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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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Structural properties.(A) Density plots of packing fraction and average individual speed (averaged per frame) as functions of rotation Op and polarization Or for 150 fish. (B) The plot illustrates the correlation between individual speed and local polarization estimated in two ways from the underlying density maps (the example shown in the background is for 150 fish). The stapled curves are produced by averaging across individual speeds for each value of the order parameter; the solid curves from averaging across the order parameter values for each individual speed. The local polarization of an individual fish is defined as the polarization Op restricted to the area inside a circle with radius 15.6 cm (approximately 3 body lengths) centered at the individual fish. (C) Average individual speed at different radial positions in the milling state. (D) Average rotational order parameter at the same positions. The radial division of the milling state in (C) and (D) is constructed by centering six shells outside each other, where the outermost shell has a radius defined by the distance from the group's center of mass to the median distance of the five most peripheral fish (see illustration in Fig. S11). The width of each shell is the radius of the outer shell divided by six. The averages are calculated for each shell, where the outer shell even includes peripheral fish.
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pcbi-1002915-g007: Structural properties.(A) Density plots of packing fraction and average individual speed (averaged per frame) as functions of rotation Op and polarization Or for 150 fish. (B) The plot illustrates the correlation between individual speed and local polarization estimated in two ways from the underlying density maps (the example shown in the background is for 150 fish). The stapled curves are produced by averaging across individual speeds for each value of the order parameter; the solid curves from averaging across the order parameter values for each individual speed. The local polarization of an individual fish is defined as the polarization Op restricted to the area inside a circle with radius 15.6 cm (approximately 3 body lengths) centered at the individual fish. (C) Average individual speed at different radial positions in the milling state. (D) Average rotational order parameter at the same positions. The radial division of the milling state in (C) and (D) is constructed by centering six shells outside each other, where the outermost shell has a radius defined by the distance from the group's center of mass to the median distance of the five most peripheral fish (see illustration in Fig. S11). The width of each shell is the radius of the outer shell divided by six. The averages are calculated for each shell, where the outer shell even includes peripheral fish.

Mentions: Examining the relationship between the mean speed of individuals in the group and the ‘packing fraction’ (a measure of the density of individuals within the group) and the order parameters Op and Or, we observe that low speed is associated with the group being relatively dense and both locally- and globally-disordered (the swarm state). The two locally-ordered (milling and polarized) states are characterized by higher mean speed and a decreased packing fraction (see Fig. 7A for group size of 150 fish; this relationship is common among all group sizes as shown in Fig. S8). Consequently the relationship between density and order is the opposite of that predicted by the most studied models of grouping behavior, notably the Vicsek model [15]; although we note that such simple models have been extremely useful in developing understanding of group dynamics for other animal aggregates, such as locusts [14], and other species of schooling fish [23].


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)

Structural properties.(A) Density plots of packing fraction and average individual speed (averaged per frame) as functions of rotation Op and polarization Or for 150 fish. (B) The plot illustrates the correlation between individual speed and local polarization estimated in two ways from the underlying density maps (the example shown in the background is for 150 fish). The stapled curves are produced by averaging across individual speeds for each value of the order parameter; the solid curves from averaging across the order parameter values for each individual speed. The local polarization of an individual fish is defined as the polarization Op restricted to the area inside a circle with radius 15.6 cm (approximately 3 body lengths) centered at the individual fish. (C) Average individual speed at different radial positions in the milling state. (D) Average rotational order parameter at the same positions. The radial division of the milling state in (C) and (D) is constructed by centering six shells outside each other, where the outermost shell has a radius defined by the distance from the group's center of mass to the median distance of the five most peripheral fish (see illustration in Fig. S11). The width of each shell is the radius of the outer shell divided by six. The averages are calculated for each shell, where the outer shell even includes peripheral fish.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002915-g007: Structural properties.(A) Density plots of packing fraction and average individual speed (averaged per frame) as functions of rotation Op and polarization Or for 150 fish. (B) The plot illustrates the correlation between individual speed and local polarization estimated in two ways from the underlying density maps (the example shown in the background is for 150 fish). The stapled curves are produced by averaging across individual speeds for each value of the order parameter; the solid curves from averaging across the order parameter values for each individual speed. The local polarization of an individual fish is defined as the polarization Op restricted to the area inside a circle with radius 15.6 cm (approximately 3 body lengths) centered at the individual fish. (C) Average individual speed at different radial positions in the milling state. (D) Average rotational order parameter at the same positions. The radial division of the milling state in (C) and (D) is constructed by centering six shells outside each other, where the outermost shell has a radius defined by the distance from the group's center of mass to the median distance of the five most peripheral fish (see illustration in Fig. S11). The width of each shell is the radius of the outer shell divided by six. The averages are calculated for each shell, where the outer shell even includes peripheral fish.
Mentions: Examining the relationship between the mean speed of individuals in the group and the ‘packing fraction’ (a measure of the density of individuals within the group) and the order parameters Op and Or, we observe that low speed is associated with the group being relatively dense and both locally- and globally-disordered (the swarm state). The two locally-ordered (milling and polarized) states are characterized by higher mean speed and a decreased packing fraction (see Fig. 7A for group size of 150 fish; this relationship is common among all group sizes as shown in Fig. S8). Consequently the relationship between density and order is the opposite of that predicted by the most studied models of grouping behavior, notably the Vicsek model [15]; although we note that such simple models have been extremely useful in developing understanding of group dynamics for other animal aggregates, such as locusts [14], and other species of schooling fish [23].

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
Related in: MedlinePlus