Social interactions, information use, and the evolution of collective migration.
ABSTRACT Migration of organisms (or cells) is typically an adaptive response to spatiotemporal variation in resources that requires individuals to detect and respond to long-range and noisy environmental gradients. Many organisms, from wildebeest to bacteria, migrate en masse in a process that can involve a vast number of individuals. Despite the ubiquity of collective migration, and the key function it plays in the ecology of many species, it is still unclear what role social interactions play in the evolution of migratory strategies. Here, we explore the evolution of migratory behavior using an individual-based spatially explicit model that incorporates the costs and benefits of obtaining directional cues from the environment and evolvable social interactions among migrating individuals. We demonstrate that collective migratory strategies evolve under a wide range of ecological scenarios, even when social encounters are rare. Although collective migration appears to be a shared navigational process, populations typically consist of small proportions of individuals actively acquiring directional information from their environment, whereas the majorities use a socially facilitated movement behavior. Because many migratory species face severe threat through anthropogenic influences, we also explore the microevolutionary response of migratory strategies to environmental pressures. We predict a gradual decline of migration due to increasing habitat destruction and argue that much greater restoration is required to recover lost behaviors (i.e., a strong hysteresis effect). Our results provide insights into both the proximate and ultimate factors that underlie evolved migratory behavior in nature.
Article: How and why do insects migrate?[show abstract] [hide abstract]
ABSTRACT: Countless numbers of insects migrate within and between continents every year, and yet we know very little about the ultimate reasons and proximate mechanisms that would explain these mass movements. Here we suggest that perhaps the most important reason for insects to migrate is to hedge their reproductive bets. By spreading their breeding efforts in space and time, insects distribute their offspring over a range of environmental conditions. We show how the study of individual long-distance movements of insects may contribute to a better understanding of migration. In the future, advances in tracking methods may enable the global surveillance of large insects such as desert locusts.Science 09/2006; 313(5788):794-6. · 31.20 Impact Factor
[show abstract] [hide abstract]
ABSTRACT: Because areas suitable for growth and reproduction are often ephemeral, a primary selective force in the evolution of migratory behavior in insects is the need to colonize new habitats. However, both migration itself and flight capability reduce present reproductive success. Thus the long-term fitness benefit of migration, the colonization of new habitats, is balanced by a short-term reduction in fitness, the result being that variation for migratory ability is preserved in a population. Migration is but one component of a wide suite of functionally connected traits that together form a migratory syndrome. Genetic variation is found in all components of the migratory syndrome, and selection for migration results in a change in the frequency of expression of these components, which can be analyzed and predicted using the mathematics of quantitative genetics. We illustrate this evolutionary interplay with the example of the evolution of wing dimorphism in the sand cricket.BioScience 01/2007; · 4.62 Impact Factor
[show abstract] [hide abstract]
ABSTRACT: Many populations of large herbivores migrate seasonally between discrete home ranges. Current evidence suggests that migration is generally selected for as a means of enhancing access to high quality food and/or reducing the risk of predation. The relative importance of these alternative selection pressures should depend on the demographic circumstances facing a given population. Seasonal migration also has important implications for the structure and dynamics of large herbivore communities. Migrants should tend to be regulated by food availability, while residents should tend to be regulated by predators As a result, migrants should often outnumber residents by a considerable margin - a pattern seen in several tropical and temperate ecosystems. Differences in the mode of regulation could also imply that competition for resources will be weak in purely resident assemblages, but strong in communities dominated by migrants. Continual grazing by resident herbivores can sometimes lead to degeneration of vegetation, while systems supporting migrants are apparently more resilient. This implies that migration can have an important impact on the long-term persistence of plant-herbivore systems, particularly in areas with slow rates of vegetation regeneration.Trends in Ecology & Evolution 09/1988; 3(9):237-41. · 15.75 Impact Factor
Social interactions, information use, and the
evolution of collective migration
Vishwesha Guttal1and Iain D. Couzin1
Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, 08544
Edited* by Simon A. Levin, Princeton University, Princeton, NJ, and approved July 19, 2010 (received for review May 17, 2010)
Migration of organisms (or cells) is typically an adaptive response to
spatiotemporal variation in resources that requires individuals to
detect and respond to long-range and noisy environmental gra-
dients. Many organisms, from wildebeest to bacteria, migrate en
masse in a process that can involve a vast number of individuals.
