> "How do animals use the behavior of others to make more accurate ...
> "In making decisions between two alternative routes, we found tha...
Quorum decision-making facilitates information
transfer in fish shoals
Ashley J. W. Ward*
†
, David J. T. Sumpter
‡
, Iain D. Couzin
§
, Paul J. B. Hart
¶
, and Jens Krause
储
*Centre for Mathematical Biology, School of Biological Sciences, University of Sydney, Sydney, New South Wales 2006, Australia;
‡
Mathematics Department,
Uppsala University, 751 06 Uppsala, Sweden;
§
Department of Ecology and Evolutionary Ecology, Princeton University, Princeton, NJ 08544;
¶
Department of
Biology, University of Leicester, Leicester LE1 7RH, United Kingdom; and
储
Faculty of Biological Sciences, University of Leeds, Leeds LS2 9JT, United Kingdom
Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved March 21, 2008 (received for review October 31, 2007)
Despite the growing interest in collective phenomena such as ‘‘swarm
intelligence’’ and ‘‘wisdom of the crowds,’’ little is known about the
mechanisms underlying decision-making in vertebrate animal groups.
How do animals use the behavior of others to make more accurate
decisions, especially when it is not possible to identify which individ-
uals possess pertinent information? One plausible answer is that
individuals respond only when they see a threshold number of
individuals perform a particular behavior. Here, we investigate the
role of such ‘‘quorum responses’’ in the movement decisions of fish
(three-spine stickleback, Gasterosteus aculeatus). We show that a
quorum response to conspecifics can explain how sticklebacks make
collective movement decisions, both in the absence and presence of
a potential predation risk. Importantly our experimental work shows
that a quorum response can reduce the likelihood of amplification of
nonadaptive following behavior. Whereas the traveling direction of
solitary fish was strongly influenced by a single replica conspecific,
the replica was largely ignored by larger groups of four or eight
sticklebacks under risk, and the addition of a second replica was
required to exert influence on the movement decisions of such
groups. Model simulations further predict that quorum responses by
fish improve the accuracy and speed of their decision-making over
that of independent decision-makers or those using a weak linear
response. This study shows that effective and accurate information
transfer in groups may be gained only through nonlinear responses
of group members to each other, thus highlighting the importance of
quorum decision-making.
behavior 兩 collective decision-making 兩 schooling 兩 shoaling 兩 social
A
nimal groups, including humans, often exhibit complex dy-
namic patterns that emerge from local interactions among
group members (1–4). This collective behavior is of particular
interest when individuals with limited personal information use
cues and signals provided by others to decide on a course of action
(5, 6).
It has been suggested that the accuracy of decision-making
increases with group size (7). However, our understanding of
exactly how behavioral interactions scale to collective properties,
and the consequences of this process to individual survival, are
limited because of the difficulty inherent in addressing the com-
plicated feedbacks that arise from repeated social interactions (1, 4,
8): individuals both create and are influenced by their social context
(9, 10). In many social interactions, it may not be possible to identify
which individuals, if any, possess pertinent information. Simply
copying the behavior of others indiscriminately may lead to cas-
cades of information transfer where the nonadaptive behavior of
single animals or small numbers of individuals is reproduced by
many other individuals at no benefit to the copiers (11–13).
Recent advances in understanding collective decision-making
have mostly come from studies of eusocial and gregarious insects (6,
14–19). These studies have emphasized the importance of quorum
response s, where animals’ probability of exhibiting a particular
behavior is an increasing function of the number of conspecifics
already performing this behavior (4). For example, cockroaches rest
longer under shelters containing other re sting cockroaches. As a
result, they make a ‘‘consensus decision’’ to shelter together under
one of two identical shelters (15). In Temnothorax ants, the prob-
ability that an ant will carry other ants to a new nest increases with
the population of ants at that new nest (6). This quorum response
can amplify the differences in the populations at two alternative
nests, usually leading to preferential choice of the better of the two
(17, 20).
