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Social Foraging by Honey Bees:
Mechanisms, Dynamics, and Adaptation
Kevin M. Passino
Dept. Electrical and Computer Eng.
Ohio State University
2015 Neil Avenue, Columbus, OH 43210
December 14, 2006
Abstract
There are two main components of social foraging by honey bees. First, there is a continuous adjustmentof the number of bees dedicated to exploitation of forage sites and wide-area exploration for new forageoptions. Second, there is a persistent reallocation of the proportions of the foraging workforce dedicated toexploiting each site in order to match unpredictably changing relative forage site profitabilities. In this paperwe develop a model of bees’ foraging for nectar that can represent individual-level sensing, decision-making,and communication mechanisms, along with known restrictions on information flow in the hive. Our modelalso represents that as additional bees forage at a site its profitability generally degrades due to exploitationcompetition; hence, we can study the intimate coupling between the dynamics of forage site profitability andthe hive’s reallocation of foragers. To illustrate these dynamics, we first show how in spite of the fact that nobee can know the profitability of all sites, the hive achieves an approximate “ideal free distribution” (IFD) offoragers with an allocation of foragers proportional to relative site profitability. We identify the mechanismsunderlying the achievement of the emergent distribution and the hive’s ability to completely ignore foragesites of relatively inferior quality. We explain how the mechanisms lead to fast reallocations when there aresudden and significant site profitability changes or new site discoveries. Next, we show that natural selectionseems to have settled on values of the behavioral parameters representing the dance strength determinationrule that balance the desire to maximize nectar intake, yet minimize the time-energy investments in dancing.We show that achievement of this balance allows for individual-level forage site profitability assessment er-rors since the foraging process effectively filters such errors at the colony level. Finally, we identify the closerelationships between the dynamics of social foraging and nest-site selection by honey bees since it providesan excellent example of how changing a single individual-level behavioral rule can result in dramaticallydifferent emergent group behavior. Building on these relationships we briefly discuss how social foraging canbe viewed as a group cognition process.
Keywords: Apis mellifera, honey bee, foraging, group decision making, collective decision making.
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1 Introduction
The aggregate behavioral properties of a group of social animals are typically emergent phenomenathat arise from many individual decisions that are made with local inaccurate information andlimited inter-individual cues or signals (reviewed by Camazine et al. 2001). Examples where theemergent behavior is thought of as a “group-level decision” include selecting the best travel route(Deneubourg and Goss 1989), choosing among food sources by ants (Beckers et al. 1990) and bees(Seeley 1995), and deciding where to nest by ants (Mallon et al. 2001; Pratt et al. 2002) and bees(see reviews in (T. Seeley and Visscher 2004; T. D. Seeley et al. 2006)). Here, we introduce amathematical model of social foraging by honey bees and use it to investigate how local poorly-informed decisions by bees, mediated by limited inter-bee communications, can lead to fast yeteffective allocations of nectar foragers in a changing foraging environment.
Several models of social foraging by honey bees have been published. A differential equationmodel of functional aspects of dynamic labor force allocation of honey bees is developed and val-idated for one set of experimental conditions in (Seeley et al. 1991; Camazine and Sneyd 1991).Sumpter and Pratt (2003) introduced a generic nonlinear differential equation model that can rep-resent social foraging processes in both bees and ants. Like (Sumpter and Pratt 2003), our model ofrecruitment uses the idea from (Seeley et al. 1991; Camazine and Sneyd 1991) that dance strengthproportioning on the dance floor shares some characteristics with the evolutionary process (e.g.,with fitness corresponding to forage site profitability and reproduction to recruitment as discussedby Seeley (1995)). Here we make such connections more concrete by modeling the bee recruitmentprocess in an analogous manner to how survival of the fittest and natural selection are modeled ingenetic algorithms using a stochastic process of “fitness proportionate” selection (Mitchell 1996).The manner in which we model the nectar unloading process uses ideas from the probability mod-els and experiments in (Seeley and Tovey, 1994; Seeley 1995) and is consistent with the analysisin (Anderson and Ratnieks 1999) on how the random unload wait-time information is used. Todetermine the proportion of foragers that should explore versus exploit known forage sites, we usethe experimental results in (T. Seeley 1983) to model “recruitment to exploring” as being inverselyproportional to the wait time to find a recruiting dancer. The optimal proportion of explorers andresters is shown in (Anderson 2001) to depend critically on the profitability of available forage,and the ability to find it. Our simulations confirm this by showing that the hive will regulate therelative proportioning of the roles foragers take depending on forage conditions. Here, unlike in(Anderson 2001), the mechanisms underlying the allocations will be clear, and we will illustratethe emergent allocation dynamics.