Despite the ubiquity of collective migration, and the key function it
playsintheecology ofmanyspecies,itisstillunclearwhatrole social
interactions play in the evolution of migratory strategies. Here, we
explore the evolution of migratory behavior using an individual-
based spatially explicit model that incorporates the costs and
benefits of obtaining directional cues from the environment and
evolvable social interactions among migrating individuals. We
range of ecological scenarios, even when social encounters are rare.
Although collective migration appears to be a shared navigational
process, populations typically consist of small proportions of indi-
viduals actively acquiring directional information fromtheir environ-
ment, whereas the majorities use a socially facilitated movement
behavior. Because many migratory species face severe threat
through anthropogenic influences, we also explore the microevolu-
We predict a gradual decline of migration due to increasing habitat
destruction and argue that much greater restoration is required to
recover lost behaviors (i.e., a strong hysteresis effect). Our results
provide insights into both the proximate and ultimate factors that
underlie evolved migratory behavior in nature.
availability, to escape from competition, and/or to reach
newer habitats, etc. (1–8). To migrate, both uni- and multicellular
organisms have evolved the ability to detect and respond to di-
rectional cues in the environment. This ability, in species such as
passerine birds and in many groups of vertebrates and insects, may
correspond to magnetoreceptivity (9), odor taxis (10), or tracking
changes in resource distributions (11). In bacteria and cells, di-
rectional information may result from an ability to respond to
thermal, chemical, or electromagnetic gradients (12).
It has been suggested that individual organisms can be seen as
information processing units (13) and that interactions among
organisms can provide collective benefits (14–20). For example, if
each individual is error prone in its detection of the migratory di-
rection, grouping may facilitate the spontaneous averaging of in-
dividual measurements, leading to improved navigation ability,
a property known as the “many wrongs principle” (16). In many
navigating groups, however, participants are mixed, such that
nearby individuals who may share these potential benefits are of
low relatedness. Even in migrating ungulates where family mem-
bers often maintain cohesion, and can thus be thought of as
a functional unit for selection, relatedness between nearby family
groups can be low (21). It remains unclear, therefore, how indi-
viduals optimize tradeoffs between costs and benefits of migration
and thus how, and under what ecological conditions, different mi-
gratory strategies evolve.
igration is often an adaptive response to changes in resource
Here, we develop an individual-based, spatially explicit evolu-
tionary model of organismal movement and social interactions and
use this to investigate migratory strategies under a wide range of
densities and cost-benefit structures that represent diverse eco-
logical scenarios. We also explore how habitat fragmentation and
changes in population density over relatively short ecological time
scales, such as those induced by anthropogenic influence (22–24),
may be expected to affect migratory behavior.
Model for the Evolution of Migration
about the appropriate migratory direction by exploiting environ-
mental features such as orienting using geomagnetic field cues (9)
by an evolvable parameter ωgi(henceforth referred to as “gradient
detection ability”), where i refers to the index of the focal in-
dividual. A solitary individual in the absence of such an ability, i.e.,
travel probabilistically more accurately along the environmental
gradient. Thus they accumulate migratory benefits, defined as the
migratory direction that asymptotically reaches a maximum value
(Fig. 1A and SI Appendices A and B).
We assume that individuals incur costs that increase mono-
tonically with their ωgi(Fig. 1B) because of properties such as en-
ergy expenditure involved (25) and/or associated costs such as
reduced predator vigilance during the gradient detection process.
butthespecific form ofthecost function chosen does not affect the
qualitative nature of the results (see SI Appendices A, B, and C for
details of model implementation and Appendix D for comments on
generality with respect to cost function).
An evolvable “sociality” trait, denoted by ωsi, represents the
possibility of social interactions (26), specifically, being attracted
toward and aligning direction of travel with nearby individuals (17,
27). This can be facilitated by vision (and/or other sensory modali-
ties) in insects and vertebrates or through more local mechanisms
such as adhesion, contact forces, and/or chemical signaling in bac-
teria or cells (28). We assume that this ability comes at a cost that
increases monotonically with ωsi.
preference to travel along the migratory gradient and their social
respective evolvable traits, ωgiand ωsi(17). Depending on the value
of these traits, individuals can exhibit a wide range of motion in-
cluding random walk (low ωgiand low ωsi), solitary migration (large
Author contributions: V.G. and I.D.C. designed research; V.G. performed research; V.G.
