A common food source for many ant species is honey-dew secreted by aphids (Addicott, 1979; Volkl et al., 1999;Novgorodova, 2004) and a common (and sometimes spectac-ular) foraging behaviour is aggregation by mass recruitment(Holldobler and Wilson, 1990; Beckers et al., 1992; Sumpterand Pratt, 2003). Aggregation describes the gathering of antsin large numbers at a food site distant from the nest and massrecruitment occurs via a pheromone trail: incoming (and someoutgoing) ants leave a pheromone trail that recruits and leadsants in the nest to the food site (Calenbuhr and Deneubourg,1992; Calenbuhr et al., 1992).
There are other forms of recruitment (Cassill, 2003)and other food types exploited by ants (Holldobler andWilson, 1990) but in this paper we shall focus on mass
recruitment via pheromone trail and the exploitation ofa single food source consisting of an aphid colony (or anexperimental surrogate, consisting of a large sugar droplet).The data used shall come primarily from Lasius niger, but
we shall also use physiological parameters from Solenopsisinvicta.
Given this type of food source and this type of recruitment,the ant colony faces three problems concerning the allocationof its foraging labour force: (i) how many ants to allocate to for-aging, as opposed to other activities within the nest; (ii) whichfood source of the same sugar type to explore, comparing itsdistance and size; (iii) which food source to exploit if differentsugar types are available.
Problem (ii) above has been successfully explored inBeckers et al. (1992) and Mailleux et al. (2000, 2003): uponreaching the foraging ground an ant tries to ingest a desiredvolume, and if it succeeds in doing so it lays a pheromone trailon the homeward trip. This simple rule suffices to choose thebest food source (in terms of distance and size) and also to allo-cate the optimal number of recruits to that food source (when
the food source is crowded ants cannot reach the desired vol-ume and no longer lay the pheromone trail).
Problem (iii) is more difficult to explore since it involvesknowing how ants respond to different recruitment stimuli
recruitment.In Section 2 we present a behavioural model that formal-
izes these rules. In Section 3 we derive the implications ofthe behavioural model at the colony level. In Section 4 we
0 ( 2 0 0 7 ) 384–392 385
compare the predictions of the model against empirical obser-vations. Section 5 discusses the results and implications ofthe model.
2. Model of individual behaviour
We consider that regarding foraging, ants belong to one ofthree castes. Scouts are 2–5% of population in a maturecolony (1000–10,000 individuals) and are dedicated foragers(Holldobler and Wilson, 1990). Remaining ants are nonscouts,of which a fixed number are facultative foragers and theremaining are nonforagers (Mailleux et al., 2003). It is the avail-ability of facultative foragers to recruitment that sets the totalnumber of foragers.
This classification of castes is a simplification of realityconvenient for the purposes of this study, for the study of otheraspects of ants’ behaviour, other castes (nurses, patrollers,etc.) might need to be defined. We also do not worry aboutthe problem of caste differentiation: within a time horizon ofa few days the number of ants in each caste can be consideredfixed.
In the model, relative volume is the key variable thatdetermines ant’s behaviour. Let relative volume, v, be the ratiobetween ingested volume (gaster content) and desired volume(the volume ingested by the ant when allowed to drink adlibitum). These volumes refer to liquid food stored by the antand not yet digested. This liquid food can be freely exchangedbetween ants through trophallaxis (Holldobler and Wilson,1990).
The model assumes that an ant is either busy or idle. Whenbusy, its relative volume decreases at a constant rate, due tometabolic activity. When idle its relative volume remains con-stant, as metabolic activity is assumed to be negligible.
The existence of a behavioural link between starvation andidleness is reasonable because it slows down the metabolicactivity of the colony as the starvation period increases,increasing the length of time it can survive until a new foodsource is discovered.
The remainder of this section completes the individualbehaviour model.
2.1. Recruitment and starvation
Cassill and Tschinkel (1999) subject ants to prolonged starva-tion and observe how their relative volume change. Initially vdecreases linearly but from a certain time onward v becomesalmost constant (Cassill and Tschinkel, 1999, p. 303, Fig. 1).This pattern corresponds to a high (respectively low) proba-bility of the ant being busy if v is high (respectively low), witha sudden switch for an intermediate value of v.
