Download Specialization and evolutionary branching within

Specialization and evolutionary branching
within migratory populations
Colin J. Torney1, Simon A. Levin, and Iain D. Couzin
Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544
collective behavior ∣ evolutionary dynamics ∣ social information
A
s a generalized process, animal migration is observed over a
broad range of temporal and spatial scales, from the diurnal,
vertical migration of lake dwelling algae (1), to the annual interoceanic movements of the humpback whale (2). Animals typically
navigate these routes through the use of environmental cues and
various sensory modalities, responding to nutrient or thermal
gradients, magnetic or electric fields, odor cues, or visual markers
(3). Detecting and processing these sources of information is
nontrivial and doing so requires an investment either in time
and energy or in the physiological cost of developing cognitive
or sensory capacity (4).
The potential cost associated with the acquisition of information raises several issues in the context of leadership dynamics
and social interaction. Previous works have shown how sociality
can facilitate or even enable the use of environmental cues, both
theoretically (5–7) and empirically (8, 9). In ref. 10, Couzin et al.
demonstrated that small groups of individuals can affect the direction choice of large numbers of uninformed followers, indeed
it was shown that for a fixed proportion of leaders the accuracy in
terms of the group direction increases as the size of the group
grows. If considering collective migration alongside these studies
of social interaction, the relation between the costly acquisition
of personal information, which drives the migration, and the uninformed following of others may be mapped to a producerscrounger dynamic (11). In this situation the commodity being
generated is information (12) and it is available via two sources,
directly from the environment (personal information) or through
observations of the behavior of conspecifics (social information) (4).
www.pnas.org/cgi/doi/10.1073/pnas.1014316107
In a recent study (13) this process was examined using an
individual based model governed by localized rules of attraction,
alignment etc., with differing degrees of independence and sociality. This work showed that, under certain conditions, specialized
groups of leaders form. The challenge in understanding and
classifying models of this type lies in identifying an appropriate
methodology for the abstraction of interindividual interactions
(i.e., amenable to analysis of the evolutionary process but still
able to capture the higher level effects of low level behavior). This
abstraction is required if complex, interacting systems are to be
mapped to a framework that is compatible with the mathematical
study of evolutionary dynamics. In this work we employ a unique
mean-field approximation of these social interactions to analyze
the frequency dependent selection dynamics created via the
collective migratory process.
Model Description
In our model, individuals are able to access an imperfect source
of information indicating the correct migration route. It is assumed that the effective utility of the environmental cue is an
increasing, but saturating, function of the level of investment
made in its detection. This represents the diminishing returns that
may be associated to variable behavioral or physiological parameters, such as the frequency direction choice is evaluated, or
the exploration of alternative routes. The distance travelled in
the correct direction over a fixed time interval is then continuously mapped to a linear fitness benefit, which may result from
long range improvements in mean abiotic conditions or nutrient
availability, reduced risk from nocturnal predators by reaching
safe havens earlier, first access to preferred breeding or grazing
grounds etc.
A lone individual is then faced with the simple challenge of
optimizing the level of return versus cost of investment, however
an individual within a group may also choose to follow others and
it is assumed that imitation of direction choice incurs no cost.
(Note: The validity of this assumption is dependent on the cost
to sociality being less than the cost of obtaining personal information but not necessarily zero.)
To describe and quantify the social information source we note
an equivalence between the interacting, migratory population defined by an infinite-dimensional system of orientation headings,
and a collection of coupled chemical or biological oscillators.
This leads us to an approach proposed by Kuramoto (14, 15)
in his study of the onset of synchronization in large-scale coupled
oscillator systems. By using Kuramoto’s method, the population
is described by two key properties; the average heading of the
group and the degree of coherence; i.e. the level of consensus
around the average heading.
The result is two well-defined sources of information, an external fixed source that represents an environmental cue, and
Author contributions: C.J.T., S.A.L., and I.D.C. designed research; C.J.T. performed research;
C.J.T., S.A.L., and I.D.C. analyzed data; and C.J.T., S.A.L., and I.D.C. wrote the paper.
The authors declare no conflict of interest.
1
To whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1014316107/-/DCSupplemental.
PNAS Early Edition ∣
1 of 6
APPLIED
MATHEMATICS
Understanding the mechanisms that drive specialization and speciation within initially homogeneous populations is a fundamental
challenge for evolutionary theory. It is an issue of relevance for significant open questions in biology concerning the generation and
maintenance of biodiversity, the origins of reciprocal cooperation,
and the efficient division of labor in social or colonial organisms.