de Vries and Biesmeijer (1998) introduce an “individual-oriented” model of social foraging andvalidate it against one set of experimental conditions as was done in (Seeley et al. 1991; Camazineand Sneyd 1991). More recently, de Vries and Biesmeijer (2002) expanded and improved themodel in (de Vries and Biesmeijer 1998) (e.g., taking into account the findings in (T. D. Seeleyand Tovey 1994)) and studied how the number of foragers visiting two equally profitable sitescan diverge (“symmetry breaking”), cross inhibition (as studied in (Seeley 1995), p. 143), and fordifferent currencies what they call an “equal harvest rate distribution” (EHD) where patch sizeprovides negative feedback for recruitment (i.e., their patch degradation is driven by “interference”between foraging bees (Parker and Sutherland 1986; Giraldeau and Caraco 2000)). Additionaldetailed models of the social foraging process are found in (Cox and Myerscough 2003; Schmickl andCrailsheim 2003). These models quantify most of the features of the other models and additionalcharacteristics, and like others rely on the work in (T. Seeley et al. 1991; Camazine and Sneyd 1991).
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Moreover, Cox and Myerscough (2003) include polyandry-induced individual bee differences anduse simulations to show that a heterogeneous colony can reduce the variance in the average amountof nectar collected.
Bartholdi et al. (1993) study the pattern of forager allocation and the optimality of it. In relatedwork, Dukas and Edelstein-Keshet (1998) study the spatial distribution of solitary and social foodprovisioners under different currency assumptions. The work in (Bartholdi et al. 1993) and (Dukasand Edelstein-Keshet 1998) identify connections to the concept of the “ideal free distribution”(IFD) (Fretwell and Lucas 1970). The work in (de Vries and Biesmeijer 2002) studies conditionsunder which their EHD (see above) is reached and discuss the impact of currency and relationsto the IFD. Experimental evidence that the honey bees seem to achieve an IFD of foragers hasbeen found in a number of experimental scenarios (e.g., in (Seeley 1986; Seeley et al. 1991; Seeleyand Towne 1992)). Of course, for such studies hive information-flow restrictions and physical beetravel constraints make perfect achievement of the distribution impossible (indeed for bees the“ideal” and “free” assumptions (Fretwell and Lucas 1970) certainly do not hold). Hence, onlyapproximate achievement of matching the relative numbers of employed foragers to relative siteprofitability is viewed as achievement of an IFD. Moreover, there is the practical matter that thetheory of the IFD (Fretwell and Lucas 1970; Giraldeau and Caraco 2000) normally assumes thatthere is a large (theoretically infinite) number of foragers so that forager density is a continuousvariable. Otherwise, achievement of equal fitness is not always possible since it depends on therelative number of sites and foragers. In experiments, even relatively few bees have been found toachieve an “IFD.” Here, consistent with experimental work, our simulations result in achievement ofa distribution that is close to what the literature refers to as an IFD. We sidestep using a detailedcharacterization of energetics and currency (since there is not enough experimental evidence tojustify which currency to use) in favor of using a generic measure of forage site profitability. Whilethis lessens the predictive power of our model, we are still able to make conclusions about theessential underlying mechanisms that lead to an emergent IFD, and we can match the qualitativebehavior found in a wide array of experiments. Unlike in (de Vries and Biesmeijer 2002) wherethe interference-type model is used, we use a traditional IFD approach with “suitability functions”(Fretwell and Lucas 1970) that represent how patch quality degrades as more foragers arrive at apatch (we have also tested suitability functions that represent “interference” and get qualitativelysimilar results). Unfortunately, there is no experimental work to fully justify either our approachor the approach in (de Vries and Biesmeijer 2002); however, both provide plausible representationsfor how bees’ exploitation of forage sites degrades the profitability of those sites.
