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J. theor. Biol. (2002) 215, 481–489doi:10.1006/jtbi.2001.2518, available online at http://www.idealibrary.com on
Task Partitioning in a Ponerine Ant
Guy Theraulaznw, Eric Bonabeauzy, RicardV. Sol!eO, Bertrand Schatzn
and Jean-Louis Deneubourgz
nLaboratoire d’Ethologie et Cognition Animale, CNRS - ERS 2382, Universit !e Paul Sabatier, 118 routede Narbonne, 31062 Toulouse C !edex, France zEuroBios, 9, rue de Grenelle, 75007 Paris, France ySantaFe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA OComplex Systems Research Group,Department of Physics, FEN-UPC, Campus Nord B4, 08034 Barcelona, Spain zCenter for NonlinearPhenomena and Complex Systems, Universit !e Libre de Bruxelles, C.P. 231, Campus Plaine, B-1050
Brussels, Belgium
(Received on 11 July 2001, Accepted in revised form on 13 November 2001)
This paper reports a study of the task partitioning observed in the ponerine ant Ectatommaruidum, where prey-foraging behaviour can be subdivided into two categories: stinging andtransporting. Stingers kill live prey and transporters carry prey corpses back to the nest.Stinging and transporting behaviours are released by certain stimuli through responsethresholds; the respective stimuli for stinging and transporting appear to be the number oflive prey and the number of prey corpses. A response threshold model, the parameters ofwhich are all measured empirically, reproduces a set of non-trivial colony-level dynamicalpatterns observed in the experiments. This combination of modelling and empirical workconnects explicitly the level of individual behaviour with colony-level patterns of workorganization.
r 2002 Elsevier Science Limited. All rights reserved.
1. Introduction
Understanding the regulation of division oflabour in insect colonies is essential for under-standing the evolution of social behaviour (Oster& Wilson, 1978; Bourke & Franks, 1995).Division of labour refers to the division of theworkforce among the range of tasks performedin the colony. Recently, attention has beendirected towards the way in which workersactually performed a given task. The so-calledtask partitioning, whereby a piece of work isdivided among two or more workersFsuch asthe partitioning of the collection of a load of
wAuthor to whom correspondence should be addressed.E-mail: [email protected]
0022-5193/02/$35.00/0
forage between a forager and a storer or atransporterFhas been recognized as an impor-tant form of work organization in social insects(Jeanne, 1986; Ratnieks & Anderson, 1999).Task partitioning may also exist in the absenceof division of labour as it was pointed out byRatnieks & Anderson (1999; see also Andersonet al., 2001). What is particularly interestingabout task partitioning is that it appears torequire a higher degree of coordination andperhaps more complex exchanges of informationamong nestmates than other patterns of divisionof labour.
In a previous paper (Bonabeau et al., 1998),we have conjectured that a simple res-ponse threshold model (Robinson, 1987, 1992;
r 2002 Elsevier Science Limited. All rights reserved.
Plowright & Plowright, 1988; Bonabeau et al.,1996, 1998) could account for task partitioningwhen the stimulus for subtask 2 (for example,storing a load of forage) increases as a result ofworkers performing subtask 1 (for example,foraging). The idea behind the response thresh-old model is simple: when the intensity of astimulus associated with a task exceeds theresponse threshold of a worker, that workerengages in task performance with a high prob-ability in response to the stimulus. Task perfor-mance reduces the intensity of the stimulus,thereby decreasing the probability thatother workers engage in the same task. Ifperforming subtask 1 not only decreases stimu-lus intensity for subtask 1 but also increasesstimulus intensity for subtask 2, what should beobserved is a form of task partitioning where thenumber of workers performing subtask 2 in-creases as subtask 1 is being accomplished. Thisprediction is yet to be tested in three importantways:
(1) identify a species in which such a pheno-menon is observed,
(2) show the existence of a response thresholdmechanism involved in the phenomenon, and
(3) measure empirically the model’s param-eters. Validating this model would clearly estab-lish for the first time the connection betweenindividual behaviour and coordinated colony-level behaviour in the context of the regulationof division of labour.