Several mathematical frameworks have been developed to address
this question and models based on evolutionary game theory or
the adaptive dynamics of phenotypic mutation have demonstrated
the emergence of polymorphic, specialized populations. Here we
focus on a ubiquitous biological phenomenon, migration. Individuals in our model may evolve the capacity to detect and follow
an environmental cue that indicates a preferred migration route.
The strategy space is defined by the level of investment in acquiring personal information about this route or the alternative
tendency to follow the direction choice of others. The result is a
relation between the migratory process and a game theoretic dynamic that is generally applicable to situations where information
may be considered a public good. Through the use of an approximation of social interactions, we demonstrate the emergence of a
stable, polymorphic population consisting of an uninformed subpopulation that is dependent upon a specialized group of leaders.
The branching process is classified using the techniques of adaptive
dynamics.
ECOLOGY
Contributed by Simon A. Levin, September 28, 2010 (sent for review April 30, 2010)
a social source dependent on the behavior of the population. Individuals can then freely evolve sociality, meaning aggregation
and the imitation of others, or leadership, the tendency to acquire
information and respond autonomously.
For simplicity we assume that if θ defines an individual’s orientation relative to a fixed axis and it moves at a constant speed
in this direction, the orientation θ ¼ 0 corresponds to the heading
associated with maximum benefit; i.e. along the defined migration route. The quality of the environmental cue is controlled
by a noise term in the dynamics of an individual’s orientation
heading, representing fluctuating external fields or internal errors
in their detection. Increased investment in detection reduces this
noise and brings the orientation toward the optimal direction
θ ¼ 0. This leads to an Ornstein–Uhlenbeck process of the form
A
dθg ¼ −xg θdt þ σdW t ;
B
[1]
where xg is the evolutionarily tunable parameter that defines the
degree of investment into gradient detection, dW t is a standard
Wiener process and σ encompasses the noise inherent in detecting the gradient. Hence, increasing the amount of investment
made means greater average migration speed.
Considering first the situation for a single individual, we use a
Fokker–Planck (16) representation of Eq. 1,
∂ρ
∂
σ 2 ∂2 ρ
¼ xg ðθρÞ þ
;
∂t
∂θ
2 ∂θ2
rffiffiffiffiffiffiffiffi 2
xg −xg2θ
e σ :
πσ 2
[3]
This equation represents an approximation of the distribution
only. It is exact in the limit of xg ∕σ 2 → ∞, however it remains
a good approximation even for small values of xg (see Fig. 1
for a comparison of the average migration speed calculated using
this approximation and the numeric values, and SI Text for comparisons of Eq. 3 to the actual distribution). Effectively, by using
Eq. 3, we are neglecting the periodic boundaries, assuming ρðθÞ is
defined on an infinite domain, and then extracting the only region
that is meaningful to our system θ ∈ ½−π;π.
Using this density function, the distance traveled within a given
period can be calculated by projecting the density at each point
on to the optimum directional vector. This is then normalized to
give a mean velocity along the migration route that lies in the
interval (0,1)
Z
2
−σ
:
ρðθÞ cosðθÞdθ ≈ exp
4xg
−π
π
[4]
If ρðθÞ were defined by a Dirac delta function then the individual
would spend all its time perfectly aligned along the correct heading and the average migration speed would be maximized. In the
absence of any cost for personal information xg ¼ ∞ is an evolutionary attractor and all individuals increase their investment so
that ρðθÞ → δðθÞ. In our model the average migration speed is
scaled by a reducing factor dependent on the level of investment
xg . The fitness, F associated to a given strategy is
2 of 6 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1014316107
0.8
0.6
rx
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
x
0.4
0.2
0
∂sx(y)
––––– -0.2
∂y
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0.4
-0.6
[2]
from which it is possible to obtain the probability density function
ρðθÞ that defines the time averaged distribution in θ for a given
value of investment. A stationary solution to Eq. 2 can be found if
it is assumed ρðπÞ → 0; i.e., an individual spends a vanishingly
small amount of time moving opposite to the desired direction. In
this case the periodic boundary conditions due to the angular
rotation may be neglected and an analytical approximation of
the probability density function can be obtained as
ρðθÞ ¼
1
-0.8
0
0.2
0.4
x
Fig. 1. (A) Phase coherence r x as a function of x, lines are analytical result,
points are from the numerics, black dash-dot (asterisk) is for a lone individual
(the asocial case of Eq. 4), red solid (square) η ¼ 2, green dash (circle) η ¼ 2.5,
blue dot (triangle) η ¼ 3. Outline shapes, average migration speed; filled
shapes, population order parameter modulus r, which may not be centered
x ðyÞ
as a function of x.
around the correct direction. (B) Local fitness gradient ∂s∂y
Lines are different values of cost, negative values indicate the population will
move toward the left, positive to the right.