Although it is known that vertebrates do use social cues provided
by conspecifics in making decisions about foraging (21–23), coop-
eration (24, 25), the timing of activities (9) and during navigation
(26), less is known about how individual decisions contribute to, and
are influenced by, collective patterns of behavior and how these
decisions in turn facilitate individual-level adaptive response s to an
uncertain environment (4). Fish provide a promising test-bed for
new theories about collective behavior (27); aggregations of fish
may display complex collective patterns yet, because of their small
size and easy maintenance, are amenable to experimental studies.
In this study, we presented three-spine sticklebacks with a choice of
moving to one of two refugia in a simple Y-maze (see Fig. 1).
Resin-cast, replica sticklebacks, which were attached to a motor and
were towed through the maze to one of the refugia, were used in
order that we could examine their effect on the movement decisions
of the experimental fish. In the second phase of the study, we added
a potential predation threat (a resin-cast replica of a sympatric
predator) to one arm of the maze to manipulate the costs of
following. By testing experimental fish in different group sizes (one,
two, four, and eight fish) and with or without one or more replica
conspecifics, we investigated the role of quorum responses in the
movement decisions of fish.
We used a simulation model to explore whether quorum re-
sponses allow for more accurate decisions than those made by
individuals acting independently of conspecifics and how decision-
making accuracy and speed change with group size. Our model is
based on the hypothesis that the propensity to go either left or right
in the maze increases as a function of the number of individuals that
have recently gone left and right. In the absence of other individuals
having taken a particular direction, uncommitted individuals
choose the left right direction with a ratio l:r, i.e., they have a
constant probability l/(l ⫹ r) of choosing the left option and r/(l ⫹
r) of choosing the right option.
In the presence of conspecifics, an individual’s probability of
going left increases as a function of the number of individuals that
have gone left in the last T time steps and decreases with the number
that have gone either right or remain uncommitted. In addition to
T, two parameters determine the shape of this response: a, which
Author contributions: A.J.W.W. and J.K. designed research; A.J.W.W. and D.J.T.S. per-
formed research; A.J.W.W., D.J.T.S., I.D.C., P.J.B.H., and J.K. analyzed data; and A.J.W.W.,
D.J.T.S., and J.K. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
†
To whom correspondence should be addressed. E-mail: ashleyjwward@gmail.com.
This article contains supporting information online at www.pnas.org/cgi/content/full/
0710344105/DCSupplemental.
© 2008 by The National Academy of Sciences of the USA
6948– 6953
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is the spontaneous accept rate, and k, which is the steepness of
response. Specifically, the probability of an individual going left on
time step t ⫹ 1is
al ⫹ 共m ⫺ al兲
共L共t兲 ⫺ L共t ⫺ T兲兲
k
U共t兲
k
⫹ 共L共t兲 ⫺ L共t ⫺ T兲兲
k
⫹ 共R共t兲 ⫺ R共t ⫺ T兲兲
k
,
[1]
where m is the maximum probability of committing to a decision;
U(t) is the number of uncommitted individuals at time t; and L(t)
and R(t) are, respectively, the total number of individuals that have
gone left and right by time t. A similar commitment probability is
set for going right, i.e., the probability of going right on time step
t ⫹ 1is
ar ⫹ 共m ⫺ ar兲
共R共t兲 ⫺ R共t ⫺ T兲兲
k
U共t兲
k
⫹ 共L共t兲 ⫺ L共t ⫺ T兲兲
k
⫹ 共R共t兲 ⫺ R共t ⫺ T兲兲
k
.
[2]
We show examples of Eq. 2 for different parameter values in
supporting information (SI) Fig. S1. When k ⫽ 1, the probability of
going left is proportional to the number that have previously gone
left. This weak linear response can be contrasted with that for larger
values of k, where once a threshold number of individuals have gone
left there is a dramatic increase in the probability of going left. We
call the case where k ⬎ 1 a quorum response, because the
probability of making a particular movement decision increases
once a quorum is met. In the case of Eq. 1, the quorum size is
determined by the size of the group of uncommitted individuals and
the number taking the alternative direction. The parameter a
determines the speed of decision when very few individuals have
made a decision and U(t) together with R(t) determine the quorum
threshold size.