Mathematically, this empirical activity pattern can be gen-erated as follows. Imagine that at any point in time a non-scout of relative volume v generates a random number X,taken from a normal distribution bounded between 0 and1. If X < v, the ant is busy, otherwise it is idle. The applica-
e c o l o g i c a l m o d e l l i n g
within the nest. Cassill (2003) explored this issue and observedthat (for S. invicta at least) the strict assumption of massrecruitment induced by a pheromone trail does not hold. Thescout returning to the nest actively tries to advertise food tothe nestmates and employs diverse recruitment signals, fur-thermore, the advertisement effort and the diversity of recruit-ment signals increases with food quality.
In this paper problem (i) is addressed. In the context of massrecruitment and foraging on a single liquid sugar food source,what is the number of workers that the colony should makeavailable for recruitment?
Cassill (2003) observed that potential recruits were by andlarge unemployed (i.e., inactive at the moment of recruit-ment) hungry ants. Mailleux (2002, pp. 100–103) subjected antcolonies to prolonged starvation and observed that numberof ants searching for food (scouts) is almost independent ofthe starvation period but that the number of recruits (activeforagers once a food source has been discovered) increaseswith the length of the starvation period up to a maximumvalue.
In the present paper we draw evidence from these andother sparse empirical evidence to build a simple model oflabour and food distribution within the colony such that,with each ant following simple decision rules based on itscurrent physiological state, the empirically observed colony-level behaviour emerges. To our knowledge no other suchmodel of individual recruitment behaviour has previouslybeen reported. Instead, recruitment models directly assume agiven colony-level response independent of the state of indi-viduals (Sumpter and Pratt, 2003).
The assumptions of the model are the following. Thereare three castes of ants—scouts: very active individuals, spe-cialized in searching and retrieving food; nonforagers: alwaysremain inside the nest, include nurses, brood, sexual indi-viduals, the queen and other specialized workers; facultativeforagers: typically remain inside the nest but can becomerecruits.
The foraging state of each ant is determined by its gastercontent. Scouts are busy if its gaster content is below a criticalvolume, otherwise they remain idle inside the nest. Nonfor-agers and facultative foragers are busy if its gaster content isabove a critical volume, otherwise they are idle. Idle facultativeforagers are potential recruits.
Food is exchanged between individuals via trophallaxis(mouth-to-mouth liquid sugar exchange). Ants store liquidfood (not yet digested) in their gaster and, if solicited, trans-fer it to other ants. We assume that a “priority rule” in foodsupply is achieved by making the propensity to donate and todemand food in a trophallaxis event vary between individuals.Such a priority rule ensures that facultative foragers becomestarved – hence potential recruits – before nonforagers becomeinactive.
With the rules defined above (facultative foragers becomepotential recruits if gaster content is low and there is a priorityrule in trophallaxis), there is a direct link between the degreeof satiation of the colony and the number of ants available for
tion of this model to the data of Cassil and Tschinkel (forS. invicta) yields the following parameter values: metabolicrate 0.4 day−1; mean 0.2 and S.D. 0.05 for random variable X.The mean of X is the critical activity volume. The probability
386 e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 384–392
t) an
Fig. 1 – Probability of being busy for nonscouts (lef
of being busy, as a function of gaster content, is displayed inFig. 1.
A link between relative volume and potential recruitmentis to be expected. The motivation for this hypothesis is that afacultative forager faces a time constraint (it can forage or dosomething else, not both at a time) and it needs a cue to decidewhich task to pursue. This cue can be relative volume since asgaster content decreases the potential reward for foraging ishigher. Furthermore, emptier individuals can retrieve largeramounts of food.
Thus, we propose that only idle facultative foragers are poten-tial recruits. This way, if relative volume is below (respec-tively above) the threshold level, the facultative forageris almost surely available (respectively not available) forrecruitment. This hypothesis is consistent with the obser-vations that recruitment is weak when the colony is sati-ated and strong otherwise (Mailleux, 2002, pp. 100–103)and that foragers are typically starved (Mailleux, 2002, pp.108–110). At the individual level behaviour, Cassill (2003, p.447, Fig. 6) shows that while idle and hungry individualsare very likely to be recruited, busy and satiated individualsare not.
The activity state of ants belonging to different castes issummarized in Table 1.
We further propose that the total number of facultative for-agers is smaller than the total of ants (that is, there is a posi-
tive number of nonforagers). This hypothesis is biologicallyreasonable since there are elements in the colony (queen,brood and nurses) that never forage. Cassill (2003, p. 447,Fig. 7) shows that some ants (even many idle individu-
Table 1 – Activity state of ants belonging to differentcastes
Busy Idle
Scouts Foraging Not foragingFacultative foragers Not foraging Potential recruitsNonforagers Never forage
Potential recruits forage if and only if stimulated by pheromone trail(or direct recruitment).
d scouts (right), as a function of relative volume, v.
als) are never recruited, strengthening the validity of thisassumption.
