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Collective states, multistability and transitional behavior in schooling fish.

Tunstrøm K, Katz Y, Ioannou CC, Huepe C, Lutz MJ, Couzin ID - PLoS Comput. Biol. (2013)

Bottom Line: 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.Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups.Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America. tunstrom@princeton.edu

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.

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Transition patterns for 150 fish.(A) Density plot of the smallest distance from the center of mass of the fish shoal to the tank boundary as a function of rotation and polarization (dmin = 26 and dmax = 52 cm). The overlaid arrows are the averaged trajectories of all transitions in the rotation-polarization phase space. (B–D) Density plots of transitions from polarized to milling state (B), from polarized to swarm state (C) and from milling to swarm state (D). Overlaid the density plots are the corresponding velocity fields of the transition data (in the rotation-polarization phase space). Plots of the reverse transitions and group sizes 30, 70 and 300 are provided in SI.
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pcbi-1002915-g005: Transition patterns for 150 fish.(A) Density plot of the smallest distance from the center of mass of the fish shoal to the tank boundary as a function of rotation and polarization (dmin = 26 and dmax = 52 cm). The overlaid arrows are the averaged trajectories of all transitions in the rotation-polarization phase space. (B–D) Density plots of transitions from polarized to milling state (B), from polarized to swarm state (C) and from milling to swarm state (D). Overlaid the density plots are the corresponding velocity fields of the transition data (in the rotation-polarization phase space). Plots of the reverse transitions and group sizes 30, 70 and 300 are provided in SI.

Mentions: In order to reveal more clearly the role of boundary effects on state transitions we use our extensive time series to characterize the typical nature of transitions and relate these to whether the group tends to be relatively close to, or far from, the boundary. We present data for 150 fish in Fig. 5 (for other group sizes see Fig. S5). In Fig. 5A the arrows represent the average trajectories that groups take through Op and Or space when transitions occur and, unlike Fig. 2, the density plot now depicts the distance db from the center of mass of the group to the closest point at the tank boundary; red colors represent a relatively large distance and blue colors relative proximity to the boundary. A more detailed view of the transition dynamics is presented in Figs. 5B–D. Here, for each of the transitions from polar to milling, polar to swarm and milling to swarm state, the average trajectories are plotted as a velocity field in the Op and Or phase space overlaid on the density plot showing the distribution of trajectories (for the reverse transitions and other group sizes see Figs. S6 and S7).


Collective states, multistability and transitional behavior in schooling fish.

Tunstrøm K, Katz Y, Ioannou CC, Huepe C, Lutz MJ, Couzin ID - PLoS Comput. Biol. (2013)

Transition patterns for 150 fish.(A) Density plot of the smallest distance from the center of mass of the fish shoal to the tank boundary as a function of rotation and polarization (dmin = 26 and dmax = 52 cm). The overlaid arrows are the averaged trajectories of all transitions in the rotation-polarization phase space. (B–D) Density plots of transitions from polarized to milling state (B), from polarized to swarm state (C) and from milling to swarm state (D). Overlaid the density plots are the corresponding velocity fields of the transition data (in the rotation-polarization phase space). Plots of the reverse transitions and group sizes 30, 70 and 300 are provided in SI.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3585391&req=5

pcbi-1002915-g005: Transition patterns for 150 fish.(A) Density plot of the smallest distance from the center of mass of the fish shoal to the tank boundary as a function of rotation and polarization (dmin = 26 and dmax = 52 cm). The overlaid arrows are the averaged trajectories of all transitions in the rotation-polarization phase space. (B–D) Density plots of transitions from polarized to milling state (B), from polarized to swarm state (C) and from milling to swarm state (D). Overlaid the density plots are the corresponding velocity fields of the transition data (in the rotation-polarization phase space). Plots of the reverse transitions and group sizes 30, 70 and 300 are provided in SI.
Mentions: In order to reveal more clearly the role of boundary effects on state transitions we use our extensive time series to characterize the typical nature of transitions and relate these to whether the group tends to be relatively close to, or far from, the boundary. We present data for 150 fish in Fig. 5 (for other group sizes see Fig. S5). In Fig. 5A the arrows represent the average trajectories that groups take through Op and Or space when transitions occur and, unlike Fig. 2, the density plot now depicts the distance db from the center of mass of the group to the closest point at the tank boundary; red colors represent a relatively large distance and blue colors relative proximity to the boundary. A more detailed view of the transition dynamics is presented in Figs. 5B–D. Here, for each of the transitions from polar to milling, polar to swarm and milling to swarm state, the average trajectories are plotted as a velocity field in the Op and Or phase space overlaid on the density plot showing the distribution of trajectories (for the reverse transitions and other group sizes see Figs. S6 and S7).

Bottom Line: 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.Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups.Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.

View Article: PubMed Central - PubMed

Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, United States of America. tunstrom@princeton.edu

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.

Show MeSH