and I.D.C. analyzed data; and V.G. and I.D.C. wrote the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
1To whom correspondence may be addressed. E-mail: firstname.lastname@example.org or
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
| September 14, 2010
| vol. 107
| no. 37www.pnas.org/cgi/doi/10.1073/pnas.1006874107
large ωgiand moderate to large ωsi). Consequently, they acquire
a fitness corresponding to the migratory benefits minus the costs
incurred. Individuals are assumed to reproduce with a probability
proportional to their relative fitness and pass on their traits to their
Evolution of Migratory Strategies
Individuals optimize tradeoffs between the benefits of migration
and the costs involved in the migratory gradient detection. For
solitary individuals, the fitness does not depend on the strategy of
other individuals; hence, the evolutionary stable strategy of the
gradient detection ability is same as the value ωgthat optimizes the
fitness (Fig. 1C and SI Appendix B).
We now consider populations in which individuals may encoun-
ter each other. Under a very broad range of parameter conditions,
we find that populations evolve two coexisting frequency-dependent
strategies, with both the strategies being equivalent in terms of
fitness (Fig. 2 A–C and SI Appendix D). In one mode, individuals
have a relatively high gradient detection ability with a weak so-
ciality trait (referred to as “leaders”). In the other mode, indi-
viduals have an extremely weak or nonexistent gradient detection
ability and possess strong social interactions (referred to as
“social individuals”) (Fig. 2C).
In this population, social individuals are locally attracted to each
other and to leaders, forming groups. Leaders preferentially move
in the direction of the gradient and are less influenced by others
because of their relatively weak social tendency. Consequently,
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Gradient detection ability ( g) Gradient detection ability ( g)
Gradient detection ability ( g)
Migratory benefits (b)
Cost of gradient detection (cg)
Migratory fitness (f)
calculations of migratory benefits, b, gained by a solitary individual (SI Appendix B). (B) Cost of gradient detection is given by cg= pg(exp(ωg/ωgc) − 1) and (C)
comparison between numerical simulations and analyticalcalculations of the individualfitness (f = b − cg). Parameters: size ofpopulation N = 1, pg= 0.75, ωgc= 4.0,
strength of noise in perception σr= 0.1.
independent of the initial conditions (SI Appendix C). (C) A 2D histogram of the evolved state at the 1,500th generation demonstrates the relatively small
proportion of leaders (high, ωg; low, ωs) and a majority of social individuals (low, ωg; high, ωs). (D) The group composition is quantified by the proportion of leaders
as a function of group size. Parameters: N = 16384, ρ = 7.0 × 10−3individuals per BL2, pg= 0.75, cost of sociality ps= 0.0 and σr= 1.00. The other parameter values
are in SI Appendix A.
The evolution of collective migration. (A and B) The evolution of the gradient detection ability, ωg, and sociality trait, ωs, respectively. This result is largely
Guttal and Couzin PNAS
| September 14, 2010
| vol. 107
| no. 37
composite groups consisting of both social individuals and leaders
the whole group typically acquires a directed motion up the gra-
dient. We note that, to an observer, it would likely appear that all
individuals are actively climbing the gradient.
As a consequence of this complex spatiotemporal dynamic, we
see a fission–fusion process at the population level where groups
constantly merge and split during the collective migration (as seen
in many natural populations) (30; Movies S1 and S2). Couzin et al.
(17)showedthattheproportionofleaders neededtoguidea group
to the desired destination with a given accuracy decreases with
increasing group size (17). Here, we reveal that this leadership
the dynamics of groups merging and splitting (Fig. 2D).
Density and Cost Structure
To test the generality of our results, we investigate the evolved
states under a wide range of ecological scenarios (Fig. 3 and SI
Appendix E). Notably, population density can determine, in large
part, how often individuals encounter one another. The costs of
gradient detection may be species- and/or environment-specific.
Additionally, factors such as group size and/or spatial position in
example, individuals (typically leaders) who either tend to occupy
may be more susceptible to predation (31) or pay higher energetic
costs through increased vigilance (32). They might also fail to ex-
ploit socially facilitated environmental change, such as moving
where others have trampled through vegetation, as in ungulates
(33). Within our model, such species-specific details can be ap-
proximated by rescaling the effective cost incurred while perform-
ing gradient detection.
