A mathematical model of herding in horse-harem group
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ARTICLE
A mathematical model of herding in horse‑harem group Clark Kendrick Go1 · Monamie Ringhofer2 · Bryan Lao1 · Takatomi Kubo1 · Shinya Yamamoto2 · Kazushi Ikeda1 Received: 8 March 2020 / Accepted: 12 June 2020 © Japan Ethological Society 2020
Abstract In animal groups, individual interactions achieve coordinated movements to maintain cohesion. In horse-harem groups, herding is a behaviour in which stallions chase mares from behind; it is considered to assist with group cohesiveness. The mechanisms of the group cohesion were studied using the methods of drone filming and video tracking during herding and two phases of interactions were found based on the mares’ timing of movement initiation. The study shows that mares that move first are those nearest to the stallion; while the movement initiation of the later mares is determined by the distance from the nearest moving mare. Thus, as a second step to the full understanding of group cohesion, we propose a mathematical model of mares herded by a harem stallion, which is a modification of a sheep model during shepherding. Our model is a linear combination of the five components: inertia, repulsion from the stallion, short-range repulsion, synchronisation attraction, and attraction to the centre of the group. We tune the parameters of our proposed models based on the data and successfully reproduce the movements and directional trends of the mares. Keywords Math model · Herding · Interaction of individuals · Movement · Horses
Introduction Many species across the animal kingdom exhibit different forms of collective motions under various circumstances. Some maintain a cohesive group through subtle interactions among conspecifics (Ballerini et al. 2008; Nagy et al. 2010; Petit et al. 2009) and others perform group aggregation and evasion in the face of a threat to lower the risk of predation (Hamilton 1971; King et al. 2012; Strömbom et al. 2014). To examine their strategies, animal groups have been tracked remotely with high spatial and temporal resolutions (Herbert-Read 2016) using advanced technologies such as GPS technology (Ákos et al. 2014; Nagy et al. 2010; Strandburg-Peshkin et al. 2015), imaging algorithms (Grünbaum et al. 2005; Lukeman et al. 2010; Yamanaka and Takeuchi 2018) and unmanned aerial vehicles (Christie et al. 2016; Inoue et al. 2018). For example, GPS devices attached to animals uncovered leader–follower relationships within the group during motion, that is, certain individuals within a group of pigeons contribute more to the decision-making * Kazushi Ikeda [email protected] 1
Nara Institute of Science and Technology, Ikoma, Japan
Kyoto University, Kyoto, Japan
2
during their flight; while others consistently copy movements (Nagy et al. 2010). In another example, roles in a group of pet dogs may change while out walking, but the relationships among conspecifics are stable in the long term (Ákos et al. 2014). In many other animal groups, leadership and decision-making are determined by the fitness of the group in mind (Fischhoff e
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