2.2. Trophallaxis
The relative volume of an ant is determined by its trophallacticexchanges. We propose the following model for the process oftrophallaxis.
If two ants encounter inside the nest (encounter rate higherthan 1 s−1), each ant tests whether to offer, to demand or todo nothing. If one decides to offer and the other to demand,food exchange occurs at a rate of 0.01 �l s−1 (similar to drinkingspeed, Mailleux et al., 2000).
Decision to offer and to demand are independent Bernoullitrials. The probability of success of an ant’s decision to offerequals relative volume, v (Cassill and Tschinkel, 1999, p. 304,Fig. 3). This formalizes the intuitive idea that the more satiatedan ant is the more likely it is to offer.
The probability of success of an ant’s decision to demand isdifferent for scouts and nonscouts. Scouts are addressed in thefollowing subsection. In the case of nonscouts the probabilityof successful demand is (1 − va)1/a, with constant a defined as−ln 2/lnvd and 0 < vd < 1. This can be visualized in Fig. 2.
The probability of successful demand of nonscouts isderived from theoretical considerations. When the relativevolume of the nonscout is 0 (the nonscout is fully starved),its probability of demand must be 1, otherwise the nonscoutwould commit suicide. When the relative volume is 1 (thenonscout is fully satiated), its probability of demand mustbe 0, because it cannot store more food. Between those twoextremes, the demand function should be monotonic, as it isreasonable that the propensity to demand increases with star-vation.
The family of functions defined by (1 − va)1/a is interest-ing because it follows the required conditions, it is mathe-matically tractable and an easy interpretation can be givento parameter vd, the equilibrium demand volume, defined bya = −ln 2/lnv . The equilibrium demand volume is the rela-
d
tive volume at which an ant’s probability to demand and tooffer is the same.
A multivalued distribution of vd among nonscouts definesa priority rule, as nonscouts with a higher vd are, in average,
e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 384–392 387
Fig. 2 – Probabilities to offer and demand in trophallaxis, as a function of relative volume, v. Top left is probability to offer,all ants. Top right is probability to demand, scouts. Bottom is probability to demand, nonscouts: the three curvescorrespond to different values of equilibrium demand volume, vd.
mtt
2
Faimemtussat
aits
wa
rium demand volumes, v , and we consider a broad class of
ore likely to demand food from other ants. Thus, we expecthat the set of facultative foragers is the set of nonscouts withhe lower values of vd.
.3. Scouts
acultative foragers are recruited only when there is alreadypheromone trail connecting the nest to the food source. It
s scout’s task to discover food sources and to initiate recruit-ent. Scouts’ behaviour regarding starvation must be differ-
nt from that of nonscouts, because it is when the colony isost starved (and the number of potential recruits is higher)
hat the discovery of food sources is most required. That is,nlike nonscouts, scouts must be busy when the colony istarved. Furthermore, it was observed experimentally thatcouts are always starved, irrespective of the degree of sati-tion of the colony (Mailleux, 2002). These observations leado the following hypotheses.
Regarding trophallaxis, when a scout encounters anothernt, the probability of successful demand is 1 (respectively 0)f v < 0.1 (respectively v ≥ 0.1). This demand function ensureshat the relative volume of scouts is reset at 0.1 each time a
cout returns to the nest. This can be seen in Fig. 2.
Busy scouts remain outside the nest, searching for foodhile idle scouts remain inside. Scouts remain idle if v > 0.1,
s shown in Fig. 1. If v < 0.1, they alternate between the busy
and the idle state. When in one state they have a constantprobability per unit time of changing to the other state. Aconstant give-up probability per unit time implies that the dis-tribution of time spent in each state is exponential (which isroughly consistent with the empirical evidence, Mailleux etal., 2000).
The behaviour of scouts is important for foraging but notto determine the number of individuals available for recruit-ment, which is the topic of the following section. Scouts aredealt with again in Section 4.