We begin by exploring the evolved migratory strategies as
a function of density (ρ) and gradient detection cost ( pg) but for
a fixed and relatively small value of social cost ( ps). In extremely
low-density populations, where the probability of encountering
others is negligible and/or when it is inexpensive to evolve the
or no sociality, unlike those in Fig. 2A, because of costs associated
with social interactions. As a result, we find solitary migration. At
the other extreme, when densities are so high that frequent colli-
sionsamongindividuals inhibitmigration and/orwhenthegradient
use gradient information leading to resident (i.e., nonmigratory)
populations (Fig. 3 A and B, dark regions).
There is, however, a very large intermediate region ofparameter
space where leaders and social individuals coexist and populations
exhibit collective migration (Fig. 3A, yellow-red region). We note
that collective migration evolves even at very low densities where
where BL is the typical body length. Thus, even for species that are
not considered traditionally to migrate collectively, social inter-
actions may still play an important role.
The costs of social interactions ( ps) may typically be relatively
small because they are facilitated by an already necessary machin-
ery, such as vision, or physical forces, such as contact/friction (28).
However, larger group sizes can lead to increased competition for
resources among group members. These features can be included
by rescaling the cost of sociality; for example, psis larger when the
competition for resources is high. In SI Appendix E, we show
evolved migratory strategies for a range of gradient detection ( pg)
andsocialcosts ( ps).Dependingonthevalue ofthesecosts relative
to the migratory benefits, we find three qualitatively different mi-
gratory states of solitary migration (zero to moderate pgand small
to high ps), resident populations (high pg), and collective migration
(a large intermediate region, as in Fig. 3).
Thus far, we assumed that individuals use the same strategy at all
times within a generation. We now consider more intricate and
dynamic strategies; individuals might be able to modify their in-
contexts. For instance, to avoid being exploited by social individu-
als, and/or to exploit others with gradient detection ability, an in-
dividual may not perform gradient detection when the local
condition is crowded, despite possessing a very high gradient de-
tection ability (ωgi). This can be facilitated by a quorum sensing
ability in cells or microorganisms (34) or, more generally, as
a consequence of responding to the state of the local environment.
Even under such scenarios, we find that the frequency-dependent
coexisting strategies of leaders, who use gradient detection almost
all of the time, and social individuals, who very rarely do, remains
evolutionarily stable (SI Appendices F and G). In other words,
migratory individuals in our model do not evolve context-de-
pendent interactions even when given the possibility to do so.
Density ( ): Individuals per BL2
Density ( ): Individuals per BL2
05 1005 10
Cost of gradient detection (pg) Cost of gradient detection (pg)
Population migratory ability
afunction ofdensity(ρ),and costofgradientdetection(pg).Notethatdensityismeasuredin unitsof individualsper BL−2.Populationmigratoryabilityis defined as
the migratory benefits averaged over all individuals in the population. (A) There is a clearly demarcated individual-migration state (bright region), a collective-
migration state (yellow-red regions), and a no-migration state (dark region) with sharp changes in the proportion of leaders between these evolved migratory
states. (B) Corresponding changes inpopulation migratory ability is relatively gradual. The density is on a log-axis covering nearly fiveorders of magnitude and the
collective migration state occurs for densities where the interactions between individuals are very rare. Parameters: N = 320, ps= 1.0, and σr= 0.10. The other
parameter values are in SI Appendix A.
Evolvedmigratorystrategiesunder different ecologicalconditions.(A)Theproportionofleadersin,and(B)themigratoryabilityof, evolvedpopulationsas
| www.pnas.org/cgi/doi/10.1073/pnas.1006874107Guttal and Couzin
Impact of Environmental Pressures
Anthropogenic pressures can significantly influence population
density, as seen in the steep decline of American bison (Bison
bison), and even result in extinction, as occurred with passenger
pigeon (Ectopistes migratorius) (1, 22–24). This is despite em-
pirical studies that provide evidence for rapid microevolutionary
changes in migratory patterns, for example in birds, within de-
cadal time scales (1, 35, 36). Here we investigate the impact of
habitat fragmentation and changes in population density on
In migratory species, as habitat fragmentation increases, indi-
viduals have to travel disproportionately larger distances to reach
suitable habitats [because of, for example, a reduced frequency of
encountering stop-over or refueling sites (1)] and thus to accumu-
late migratory benefits. We implement this by assuming that the
benefit bisa nonlinearfunction ofthe average distance migratedd,
corresponding to a contiguous habitat. The larger the value of β,
the larger the nonlinearity, and, hence, the organism must cover
linearity, for example, may correspond to the need of some bird
small changes in the habitat fragmentation (β) and allow adapta-
tion of traits, ωgiand ωsi, for a small number of generations, ng, to
account for the relatively short ecological time scales. This is in
contrast to our previous focus on robust evolutionary stable states
that could not be invaded by other mutant strategies and that are
how our results are affected by different values of ng.
