Propagating waves, information transfers of direction of travel in collective groups, have been observed in animal groups of insects, birds, fish, and mammals. Nevertheless, although many previously proposed models of group behaviors have elucidated various aspects of collective motion, none has directly shown the propagating wave constructively. These models consisted of flocking algorithms in which individuals modify their positions or velocities through average responses to their neighbors. The algorithms involve the function of diluting local fluctuations, individual motions, or social cues that initiate coherent decision-making of where to travel and which spread through a group in the form of a wave. The present study challenged physics-inspired models based solely on average interaction and instead proposed a combination with pair interaction: the ‘copy’ mechanism. By the mechanism, individuals specially attend and mimic the motion of the largest turning neighbor. The model comprises three modes (base, copy, and align modes) that sequentially switch among themselves, depending on the degree of variance in direction. The model therefore involves propagating waves that produce rapid collective responses and quick turning motions of a group. This proposal is an attempt to uncover the mechanisms of self-organized waves in simulation studies of coordinated groups, without explicit signals such as alarm calls. Understanding such mechanisms is expected to contribute to the ‘collective mind’ metaphor, answering the question of how animal groups obtain higher-order computational capabilities from local inter-individual interactions.
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