2 Social Foraging Model
Modeling social foraging for nectar involves representing the environment, activities during beeexpeditions (exploration or foraging), unloading, dance strength decisions, explorer allocation, re-cruitment on the dance floor, and accounting for interactions with other hive functions. Theexperimental studies we rely on are summarized in (Seeley 1995). Our primary sources for con-structing components of our model are as follows: dance strength determination, dance threshold,and unloading area (Seeley and Towne 1992; Seeley 1994; Seeley and Tovey 1994); dance floorand recruitment rates (Seeley et al. 1991); and explorer allocation and its relation to recruitment(Seeley 1983; Seeley and Visscher 1988).
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2.1 Landscape of Foraging Profitability
We assume that there are a fixed number of B bees involved in foraging. For i = 1, 2, ..., B bee iis represented by θi ∈ ℜ2 which is its position in two-dimensional space. Here, we let B = 200,a relatively low value, but one that is in the range of values used in some experiments (Seeley1995). During foraging, bees sample a “foraging profitability landscape” which we think of as aspatial distribution of forage sites with encoded information on foraging profitability that quantifiesdistance from hive, nectar sugar content, nectar abundance, and any other relevant site variables.The foraging profitability landscape is denoted by Jf (θ). It has a value Jf (θ) ∈ [0, 1] that isproportional to the profitability of nectar at a location specified by θ ∈ ℜ2. Hence, Jf (θ) = 1represents a location with the highest possible profitability, Jf (θ) = 0 represents a location with noprofitability, and 0 < Jf (θ) < 1 represents locations of intermediate profitability. For θ = [θ1, θ2]
⊤,the θ1 and θ2 directions for our example foraging area are for convenience scaled to [−1, 1] sincethe distance from the hive is assumed to be represented in the landscape. We assume the hive isat [0, 0]⊤.
We do not calibrate the model in terms of energetics and currency used for foraging decisions.Experiments have not been conducted to fully quantify the effects of: (i) time of day and season sincethe hive seems to employ foraging strategies to optimize the net rate of energy gain or efficiency ofenergetic gain at different times (Seeley 1995) and it is not clear how the bees switch between thesetwo strategies (or indeed if other strategies are used, such as those based on variance to achieverisk-sensitive behavior (Giraldeau and Caraco 2000)); (ii) energetic expenditure for flight to a sitewhen the dependency on wind and temperature conditions is taken into account; (iii) energeticexpenditure to search for and find both new and existing forage sites bees have been recruitedto; (iv) energetic content of the nectar of multiple simultaneously available species of flowers andenergetic expenditure to gather nectar from a site which depends on flower density, obstructions,wind, and other foragers; and (v) knowledge of the interactions of all these characteristics over alarge region that is searched and exploited during foraging. It is for this reason that we take adifferent approach here. We assume that Jf (θ) represents the overall profitability at location θ andthis can include many of the factors above. This assumption clearly decreases the predictive powerof the model; however, it allows us to easily set numeric values for several behavioral parametersin terms of experiments, and still allows us to represent general qualitative behavior of the socialforaging process and validate this behavior with respect to experiments.