3. Analysis of colony behaviour
In this section we derive the implications of the behaviouralmodel of Section 2 at the colony level. We are interestedin determining the number of potential recruits, for a givendegree of satiation of the colony. To do so, we first derive anexpression for the equilibrium volume of an arbitrary ant (inthis section we only consider nonscouts). The distribution ofequilibrium volumes depends on the distribution of equilib-
d
such distributions. Afterwards, we consider the effect of thephysiological parameters (metabolic rate and distribution ofcritical activity volume) on the colony-level energy consump-tion rate. Finally, we derive the number of potential recruits.
i n g 2 0 0 ( 2 0 0 7 ) 384–392
Fig. 3 – Equilibrium relative volume of a nonscout, v, as a
388 e c o l o g i c a l m o d e l l
3.1. Energy distribution
Since metabolic rate (∼10−6 �s−1)1 is much lower than therate of food exchange via trophallaxis (∼10−2 �s−1, assumedsimilar to drinking speed, Mailleux et al., 2000) and contactsbetween ants in the nest are frequent (more than 1 s−1, visualobservation) it is reasonable to assume that energy distribu-tion in the nest is in steady state determined by the trophal-lactic exchanges.
Let be the rate at which food is transferred duringtrophallaxis. Recall from Section 2.3 that the probability thatant i demands is vi and the probability that an ant j offersis (1 − vaj
j)1/aj , where vi and vj are the relative volumes of
ants i and j (and ai and aj are the respective parameters).The expected flow from j to i during a trophallaxis eventis vi(1 − vaj
j)1/aj (≤ since neither i necessarily demands
nor j necessarily offers). The expected flow from i to j is vj(1 − vai
i)1/ai .
The total flow of food to and from ant i via trophallaxis istherefore given by
vi
∑j�=i
�j(1 − vajj )1/aj and (1 − vaii )1/ai∑j�=i
�jvj,
where �j denotes the probability that ant i interacts with antj, and summation is carried over all ants in the colony.
If metabolic costs are neglected, vi changes until the totalflow of food to and from ant i cancel each other. Furthermore,if we assume that ant i is equally likely to meet any other antj (that is, if �j is the same for any j), the following conditionmust hold:
vi
∑j�=i
(1 − vajj )1/aj = (1 − vaii )1/ai∑j�=i
vj.
The former equation can be transformed into
vi(1 − vai
i)1/ai
=∑
j�=ivj∑j�=i(1 − vaj
j)1/aj
.
Now, if we assume that the number of ants is large, the sum-mations on the right-hand side of the last equation remainunaffected by the addition of the terms referring to ant i. Inthis case, the right hand side becomes independent of i. Sincethe same condition must hold for any ant i, it follows that:
vi(1 − vai
i)1/ai
= constant. (1)
Let v, vd and a = −ln 2/lnvd be defined for a generic ant. Eq.(1) implies that v = vKd for any ant, where K is a constant.
Constant K is determined by solving the normalization con-dition:
∫vKd�(vd) dvd = V, (2)
1 Calculated assuming desired volume of 1 �l (Mailleux et al.,2000) and relative volume consumption rate of 0.4 day−1 (Cassilland Tschinkel, 1999), corresponding to a sugar concentrationfound in nature (Volkl et al., 1999).
function of average relative volume, V, for different valuesof equilibrium demand volume, vd.
where V is average relative volume of the colony and �(vd) is theprobability density function of vd.
3.2. Priority rule
Recall that �(vd) is yet unspecified. Let us first consider thatvd has uniform distribution in (0,1). In this case, solving thenormalization condition (Eq. (2)), one obtains K = 1/V − 1. Therelative volume of an ant becomes:
v = v1/V−1d . (3)
Eq. (3) shows that v always rises with V, for any ant, and thatants ranking low in the priority hierarchy (that is, with a lowvd) always have a lower v than other ants. Fig. 3 shows thedynamics of Eq. (3).
Let �(v) be the probability density function of v. Since �(v) =�(vd) dvd/dv, one obtains:
�(v) = (V/(1 − V))vV/(1−V)−1. (4)
The behaviour of �(v) as a function of V is rather intuitive:when the colony is full (V = 1), all individual ants are full (�(v)takes value 1 if v = 1 and 0 otherwise); if the colony is rel-atively full (V > 0.5), most individuals are full (�(v) increasesmonotonically with v); if V = 0.5, �(v) is uniform; if the colonyis relatively empty (V < 0.5), most individuals are empty (�(v)decreases monotonically with v); when the colony is empty(V = 0), all individuals are empty (�(v) takes value 1 if v = 0 and0 otherwise). Fig. 4 shows the dynamics of Eq. (4).