The present paper is aimed at showing that themodel does explain a pattern of task partitioningamong hunters of the neotropical ponerine antEctatomma ruidum. In this species, huntingbehaviour can be subdivided into two categories:stinging, that is, killing live prey, and transport-ing, that is, carrying prey corpses back to thenest (Schatz et al., 1996; Schatz, 1997). Such abehaviour of partitioned stinging and transport-ing is also observed in Pachycondyla caffraria(Agbogba & Howse, 1992). We present a strongevidence that the stinging and transportingbehaviours are released by certain stimulithrough response thresholds; the respectivestimuli for stinging and transporting appear tobe the number of live prey and the number ofprey corpses. In addition, we have been able to
measure experimentally all the model’s para-meters. When the empirical values of the model’sparameters are used, the model reproduces thevarious dynamical patterns observed in theexperiments with a remarkable accuracy.
Section 2 introduces the experimental obser-vations. The model is described in more math-ematical detail in Section 3. The additionalexperimental measurements suggested by themodel are described in Section 4. The results ofsimulating the model with empirical parametervalues are given in Section 5. Finally, theevolutionary implications, the limitations, andthe possible extensions of the model are dis-cussed in Section 6.
2. Experiments
2.1. METHODS
Ectatomma ruidum is common in coffee orcocoa plantations where it preys upon a widevariety of arthropods (Lachaud, 1990). Coloniesof E. ruidum, collected in the Mexican state ofChiapas, were reared in plaster nests and placedin an experimental room under controlledconditions (temperature: 25711C; humidity:6075%; photoperiod: 12 : 12 L/D). Fruit flies(Drosophila melanogaster) were offered to thecolony as prey. The experimental set-up wascomposed of a foraging area (30� 30 cm) con-nected at one end with the nest and at the otherend with the hunting area consisting of a circularPetri dish (diameter: 9 cm, height: 1 cm). The preywere provided in the hunting area. Four colonieswere studied with five different quantities of prey(colony with 60 workers: 50, 80, 120, 150, and200 prey; colony with 130 workers: 80, 150, 200,250, and 300 prey; colony with 240 workers: 80,150, 250, 400, and 500 prey; colony with 350workers: 100, 200, 300, 400, and 500 prey).During preliminary observations, all workersperforming any kind of predatory act weremarked individually. During the experiments,all the behavioural acts of all the marked antswere recorded every 10min over a 2-h period.
2.2. RESULTS
As has been previously reported (Schatz et al.,1996; Schatz, 1997), two main types of behaviour
can be clearly distinguished in E. ruidum hunters:stinging live prey and transporting dead prey.The typical sequence of attack of a stinger hasbeen described by Lachaud (1990) and Schatzet al. (1997). Once the prey is motionless it ismost often dropped on the ground. Dead preylying on the ground are picked up by transpor-ters and taken to the nest. Active solicitation of astinger by a transporter for prey transfer mayalso be observed. Transporters usually comeback to the hunting area. Workers can exhibitboth behaviours within an experiment. Figure 1shows the number of live and dead prey as afunction of time as well as the fraction ofworkers observed in either the stinger or thetransporter state, for three experimental situa-tions which are representative of the main,observed dynamical patterns [130 workers and80 prey (12 hunters involved), 130 workers and250 prey (15 hunters involved), 240 workersand 500 prey (22 workers involved)]. Thedynamical patterns can be subdivided into twomain categories: those where the initial number
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Fig. 1. Fractions of colony workers that are stingers or ta function of time for three different colony sizes and initial nu(c, d) Colony size=130, number of prey=250. (e, f ) Colony siztransporters. (b, d, f ) ( ) live prey; ( ) dead bodies.
of live drosophilas is relatively small (of theorder of 50 – 80) and those where the initialnumber of live drosophilas is relatively large (upto 500). In the first case [Fig. 1(a), (b)], thefraction of stingers first increases and thenrapidly decreases, the fraction of transportersincreases with a time delay with respect to thefraction of stingers, and then decreases slowly;the number of prey decreases quickly, while thenumber of corpses available for transport firstincreases and then decreases, but never becomeslarge. By contrast, in the second case [Figs. 1(c),(d) and (e), (f)], the fraction of stingers increasesand reaches a stable plateau, the fraction oftransporters increases with a small time delayand reaches a similar plateau; the number of liveprey decreases steadily with time and the numberof corpses in the arena increases steadily.
