Modelling nematode movement using time-fractional dynamics

We use a correlated random walk model in two dimensions to simulate the movement of the slug parasitic nematode Phasmarhabditis hermaphrodita in homogeneous environments. The model incorporates the observed statistical distributions of turning angle and speed derived from time-lapse studies of individual nematode trails. We identify strong temporal correlations between the turning angles and speed that preclude the case of a simple random walk in which successive steps are independent. These correlated random walks are appropriately modelled using an anomalous diffusion model, more precisely using a fractional sub-diffusion model for which the associated stochastic process is characterised by strong memory effects in the probability density function.     

Random movement pattern of the sea urchin Strongylocentrotus droebachiensis

We describe the fine-scale movement of the sea urchin Strongylocentrotus droebachiensis based on analyses of video recordings of undisturbed individuals in the two habitats which mainly differed in food availability, urchin barrens and grazing front. Urchin activity decreased as urchin density increased. Individuals alternated between moving and being stationary and their behaviour did not appear to be affected by either current velocity (within the range from 0 to 15 cm s^− 1) and temperature (2.3 to 6.0 °C). Movement of individuals at each location was compared to that predicted by a random walk model. Mean move length (linear distance between two stationary periods), turning angle and net squared displacement were calculated for each individual. The distribution of turning angles was uniform at each location and there was no evidence of a relationship between urchin density and either move length or urchin velocity. The random model predicted a higher dispersal rate at locations with low urchin densities, such as barrens habitats. However, the movement was sometimes greater or less than predicted by the model, suggesting the influence of local environmental factors. The deviation of individual paths from the model revealed that urchins can be stationary or adopt a local (displacement less than random), random or directional movement. The net daily distance displaced on the barrens, predicted by a random walk model, was similar to the observed movement recorded in our previous study of tagged urchins at one site, but less than that observed at a second site. We postulate that the random dispersal of urchins allows individuals on barrens to reach the kelp zone where food is more abundant although the time required to reach the kelp zone may be considerable (months to years). Urchins decrease their rate of dispersal once they reach the kelp zone so that they likely remain close to this abundant food sources for long periods.     

One size does not fit all: flexible models are required to understand animal movement across scales

1. Large data sets containing precise movement data from free-roaming animals are now becoming commonplace. One means of analysing individual movement data is through discrete, random walk-based models. 2. Random walk models are easily modified to incorporate common features of animal movement, and the ways that these modifications affect the scaling of net displacement are well studied. Recently, ecologists have begun to explore more complex statistical models with multiple latent states, each of which are characterized by a distribution of step lengths and have their own unimodal distribution of turning angles centred on one type of turn (e.g. reversals). 3. Here, we introduce the compound wrapped Cauchy distribution, which allows for multimodal distributions of turning angles within a single state. When used as a single state model, the parameters provide a straightforward summary of the relative contributions of different turn types. The compound wrapped Cauchy distribution can also be used to build multiple state models. 4. We hypothesize that a multiple state model with unimodal distributions of turning angles will best describe movement at finer resolutions, while a multiple state model using our multimodal distribution will better describe movement at intermediate temporal resolutions. At coarser temporal resolutions, a single state model using our multimodal distribution should be sufficient. We parameterize and compare the performance of these models at four different temporal resolutions (1, 4, 12 and 24 h) using data from eight individuals of Loxodonta cyclotis and find support for our hypotheses. 5. We assess the efficacy of the different models in extrapolating to coarser temporal resolution by comparing properties of data simulated from the different models to the properties of the observed data. At coarser resolutions, simulated data sets recreate many aspects of the observed data; however, only one of the models accurately predicts step length, and all models underestimate the frequency of reversals. 6. The single state model we introduce may be adequate to describe movement data at many resolutions and can be interpreted easily. Multiscalar analyses of movement such as the ones presented here are a useful means of identifying inconsistencies in our understanding of movement.     

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.     

Sampling rate effects on measurements of correlated and biased random walks

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.     

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.     