In the previous calculations, the distribution of the demandequilibrium volume, vd, was assumed uniform in (0,1). Nowconsider the more general case where the distribution den-
sity is given by (�+ 1)v�d, with � real and vd taking values in(0,1). This distribution reduces to the uniform distribution if� is 0, is monotonically increasing for positive � and mono-tonically decreasing for negative �. By solving Eq. (2) again,
e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 384–392 389
Fig. 4 – Probability density function of the number ofnonscouts of relative volume v, for different values ofa
wvdii
s0vfoeeist
ef1da
a
asi
3
IP
tnew
Fig. 5 – Average relative volume, V, and the fraction of idleindividuals, �, after starvation period t (days). Metabolic
−1
dVdt
= �(1 − vV/(1−V)c ). (6)
verage relative volume, V.
e find that K is now given by K = (�+ 1)(1/V − 1). The relativeolume of an arbitrary ant is no longer given by Eq. (3) but theensity function of relative volume is still given by Eq. (4). This
mplies that the distribution of relative volume in the colonys independent of �.
Consider an arbitrary ant being recruited and compare theituations of parameter � being different from and equal to. Its relative volume is the same. Its equilibrium demandolume is not necessarily the same, but that influences theoraging behaviour of the recruited ant is only in the lengthf the trophallaxis period after its return to the nest (a lowerquilibrium volume implies that trophallaxis is longer). How-ver, since that happens after the ant returned to the nest, its not important for the dynamics of foraging. Hence, the con-ideration of the uniform distribution is a good approximationo any power-law distribution.
Considering the typical problems of parameter fitting frommpirical data, it is reasonable to assume that any monotonicunction can be approximated by a function of the form (�+)v�d. Therefore, we are led to conclude that as long as thatistribution is monotonic its precise shape does not affect insignificant way the foraging behaviour of the colony.
If vd does not possess a monotonic distribution, a simplenalytical relation between �(v) and V can no longer be found.
In the remainder of the paper we consider that vd possessesmonotonic distribution and, therefore, for the purpose of
tudying the dynamics of foraging, we further consider that its well approximated by the uniform distribution.
.3. Idle individuals
n Section 2 we assume that each ant is busy a proportion(X < v) of its time, where X is a normal distribution (recall
hat in Section 3 we only consider nonscouts). Given the largeumber of ants in a colony, this is equivalent to assuming thatach ant is busy with probability 1 if v > vc and 0 if v ≤ vc,ith the critical activity volume, vc, being normally distributed
rate of 0.4 day and mean critical activity volume of 0.2.
(across ants) with the same parameters as X. The probabilitythat a randomly chosen ant of volume v is busy is given byP(vc < v).
Average relative volume after a period of starvation oflength t, V(t), is a solution of
dV(t)dt
= −�(
1 −∫ 1
0
�(vc) dvc
∫ vc
0
�(v) dv
).
In the above �(·) is a probability density function and � ismetabolic rate. The double integral is the fraction of idle non-scouts. We assume that the distributions of vd and vc areindependent.
If probability density �(v) is given by Eq. (4), integral∫ vc
0�(v) dv becomes vV/(1−V)
c . Integral∫ 1
0�(vc)vV/(1−V)
c dvc, with�(vc) normally distributed is difficult to solve analytically.
We performed the numerical integration of Eq. (3) for plau-sible parameter values (metabolic rate 0.1–0.8 day−1; averagecritical volume 0.1–0.4; standard deviation of critical volume0.01–0.1) and observed that assuming standard deviation 0changes results very little except when the fraction of idle indi-viduals is very low. For purposes of studying recruitment, inthe latter situation a large error is acceptable since the mag-nitude of the number of idle individuals is always small. Withthis assumption, the fraction of idle individuals, �, becomes:
� = vV/(1−V)c , (5)
where vc is average critical volume (0.2 for S. invicta, Cassill andTschinkel, 1999).
The dynamics of average volume under starvation cantherefore be simplified to
The curves of the degree of satiation of the colony and of thefraction of idle individuals for the parameter values of Solenop-sis rubra (Section 2.1) are presented in Fig. 5.
390 e c o l o g i c a l m o d e l l i n g
Fig. 6 – Fraction of facultative foragers available for
recruitment as a function of average relative volume, V.
3.4. Number of potential recruits
In conclusion, if the distribution of vd is monotonic and V(0),� and vc are known, V(t) can be computed from Eq. (6).
If we assume that facultative foragers are the first non-scouts to become starved, the fraction of potential recruitsfrom all ants in the colony, �r, is given by
�r = min{vV/(1−V)c ; fm
}, (7)
where fm is the fraction of facultative foragers in the colony.The relative volume of a recruit is smaller or equal to v1/V−1
c .From Fig. 6, which plots Eq. (7), it is clear that when V is
large the fraction of nonscouts is very small. Therefore, aftera short period of starvation no mass recruitment can occur. IfV falls below a lower threshold defined implicitly by vV/(1−V)
c >
fm, the recruitment flow becomes independent of V. If vc andfm are both 0.2, the recruitment flow becomes independent ofV if V < 0.5, which corresponds to a starvation period of 2 days(see Fig. 6). This choice of value for fm is therefore consistentwith the observation that recruitment flow reaches the samemagnitude when starvation is longer than 2 days (Mailleux,2002; Portha et al., 2002).