3. The Model
The idea behind the model (Bonabeau et al.,1998) is simple. When live drosophila are
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ransporters, and numbers of live prey and dead bodies asmber of prey. (a, b) Colony size=130, number of prey=80.e=240, number of prey=500. (a, c, e) ( ) stingers; ( )
G. THERAULAZ ET AL.484
presented to the colony, stinger behaviour isactivated because the number of live preyexceeds the stinging threshold of workers.Workers start killing the prey, which leads tothe presence of an increasing number of corpsesto be transported to the nest. Transporterbehaviour is stimulated as the number of corpsesexceeds the transporting threshold of workers.This may explain the apparent task successionobserved in the experiments with a small numberof prey [Fig. 1(a), (b)]. When there is a largenumber of prey, the killing rate is not sufficientto take the stimulus below the stinging thresholdof workers within the time frame of the exp-eriment, so that the number of stingers is notobserved to decrease (Figs. 1(c), (d) and (e), (f)].
Figure 2 gives a logical sketch of the model.Let S be the total number of workers and N thetotal number of potential hunters, that is, theequivalent of the number of workers thatexhibits a hunting behaviour (stinging and/orprey transportation) at least once during anexperiment. We assume that N is given: deter-mining how this number varies as a function ofcolony size requires another model. Let xi be thefraction of these N workers engaged in perform-ing task i (i ¼ 0: non-hunting activity; i ¼ 1:stinging, i ¼ 2: transporting), n1 the number oflive prey in the hunting area, n2 the number ofprey corpses in the hunting area, ti the averagetime spent performing hunting task i before
n1 n2
x0 �1x2x1 �2
�1
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�2
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Fig. 2. Transitions between different states in the task-partitioning model. Two basic ‘‘compartments’’ are defined:variables including stimuli (n1: number of live prey; n2:corpses) and tasks (x1, x2), with x0=1�x1�x2 (see text).( ) arrows indicate transitions in population values and( ) arrows indicate interactions that allow transitions tooccur. The square boxes introduce the presence of thresholdfunctions weighting the effect of interactions.
switching to a non-hunting task or beforebecoming inactive, a1 the efficiency of task 1performance, a2 the efficiency of task 2 perfor-mance. a1 is also the rate of corpse productionper stinger per time unit: the rate of increase ofn2 is proportional to a1. Finally, let f1=Nx1/S bethe fraction of workers in the colony that are inthe stinger state and f2=Nx2/S the fraction ofworkers in the colony that are in the transporterstate.
The response threshold model (Bonabeauet al., 1996, 1998) assumes that the probabilitythat a worker responds to the perceived presenceof n1 live prey by becoming a stinger is equal toc1ð1� 10�n1=y1Þ where y1 is the worker’s responsethreshold associated with the stinging task andc1 is a positive rate: in other words, when n1cy1,the worker is more likely to respond. Similarly,the probability that a worker responds tothe presence of n2 prey corpses in the huntingarea by becoming a transporter is equal toc2ð1� 10�n2=y2Þ; where y2 is the response thresh-old associated with the transporting task and c2is a positive rate. In reality, the stimuli thatpotential stingers or transporters respond to arecertainly more complex but they can be ab-stracted away and included into the variables n1and n2 as a first approximation. The values of c1,y1, t1, a1, c2, y2, t2, and a2 need to be estimatedexperimentally. The dynamics of x1, x2, n1 and n2are described by
@tx1 ¼ c1 1� 10�n1=y1� �
1� x1 � x2ð Þ �x1t1; ð1Þ
@tx2 ¼ c2 1� 10�n2=y2� �
1� x1 � x2ð Þ �x2t2; ð2Þ
@tn1 ¼ �a1Nx1; ð3Þ
@tn2 ¼ a1Nx1 � a2Nx2; ð4Þ
where @t denotes derivatives with respect to time.The first right-hand term of eqn (1) expressesthat new stingers are ‘‘recruited’’ from thefraction (1�x1�x2) of potential hunters whichare not yet stingers or transporters, and thatsuch workers are recruited with probabilityc1ð1� 10�n1=y1Þ per time unit. The second
TASK PARTITIONING IN A PONERINE ANT 485
right-hand term of eqn (1) reflects the fact thatstingers give up hunting after spending a time t1on average in the stinger state. Equation (2),which describes the dynamics of the fraction oftransporters, x2; is very similar in structure andmeaning to eqn (1). Notice that no directtransition from stinging to transporting or fromtransporting to stinging is included in theequations; individuals can, and are observed to,perform the two tasks within an experiment, butthis requires a transition to the non-hunter statefirst. Equation (3) states that the number of liveprey, in the absence of an inflow of such a preyinto the arena, decreases at a rate aNx1: if thereare Nx1 stingers within a time unit (x1 is thefraction of stingers and N the number ofpotential hunters, so that Nx1 is the totalnumber of stingers), each killing a1 prey withinthat time unit, then a total of a1Nx1 prey is killedduring that time unit. Equation (4) states thatthe a1Nx1 killed prey become corpses that can betransported to the nest at a rate a2Nx2: if thereare Nx2 transporters within a time unit (x2 is thefraction of transporters and N the number ofpotential hunters, so that Nx2 is the totalnumber of transporters), each successfully trans-porting a2 dead bodies to the nest within thattime unit, a total of a2Nx2 dead bodies aresuccessfully carried to the nest during that timeunit.
