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.

Animals often travel in groups, and their navigational decisions can be influenced by social interactions. Both theory and empirical observations suggest that such collective navigation can result in individuals improving their ability to find their way and could be one of the key benefits of sociality for these species. Here, we provide an overview of the potential mechanisms underlying collective navigation, review the known, and supposed, empirical evidence for such behaviour and highlight interesting directions for future research. We further explore how both social and collective learning during group navigation could lead to the accumulation of knowledge at the population level, resulting in the emergence of migratory culture. This article is part of the theme issue ‘Collective movement ecology’.

Advances 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.

Traditional studies of animal navigation over both long and short distances have usually considered the orientation ability of the individual only, without reference to the implications of group memb...

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.