4. Application to recruitment
In the present section we test the predictions of the modelagainst empirical data provided by Mailleux (2002, pp. 94–110).The experimental setting can briefly be summarized as fol-lows. An initially well-fed colony of circa 1000 individuals isfirst subjected to a period of starvation of 1, 4 and 8 days andafterwards presented with an abundant food source at a closedistance from the nest entrance. The number of ants arrivingat the food source during 1 h after the food source had beendiscovered is recorded.
Following a well-established tradition (Sumpter and Pratt,2003), we model the ants’ behaviour through a set of differ-ential equations, taking the number of ants as a continuous
variable.
The colony consists of 20 scouts and 1000 nonscouts, ofwhich 200 are potential foragers (i.e., fm is 0.2). Let all antshave the same desired volume, taking value 1 �l (Mailleux et
2 0 0 ( 2 0 0 7 ) 384–392
al., 2000). Let us make a change of notation and let now V betotal ingested volume (not relative). Total desired volume of thenest is 1000 �l.
Assume that both scouts and nonscouts can be in one ofthree states. Scouts can be outside the nest (searching for foodor foraging), in the nest performing trophallaxis or in the nestidle. Nonscouts can be outside the nest foraging (recruits), inthe nest performing trophallaxis or in nest busy with nonfor-aging activities (when they are assumed to have an equilib-rium relative volume). Let sf and nf be the number of scoutsand recruits foraging, let st and nt be the number of scoutsand nonscouts performing trophallaxis and si be the numberof idle scouts. Assume that all these can take real non-negativevalues (a noninteger number of ants can be thought of as anaverage value).
The time that an ant (scout or nonscout) takes to go to thefood source, feed and return is of 300 s (Mailleux, 2002, pp.100–103). The food source is abundant so there is no conges-tion. Ants leaving the nest carry 0.1 �l of ingested volume (halfof the average critical volume). Ants return full and lay 1 unitof pheromone during the return trip. Pheromones evaporateat a constant rate of 1/40 min−1 (Beckers et al., 1992).
The trophallaxis process is simplified as follows: all sati-ated ants returning to the nest offer (st and nt) but only idleants in the nest demand with trophallaxis rate of 0.01 �l s−1
(similar to drinking speed, Mailleux et al., 2000). The fraction �of idle nonscouts in the colony is given by Eq. (5). The net ratesof energy transfer from st and nt to the bulk of the colony at anymoment in time are 0.01�st and 0.01�nt �l s−1, respectively.
Facultative foragers offer until they attain their equilibriumvolume, moment at which they are “absorbed” by the bulk ofthe colony. Therefore, for each �l of volume transferred fromnt to the bulk of the colony there is one facultative forageralso returning to the set of nonforaging ants. Scouts offer fooduntil their ingested volume reaches 0.1 �l and then becomeidle inside the nest. Therefore, for each 0.9 �l of volume trans-ferred from st to the bulk of the colony there is one scout thatbecomes idle.
Scouts leave the idle state at a rate of 1/70 s−1 (Mailleux,2002, pp. 100–103). The probability than an idle facultativeforager is recruited is a hyperbolic function of the totalpheromone concentration (Beckers et al., 1992), hence, theflow of recruited nonscouts is given by
R(t) = rmin{�V,200 − (nf + nt)} c
c+ ck(8)
In Eq. (8), the term between brackets comes from Eq. (7), cdenotes pheromone concentration, r and ck are parameters,which we fitted empirically as 0.025 and 100 s−1.
The dynamic system is summarized in Table 2.The system summarized in Table 1 was solved numerically,
assuming initial degree of satiation of the colony, V(0), to takethe values of 600, 100 and 50 �l (roughly corresponding to 1,4 and 8 days of starvation, given by Eq. (6)). Fig. 7 shows thenumber of ants that arrive at the food source during intervals
of 5 min, for the several theoretical and empirical situations.
We can see that the predictions of the model agreeroughly with the empirical data. After 1 day of starvation norecruitment takes place, after 4 and 8 days of starvation the
The explanation of variables can be found in Section 4.