F ¼ exp
2
−σ
exp½−cx2g ;
4xg
[5]
where a reducing factor has been introduced with a strength
defined by c and a quadratic dependence on xg so that higher
values of investment become increasingly costly. As an additive
fitness differential (i.e., in units of fitness), this cost is defined as
2
2
exp½−σ
4xg ð1 − exp½−cxg Þ. Our choice of this cost function is motivated by analytical tractability; the results do not appear to depend on its exact form. For example, numerical simulations
with linear additive cost demonstrate equivalent behavior (see
also ref. 13).
Because individuals in the asocial case are noninteracting, the
optimum value of xg can be easily found as
2 1
σ 3
xg ¼
:
[6]
8c
In the absence of any social interaction, there can be no frequency
dependent effects and xg represents an evolutionary stable strategy that all initial strategy values will evolve toward. However social interactions dramatically alter this situation as the direction
choice of others is another source of information that can be
exploited through copying, without expending energy in detecting
environmental cues directly.
Exactly how this second source of information will be accessed
is dependent on the system in question, but for many migratory
systems the degree of ordering and the average orientation within
Torney et al.
where r is a measure of the degree of alignment or phase coherence ranging from zero to one, and ψ is the average orientation.
It is therefore possible to define a process equivalent to Eq. 1
where an individual may decide to pay attention to social information with the amount of noise being dependent on r. Precisely
how the quality of information relates to r is unknown but it can
be assumed to be a decreasing function, with the more disordered
the population the more difficult it is to estimate the correct direction. Here we select the simplest function, a linear relation
between the variance of the noise term and the population phase
coherence, r,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dθs ¼ −xs ðθ − ψÞdt þ η ð1 − rÞdW t ;
[8]
where xs represents the degree to which an individual observes
and imitates its neighbors and η measures the difficulty of extracting information from social interactions. The magnitude of η
determines how the noise term scales with the modulus of the
order parameter and so may represent several factors such as
the population density or the interaction range of the organisms.
The variables xg and xs therefore define a trait space through
which a population may evolve. A given strategy is based on a
combination of using both social and environmental information,
so that Eqs. 1 and 8 are combined through a weighted average in
proportion to the investment allocation made
dθt ¼
xg dθg þ xs dθs
;
xg þ xs
[9]
where dθg and dθs are defined by Eqs. 1 and 8 respectively.
Finally, the two dimensional trait space defined by xs and xg is
reduced to one dimension by constraining the two traits to be negatively correlated so that xs ¼ 1 − xg . This space is restricted to
½0;1 and an individual is defined by a single value. Dropping subscripts we refer to the evolvable trait as x where x ¼ xg ; i.e., x ¼ 0
corresponds to a behavior where an individual pays no attention
to any environmental gradients and instead solely copies the direction choice of others. Similarly x ¼ 1 corresponds to an asocial
individual, with maximum investment in detecting environmental cues.
Evolutionary Dynamics
Before studying the dynamics of the model, it is necessary to define an appropriate region in parameter space in which results
will be of interest. Specifically we note that a condition for evolving higher environmental information processing capacity is that
there should be an increase in mean velocity as more investment
is made.
Torney et al.
Invasibility Analysis
To obtain an analytical description of the evolutionary dynamics
we first make the assumption ψ ≃ 0. This is true if x > 0 (after
PNAS Early Edition ∣
3 of 6
ECOLOGY
−π
The system defined by Eqs. 7 and 8 exhibits a transition to
a highly aligned state analogous to that observed in the selfpropelled particle model of Vicsek et al. (17). In our model,
as the social noise term η is reduced a transition occurs from
a disordered, unaligned population, to a perfectly aligned state.