Fitting the model to the data allows us to determine the form of
the response of fish to conspecifics. In particular, we can distinguish
between three alternative hypotheses: no response to conspecifics
(indicated by m ⬇ ar ⬇ al); weakly linear response (m ⬎ ar and k ⬇
1); and quorum re sponse (m ⬎ ar and k ⬎ 1). Details of model
fitting are presented in Materials and Methods.
Results
Response of Test Fish to Replica Conspecific(s). Across all treatments
and group sizes, fish consistently took the direction initiated by the
majority of the replica conspecifics within the Y-maze (Fig. 2),
although the tendency to follow a single replica declined as group
size increased (Fig. 2c). Solitary fish and focal fish in groups of two
showed a significant tendency to follow a single replica (solitary:
P ⬍ 0.001; n ⫽ 20; group of two: P ⫽ 0.003; n ⫽ 20), whereas focal
fish in groups of four and eight did not (group of four: P ⬎ 0.05; n ⫽
20; group of eight: P ⬎ 0.05; n ⫽ 20). The addition of a second
replica produced a significant tendency to follow in the larger group
sizes (group of four: P ⬍ 0.0001; n ⫽ 20; group of eight: P ⫽ 0.003;
n ⫽ 20).
The greater the relative difference between the numbers of
replicas going in each direction, the more biased the test fish were
toward the majority direction. In cases where an equal number of
leaders traveled in each direction (Fig. 2 a and b), there was a
U-shaped distribution of how many fish went left or right. The
U-shaped distribution, contrasted with a binomial distribution with
a single peak, indicates that the fish responded not only to the
replicas but also to each other in making a decision of which
direction to take. In cases where different numbers of replicas went
in each direction, the distribution was J-shaped. There was thus a
strong tendency for consensus among the fish, even when they did
not take the direction of the majority of replicas. Aside from this
bias toward the majority of replicas, there was no intrinsic bias to
the left or right refuge either in the trials with a single fish and no
replica leader (P ⬎ 0.8, n ⫽ 20) or in the trials with multiple fish
and equal numbers of leaders in each direction (Fig. 2 a and b).
Fitting the model to the data indicates a quorum-like response to
conspecifics and replica fish (Fig. 2). The parameters l ⫽ r ⫽ 1 were
set in accordance to the lack of directional bias in single fish
experiments. The best fit of model to the data for the other
parameters was T ⫽ 40, a ⫽ 0.0078, and k ⫽ 3.2 (Fig. 3). The model
fitted well provided a ⬍ 0.0078, 2 ⱕ k ⱕ 4 and T ⬎ 40 (Fig. S2a).
The model did not fit well for a linear response (i.e., k ⫽ 1) or if the
fish were assumed to make decisions independently of others. The
best fit value of T ⫽ 40 was larger than the average time taken for
all fish to make a decision in simulations. For simulations with a ⫽
0.0078 and k ⫽ 3.2, the number of time steps, averaged over all
treatments, before all fish made a decision was 12.7 ⫾ 6.4 steps.
Response of Test Fish to Replica Conspecific(s) in Presence of Replica
Predator. In the previous trials, following the replica conspecific(s)
comes at little or no cost. Therefore, we designed a second set of
trials that involved the placement of a replica predator in one of the
arms of the Y-maze to explore whether this potential cost changed
the decision-making dynamics. In treatments with no replica con-
specifics, fish in all group sizes strongly avoided the predator and
moved to the alternative goal zone (Fig. 4a); in trials with single fish
and no replicas, only one of 20 fish swam past the predator, which
Fig. 1. Experimental set-up.
Ward et al. PNAS
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could be interpreted as an ‘‘error rate’’ of 0.05. In experiments
where a single replica conspecific moved past the predator, solitary
fish were significantly likely to follow (P ⬍ 0.001; n ⫽ 20), whereas
focal fish in groups of two showed no clear trend and fish in larger
groups largely disregarded the replica and moved to the alternative
goal zone (group of 4: P ⬍ 0.001, n ⫽ 20; group of 8: P ⬍ 0.001, n ⫽
20). Where two or three replica conspecifics moved past the
predator, both solitary fish (P ⬍ 0.001, n ⫽ 20) and focal fish in a
group of two (P ⫽ 0.04, n ⫽ 20) were significantly likely to follow
past the predator, whereas focal fish in larger groups no longer
showed any clear trend.
