Edward A. Codling

Random walk models in biology

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

Sampling rate and misidentification of Lévy and non-Lévy movement paths

A large number of empirical studies have attributed Lévy search patterns to the foraging movements of animals. Typically, this is done by fitting a power-law distribution with an exponent of 1 < mu < or = 3 to the observed step lengths. Most studies record the animals location at equally spaced time intervals, which are sometimes significantly longer than the natural time scale of the animals movements. The collected data thus represent a subsample of the animals movement. In this paper, the effect of subsampling on the observed properties of both Lévy and non-Lévy simulated movement paths is investigated. We find that the apparent properties of the observed movement path can be sensitive to the sampling rate even though Lévy search patterns are supposedly scale-independent. We demonstrate that, in certain contexts and dependent on the sampling rate used in observation, it is possible to misidentify a non-Lévy movement path as being a Lévy path. We also demonstrate that a Lévy movement path can be misidentified as a non-Lévy path, but this is dependent on the value of mu of the original simulated path, with the greatest uncertainty for mu = 2. We discuss the implications of these results in the context of studies of animal movements and foraging behavior.

Human exit route choice in virtual crowd evacuations

The collective behaviour of human crowds emerges from the local interactions of individuals. To understand human crowds we therefore need to identify the behavioural rules individual pedestrians follow. This is crucial for the control of emergency evacuations from confined spaces, for example. At a microscopic level we seek to predict the next step of pedestrians based on their local environment. However, we also have to consider ‘tactical-level’ individual behaviour that is not an immediate response to the local environment, such as the choice between different routes to exit a building. We used an interactive virtual environment to study human exit route decisions in simulated evacuations. Participants had to escape from a building and had to choose between different exit routes in the presence of evacuating simulated agents. We found no inherent preference for familiar routes, but under a stress-inducing treatment, subjects were more likely to display behaviour in their route choice that was detrimental to their evacuation time. Most strikingly, subjects were less likely to avoid a congested exit by changing their original decision to move towards it under this treatment. Age and gender had clear effects on reaction times in the virtual environment.

Navigation in human crowds; testing the many-wrongs principle

The ‘many-wrongs principle’ predicts that animal group cohesion can cause groups to navigate more accurately than singletons. Recent theoretical work using individual-based simulations and several empirical studies of bird flock behaviour support this principle. However, for real animal groups it remains unclear what key factors are involved and whether group cohesion alone can act to produce the effect. We tested model predictions using human participants in a large circular arena. They were tested alone and in groups of two, three, six and 10, in three trials. For each trial, individuals were instructed to stay together and approach a preset but unmarked target on the arena perimeter. The target instruction included a degree of directional uncertainty of 22.5°, 67.5° or 112.5°. Individual directional uncertainty was equal for each group member within a trial, but differed between trials. As expected, we found that groups comprising individuals with lower directional uncertainty navigated more accurately. Group navigational accuracy increased with group size but only between singletons and groups of 10 and only when individuals had a high directional uncertainty of 112.5°. This study provides evidence in human groups that group cohesion can increase navigational accuracy but that this effect is restricted to larger group sizes and when individual directional uncertainty is high.

Diffusion about the mean drift location in a biased random walk.

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a two-dimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. We demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. We relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement.

Distinguishing social from nonsocial navigation in moving animal groups.

Many animals, such as migrating shoals of fish, navigate in groups. Knowing the mechanisms involved in animal navigation is important when it comes to explaining navigation accuracy, dispersal patterns, population and evolutionary dynamics, and consequently, the design of conservation strategies. When navigating toward a common target, animals could interact socially by sharing available information directly or indirectly, or each individual could navigate by itself and aggregations may not disperse because all animals are moving toward the same target. Here we present an analysis technique that uses individual movement trajectories to determine the extent to which individuals in navigating groups interact socially, given knowledge of their target. The basic idea of our approach is that the movement directions of individuals arise from a combination of responses to the environment and to other individuals. We estimate the relative importance of these responses, distinguishing between social and nonsocial interactions. We develop and test our method, using simulated groups, and we demonstrate its applicability to empirical data in a case study on groups of guppies moving toward shelter in a tank. Our approach is generic and can be extended to different scenarios of animal group movement.