(See SI Text for figures showing this transition.) There exists a
bifurcation point where the stable, disordered state disappears
and the only attractor is at r ¼ 1. By matching derivatives this
point can be found as η ¼ 2.
If η < 2, the population has a high level of ordering even in the
absence of directional information. For nonzero x the result is a
separation of time scales. Firstly the population becomes highly
ordered, then due to the asymmetry created by x > 0, it will converge on the optimum direction (see SI Text). In this situation
there is little incentive for greater investment and migration ability is weak due to long transient effects. A necessary condition
for evolving toward interior solutions in the region x ∈ ½0;1 is
that η > 2.
Incorporating this constraint, simulations are performed with
an initially homogeneous, monomorphic population in which
mutations around the mean phenotype occur. Individuals perform the migratory process until an accurate picture of fitness
is obtained. The fitness, average migration speed less cost paid,
is calculated and a standard roulette wheel selection algorithm
(18) is used to determine which strategies will proceed to the next
generation.
In this way the population moves through the trait space.
Mutations that are beneficial result in increased fitness and higher than average reproduction. In this case the initially rare mutant
will invade the resident population and the mean phenotype will
shift. The evolutionary dynamics are governed by the selection
gradient, the rate of change of fitness with distance from the resident phenotype. If the selection gradient is positive (negative)
the trait value will increase (decrease). A selection gradient that
is zero corresponds to a singular strategy that can be classified as
an evolutionary stable strategy (ESS) (19, 20) and/or a convergence stable strategy (CSS) (21, 22).
An ESS represents an evolutionary end point for a population
because, by definition, once it has been established it cannot be
invaded. A CSS represents an attracting point, toward which a
population will evolve. These categories are neither mutually
exclusive or interdependent. A population will necessarily move
toward a CSS but this point may be an ESS, in which case no
further evolutionary change will occur, or it may not, meaning
there exists the potential for disruptive selection and branching
into a polymorphic population.
Numerical simulations were performed using large populations of individuals (65,536) that process environmental information according to Eq. 1 and respond to neighbors via Eq. 8 and the
global order parameter. The resulting dynamic is qualitatively
equivalent to simulations that explicitly include localized interactions (13).
Fig. 2 shows how population traits change over time and illustrates the presence of an attracting convergence stable strategy at
which evolutionary branching may occur. Two stable outcomes to
the dynamics are possible. The first is that the population will
evolve to a single evolutionary stable monomorphic strategy.
The second situation occurs when the convergence stable point
is evolutionarily unstable. In this case, the population will spontaneously branch and form a specialized, polymorphic population
consisting of an informed group that lead and drive the migration
and a social group, dependent on these leaders and the directional information they provide, that survive only by following
their neighbors.
APPLIED
MATHEMATICS
the population will be the key parameters that determine how
useful and accessible the social information is. When the population is disordered, little information about the average heading
can be obtained. As global ordering increases the quality of information increases, and even short-range, local observations
of near neighbors give an accurate picture of the global mean
orientation.
The degree of ordering and the average orientation can be
encapsulated by a single complex order parameter (14). In a
monomorphic population the system is essentially ergodic, therefore, for sufficiently large population sizes, the time averaged
orientation distribution defined by ρðθÞ is equivalent to the instantaneous probability density function averaged over the population, and Eq. 4 can be related to the complex order parameter
of the Kuramoto model
Z π
reiψ ¼
ρðθÞeiθ dθ;
[7]
2000
0
A
500
Generation
1500
1000
1000
1500
When considering the possible invasion of mutant phenotypes
we define the resident population as having trait value x with
population level order parameter, r x . The time averaged orientation of a mutant will be different to that of the resident population, but because there are only small numbers of mutants within
a large population, the ensemble average will not be affected. A
mutant with trait value y will pay the cost associated with this
value but will be able to use the social information contained
in a population with trait value x. For the time averaged distribution this gives
Z
500
2000
2 2
y σ þ ð1 − yÞ2 η2 ð1 − r x Þ
:
ρy ðθÞ cosðθÞdθ ¼ exp −
4ð2y2 − 2y þ 1Þ
−π
π
[12]
The differential fitness of mutant y in a resident population x is
2500
0
0
0.2
0.4
0.6
0.8
2000
0
B
sx ðyÞ ¼ Fðy;xÞ − Fðx;xÞ;
1.0
[13]
where
2 2
y σ þ ð1 − yÞ2 η2 ð1 − r x Þ
exp½−cy2 :
Fðy;xÞ ¼ exp −
4ð2y2 − 2y þ 1Þ
500
[14]
Generation
1500
1000
1000
1500
500
2000
2500
the population will evolve to increasing values of x, and if this
quantity is negative it will decrease. The evolutionary singular
strategies are defined by
∂Fðy;xÞ ¼ 0.
