allows single fluorophores to be observed inside living cells. This advance in live-cell imaging means that the spatial and temporal dynamics of individual molecules can be measured directly. Because of the stochastic nature of single molecule behavior a statistically meaningful number of individual molecules must be detected and their separate trajectories in space and time stored and analyzed. Here, we describe digital image processing methods that we have devised for automatic detection and tracking of hundreds of molecules, observed simultaneously, in vitro and within living cells. Using this technique we have measured the diffusive behavior of pleckstrin homology domains bound to phosphoinositide phospholipids at the plasma membrane of live cultured mammalian cells. We found that mobility of these membrane-bound protein domains is dominated by mobility of the lipid molecule to which they are attached and is highly temperature dependent. Movement of PH domains isolated from the tail region of myosin-10 is consistent with a simple random walk, whereas, diffusion of intact PLC-delta1 shows behavior inconsistent with a simple random walk. Movement is rapid over short timescales but much slower at longer timescales. This anomalous behavior can be explained by movement being restricted to membrane regions of 0.7 microm diameter.