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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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Density plots of polarization vs. rotation from experiments.The data shown are averaged over all replicates for each of the groups of 30, 70, 150, and 300 golden shiners. The order parameter space is divided into four regions—swarm (S), polarized (P), milling (M), and transition (T)—each being characterized by the dominant dynamical state of the fish school in that particular region. Different values of pmin and pmax were used for each group size to emphasize the density patterns and regions with no data are colored black. The insert in the 30 fish plot shows the density plot from an experiment with 30 fish and the tank area reduced to one tenth of the original.
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pcbi-1002915-g002: Density plots of polarization vs. rotation from experiments.The data shown are averaged over all replicates for each of the groups of 30, 70, 150, and 300 golden shiners. The order parameter space is divided into four regions—swarm (S), polarized (P), milling (M), and transition (T)—each being characterized by the dominant dynamical state of the fish school in that particular region. Different values of pmin and pmax were used for each group size to emphasize the density patterns and regions with no data are colored black. The insert in the 30 fish plot shows the density plot from an experiment with 30 fish and the tank area reduced to one tenth of the original.

Mentions: To demonstrate more clearly these dynamically-stable states, in Fig. 2 we consider the proportion of time groups spend in different regions of the two-dimensional phase space spanned by the order parameters Op and Or (red representing more time spent in a given region and blue the least time; black areas signify regions of the phase space not visited by groups in our experiments). While all three states of motion were manifest in all groups, there are also visible differences relating to increasing group size. For the smallest group size of 30 fish we see that the polarized group state predominates (high Op and low Or). Only rarely did groups of this size exhibit swarm behavior (low Op and low Or), and even less frequently did they adopt the rotating group state (low Op and high Or). The fluctuations in the order parameters are also most frequent for this group size (Fig. 1B). For a group size of 70 fish the frequency of transitions decreases and the collective states corresponding to the three dynamically-stable states become clearly distinguishable as ‘hotspots’: the polarized state is no longer dominant, with milling and swarm behavior also being common. As group size is increased further, to 150 and then 300 fish, groups spend most of their time milling, displaying fewer transitions into (and among) the polar state and swarm state. For all group sizes the milling state has an equal probability of rotating clockwise and counter-clockwise, i.e. groups did not exhibit a handedness (see Fig. S1).


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)

Density plots of polarization vs. rotation from experiments.The data shown are averaged over all replicates for each of the groups of 30, 70, 150, and 300 golden shiners. The order parameter space is divided into four regions—swarm (S), polarized (P), milling (M), and transition (T)—each being characterized by the dominant dynamical state of the fish school in that particular region. Different values of pmin and pmax were used for each group size to emphasize the density patterns and regions with no data are colored black. The insert in the 30 fish plot shows the density plot from an experiment with 30 fish and the tank area reduced to one tenth of the original.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002915-g002: Density plots of polarization vs. rotation from experiments.The data shown are averaged over all replicates for each of the groups of 30, 70, 150, and 300 golden shiners. The order parameter space is divided into four regions—swarm (S), polarized (P), milling (M), and transition (T)—each being characterized by the dominant dynamical state of the fish school in that particular region. Different values of pmin and pmax were used for each group size to emphasize the density patterns and regions with no data are colored black. The insert in the 30 fish plot shows the density plot from an experiment with 30 fish and the tank area reduced to one tenth of the original.
Mentions: To demonstrate more clearly these dynamically-stable states, in Fig. 2 we consider the proportion of time groups spend in different regions of the two-dimensional phase space spanned by the order parameters Op and Or (red representing more time spent in a given region and blue the least time; black areas signify regions of the phase space not visited by groups in our experiments). While all three states of motion were manifest in all groups, there are also visible differences relating to increasing group size. For the smallest group size of 30 fish we see that the polarized group state predominates (high Op and low Or). Only rarely did groups of this size exhibit swarm behavior (low Op and low Or), and even less frequently did they adopt the rotating group state (low Op and high Or). The fluctuations in the order parameters are also most frequent for this group size (Fig. 1B). For a group size of 70 fish the frequency of transitions decreases and the collective states corresponding to the three dynamically-stable states become clearly distinguishable as ‘hotspots’: the polarized state is no longer dominant, with milling and swarm behavior also being common. As group size is increased further, to 150 and then 300 fish, groups spend most of their time milling, displaying fewer transitions into (and among) the polar state and swarm state. For all group sizes the milling state has an equal probability of rotating clockwise and counter-clockwise, i.e. groups did not exhibit a handedness (see Fig. S1).

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