[16]
∂y y¼x
0
0
0.2
Increasing
sociality
0.4
0.6
Trait (x)
0.8
1.0
Increasing
cue detection
Fig. 2. Branching process for σ ¼ 1, η ¼ 2.5, c ¼ 0.9 (A) and c ¼ 0.5 (B).
Generation number is y axis and trait value x is x axis. Red line is the critical
value x C left of which branching occurs.
relaxation of transients) and is entirely accurate when some level
of migration ability is attained. The orientation of an individual is
then described by the combination of its tendency to follow others
and the environmental cue
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dθt ¼ −ð2x2 − 2x þ 1Þθdt þ x2 σ 2 þ ð1 − xÞ2 η2 ð1 − r x ÞdW t :
[10]
Again introducing the approximation ρðπÞ → 0 the average
migration speed (which is equivalent to the phase coherence because ψ ≃ 0) is found as
2 2
x σ þ ð1 − xÞ2 η2 ð1 − r x Þ
:
rx ¼ exp −
4ð2x2 − 2x þ 1Þ
[11]
A comparison of the values of the implicitly defined value r x as
a function of x from the individual based simulations and the
analytical approximation is shown in Fig. 1. Conceptually the
presence of r x on both sides of Eq. 11 can be related to the time
averaged orientation of an individual (left hand side) and the
instantaneous ensemble average of the population (right hand
side). It is this distinction that enables us to approximate the
fitness of a single mutation in the resident population.
4 of 6 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1014316107
This value defines the long term exponential growth rate of the
mutant in an environment containing individuals with phenotype
x, therefore its gradient with respect to y defines the direction in
trait space in which the population will move. If
∂sx ðyÞ ∂Fðy;xÞ ¼
> 0;
[15]
∂y y¼x
∂y y¼x
In Fig. 1 the selection gradient is shown as a function of the resident trait value, illustrating the presence of a convergent stable
point for intermediate values of the cost parameter. For cost values that are too high, the population collapses to a nonmigratory
state with zero investment, whereas for low values an informed,
asocial population emerges (x ¼ 1). In the following analysis,
we neglect these scenarios and focus on the case where interior
convergent stable strategies exist.
Numerical simulations suggest that the solution converges to a
singular point. Still, however, that point may not be an ESS if
∂2 Fðy;x Þ > 0.
[17]
∂y2 y¼x
In this case, evolutionary branching may occur (23, 24). It is
straightforward to calculate the conditions for this branching
(see SI Text) as
0 < x <
pffiffiffi
5− 7
:
6
[18]
No closed form solution for x in terms of the parameters of the
system can be obtained; however, it is fully defined by Eqs. 16 and
11. This leads to a function with transcendental terms that behaves as a quadratic in the region x ∈ ð0;1Þ and tends to −∞
as x → 1. Although the roots cannot be obtained
directly the sign
pffiffi
of this function at the critical value xC ¼ 5−6 7 will show whether
the root lies to the left or right and hence whether the singular
Torney et al.
B
0
1
D
mutant y
C
0
0
1
0
resident x
1
resident x
Fig. 3. Pairwise invasibility plots. Dark regions represent positive differential
fitness (mutant will invade), light regions negative (mutant will not invade).
(A, B) Numerical result, (C, D) Analytical approximation. For (A) and (C) c ¼ 0.4
(no branching). For (B) and (D) c ¼ 0.9 (branching occurs). The population resides on the diagonal, mutations occur along the vertical axis and the sign of
the fitness differential determines if it will invade. Invasion results in this mutant becoming the resident and the population shifts along the line y ¼ x
until it reaches a convergence stable strategy, x . If branching is to occur once
a CSS is reached, it is required that a line drawn vertically from x will immediately encounter a region of positive differential fitness. Numerical results
are obtained by evaluating the average fitness of a single individual in the
resident population, analytical results stem from Eq. 13.