Social Aggregation in Pea Aphids: Experiment and Random Walk Modeling

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid''s nearest neighbor. For large nearest neighbor distances, when an aphid is essentially isolated, its motion is ballistic with aphids moving faster, turning less, and being less likely to stop. In contrast, for short nearest neighbor distances, aphids move more slowly, turn more, and are more likely to become stationary; this behavior constitutes an aggregation mechanism. From the experimental data, we estimate the state transition probabilities and correlated random walk parameters as a function of nearest neighbor distance. With the individual-level model established, we assess whether it reproduces the macroscopic patterns of movement at the group level. To do so, we consider three distributions, namely distance to nearest neighbor, angle to nearest neighbor, and percentage of population moving at any given time. For each of these three distributions, we compare our experimental data to the output of numerical simulations of our nearest neighbor model, and of a control model in which aphids do not interact socially. Our stochastic, social nearest neighbor model reproduces salient features of the experimental data that are not captured by the control.     

A model of animal movements in a bounded space

Most studies describing animal movements have been developed in the framework of population dispersion or population dynamics, and have mainly focused on movements in open spaces. During their trips, however, animals are likely to encounter physical heterogeneities that guide their movements and, as a result, influence their spatial distribution. In this paper, we develop a statistical model of individual movement in a bounded space. We introduced cockroaches in a circular arena and quantified accurately the behaviors underlying their movement in a finite space. Close to the edges, we considered that the animals exhibit a linear movement mode with a constant probability per unit time to leave the edge and enter the central zone of the arena. Far from the walls cockroaches were assumed to move according to a diffusive random walk which enabled us to overcome the inherent problem of the quantification of the turning angle distribution. A numerical model implementing the behavioral rules derived from our experiments, confirms that the pattern of the spatial distribution of animals observed can be reliably accounted for by wall-following behaviors combined with a diffusive random walk. The approach developed in this study can be applied to model the movements of animals in various environment under consideration of spatial structure.     

Dynamical aspects of animal grouping: Swarms,schools, flocks,and herds

Grouping of animals is a natural phenomenon in which a number of animal individuals are involved in movement as forming a group. Examples are insect swarms and fish schools. In this article an attempt is made to describe the motion of grouping individuals kinematically as distinct from simple diffusion or random walk, to model the grouping on the basis of dynamics of animal motion, and to interpret the grouping from the standpoint of advection-diffusion processes. Also presented is dynamical modeling for the group size distribution as a result of amalgamation and splitting processes of groups.Examples of animal grouping are described in detail. They are insect swarms, zooplankton swarms, fish schools, bird flocks, and mammal herds. The presented mathematical models are compared with data of these animal groupings.     

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.     

A diffusion-based theory of organism dispersal in heterogeneous populations

We develop a general theory of organism movement in heterogeneous populations that can explain the leptokurtic movement distributions commonly measured in nature. We describe population heterogeneity in a state-structured framework, employing advection-diffusion as the fundamental movement process of individuals occupying different movement states. Our general analysis shows that population heterogeneity in movement behavior can be defined as the existence of different movement states and among-individual variability in the time individuals spend in these states. A presentation of moment-based metrics of movement illustrates the role of these attributes in general dispersal processes. We also present a special case of the general theory: a model population composed of individuals occupying one of two movement states with linear transitions, or exchange, between the two states. This two-state "exchange model" can be viewed as a correlated random walk and provides a generalization of the telegraph equation. By exploiting the main result of our general analysis, we characterize the exchange model by deriving moment-based metrics of its movement process and identifying an analytical representation of the model's time-dependent solution. Our results provide general and specific theoretical explanations for empirical patterns in organism movement; the results also provide conceptual and analytical bases for extending diffusion-based dispersal theory in several directions, thereby facilitating mechanistic links between individual behavior and spatial population dynamics.     

Modeling the role of myosin 1c in neuronal growth cone turning

We addressed the mechanical basis for how embryonic chick dorsal root ganglion growth cones turn on a uniform substrate of laminin-1. Turning is significantly correlated with lamellipodial area but not with filopodial length. We assessed the lamellipodial contribution to turning by asymmetric micro-CALI of myosin isoforms that causes localized lamellipodial expansion (myosin 1c) or filopodial retraction (myosin V). Episodes of asymmetric micro-CALI of myosin 1c (or myosin 1c and V together) caused significant turning of the growth cone. In contrast, repeated micro-CALI of myosin V or irradiation without added antibody did not turn growth cones. These findings argue that lamellipodia and not filopodia are necessary for growth cone turning. To model the role of myosin 1c on growth cone turning, we fitted the measured trajectories from asymmetric micro-CALI of myosin 1c-treated and untreated growth cones to the persistent random walk model. The first parameter in this equation, root-mean-square speed, is indistinguishable between the two data sets whereas the second parameter, the persistence of motion, is significantly increased (2.5-fold) as a result of asymmetric inactivation of myosin 1c by micro-CALI. This analysis demonstrates that growth cone turning results from an increase in the persistence of directional motion rather than a change in speed. Taken together, our results suggest that myosin 1c is a molecular correlate for directional persistence underlying growth cone motility.     

