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Polarity, group velocity, and inter-individual spacing are characteristics of fish schools that strongly affect individual school members. However, these characteristics are group-level 'emer- gent properties': collective outcomes of behavioral interactions among members, not under direct control of any single member. The relationships between members' behaviors and the emergent group properties they produce are complex and poorly understood. In this study, we quantified 3D trajectories of all individual fish within 4- and 8-fish populations of Danio aequipinnatus, using stereo videography and a computerized tracking algorithm. We compared group polarity, group speed, and mean nearest-neighbor distances of schools within these populations to a simulation model that explored how fish responded to attraction/repulsion, alignment and random forces. Real fish exhi- bited a high degree of temporal variability in both polarity and group speed. Polarity and speed of simulated schools depended very strongly on the strength of the alignment force. Time-averaged polarity of real fish schools was most similar to simulated schools when alignment force was 1 to 5% of the attraction/repulsion force. For both real and simulated fish, a clear relationship existed between group speed and polarity: polarized groups were faster than non-polarized groups. We pro- pose a multi-dimensional state space where several emergent property statistics are represented along the axes, and suggest certain 'preferred' ranges of state space within which animal groups tend to localize, and in which they can sustain distinct types of regular architecture.

Individual behavior and emergent properties of fish schools: a comparison of observation and theory