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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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Statistics of state transitions.(A) Fraction of time spent in the different dynamical states shown for each group size. The error bars are showing the standard deviation measured across replicates. The group of 30 fish is predominantly in the polarized state, but less time is spent in this state with increasing group size (GLM: F1,22 = 36.21, P = 4.6e-06). As the group size increases, the groups gradually spend more time in the milling state (GLM: F1,22 = 19.31, P = 0.00023). The amount of time spent in the transition regime is high, but constant, for all group sizes (GLM: F1,22 = 0.67, P = 0.42). Across all group sizes (GLM: F1,22 = 0.053, P = 0.82) little time is spent in the swarm state. (B) Fraction of transitions from one state to another for the different groups. The error bars are showing the standard deviation measured across replicates. For all group sizes, the polar state predominantly transitions into the swarm state compared to the milling state (GLMM: F1,23 = 58.77, P<0.0001). The swarm state is dominated by transitions into the polarized state (GLMM: F1,23 = 55.69, P<0.0001). Although these transitions are consistent across group sizes, there is a significant interaction between group size and the frequencies of transitioning from the milling state to the swarm and polarized states (GLMM: F1,22 = 13.30, P = 0.0014). While the milling state tended to transition into the polar state, for 300 fish there was a roughly equal probability of transitioning to the polar or swarm states. (C) Rank plots showing the probability of being in a state longer than time Ts before moving into a different regime. There was no significant difference between group sizes in the persistence of the polar state (GLMM: F1,22 = 0.58, P = 0.45), although group size increased the persistence of the transition (F1,22 = 24.52, P = 1e-04) and milling states (F1,22 = 17.54, P = 4.0e-04), and to a lesser degree, the swarm state (F1,22 = 5.31, P = 0.031).
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pcbi-1002915-g004: Statistics of state transitions.(A) Fraction of time spent in the different dynamical states shown for each group size. The error bars are showing the standard deviation measured across replicates. The group of 30 fish is predominantly in the polarized state, but less time is spent in this state with increasing group size (GLM: F1,22 = 36.21, P = 4.6e-06). As the group size increases, the groups gradually spend more time in the milling state (GLM: F1,22 = 19.31, P = 0.00023). The amount of time spent in the transition regime is high, but constant, for all group sizes (GLM: F1,22 = 0.67, P = 0.42). Across all group sizes (GLM: F1,22 = 0.053, P = 0.82) little time is spent in the swarm state. (B) Fraction of transitions from one state to another for the different groups. The error bars are showing the standard deviation measured across replicates. For all group sizes, the polar state predominantly transitions into the swarm state compared to the milling state (GLMM: F1,23 = 58.77, P<0.0001). The swarm state is dominated by transitions into the polarized state (GLMM: F1,23 = 55.69, P<0.0001). Although these transitions are consistent across group sizes, there is a significant interaction between group size and the frequencies of transitioning from the milling state to the swarm and polarized states (GLMM: F1,22 = 13.30, P = 0.0014). While the milling state tended to transition into the polar state, for 300 fish there was a roughly equal probability of transitioning to the polar or swarm states. (C) Rank plots showing the probability of being in a state longer than time Ts before moving into a different regime. There was no significant difference between group sizes in the persistence of the polar state (GLMM: F1,22 = 0.58, P = 0.45), although group size increased the persistence of the transition (F1,22 = 24.52, P = 1e-04) and milling states (F1,22 = 17.54, P = 4.0e-04), and to a lesser degree, the swarm state (F1,22 = 5.31, P = 0.031).

Mentions: As we previously saw in Fig. 2, the proportion of time spent in the polarized state decreases strongly with group size from 0.57 to 0.26, 0.18 and 0.05 (30 to 300 fish shoals, respectively, light blue columns in Fig. 4A). Likewise, the proportion of time in the milling state increases with group size from 0.03 to 0.18, 0.36 and 0.45 (yellow columns Fig. 4A). The group size has however little effect on the proportion of time spent in the swarm state or in the transition region, which varies between 0.09 and 0.11 (dark blue columns Fig. 4A), and 0.31 and 0.44 (brown columns in Fig. 4A), respectively. The fraction of transitions between states, as illustrated by Fig. 4B, exhibit little variation between the group sizes. The only visible trend is a small increase in the number of transitions between the milling state and the swarm state (see also Fig. S4 for alternative graphics).


