Modelling the movement of a soil insect

We use a linear autoregressive model to describe the movement of a soil-living insect, Protaphorura armata (Collembola). Models of this kind can be viewed as extensions of a random walk, but unlike a correlated random walk, in which the speed and turning angles are independent, our model identifies and expresses the correlations between the turning angles and a variable speed. Our model uses data in x- and y-coordinates rather than in polar coordinates, which is useful for situations in which the resolution of the observations is limited. The movement of the insect was characterized by (i) looping behaviour due to autocorrelation and cross correlation in the velocity process and (ii) occurrence of periods of inactivity, which we describe with a Poisson random effects model. We also introduce obstacles to the environment to add structural heterogeneity to the movement process. We compare aspects such as loop shape, inter-loop time, holding angles at obstacles, net squared displacement, number, and duration of inactive periods between observed and predicted movement. The comparison demonstrates that our approach is relevant as a starting-point to predict behaviourally complex moving, e.g. systematic searching, in a heterogeneous landscape.     

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.     

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.     

The random walk of Azospirillum brasilense

The bacterium Azospirillum brasilense has been frequently studied in laboratory experiments. It performs movements in space where long forward and backward runs on a straight line occur simultaneously with slow changes of direction of the line. A model is presented in which a correlated random walk on a line is joined to diffusion on a sphere of directions. For this transport system, a hierarchy of moment approximations is derived, ranging from a hyperbolic system with four dependent variables to a scalar damped wave equation (telegraph equation) and then to a single diffusion equation for particle density. The original parameters are compounded in the diffusion quotient. The effects of these parameters, such as particle speed or turning rate, on the diffusion coefficient are discussed in detail.     

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.     

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 generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999     

A Turing model with correlated random walk

 If in the classical Turing model the diffusion process (Brownian motion) is replaced by a more general correlated random walk, then the parameters describing spatial spread are the particle speeds and the rates of change in direction. As in the Turing model, a spatially constant equilibrium can become unstable if the different species have different turning rates and different speeds. Furthermore, a Hopf bifurcation can be found if the reproduction rate of the activator is greater than its rate of change of direction, and oscillating patterns are possible. Received 24 February 1995; received in revised form 6 September 1995     

Modeling the spread of Phytophthora

We consider a model for the morphology and growth of the fungus-like plant pathogen Phytophthora using the example of Phytophthora plurivora. Here, we are utilizing a correlated random walk describing the density of tips. This random walk incorporates a delay in branching behavior: newly split tips only start to grow after a short while. First, we question the effect of such a delay on the running fronts, for uniform- as well as non-uniform turning kernels. We find that this delay primarily influences the slope of the front and therewith the way of spatial appropriation, and not its velocity. Our theoretical predictions are confirmed by the growth of Phytophthora in concrete experiments performed in Petri dishes. The second question addressed in this paper, concerns the manner tips are interacting, especially the point why tips stop to grow “behind” the interface of the front, respectively in confrontation experiments at the interface between two colonies. The combination of experimental data about the spatially structured time course of the glucose concentration and simulations of a model taking into account both, tips and glucose, reveals that nutrient depletion is most likely the central mechanism of tip interaction and hyphal growth inhibition. We presume that this is the growing mechanism for our kind of Phytophthora in infected plant tissue. Thus, the pathogen will sap its hosts via energy depletion and tissue destruction in infected areas.     

Modeling Group Formation and Activity Patterns in Self-Organizing Collectives of Individuals

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.     

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.     

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.     

Human Mammary Epithelial Cells Exhibit a Bimodal Correlated Random Walk Pattern

Background

Organisms, at scales ranging from unicellular to mammals, have been known to exhibit foraging behavior described by random walks whose segments confirm to Lévy or exponential distributions. For the first time, we present evidence that single cells (mammary epithelial cells) that exist in multi-cellular organisms (humans) follow a bimodal correlated random walk (BCRW).

Methodology/Principal Findings

Cellular tracks of MCF-10A pBabe, neuN and neuT random migration on 2-D plastic substrates, analyzed using bimodal analysis, were found to reveal the BCRW pattern. We find two types of exponentially distributed correlated flights (corresponding to what we refer to as the directional and re-orientation phases) each having its own correlation between move step-lengths within flights. The exponential distribution of flight lengths was confirmed using different analysis methods (logarithmic binning with normalization, survival frequency plots and maximum likelihood estimation).

Conclusions/Significance

Because of the presence of non-uniform turn angle distribution of move step-lengths within a flight and two different types of flights, we propose that the epithelial random walk is a BCRW comprising of two alternating modes with varying degree of correlations, rather than a simple persistent random walk. A BCRW model rather than a simple persistent random walk correctly matches the super-diffusivity in the cell migration paths as indicated by simulations based on the BCRW model.     

Exciton percolation III. Stochastic and coherent migration in binary and ternary random lattices

We develop a method for calculating energy migration in random heterogeneous aggregates, with potential application to the primary process in photosynthetic units. A Monte Carlo technique is employed to study several types of random walk motion in a random binary lattice. Our computations include 2 and 3 dimensional lattices of different topology and employ correlated steps with a Gaussian distribution of directional memory. The effects of the characteristics of the motion and its parameters are displayed and discussed. The lower threshold for efficient visitation by the walker is given by the critical percolation concentration. However, a higher threshold is found in the case of coherent motion. This new “turning point” appears to play an important role in the process of exciton transport. The exciton percolation formalism is utilized, giving results for ternary random lattices where the third component is very dilute and acts as a sensor. The results are applied to a system representing the ¹1B_2u, naphthalene exciton dynamics in an isotopic and chemically mixed crystal, which by itself is supposed to mimic the exciton transport in the photosynthetic units of green plants. Physically reasonable parameters, trends and limits are discussed. Also, an analytical solution is derived and tested for a physically reasonable limit of semicoherent motion in a perfect lattice. The ramification of this work on bioexciton transfer is discussed, especially concerning the light harvesting units in green plants. It leads to a simple minded model that rationalized the ratio of antenna to active-center molecules. Our most important result is that incoherent exciton transfer, i.e. simple random walk, is the most efficient process for significantly heterogeneous aggregates.     

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.     

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.     

Intraspecific variation in movement patterns: modeling individual behaviour in a large marine predator

In large marine predators, foraging entails movement. Quantitative models reveal how behaviours can mediate individual movement, such that deviations from a random pattern may reveal specific search tactics or behaviour. Using locations for 52 grey seals fitted with satellite-linked recorders on Sable Island; we modeled movement as a correlated random walk (CRW) for individual animals, at two temporal scales. Mean move length, turning angle, and net squared displacement (R²_n: the rate of change in area over time) at successive moves over 3 to 10 months were calculated. The distribution of move lengths of individual animals was compared to a Lévy distribution to determine if grey seals use a Lévy flight search tactic. Grey seals exhibited three types of movement as determined by CRW model fit: directed movers – animals displaying directed long distance travel that were significantly underpredicted by the CRW (23% of animals); residents – animals remaining in the area surrounding Sable Island that were overpredicted by the model (29% of animals); and correlated random walkers – those (48% of animals) in which movement was predicted by the CRW model. Kernel home range size differed significantly among all three movement types, as did travel speed, mean move length, mean R²_n and total distance traveled. Sex and season of deployment were significant predictors of movement type, with directed movers more likely to be male and residents more likely to be female. Only 30% of grey seals fit a Lévy distribution, which suggests that food patches used by the majority of seals are not randomly distributed. Intraspecific variation in movement behaviour is an important characteristic in grey seal foraging ecology, underscoring the need to account for such variability in developing models of habitat use and predation.