Analyzing fish movement as a persistent turning walker

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein–Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.     

From random walks to informed movement

The analysis of animal movement is a large and continuously growing field of research. Detailed knowledge about movement strategies is of crucial importance for understanding eco‐evolutionary dynamics at all scales – from individuals to (meta‐)populations. This and the availability of detailed movement and dispersal data motivated Nathan and colleagues to published their much appreciated call to base movement ecology on a more thorough mechanistic basis. So far, most movement models are based on random walks. However, even if a random walk might describe real movement patterns acceptably well, there is no reason to assume that animals move randomly. Therefore, mechanistic models of foraging strategies should be based on information use and memory in order to increase our understanding of the processes that lead to animal movement decisions. We present a mechanistic movement model of an animal with a limited perceptual range and basic information storage capacities. This ‘spatially informed forager’ constructs an internal map of its environment by using perception, memory and learned or evolutionarily acquired assumptions about landscape attributes. We analyse resulting movement patterns and search efficiencies and compare them to area restricted search strategies (ARS) and biased correlated random walks (BCRW) of omniscient individuals. We show that, in spite of their limited perceptual range, spatially informed individuals boost their foraging success and may perform much better than the best ARS. The construction of an internal map and the use of spatial information results in the emergence of a highly correlated walk between patches and a rather systematic search within resource clusters. Furthermore, the resulting movement patterns may include foray search behaviour. Our work highlights the strength of mechanistic modelling approaches and sets the stage for the development of more sophisticated models of memory use for movement decisions and dispersal.     

Patch exploitation in two dimensions: from Daphnia to simulated foragers

We explore the variability that animals display in their movement choices as they forage in a finite-sized food patch with a uniform food distribution, and present a framework for how these choices may be adjusted to optimize foraging efficiency. Inspired by experimental studies of the zooplankton Daphnia, we model foraging animals as “agents” moving in two dimensions in repeated and successive sequences of hops, pauses, and turns. For Daphnia and other species, critical movement parameters such as hop lengths, pause times, and turning angles are typically reported as probability density functions. Similarly, the agents in our simulations choose their movement parameters at random from such distributions. Each distribution is defined by a characteristic width, which we interpret as a “noise width,” available to be tuned for increased foraging efficiency. We investigate the sensitivity of the system by measuring the food gathered by the agents as the turning angle and hop length noise widths are varied. In all cases, we find a maximum in food gathered at some particular value of the noise width in question, suggesting that these results can be considered robust examples of natural stochastic resonance.     

How Do Ants Make Sense of Gravity? A Boltzmann Walker Analysis of Lasius niger Trajectories on Various Inclines

The goal of this study is to describe accurately how the directional information given by support inclinations affects the ant Lasius niger motion in terms of a behavioral decision. To this end, we have tracked the spontaneous motion of 345 ants walking on a 0.5×0.5 m plane canvas, which was tilted with 5 various inclinations by rad ( data points). At the population scale, support inclination favors dispersal along uphill and downhill directions. An ant''s decision making process is modeled using a version of the Boltzmann Walker model, which describes an ant''s random walk as a series of straight segments separated by reorientation events, and was extended to take directional influence into account. From the data segmented accordingly ( segments), this extension allows us to test separately how average speed, segments lengths and reorientation decisions are affected by support inclination and current walking direction of the ant. We found that support inclination had a major effect on average speed, which appeared approximately three times slower on the incline. However, we found no effect of the walking direction on speed. Contrastingly, we found that ants tend to walk longer in the same direction when they move uphill or downhill, and also that they preferentially adopt new uphill or downhill headings at turning points. We conclude that ants continuously adapt their decision making about where to go, and how long to persist in the same direction, depending on how they are aligned with the line of maximum declivity gradient. Hence, their behavioral decision process appears to combine klinokinesis with geomenotaxis. The extended Boltzmann Walker model parameterized by these effects gives a fair account of the directional dispersal of ants on inclines.     

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.