4. Estimating Parameter Values
In order to validate the model, we now have tocheck that the response functions follow thefunctional form assumed in the model, and thevalues of several of the model’s parameters, c1;y1; t1; a1; c2; y2; t2; and a2; have to be estimated.The transition probabilities to and from thestinger and transporter states were measured asa function of the number of live prey, denoted byn1, and the number of dead prey, denoted by n2.Let Pij nkð Þ be the probability per time unit that aworker exhibits a transition from behaviouralstate i to behavioural state j as a function of nk(k ¼ 1; 2), with the following conventions: i ¼ 0corresponds to a non-hunter state, i ¼ 1 to thestinger state, and i ¼ 2 to the transporter state.These transition probabilities were measuredonly for those N workers that exhibited a
hunting behaviour (stinging or transporting)at least once during the course of an experiment.For example, P01 n1ð Þ was measured as follows:for each individual that exhibited a huntingbehaviour at least once during the experiment,we counted as to how often a transition from anon-hunter state to the stinger state wasobserved as a function of the value of n1 at thetime when the transition was observed. Ourassumption is that transition probabilities to andfrom the two hunter states depend primarily onthe levels of the stimuli associated with thesehunter states, represented by n1 and n2: This,obviously, is a simplifying assumption that doesnot take into account the physiological state ofthe colony, possible recruitment mechanisms andadditional stimuli.
Figures 3 and 4 show P01 n1ð Þ; the averageprobability of transition from state 0 to state 1per minute per individual as a function of n1; andP02 n2ð Þ; the average probability of transitionfrom state 0 to state 2 per minute per individualas a function of n2: Error bars along the y-axiscorrespond to averaging over individuals andexperiments (in that order), and error bars alongthe x-axis correspond to averaging over databins (see figure legends for more details). Thedotted lines represent the best fits of the typeci 1� 10�ni=yi� �
; where yi is a threshold and ci is atransition rate. One hypothesis that mightexplain why this type of functional form isobserved is, roughly, the following: if, at eachencounter with a live prey item, a worker has afixed probability r of responding to the item,then the probability that the worker will notrespond to the first n1 encountered items isgiven by 1� rð Þn1 ; and the probability that itwill respond within the n1 encounters is given by1� 1�rð Þn1¼ 1� 10n1log 1�rð Þ ¼ 1� 10�n1=y1 withy1 ¼ �1=logð1� rÞ: This hypothesis, which hasbeen introduced by Chr!etien (1996) to explainthe formation of cemeteries in the ant Lasiusniger, remains to be tested for the presentsituation. For P01 n1ð Þ; we find c1 ¼ 0:0553min�1
�1 and y1 ¼ 109:7. For P02 n2ð Þ; we findc2 ¼ 0:0386min�1 and y2 ¼ 15:9. Notice thatthe dynamics of n1 and n2 are obviouslycorrelated in these experiments, while we areassuming that they are independent: measuringthe exact transition frequencies would require a
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Fig. 3. Average probability of transition P01 n1ð Þ from anon-hunter state to the stinger state per minute perindividual as a function of n1, the number of live prey inthe hunting area. Error bars along the y-axis correspond toaveraging over individuals and experiments (in that order),and error bars along the x-axis correspond to averagingover data bins. The following bins were used for live prey:0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70–79, 80–89, 90–99, 100–199, 200–299, 300–399, and 400–499.(. . . . . . .) line represents best fit of the typec1 1� 10�n1=y1� �
; with c1=0.0553min�1 and y1=109.7.