population to migrate reduces relatively gradually (Fig. 4B, solid
line). At high levels of habitat fragmentation, no individuals evolve
to be leaders, and therefore, the population loses its migratory
ability. Even after restoring the habitat, however, a population’s
migratory ability does not recover at the same habitat quality at
whichitdeclined;i.e., itshowsstronghysteresis, ormemory,effects
(Fig. 4B, dotted line). In highly fragmented habitats, a small mu-
tationinωgithat mildly alters theinformationuse does notimprove
the individual’s fitness; it requires large mutations in ωgi, exceeding
a threshold, to sufficiently enhance the information use and thus
migratory benefits that exceed the costs incurred (in ωgi). Large
mutations, however, typically do not occur on relatively short
ecological time scales. Upon substantial habitat restoration, the
required threshold change in the information use reduces and can
be reached by mutations occurring on ecological time scales and
hence migratory ability is reestablished (SI Appendix H).
We also find hysteresis effects, although less pronounced, as
a function of population density. These results are quantitatively,
but not qualitatively, affected by various choices of ng, represent-
ing different rates of change of ecological conditions; more spe-
cifically, the faster the rate of change of ecological conditions, the
lower the probability of large mutations and thus the stronger the
hysteresis effect (SI Appendix H). Note that we do not include an
explicit habitat structure where fragmentation is measured, for
example, by the extent of patchiness in the resource availability.
Instead, we approximated a plausible impact of habitat fragmen-
tation on migratory individuals by assuming that benefits are
a nonlinear function of the distance traveled. Also, our focus was
on the microevolutionary response of migratory strategies to
ecological changes but not the growth and decline/extinction of
populations themselves. Our model framework, however, can
potentially be useful in investigating combined effects of adaptive
migratory strategies together with the density-dependent growth
and mortality of populations.
Our model predicts that individuals who invest in acquiring in-
formation about the migratory direction from environmental cues
are readily exploited by others who adopt a socially facilitated
movement behavior. For a wide range of biological assumptions,
these two coexisting strategies result in collective migration with
fission–fusion process. Furthermore, even when interactions
perhaps hitherto unknown) role.
Collective migration occurs also when all individuals of a pop-
ulation evolve to use both the migratory directional information
and social cues. Migrating groups in these evolved populations
preserve their group composition over relatively long time scales.
However, this strategy is expected to occur only when the costs of
gradient information use and sociality are both negligibly small in
comparison with the benefits of migration. We also emphasize
general predictions of our model, that the ecology of species, rep-
resented by population density, habitat structure, costs, and bene-
fits of migration, determines whether populations will evolve to
a resident, a solitary migratory, or a collective migratory strategy.
Although a precise quantification of costs and benefits of in-
formation can be difficult, we suggest that evidence for (or the lack
of) a bimodal, or other such strongly skewed, population structure
in information use, as suggested here, will provide insights to un-
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
Normalized migratory distance (d
Migratory benefits (b)
Population migratory ability
Habitat fragmentation (
of habitat fragmentation (β):b = dβ. (B) The solid line shows response to increasing habitat fragmentation (i.e., increasing β, starting from β = 1). The dotted line
in SI Appendices A and H.