Fig. 7 – Flow of ants arriving at the food source (ants/5 min)vd
rf
em8drc
5
Ibsnisd
rr(
tmos
s. time t (min). Theoretical data is in columns, empiricalata is in lines.
ecruitment curves are very similar, with a maximum flow oforagers at 30–40 min after the discovery of the food source.
There are, however, differences between predictions andmpirical data: the model predicts that after 4 and 8 days theaximum flow of foragers peaks to the same value, but atdays this peak occurs later. The empirical curves, however,isplay the 8 days recruitment curve always above the 4 daysecruitment curve. Besides, all empirical recruitment curvesonverge faster than do the theoretical predictions.
. Discussion
n this paper we presented a model that determines the num-er of potential recruits in L. niger, when feeding on a liquidugar source. The core of the model are two rules: (i) there is aumber of facultative foragers that become potential recruits
f starved and (ii) facultative foragers are more likely to becometarved than nonforagers because they are more likely toonate food in a trophallaxis event.
We developed and analyzed a model based on theseules, deriving the number of facultative foragers available forecruitment as a function of average energy in the colony (Eq.7)), after an arbitrary period of starvation (Eq. (6)).
We developed a simplified recruitment model and saw that
he predictions of the behavioural model are in rough agree-
ent with the empirical data. However, since the parametersf were taken from different species (L. niger and Solenop-is rubra) and in very specific experimental conditions, the
0 ( 2 0 0 7 ) 384–392 391
numerical values presented should not be considered as beinguniversally valid.
The model also makes a prediction that has not yet beenobserved. Eq. (7) implies that for a high degree of satiationno facultative foragers are available for recruitment but thecolony can still receive energy inputs. This implies that at theend of a recruitment event only scouts are foraging.
The main assumptions involved in the derivation of Eqs.(6) and (7) are the following. The first set of assumptions isthat the number of ants is large, that the trophallaxis rate ishigh and that all ants are equally likely to meet. The secondimportant assumption is that the distribution of equilibriumdemand volumes (the relative volume at which an ant is aslikely to offer as to demand food) can be approximated by apower-law.
The first set of assumptions seems reasonable, and canbe subject to empirical testing. The second assumption wasmade due to analytical simplicity, but can in principle betested empirically. Assume that a given ant can be isolated andits gaster content measured. If this ant is allowed to interactwith other ants, for different levels of gaster content, its prob-ability to offer and to demand can be measured. If the sameprocedure is repeated for several ants, not only the curve ofequilibrium demand volumes but also the offer and demandprobability functions can be measured.
It is known that foraging behaviour upon aphid honeydewand upon animal protein differs considerably (Detrain andDeneubourg, 1995; Le Breton and Fourcassie, 2004). Also, for-aging upon aphid honeydew is influenced by previous atten-dance (Glinwood et al., 2003) and the availability of alternativeresources (Offenberg, 2001). In general, the characteristics ofthe resource influence foraging behaviour. In our model suchinfluence manifests through the parameters: critical activitythreshold, number of scouts and number of facultative for-agers. The actual values of these parameters are most likelythe result of evolutionary pressure and probably differ amongspecies, according to the properties of the resource beingexploited.
In this paper we focused on determining the number ofpotential recruits without studying in detail how the recruit-ment process occurs. The link between the model developedhere and a model exploitation of several food sources (possi-bly of different sugar types) requires a better understandingof the recruitment process, which enters our model via therecruitment function (Eq. (8)).
Cassill (2003) shows that the recruitment process is notmediated by the pheromone trail alone but that a recruiteradvertises food and exhibits recruitment displays to poten-tial recruits. Nevertheless, when the pheromone trail becomesstrong, it is reasonable to imagine that pure chemical recruit-ment takes place. In any of these circumstances (directand chemical recruitment) how does a potential recruitassess the quality of the food source to which it is beingrecruited?
A minimal assumption is that there is a pheromone “cur-rency” such that each sugar type has a different unitary value,
having the pheromone leading to richer sugars a higher value.This formulation in terms of chemical recruitment appliesequally well to the direct recruitment, with the evidence pre-sented by Cassill (2003, p. 447, Figs. 6 and 7).
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53, 131–144.
392 e c o l o g i c a l m o d e l l
An extension of the existing model to the study ofthe exploitation of several food sources (along the lines ofNicolis and Deneubourg, 1999) would require the experi-mental specification of this “pheromone currency” in termsof trail laying (or direct recruitment) and response topheromones (or direct recruitment), but offers no conceptualdifficulty.