The model was fitted to experiments in the same way as the trial
with no predator. To reflect predator avoidance we set r ⫽ 1/19 and
l ⫽ 1. We set L(0) ⫽ 0 and R(0) to be the number of replica
conspecifics going right, varied between 0 and 3. We then ran the
mathematical model 1,000 times to generate a distribution of
outcomes given the treatment stimulus. Robustness of the model
was tested as above and the model fitted best when a ⬍ 0.0078, 2 ⱕ
k ⱕ 4, and 5 ⬍ T ⬍ 10 (Fig. S2b). Fig. 4 shows the best fit of the
model for the different combinations of R(0) and L(0) to the data.
The Role of Quorums and Group Size on Accuracy. Are decisions based
on quorum responses more accurate than those made by individuals
acting independently of conspecifics? Do the fitted parameters
compare with those tuned for accurate decision-making? To an-
swer these que stions, we ran simulations where we assumed that
going left constituted the ‘‘correct’’ way to go and that going right
was an error. Specifically, we set r ⫽ 1/2 and l ⫽ 1 so that the
spontaneous probability of going left is twice that of going right.
This assumption models the case of having a cryptic predator on the
right that a solitary individual will detect only two times out of three,
i.e., a solitary individual will go left with probability 2/3. We chose
these parameters because an error one time in three is a high error
rate, modeling a case where making an accurate decision is difficult.
Model results with different error rates are qualitatively similar to
those we present here (D.J.T.S., unpublished results). We simulated
this model to test how the proportion of individuals making an error
changed as a function of k, a, and the group size, n.
For small groups, n ⫽ 2 and n ⫽ 4, there was little improvement
in the accuracy of decision-making with errors remaining close to
1/3 for all parameter values (Fig. 3). Improvements at such small
0 0.5 1
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a
b
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g
Fig. 2. Comparison of the best fit of the model with experimental data when no predator was present. The conspecific replicas going left:right are, respectively,
1:1 (a), 2:2 (b), 0:1 (c), 0:2 (d), 0:3 (e), 1:2 (f), and 1:3 (g). Data are shown as histograms, and model outcomes are represented by a solid line.
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group sizes are limited because the first individual to go left or right
always has probability 1/3 of making an error. The second individual
then gains no information by the direction of the first individual. It
is only when the spontaneous decisions of a number of individuals
have accumulated that they can begin to gain an advantage from
following the decisions of others. Thus, larger groups, n ⫽ 8 and n ⫽
16, do benefit from re sponding to the decisions of others (Fig. 3).
This improvement strongly relies on having a disproportional
response to conspecifics; that is a nonlinear quorum response.
When k ⫽ 1 there is no improvement in the proportion of errors
made, but as k increases so too does decision accuracy. Likewise, as
the spontaneous accept rate decreases, decision accuracy increases.
k and a have the opposite effect on decision-making speed and
accuracy. Speed trades off with accuracy for any set of parameter
values. Above groups of size eight or so, group size has very little
effect on speed. Large groups can make equally fast decisions as
smaller ones without paying a cost of decreased accuracy. In nature,
we would expect to find the same trend but for it to be reduced as
groups become very large because some of the individuals may not
be able to sense the direction others have taken because of
restricted perception caused by crowding.
Discussion
In making decisions between two alternative routes, we found that
fish exhibit relative choice: the probability of an individual taking
a particular direction depends on the number of conspecifics going
in each direction. Furthermore, we found that the size of the
undecided group also plays a role in the movement decision. Larger,
undecided groups decrease the probability that an individual moves
at all in either direction. We have used these observations to
propose a quorum response model where the probability of per-
forming an action increases as a function of the number of
individuals having recently taken that action and decreases as a
function of the number taking the other action or remaining
undecided. Despite its simplicity, this model accurately fits the data
for different numbers of replica fish both in the absence and
presence of a potential predator threat. Although this model is
sufficient to account for the data, other models that do not include
all aspects of relative choice are not. In particular, a model that does
not include the effect of the undecided individuals does not match
the data as accurately as the one we present here (D.J.T.S.,
unpublished data).