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Fig. 4. Average probability of transition P02 n2ð Þ from anon-hunter state to the transporter state per minute perindividual as a function of n2; the number of dead prey inthe hunting area. Error bars along the y-axis correspond toaveraging over individuals and experiments (in that order),and error bars along the x-axis correspond to averagingover data bins. The following bins were used for dead prey:0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70–79, 80–89, 90–99, 100–199, and 200–299. (. . . . . . .) line representsbest fit of the type c2 1� 10�n2=y2
� �; with c2=0.0386min�1
and y2=15.9.
G. THERAULAZ ET AL.486
separate set of experiments with live prey only ordead prey only.
Individuals tended to specialize on eitherstinging or transporting within experimentaltrials. Transitions between stinging and trans-porting were rare. All other transition probabilitiesare either very small or do not depend signifi-cantly on n1 and n2: P10 n1ð Þ; P10 n2ð Þ; P20 n1ð Þ; andP20 n2ð Þ can be considered constant with respectto n1 and n2; and P12 n1ð Þ; P12 n2ð Þ; P21 n1ð Þ; andP21 n2ð Þ are all less than or equal to 0.008min�1
for any value of n1 or n2, which means that thecorresponding transitions are almost neverobserved during the experiments and can beneglected as a first approximation. Thus, we candefine the probabilities of transition P10 and P20;or alternatively the average times t1 ¼ P�1
10 andt2 ¼ P�1
20 spent, respectively, in the stinger andtransporter states before making a transition to anon-hunter state. We find that t1 ¼ 33:2min andt2 ¼ 48:1min. Note that workers seem to spendon an average a relatively long time performing agiven task, which certainly influences the ob-served colony-level dynamics, as do responsethresholds. The large values of t1 and t2 mightresult from the fact that behavioural acts arerecorded only every 10min: more transitionscould take place within those 10min. We alsomeasured the efficiency of stinging, that is, thenumber of live prey killed per stinger per timeunit, denoted by a1; and the efficiency of trans-porting, that is, the number of corpses success-fully transported to the nest per transporter pertime unit, denoted by a2: The average over allthe experiments yields: a1 ¼ 0:33min�1 anda2 ¼ 0:22min�1.
5. Results
Equations (1)–(4) cannot be solved analyti-cally, except in special cases. Equations (1)–(4)were integrated numerically, using the estimatedvalues of the parameters, in three distinctconditions, which correspond to the threeexperimental situations described in Section 2:
(i) 130 workers (S ¼ 130, N ¼ 12), 80 live prey(n1 (t ¼ 0)¼ 80).
(ii) 130 workers (S=130, N=15), 250 live prey(n1 (t ¼ 0)¼ 250).
TASK PARTITIONING IN A PONERINE ANT 487
(iii) 240 workers (S ¼ 240, N ¼ 22), 500 liveprey (n1 (t ¼ 0)¼ 500).
In all the three cases, the initial conditions are:x1(t ¼ 0)=x2(t ¼ 0)=0 (no worker is in a huntingstate) and n2(t ¼ 0)=0 (there are initially nocorpses in the hunting area). Figure 5 shows theresults of the numerical integration and it has tobe compared to Fig. 1. In all the three cases,there is a striking similarity between the dyna-mical patterns observed during the experimentsand the dynamical patterns obtained with themodel. These different patterns arise in boththe experiments and the model by changingthe numbers of workers and hunters and/or thenumber of live prey initially available in thehunting area. The quantitative differences ob-served between the experiments and the modelare likely to result from the approximations of
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Fig. 5. Fractions of colony workers that are stingers ( f1) oprey in the hunting area as a function of time obtained from nusizes and initial number of prey (a, b). The initial number ofS=130, N=12, x1(t=0)=x2(t=0)=0, n1(t=0)=80 and n2(t=number of workers is 130 (parameters: S=130, N=15, x1(t=initial number of prey is 500 and the number of workers in1(t=0)=500 and n2(t=0)=0). For all the three conditions: adead prey, c1=0.0553min�1, c2=0.0386min�1, t1=33.2min( )n1; ( )n2.
the model (for example, assuming that n1 and n2are uncorrelated to estimate transition probabil-ities, assuming that transition probabilitiesdepend only on n1 and n2; averaging over allworkers, neglecting recruitment, inaccuracy ofsome of the estimates, etc.).