Guttal and CouzinPNAS
| September 14, 2010
| vol. 107
| no. 37
derlying selection forces. The existence of such hierarchical struc-
ture among organisms may be deduced through an analysis of
visual tracking of identifiable cells or GPS tracking of higher
brain activity in free-flying birds (38), suggesting that the study of
the future. In addition, recent advances in our understanding of
how cells infer, and respond to, the state of its environment and
quantification of associated fitness (39, 40) make cellular systems
an attractive candidate for testing our model predictions. Other
drivers of migration, which are not mutually exclusive with our
hypothesis, include predators, competition, and/or disease avoid-
ance (1–8). As we discussed previously, these can be incorporated
by rescaling the costs and benefits of gradient climbing in our
framework and/or by making species-specific modifications to
Climate change and habitat destruction can dramatically alter
resident [e.g., blackcaps (Sylvia atricapilla); ref. 36], or lost migra-
tion can reappear [e.g., eastern house finch (Carpodacus mex-
icanus); ref. 41]. Using our model, we predict a gradual decline of
migratory behavior because of habitat destruction, but, owing to
relatively short time scale of these changes, the reestablishment of
lost behaviors will require substantially greater restoration. Our
study shows that the time scales ofecological changesplaya crucial
role in determining the response of migratory species.
in costly directional information, and social individuals who navi-
gate by following others’ motion, can be mapped onto mean-field,
discrete-strategy models that exhibit producer–scrounger (PS) dy-
namics (42, 43), where producers and scroungers are similar to
leaders and social individuals, respectively. In contrast to PS
models, our approach providesa mechanistic basis for scaling from
individual-level description to higher levels of organizations and
how it feeds back to local interactions. For example, it allows us to
capture the role of nonlinear and emergent collective properties of
socially navigating groups, such as the many wrongs principle (SI
Appendix D), and that the proportion of leaders needed to guide
migratory groups in the desired direction reduces with the group
size (Fig. 2D). Additionally, we are able to provide testable pre-
dictions regarding the spatiotemporal dynamics and the composi-
tion of migratory groups (Movies S1 and S2). Furthermore, our
approach allows us to study intricate aspects of fixed vs. context-
dependent strategies (SI Appendices F and G) and the implications
of environmental structure on the evolution of migratory strategies
on both evolutionary and ecological time scales (Figs. 3 and 4).
Here, we focused on the phenomenon of migration with
a constant global gradient that leaders could detect with relatively
small errors. Would our results continue to hold when gradients/
stimuli exhibit complex stochastic spatiotemporal variations? We
note that novel collective navigational and search properties may
arise depending on the nature of social interactions and the en-
vironmental noise (14, 44). Future studies can reveal the role of
such emergent collective properties and stochasticity in an evo-
Linking patterns of aggregation to their function is a question of
fundamentalimportanceinbiology.Our study offers insights about
the adaptive significance of social cues in migratory behavior on
both evolutionary and ecological time scales. Our results also have
broader implications for studies on the evolution of taxis and/or
foraging strategies in complex fluctuating environments. More
generally, it provides a useful framework to investigate the evolu-
tionary forces that drive collective behavior over a wide range of
spatial and temporal scales.
Materials and Methods
Movies S1 and S2 show spatiotemporal dynamics of the evolved population
of Fig. 2. SI Appendix provides further details on the model implementation
and generality of our results. It contains the following subsections: SI Ap-
pendix A, details of model implementation; SI Appendix B, evolutionary
stable strategy, or optimal strategy, for a single individual; SI Appendix C,
evolutionary simulations for populations; SI Appendix D, the evolution of
bimodal strategies and generality with respect to cost function; SI Appendix
E, evolutionary outcome as a function of cost of gradient detection and cost
of sociality; SI Appendix F, a model in which individuals can use their strategy
probabilistically; SI Appendix G, a model in which individuals can use their
strategy in a context-dependent way; SI Appendix H , the microevolutionary
response of migration to habitat fragmentation and changes in population
ACKNOWLEDGMENTS. We thank Andrew Dobson, Simon Garnier, Andrew
Hartnett, Christos Ioannou, Yael Katz, Simon Levin, Michael Raghib, Daniel
Rubenstein, Colin Torney, David Wilcove, members of Couzin Laboratory,
and two anonymous referees for comments on the manuscript. We are
grateful to Hong Li, Allison Kolpas, and Linda R. Petzold for providing us
with a version of the code that implemented our swarming model on the
high performance graphics processing units (GPU) using compute unified
device architecture (CUDA) and Yael Katz for implementing the graphical
visualization. We acknowledge support from a Searle Scholar Award 08-SPP-
201 to I.D.C., Defense Advanced Research Projects Agency Grant HR0011-05-
1-0057 to Princeton University. I.D.C. also acknowledges support from
National Science Foundation Award PHY-0848755 and Office of Naval
Research Award N00014-09-1-1074.
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| www.pnas.org/cgi/doi/10.1073/pnas.1006874107 Guttal and Couzin