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)

Statistics of state transitions.(A) Fraction of time spent in the different dynamical states shown for each group size. The error bars are showing the standard deviation measured across replicates. The group of 30 fish is predominantly in the polarized state, but less time is spent in this state with increasing group size (GLM: F1,22 = 36.21, P = 4.6e-06). As the group size increases, the groups gradually spend more time in the milling state (GLM: F1,22 = 19.31, P = 0.00023). The amount of time spent in the transition regime is high, but constant, for all group sizes (GLM: F1,22 = 0.67, P = 0.42). Across all group sizes (GLM: F1,22 = 0.053, P = 0.82) little time is spent in the swarm state. (B) Fraction of transitions from one state to another for the different groups. The error bars are showing the standard deviation measured across replicates. For all group sizes, the polar state predominantly transitions into the swarm state compared to the milling state (GLMM: F1,23 = 58.77, P<0.0001). The swarm state is dominated by transitions into the polarized state (GLMM: F1,23 = 55.69, P<0.0001). Although these transitions are consistent across group sizes, there is a significant interaction between group size and the frequencies of transitioning from the milling state to the swarm and polarized states (GLMM: F1,22 = 13.30, P = 0.0014). While the milling state tended to transition into the polar state, for 300 fish there was a roughly equal probability of transitioning to the polar or swarm states. (C) Rank plots showing the probability of being in a state longer than time Ts before moving into a different regime. There was no significant difference between group sizes in the persistence of the polar state (GLMM: F1,22 = 0.58, P = 0.45), although group size increased the persistence of the transition (F1,22 = 24.52, P = 1e-04) and milling states (F1,22 = 17.54, P = 4.0e-04), and to a lesser degree, the swarm state (F1,22 = 5.31, P = 0.031).
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Related In: Results  -  Collection

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pcbi-1002915-g004: Statistics of state transitions.(A) Fraction of time spent in the different dynamical states shown for each group size. The error bars are showing the standard deviation measured across replicates. The group of 30 fish is predominantly in the polarized state, but less time is spent in this state with increasing group size (GLM: F1,22 = 36.21, P = 4.6e-06). As the group size increases, the groups gradually spend more time in the milling state (GLM: F1,22 = 19.31, P = 0.00023). The amount of time spent in the transition regime is high, but constant, for all group sizes (GLM: F1,22 = 0.67, P = 0.42). Across all group sizes (GLM: F1,22 = 0.053, P = 0.82) little time is spent in the swarm state. (B) Fraction of transitions from one state to another for the different groups. The error bars are showing the standard deviation measured across replicates. For all group sizes, the polar state predominantly transitions into the swarm state compared to the milling state (GLMM: F1,23 = 58.77, P<0.0001). The swarm state is dominated by transitions into the polarized state (GLMM: F1,23 = 55.69, P<0.0001). Although these transitions are consistent across group sizes, there is a significant interaction between group size and the frequencies of transitioning from the milling state to the swarm and polarized states (GLMM: F1,22 = 13.30, P = 0.0014). While the milling state tended to transition into the polar state, for 300 fish there was a roughly equal probability of transitioning to the polar or swarm states. (C) Rank plots showing the probability of being in a state longer than time Ts before moving into a different regime. There was no significant difference between group sizes in the persistence of the polar state (GLMM: F1,22 = 0.58, P = 0.45), although group size increased the persistence of the transition (F1,22 = 24.52, P = 1e-04) and milling states (F1,22 = 17.54, P = 4.0e-04), and to a lesser degree, the swarm state (F1,22 = 5.31, P = 0.031).
Mentions: As we previously saw in Fig. 2, the proportion of time spent in the polarized state decreases strongly with group size from 0.57 to 0.26, 0.18 and 0.05 (30 to 300 fish shoals, respectively, light blue columns in Fig. 4A). Likewise, the proportion of time in the milling state increases with group size from 0.03 to 0.18, 0.36 and 0.45 (yellow columns Fig. 4A). The group size has however little effect on the proportion of time spent in the swarm state or in the transition region, which varies between 0.09 and 0.11 (dark blue columns Fig. 4A), and 0.31 and 0.44 (brown columns in Fig. 4A), respectively. The fraction of transitions between states, as illustrated by Fig. 4B, exhibit little variation between the group sizes. The only visible trend is a small increase in the number of transitions between the milling state and the swarm state (see also Fig. S4 for alternative graphics).

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