value, x is greater or less than the value required for branching to
occur (SI Text). If
2
xC σ þ ð1 − xC Þ2 η2 ð1 − r xC Þ
r xC − exp −
< 0;
4ð2xC − 2xC þ 1Þ
[19]
then the singular strategy lies to the left of the critical value and
the population will branch. After some manipulation the condition for the emergence of a polymorphic population reduces to
η2 <
k1 c þ σ 2
2
ð1 − exp½− σ4 − k2 cÞ
;
[20]
pffiffi
pffiffi
where k1 ¼ 224 277−544 and k2 ¼ 4 277−2. The key to interpreting this
condition lies in its influence on the location of the convergence
stable strategy. If the cost c or the environmental noise value σ
are low the CSS is found at higher values of x. Individuals are
then locked into a high investment strategy because they are predominantly asocial and the advantage to exploitation through
imitation is weak compared to the loss of environmental information. Lower values of the social noise term η moves the fixed point
to the left where individuals are more social and the advantage of
imitation is greater. As individuals invade the social information
source is weakened, effectively increasing η and shifting the
attracting strategy to the right for those that are more informed.
The population therefore diverges.
Parameter values that lie either side of this condition are used
in the simulations shown in Fig. 2 with the critical value xc plotted
Discussion
The principle of natural selection appears to define a homogenizing force, leading to the survival of only a few well adapted species whose members compete fiercely for resources. This image,
however, is inconsistent with the observed abundance of species
diversity and fails to explain the cooperation, division of labor and
specialization that facilitates cohesion across multiple biological
scales, from human social groups (25) to the unicellular colonial
origins of multicellularity (26). Much progress in reconciling
these discrepancies has come through the mathematical analysis
of evolutionary dynamics (23, 27–30); however, a major difficulty
for the study of evolving populations arises from their inherent
complexity. Often they consist of many interacting components
that display emergent, collective-level phenomena that cannot
be predicted or explained by an analysis of individuals in isolation
(31–33).
Insight into the behavior of these systems has been gained
by using techniques developed in areas such as statistical physics
(17, 34, 35), but the focus has largely been on the proximate
causes of behavior (i.e., from a mechanistic, functional perspective) and not the ultimate cause (i.e., how natural selection at
lower levels has shaped these systems or how collective-level
phenomena feeds back on the fitness of individuals). In this work
we have shown that the influence of collective interactions on
evolution can be dramatic, and in the framework presented results in the formation of a subpopulation of specialized leaders
that drive a population level migration.
The catalyst for the evolutionary divergence in our model can
be found in the creation and exploitation of information about an
optimal migration route. As the production of information is
often costly, and through imitation or social facilitation may be
inadvertently shared with others, it can be equated to a public
good. The key difference to a standard public goods game (36),
is the self-dependence of the social information source (illustrated by the implicit definition of the ordering parameter
of Eq. 11) that causes a steep transition to a high migration
ability once a threshold level of investment is reached. It is this
nonlinearity, arising from collective interactions, that creates the
selection gradients required for evolutionary branching to occur,
and ultimately leads to the stable coexistence of two discrete,
interdependent populations.
ACKNOWLEDGMENTS. We thank Vishwesha Guttal, Michael Neubert,
Zoltan Neufeld, and an anonymous referee for helpful comments on the
manuscript. This work was supported by Searle Scholars Award 08-SPP-201
(I.D.C), National Science Foundation Grant PHY-0848755 (I.D.C), Office of
Naval Research Grant N00014-09-1-1074 (I.D.C), and Defense Advanced
Research Projects Agency Grant HR0011-09-1-0055 (Princeton University).
1. Sommer U, Gliwicz ZM (1986) Long range vertical migration of Volvox in tropical lake
Cahora Bassa (Mozambique). Limnol and Oceanogr 31:650–653.
2. Pomilla C, Rosenbaum HC (2005) Against the current: an inter-oceanic whale migra-
6. Torney C, Neufeld Z, Couzin ID (2009) Context-dependent interaction leads to emergent search behavior in social aggregates. Proc Natl Acad Sci USA 106:22055–22060.
7. Codling EA, Pitchford JW, Simpson SD (2007) Group navigation and the many wrongs
tion event. Biol Letters 1:476–479.
3. Gould JL (2004) Animal navigation. Curr Biol 14:R221–R224.
4. Dall SRX, Giraldeau LA, Olsson O, McNamara JM, Stephens DW (2005) Information and
its use by animals in evolutionary ecology. Trends Ecol Evol 20:187–193.