The best fit for two of the model’s three parameters, the threshold
steepness k and the spontaneous accept probability a, were similar
both in the presence and in the absence of a predator. The third
parameter T, the time frame over which the actions of others are
taken into account, differs in the absence or presence of a predation
threat. In the absence of a predator, this time frame is long and it
is the total number of individuals who have gone left or right that
determines an individual’s propensity to copy this behavior. In the
presence of a predator, the time frame is considerably shorter and
it is only the very recent actions of conspecifics that determine
directional propensity. The shorter time frame in the presence of a
predator may be explained by the fact that only the most recent
information is relevant; fish have to cope with a circumstance that
may change rapidly.
The interactions among the fish are highly nonlinear: the best fit
of the model was achieved when the threshold steepness was k ⫽
3.25 in the absence of a predator, and k ⫽ 2.25 in the presence of
a predator. Instead of reacting proportionally to the actions of
others, fish react strongly in favor of the option chosen by the
majority. It is this disproportional reaction to conspecifics that
generates the U-shaped distribution of outcomes seen when stimuli
are equal (as in Fig. 2 a and b). These distributions are characteristic
of nonlinear positive feedback in decision-making (4, 14, 28).
Nonlinear positive feedback is often associated with the possi-
bility of errors in decision-making at the collective level (11–13).
Amplification of incorrect decisions is predicted by the model and
seen in the experiments with the predator. In the absence of other
fish, a focal individual passed a predator in only one of twenty trials,
a 5% error. When two or more replicas go past the predator, the
error increases dramatically: a group of two individuals have an
error rate of 80% whereas groups of four or eight fish are misled
30–40% of the time. Interestingly, however, a single replica doe s
not lead groups of four or eight fish effectively, especially in a risky
situation. Interpreting these results in terms of our model, we see
that uncommitted individuals in larger groups only follow above a
threshold number of leaders. This threshold dramatically reduces
the probability of errors being amplified throughout a group
because, if the probability one individual makes an error is small,
say a, then the probability that two fish independently make errors
at the same time becomes very small, i.e., a
2
.
The simulation model predicts that, although we can experimen-
tally manipulate the fish to make ‘‘errors,’’ quorum responses
generally improve the accuracy of decision-making. By biasing
decisions according to the decisions of others the proportion of
errors made is reduced, with larger groups making more accurate
decisions. In the model, it is assumed that individuals cannot tell
whether any particular individual has made a ‘‘correct’’ or ‘‘incor-
rect’’ decision, but by integrating the decisions of others individuals
can improve their own accuracy. Only simulations for k ⬎ 2 showed
strong improvements in decision-making accuracy. If the response
was proportional, i.e., k ⫽ 1, then copying the actions of others
offers no improvement over independent decision-making. The
disproportionate threshold response of the real fish is therefore
likely to be essential in improving decision accuracy in such
situations. These results help explain the ubiquity of nonlinear
response thresholds across different grouping species (4, 6, 14, 19).
Individual fish use the decisions of others to improve their own
decision-making.
Positive feedback and threshold response s have been identified
as key proximate mechanisms in the decision-making of animal
5 10
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Threshold steepness: k
Spontaneous accept rate: a
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0.2
0.25
0.3
0
1
2
3
4
5
a
b
cd
e
f
g
h
Fig. 3. Accuracy and speed of individual decision-making of model simula-
tions for different group sizes (a and e) n ⫽ 2; (b and f) n ⫽ 4; (c and g) n ⫽ 8;
and (d and h) n ⫽ 16. Simulations were run 5,000 times for different combi-
nations of spontaneous accept rates, a, and the threshold steepness, k.(a–d)
The average proportion of fish making ‘‘errors’’: the darkest blue indicates
that on average 0.2 fish have made an error; darkest red indicates that on
average 0.35 have made an error. (e and f ) Log
10
(mean number of time steps
over all individuals) until an individual moves left or right. Dark red indicates
slow decisions, and blue indicates fast decisions. The black ⴛ on each panel
represents the best fit parameter values for the fish in the absence of the
predator, and the circle represents those values in the presence of the
predator.