From a modelling perspective, the present work showsthat detailed mechanistic analytical stochastic models canstill prove useful in the study of social insects as they do inother domains of ecological modelling (Beyer, 1998; Cohen etal., 1999). This is contrast with the dominant approach in thestudy of social insects, which consists either in the simula-tion of detailed mechanistic models (Mailleux et al., 2003) orthe analytical modelling of coarse phenomenological models(Nicolis and Deneubourg, 1999).
Nevertheless, in spite of the success of the “mean field”approach developed in the analysis, the behavioural modeldeveloped here (i.e., modelling of the foraging activity stateof ants as a Markov process and of the trophallaxis decisionas a Bernoulli trial) can be incorporated in a spatially explicitindividual-based model of ant behaviour inside the nest. Thatwould allow for an explicit modelling of the rich communica-tion behaviour described by Cassill (2003) and provide for yetnew insights into ants behaviour.
Acknowledgements
The author would like to thank J.-L. Deneubourg and also C.Detrain, R. Barbosa, T. Domingos, T. Sousa and an anonymousreviewer for important comments and suggestions.
e f e r e n c e s
Addicott, J.F., 1979. A multispecies aphid-ant association: densitydependence and species-specific effects. Can. J. Zool. 57,558–569.
Beckers, R., Deneubourg, J.-L., Goss, S., 1992. Trails and U-turns inthe selection of a path by the ant Lasius niger. J. Theoret. Biol.159, 397–415.
Beyer, J.E., 1998. Stochastic stomach theory of fish: anintroduction. Ecol. Model. 114, 71–93.
2 0 0 ( 2 0 0 7 ) 384–392
Calenbuhr, V., Deneubourg, J.-L., 1992. A model forosmotropotactic orientation (I). J. Theoret. Biol. 158, 359–393.
Calenbuhr, V., Chretien, L., Deneubourg, J.-L., Detrain, C., 1992. Amodel for osmotropotactic orientation (II). J. Theoret. Biol. 158,395–407.
Cassill, D.L., Tschinkel, W.R., 1999. Task selection by workers ofthe fire ant Solenopsis invicta. Behav. Ecol. Sociobiol. 45,301–310.
Cassill, D.L., 2003. Rules of supply and demand regulaterecruitment to food in an ant society. Behav. Ecol. Sociobiol.54, 441–450.
Cohen, Y., Pastor, J., Moen, R., 1999. Bite, chew, and swallow. Ecol.Model. 116, 1–14.
Detrain, C., Deneubourg, J.-L., 1995. Scavenging by PheidolePallidula: a key for understanding decision-making systems inants. Anim. Behav. 53, 537–547.
Glinwood, R., Willekens, J., Pettersson, J., 2003. Discrimination ofaphid mutualists by an ant based on chemical cues. ActaAgric. Scand. B: Soil Plant Sci. 53 (4), 177–182.
Holldobler, B., Wilson, E.O., 1990. The Ants. Belknap Press,Cambridge, MA.
Le Breton, J., Fourcassie, V., 2004. Information transfer duringrecruitment in the ant Lasius niger L. (Hymenoptera:Formicidae). Behav. Ecol. Sociobiol. 55 (3), 242–250.
Mailleux, A.-C., 2002. Les regles de comportements a l’ originedes strategies alimentaires chez Lasius niger. PhD Thesis.
Mailleux, A.-C., Deneubourg, J.-L., Detrain, C., 2000. How do antsassess food volume? Anim. Behav. 59, 1061–1069.
Nicolis, S.C., Deneubourg, J.-L., 1999. Emerging patterns and foodrecruitment in ants: an analytical study. J. Theoret. Biol. 198,575–592.
Novgorodova, T.A., 2004. The symbiotic relationships betweensants and aphids. Zhurnal Obshchei Biologii 65 (2), 153–166.
Offenberg, J., 2001. Balancing between mutualism andexploitation: the symbiotic interaction between Lasius antsand aphids. Behav. Ecol. Sociobiol. 49 (4), 304–310.
Portha, S., Deneubourg, J.-L., Detrain, C., 2002. Self-organizedasymmetries in ant foraging: a functional response to foodtype and colony needs. Behav. Ecol. 13 (6), 776–781.
Sumpter, D.J.T., Pratt, S.C., 2003. A modelling framework forunderstanding social insect foraging. Behav. Ecol. Sociobiol.
Volkl, W., Woodring, J., Fischer, M., Lorenz, M.W., Hoffman, K.H.,1999. Ant-aphid mutualism: the impact of honeydewproduction and honeydew sugar composition on antpreferences. Oecologia 118, 483–491.
