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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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Relationship between agent speed and order in constant-speed agent based simulation model with 150 agents.(A) Density plot of agent speed as function of rotation Or and polarization Op, revealing a bistable regime between the milling and the polar states for high speeds. (B) Normalized probability plot of polarization Op as function of agent speed. (C) Normalized probability plot of rotation Or as function of agent speed. The last two plots illustrate the bifurcation that occurs as the speed is increased, where the system transitions from a swarm state and is found either in a highly polarized state or in a milling state. (See Methods for simulation details).
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pcbi-1002915-g008: Relationship between agent speed and order in constant-speed agent based simulation model with 150 agents.(A) Density plot of agent speed as function of rotation Or and polarization Op, revealing a bistable regime between the milling and the polar states for high speeds. (B) Normalized probability plot of polarization Op as function of agent speed. (C) Normalized probability plot of rotation Or as function of agent speed. The last two plots illustrate the bifurcation that occurs as the speed is increased, where the system transitions from a swarm state and is found either in a highly polarized state or in a milling state. (See Methods for simulation details).

Mentions: To demonstrate the plausibility (and indeed, generality) of speed-induced transitions, we return to the model employed above, from [3]. We verify that changing individual speed does result in qualitatively the same transitional behavior seen here; swarm behavior for relatively low speed and bistable milling and parallel group motion as individual speed increases. This result holds up to 150 agents. For groups of 300 agents the milling state is dominant and no instances of the polar state are found. (see Fig. 8 for results from simulations with 150 agents and Fig. S9 for remaining group sizes. Simulation details are found in Methods).


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)

Relationship between agent speed and order in constant-speed agent based simulation model with 150 agents.(A) Density plot of agent speed as function of rotation Or and polarization Op, revealing a bistable regime between the milling and the polar states for high speeds. (B) Normalized probability plot of polarization Op as function of agent speed. (C) Normalized probability plot of rotation Or as function of agent speed. The last two plots illustrate the bifurcation that occurs as the speed is increased, where the system transitions from a swarm state and is found either in a highly polarized state or in a milling state. (See Methods for simulation details).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002915-g008: Relationship between agent speed and order in constant-speed agent based simulation model with 150 agents.(A) Density plot of agent speed as function of rotation Or and polarization Op, revealing a bistable regime between the milling and the polar states for high speeds. (B) Normalized probability plot of polarization Op as function of agent speed. (C) Normalized probability plot of rotation Or as function of agent speed. The last two plots illustrate the bifurcation that occurs as the speed is increased, where the system transitions from a swarm state and is found either in a highly polarized state or in a milling state. (See Methods for simulation details).
Mentions: To demonstrate the plausibility (and indeed, generality) of speed-induced transitions, we return to the model employed above, from [3]. We verify that changing individual speed does result in qualitatively the same transitional behavior seen here; swarm behavior for relatively low speed and bistable milling and parallel group motion as individual speed increases. This result holds up to 150 agents. For groups of 300 agents the milling state is dominant and no instances of the polar state are found. (see Fig. 8 for results from simulations with 150 agents and Fig. S9 for remaining group sizes. Simulation details are found in Methods).

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