6. Discussion
Task partitioning occurs in many species ofsocial insects. It defines an important andapparently widespread feature of work organiza-tion in social insects. The study of differentexamples reveals that this feature has actuallyevolved several times (Ratnieks & Anderson,1999). Together with an appropriate identifica-tion of its presence in a given system, aquantitative characterization of the intrinsicdynamics is needed. Previous studies involving
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r transporters ( f2), and numbers of live (n1) and dead (n2)merical integration of eqns (1)–(4) for three different colonyprey is 80 and the number of workers is 130 (parameters:0)=0) (c, d). The initial number of prey is 250 and the
0)=x2(t=0)=0, n1(t=0)=250 and n2(t=0)=0) (e, f ). Thes 240 (parameters: S=240, N=22, x1(t=0)=x2(t=0)=0,
1=0.33min�1, a2=0.22min�1, y1=109.7 live prey, y2=15.9, and t2=48.1min. (a, c, e) ( ) f1; ( )f2. (b, d, f )
G. THERAULAZ ET AL.488
division of labour in the ants Pheidole success-fully reproduced the main quantitative featuresdisplayed by experiments (Bonabeau et al.,1996). This study used a threshold model thathas been extended in our study to the analysis ofa well-defined example of task partitioning.
The response threshold model described inthis paper is simple, plausible, consistent withexperiments, relies solely on empirically mea-sured parameter values and reproduces non-trivial dynamical colony-level patterns with aminimum of assumptions. Very few models ofdivision of labour in social insects have currentlybeen tested with experimental data (Beshers &Fewel, 2001). Here, we show that the combina-tion of modelling and empirical work connectsexplicitly individual behaviour to colony-levelbehaviour through a set of response thresholds,the existence of which has been shown. Thisquantitative example clearly shows that adifference in the response thresholds associatedwith the two kinds of stimuli (live prey andcorpses) is a sufficient condition to generate taskpartitioning among workers. It also shows howthe interplay between the dynamics of thephenomenon combined with individuals’ re-sponse thresholds affects colony-level patterns.This work suggests a new methodology to studytask partitioning in social insects; it emphasizesthe importance of designing new experiments tostudy how individuals’ behavioural responseschange with the value of the stimuli associatedwith the tasks.
The next step will be to take interindividualdifferences into account in the model. In itscurrent form, the model provides an averagedescription of the hunting pattern. Averagingout interindividual differences is fine if onewishes to reproduce global patterns, but a degreeof specialization among hunters has been re-ported (Schatz et al., 1996). This specializationsuggests, within the response threshold frame-work, innate threshold differences betweenworkers and/or learning or habituation (Ther-aulaz et al., 1998). A model combining inter-individual differences and learning (Theraulazet al., 1998) could explain the transition from notask partitioning (hunters sting and transport) totask partitioning (stinging and transporting areusually performed by two distinct workers) as
the size of the group of hunters increases and ona longer time scale. A similar phenomenon wasobserved in the eusocial wasp species Polybiaoccidentalis (Jeanne, 1991) and Mischocyttarusmastigophorus (O’Donnell, 1998). This points tothe role of colony size as an important evolu-tionary parameter (Karsai & Wenzel, 1998;Bourke, 1999; Anderson & McShea, 2001), theimpact of which could be better understood witha threshold model.
This work was supported in part by a grant fromthe GIS (Groupement d’Int!er#et Scientifique) Sciencesde la Cognition to E. B and G. T. E. B. is supportedby the Interval Research fellowship at the Santa FeInstitute, G. T. by a grant from the Conseil R!egionalMidi-Pyr!en!ees and B. S. by a grant Sciences de laCognition (MESR). We wish to thank Guy Beugnonand Jean-Paul Lachaud for their help and advice.
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