Classifying area-restricted search (ARS) using a partial sum approach

Many animals perform two distinct alternating movement strategies when foraging: intensive searches with low speed and high turning to cover a small area in high detail and extensive searches with high speed and low turning to cover a large area in low detail. Observed movement paths will tend to exhibit differences in speed and correlation between these different search strategies. Identifying transitions between strategies can enable one to acquire information regarding both the distribution of resources and the underlying behavioural mechanisms performed by a foraging animal. Methods such as the moving average, first-passage time, residence time and fractal landscape methods have been used to identify behavioural states of various real and simulated foragers. We provide a review of these current methods and identify a set of common limitations associated with each procedure. We develop a new mathematical approach: the partial sum method, which is designed to avoid these limitations. A comprehensive test is undertaken to evaluate and compare the performance of the partial sum and the existing methods using a carefully constructed set of computer-generated movement paths. Each simulated track was designed to replicate the possible paths performed by an animal under different foraging conditions. Our results provide strong evidence that the partial sum method is better than existing analytical methods for identifying transitions between two different search strategies.

Classification of behaviour in housed dairy cows using an accelerometer-based activity monitoring system

BackgroundAdvances in bio-telemetry technology have made it possible to automatically monitor and classify behavioural activities in many animals, including domesticated species such as dairy cows. Automated behavioural classification has the potential to improve health and welfare monitoring processes as part of a Precision Livestock Farming approach. Recent studies have used accelerometers and pedometers to classify behavioural activities in dairy cows, but such approaches often cannot discriminate accurately between biologically important behaviours such as feeding, lying and standing or transition events between lying and standing. In this study we develop a decision-tree algorithm that uses tri-axial accelerometer data from a neck-mounted sensor to both classify biologically important behaviour in dairy cows and to detect transition events between lying and standing.ResultsData were collected from six dairy cows that were monitored continuously for 36 h. Direct visual observations of each cow were used to validate the algorithm. Results show that the decision-tree algorithm is able to accurately classify three types of biologically relevant behaviours: lying (77.42 % sensitivity, 98.63 % precision), standing (88.00 % sensitivity, 55.00 % precision), and feeding (98.78 % sensitivity, 93.10 % precision). Transitions between standing and lying were also detected accurately with an average sensitivity of 96.45 % and an average precision of 87.50 %. The sensitivity and precision of the decision-tree algorithm matches the performance of more computationally intensive algorithms such as hidden Markov models and support vector machines.ConclusionsBiologically important behavioural activities in housed dairy cows can be classified accurately using a simple decision-tree algorithm applied to data collected from a neck-mounted tri-axial accelerometer. The algorithm could form part of a real-time behavioural monitoring system in order to automatically detect dairy cow health and welfare status.

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.

Human responses to multiple sources of directional information in virtual crowd evacuations

The evacuation of crowds from buildings or vehicles is one example that highlights the importance of understanding how individual-level interactions and decision-making combine and lead to the overall behaviour of crowds. In particular, to make evacuations safer, we need to understand how individuals make movement decisions in crowds. Here, we present an evacuation experiment with over 500 participants testing individual behaviour in an interactive virtual environment. Participants had to choose between different exit routes under the influence of three different types of directional information: static information (signs), dynamic information (movement of simulated crowd) and memorized information, as well as the combined effect of these different sources of directional information. In contrast to signs, crowd movement and memorized information did not have a significant effect on human exit route choice in isolation. However, when we combined the latter two treatments with additional directly conflicting sources of directional information, for example signs, they showed a clear effect by reducing the number of participants that followed the opposing directional information. This suggests that the signals participants observe more closely in isolation do not simply overrule alternative sources of directional information. Age and gender did not consistently explain differences in behaviour in our experiments.