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.

Abstract. 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 membership. However, recent work has suggested that being in a group can significantly improve the ability of an individual to align toward and reach a target direction or point, even when all group members have limited navigational ability and there are no leaders. This effect is known as the ''many-wrongs principle'' since the large number of individual navigational errors across the group are suppressed by interactions and group cohesion. In this paper, we simulate the many-wrongs principle using a simple individualbased model of movement based on a biased random walk that includes group interactions. We study the ability of the group as a whole to reach a target given different levels of individual navigation error, group size, interaction radius, and environmental turbulence. In scenarios with low levels of environmental turbulence, simulation results demonstrate a navigational benefit from group membership, particularly for small group sizes. In contrast, when movement takes place in a highly turbulent environment, simulation results suggest that the best strategy is to navigate as individuals rather than as a group.

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’.

Background: 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. Results: Data 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.

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