Abstract
The spontaneous emergence of pattern formation is ubiquitous in nature, often arising as a collective phenomenon from interactions among a large number of individual constituents or sub-systems. Understanding, and controlling, collective behavior is dependent on determining the low-level dynamical principles from which spatial and temporal patterns emerge; a key question is whether different group-level patterns result from all components of a system responding to the same external factor, individual components changing behavior but in a distributed self-organized way, or whether multiple collective states co-exist for the same individual behaviors. Using schooling fish (golden shiners, in groups of 30 to 300 fish) as a model system, we demonstrate that collective motion can be effectively mapped onto a set of order parameters describing the macroscopic group structure, revealing the existence of at least three dynamically-stable collective states; swarm, milling and polarized groups. Swarms are characterized by slow individual motion and a relatively dense, disordered structure. Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups. The stability of the discrete collective behaviors exhibited by a group depends on the number of group members. Transitions between states are influenced by both external (boundary-driven) and internal (changing motion of group members) factors. Whereas transitions between locally-disordered and locally-ordered group states are speed dependent, analysis of local and global properties of groups suggests that, congruent with theory, milling and polarized states co-exist in a bistable regime with transitions largely driven by perturbations. Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.
Highlights
Many animal groups display coordinated motion in which individuals exhibit attraction towards others and a tendency to align their direction of travel with near-neighbors [1]
Despite the multitude of local interactions that result in coordinated group motion we demonstrate that schooling golden shiners predominantly exist in three ‘fundamental’ dynamicallystable states of the underlying dynamics: swarm, milling and polarized motion
We establish that group states, and transitional behavior, can be represented in low-dimensional space, a projection that allows us to see the path taken by groups between the three dynamically-stable states as well as to relate the collective states exhibited to properties such as group size, individual speed and perturbations to the group
Summary
Many animal groups display coordinated motion in which individuals exhibit attraction towards others and a tendency to align their direction of travel with near-neighbors [1]. Exploring the collective behavior of simulated individuals (see Methods for simulation set up and details) we find that the model produces qualitatively similar results across the range of group size in our experiments (30, 70, 150 and 300 agents), with the polarized states being dominant for the smallest group size, and an increasing proportion of the group’s time is spent in the milling state as group size increases (see Fig. 3). Another aspect of group size is the self-regulation of density. These data support the prediction of a multi-stable locally-ordered regime in which the group can transition back and forth between the polarized and milling state through stochastic and boundary-induced effects
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