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groups (4, 29). The possible importance of information transfer in
animal groups in providing functional explanations of their evolu-
tion has been highlighted many times (10, 11, 22, 28), but has not
been directly linked to proximate explanations. Making this link is
crucial to understanding the evolution of grouping. It is not
sufficient to say that group-living animals simply gain from infor-
mation transfer, because it is specifically the nonlinear response to
conspecifics that explains both the improvements in decision-
making accuracy by individuals in groups. The nonlinear response
also explains apparently maladaptive blind copying when the group
is misled by replica fish. We now refine the information transfer
hypothesis as follows: by joining a group and responding to the
actions of others according to a steep quorum response, an indi-
vidual can gain an information proce ssing advantage over solitary
individuals. Measuring the form of these response s across species
will help determine the importance of information transfer in the
evolution of group-living in these specie s.
Materials and Methods
Experiments were performed by using three-spined sticklebacks collected from
the River Welland, Leicestershire (52° 30⬘ 35⬙ N; 0° 53⬘ 18⬙ W) during late 2003. To
test the tendency of individuals to follow conspecifics in different contexts, we
constructed an experimental arena that offered a choice of two identical refugia,
both equidistant from a starting point (Fig. 1). Fish were added to this starting
point and were then guided used remote-controlled replica fish. Sticklebacks are
known to respond well to replica conspecifics (24). The stickleback replicas were
constructed from resin plaster. The mould was made of a single 40-mm stickle-
back by using Gedeo molding alginate. This method preserved considerable
detail in the finished models. Twelve replicas were then cast by using liquid resin
plaster. Once set, these replicas were painted to resemble sticklebacks, and a hole
was drilled lengthways along the dorsal surface to allow the replica to be
threaded onto the guide line.
The replica was attached to one of two guide lines extending from one end of
the arena, positioned near to a clear square Perspex box containing test fish, to
each refuge. To investigate conflict of information, we manipulated the number
of replica fish on each line. For example, we could attach an equal number of
replicas to each line, or create a bias with more replicas moving to one refuge than
the other. When multiple replicas (maximum of three per line) were placed on the
same guide line, they were positioned one body length apart and facing (subse-
quently moving toward) the refuge.
In each trial, we acclimatized test fish for 5 min within the Perspex box before
lifting it and freeing them. Simultaneously a motor started to tow the replica(s)
at a speed of one body length per second to the far end of the arena. The
experiments continued until all fish had entered the shaded goal zones or refuges
(Fig. 1). These refuges are preferred by the fish even in the absence of a moving
replica. The side at which the replica individual(s) were presented was random-
ized. In the vast majority of experiments, the fish quickly entered the goal area,
but in a few cases a single ‘‘errant’’ fish stayed frozen to a spot for ⬎100 s; these
results were discarded. All trials were filmed from above.
We investigated the behavior of sticklebacks in response to the replica con-
specifics both in the absence and presence of a potential threat in the form of a
replica predator.
Response of Test Fish to Replica Conspecific(s). Test fish were given an absolute
choice [replica(s) on one side of arena only] or a relative choice [replica(s) on both
sides]. Absolute choice was provided by 1 replica versus 0 replicas, 2 vs. 0 or 3 vs.
0. Relative choice could be divided into symmetrical stimuli: 1 vs. 1 and 2 vs. 2; and
asymmetrical stimuli: 2 vs. 1 and 3 vs. 1. Again, we randomized the side at which
the replicas were presented. We used four different group sizes of test fish: one,
two, four, and eight, with 20 replicates for each. For group sizes four and eight,
all treatments were performed, whereas for group sizes of one and two, we
performed all treatments except for absolute choice 2 vs. 0 and 3 vs. 0. The
omission of these absolute choice treatments for smaller group sizes reflected
observations that the decisions in small groups were already strongly biased by
the presence of one leader. We also performed experiments with single test fish
and no replicas to examine the bias of fish for left or right.
Response of Test Fish to Replica Conspecific(s) in Presence of Replica Predator.