5. Grünbaum D (1998) Schooling as a strategy for taxis in a noisy environment. Evol Ecol
12:503–522.
principle in models of animal movement. Ecology 88:1864–1870.
8. Tamm S (1980) Bird orientation: Single homing pigeons compared with small flocks.
Behav Ecol Sociobiol 7:319–322.
9. Steele CW, Scarfe AD, Owens DW (1991) Effects of group size on the responsiveness
of zebrafish, Brachydanio rerio (hamilton buchanan), to alanine, a chemical attractant. J Fish Biol 38:553–564.
Torney et al.
PNAS Early Edition ∣
5 of 6
ECOLOGY
as a red line. In Fig. 3, we include pairwise invasibility plots that
show how the evolutionary dynamics shift as the parameters of
the model are changed. The condition for branching is illustrated
by the presence of positive differential fitness on the vertical line
through x .
It should be noted the branching condition described is valid
for any small but finite mutation rate. Larger mutations result in a
relaxation of this condition as greater variance in the phenotype
trait frequency means populations more readily evolve into a
polymorphic state.
APPLIED
MATHEMATICS
1
mutant y
A
10. Couzin ID, Krause J, Franks NR, Levin SA (2005) Effective leadership and decisionmaking in animal groups on the move. Nature 433:513–516.
11. Barnard CJ, Sibly RM (1981) Producers and scroungers: A general model and its application to captive flocks of house sparrows. Anim Behav 29:543–550.
12. Danchin E, Giraldeau LA, Valone TJ, Wagner RH (2004) Public information: From nosy
neighbors to cultural evolution. Science 305:487–491.
13. Guttal V, Couzin ID (2010) Social interactions, information use and the evolution of
collective migration. Proc Natl Acad Sci USA 107:16172–16177.
14. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence (Springer, Berlin).
15. Strogatz SH (2000) From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20.
16. Risken H (1996) The Fokker–Planck Equation (Springer, Berlin).
17. Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75:1226–1229.
18. Bäck T (1996) Evolutionary Algorithms in Theory and Practice (Oxford University Press,
London).
19. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18.
20. Maynard Smith J (1982) Evolution and the Theory of Games (Cambridge University
Press, Cambridge, UK).
21. Eshel I (1983) Evolutionary and continuous stability. J Theor Biol 103:99–111.
22. Christiansen FB (1991) On conditions for evolutionary stability for a continuously
varying character. Am Nat 138:37–50.
6 of 6 ∣
www.pnas.org/cgi/doi/10.1073/pnas.1014316107
23. Geritz SAH, Kisdi E, Meszéna G, Metz JAJ (1997) Evolutionarily singular strategies and
the adaptive growth and branching of the evolutionary tree. Evol Ecol 12:35–57.
24. Geritz SAH, Metz JAJ, Kisdi É, Meszéna G (1997) Dynamics of adaptation and evolutionary branching. Phys Rev Lett 78:2024–2027.
25. Fehr E, Fischbacher U (2003) The nature of human altruism. Nature 425:785–791.
26. Michod RE, Roze D (2001) Cooperation and conflict in the evolution of multicellularity.
Heredity 86:1–7.
27. Doebeli M, Hauert C, Killingback T (2004) The evolutionary origin of cooperators and
defectors. Science 306:859–862.
28. Hamilton WD (1964) The genetical evolution of social behaviour, I & II. J Theor Biol
7:1–52.
29. Nowak M, May R (1992) Evolutionary games and spatial chaos. Nature 359:826–829.
30. Trivers RL (1971) The evolution of reciprocal altruism. Q Rev Biol 46:35–57.
31. Sumpter DJT (2010) Collective Animal Behavior (Princeton University Press, Princeton).
32. Levin SA (2003) Complex adaptive systems: Exploring the known, the unknown, and
the unknowable. B Am Math Soc 40:3–20.
33. Parrish JK, Edelstein-Keshet L (1999) Complexity, pattern, and evolutionary trade-offs
in animal aggregation. Science 284:99–101.
34. Buhl J, et al. (2006) From disorder to order in marching locusts. Science 312:1402–1406.
35. Cavagna A, et al. (2010) Scale-free correlations in starling flocks. Proc Natl Acad Sci
USA 107:11865–11870.
36. Gintis H (2000) Game Theory Evolving (Princeton University Press, Princeton).
Torney et al.