We examined the tendency of test fish to follow replicas that traveled into close
proximity to a replica of a common sympatric predator, a perch (Perca fluviatilis)
of 200-mm length. We randomized the side on which the replica predator was
placed. Control experiments showed that sticklebacks (alone or in groups)
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:0, group size 2
Frequency
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:0, group size 4
Proportion going right
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:0, group size 8
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:1, group size 2
Frequency
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:1, group size 4
Proportion going right
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:1, group size 8
a
b
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:2, group size 2
Frequency
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:2, group size 4
Proportion going right
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:2, group size 8
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:3, group size 2
Frequency
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:3, group size 4
Proportion going right
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0:3, group size 8
c
d
Fig. 4. Comparison of best fit of the model with experimental data when a replica predator was present. We assume that the predator is on the right and the
number of conspecific replicas going left:right are, respectively, 0:0 (a), 0:1 (b), 0:2 (c), and 0:3 (d). Data are shown in the histograms, and model outcome is
represented by the solid line.
6952
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strongly avoided this path when not under the influence of a replica stickleback.
We tested four different treatments with 0, one, two, or three replica conspecifics
each for groups of one, two, four, and eight fish. Twenty replicates were per-
formed for each.
Data Analysis. For each group size, an individual fish was selected at random
before the beginning of the trial from the video playback. The goal zone that this
fish entered was recorded and analyzed by using a binomial test with a null
expectation of no preference for either goal zone.
Model Fitting. The parameters l and r were first fit by using experimental trials
with a single fish and no replica conspecifics. The parameter m was fixed to be
0.45, so that if all of the fish had made a particular movement decision then the
probability of following this decision was large, but the total probability per time
step was never ⬎1. To test the fit of specific values of a, k, and T, we ran the
mathematical model 1,000 times to generate a distribution of outcomes for each
treatment and fish group size. We set R(0) and L(0) to be the number of replica
fish going left or right at the start in each experimental treatment. We further
assumed L(t) ⫽ R(t) ⫽ 0 for all t ⬍ 0, because no individuals go left or right before
the start of the experiment. For convenience, we assume that the majority of
replica fish traveled is always right, although the side on which the predictor
replica was presented was randomized in the experiments. By systematically
changing the parameters a, k, and T and running each combination 1,000 times,
we were able to test which parameter combinations best fit the data (Fig. S1).
Model fit was tested by comparing the distribution of individuals going right
generated by the model with that of the data over each of the seven treatments
and each group size of two, four, and eight fish. Specifically, for each treatment
and group size we calculated the sum of squares (expected
i
⫺ observed
i
)
2
where
i denotes the histogram boxes for the proportion of fish going right 0 –20, 20 – 40,
. . . , 80 –100. The sum of squares was then summed across all treatments and
group sizes to give the overall fit of the model for a particular parameter
combination.
ACKNOWLEDGMENTS. We thank Simon Levin and two anonymous referees
for their comments, which substantially improved this manuscript. Financial
support was provided by the Natural Environment Research Council (U.K.)
(A.J.W.W. and P.J.B.H.), the Engineering and Physical Sciences Research Coun-
cil (J.K. and I.D.C.), and the Royal Society (D.J.T.S. and I.D.C.).
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Ward et al. PNAS
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> "How do animals use the behavior of others to make more accurate decisions, especially when it is not possible to identify which individ- uals possess pertinent information? One plausible answer is that individuals respond only when they see a threshold number of individuals perform a particular behavior. Here, we investigate the role of such ‘‘quorum responses’’ in the movement decisions of fish (three-spine stickleback, Gasterosteus aculeatus). We show that a quorum response to conspecifics can explain how sticklebacks make collective movement decisions, both in the absence and presence of a potential predation risk. Importantly our experimental work shows that a quorum response can reduce the likelihood of amplification of nonadaptive following behavior. "
> "In making decisions between two alternative routes, we found that fish exhibit relative choice: the probability of an individual taking a particular direction depends on the number of conspecifics going in each direction. Furthermore, we found that the size of the undecided group also plays a role in the movement decision. Larger, undecided groups decrease the probability that an individual moves at all in either direction. We have used these observations to propose a quorum response model where the probability of per- forming an action increases as a function of the number of individuals having recently taken that action and decreases as a function of the number taking the other action or remaining undecided. Despite its simplicity, this model accurately fits the data for different numbers of replica fish both in the absence and presence of a potential predator threat."