An example of the type of foraging profitability landscape we will consider has four forage sitescentered at various positions that are initially unknown to the bees. For convenience, we will referto these as follows:
• Forage site 1 location: [0.5, 0.7]⊤
• Forage site 2 location: [−0.7, 0.5]⊤
• Forage site 3 location: [0.7,−0.3]⊤
• Forage site 4 location: [−0.5,−0.5]⊤
The “spread” of each site helps to characterize the size (area) of the forage site, and the heightis proportional to the nectar profitability. Here, we use cylinders with heights N j
f ∈ [0, 1] thatare proportional to nectar profitability, and the spread of each site is defined by the radius of thecylinders which we take to all be ǫf = 0.25. It must be emphasized that all foraging profitability
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information is “encoded” in the Jf map; hence, you cannot directly think of the domain of Jf as asquare foraging area for the bees. For instance, if you scaled the θ1 and θ2 axes to kilometers, thenthe Jf map would have to be appropriately deformed to represent that forage sites further awayfrom the hive are less profitable. Below, we will say that bee i, θi = [θi
1, θi2]⊤, is “at forage site 1” if
√
(θi − [0.5, 0.7]⊤)⊤(θi − [0.5, 0.7]⊤) < ǫf (1)
We use a similar approach for other sites.
2.2 Bee Roles and Expeditions
Let k be the index of the foraging expedition and assume that bees go out at one time and returnwith their foraging profitability assessments at one time (an asynchronous model with randomlyspaced arrivals and departures will behave in a qualitatively similar manner). Our convention isthat at time k = 0 no expeditions have occurred (e.g., start of a foraging day), at time k = 1 onehas occurred, and so on. All bees, i = 1, 2, ..., B, have θi(0) = [0, 0]⊤ so that initially they are atthe hive. We assume that the bees forage for 10 hours and make a foraging expedition every 10min. for a total of 60 foraging trips in one day (this only represents one scenario where forage sitesare not too far from the hive). We will conduct our simulations over this one day and not considerday-to-day effects (e.g., resumption of foraging after a previous successful day of foraging).
Let B(j, k) be the number of bees at site j at k (measured by Equation (1)). We assume thatthe profitability of being at site j, which we denote by Jj
f for a bee at a location in site j, decreasesas the number of bees visiting that site increases, and represent this by letting, for each j,
Jjf = N j
f exp
(
−1
2(B(j, k))2/σ2
f
)
where we choose σf = 50. With this representation we think of a site as a choice for the hive,with the site degrading in profitability via the visit of each additional bee, a common assumptionin theoretical ecology (e.g., in IFD theory Jj
f is called the “suitability function” (Fretwell and
Lucas 1970)). We will consider cases where the profitability parameter of a site, N jf ∈ [0, 1], will
stay constant, and other cases where it will vary over time. Increases in N jf can occur if flowers
bloom and decreases occur if flowers die.
Of the B bees involved in the foraging process, we assume that there are Bf (k) “employedforagers” (ones actively bringing nectar back from some site and that will not follow dances).Initially, Bf (0) = 0 since no foraging sites have been found. We assume that there are Bu(k) =Bo(k) + Br(k) “unemployed foragers” with Bo(k) that seek to observe the dances of employedforagers on the dance floor and Br(k) that rest (or are involved in some other activity). Initially,Bu(0) = B, which with the rules for resting and observing given below will set the number of restersand observers. We assume that there are Be(k) “forage explorers”1 that go to random positionsin the environment, bring their nectar back if they find any, and dance accordingly, but were notdedicated to the site (of course they may become dedicated if they find a relatively good site).
We ignore the specific path used by the foragers on expeditions and what specific activitiesthey perform. We assume that a bee simply samples the foraging profitability landscape once
1In the literature, sometimes these “explorers” are called “scouts” (Seeley 1995). Here, we call them explorers (i)to reserve the term scout for the bees involved in the nest-site selection process, and (ii) since then explorers in theforaging and nest-site selection processes are conceptually the same.
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on its expedition and hence this sample represents its combined overall assessment of foragingprofitability for location θi(k). It is this value that it holds when it returns to the hive. It alsobrings back knowledge of the forage location which is represented with θi(k) for the kth foragingexpedition. Let the foraging profitability assessment by employed forager (or forage explorer) i be
N i(k) =
1 if Jf (θi(k)) + wif (k) ≥ 1
Jf (θi(k)) + wif (k) if 1 > Jf (θi(k)) + wi
f (k) > ǫn0 if Jf (θi(k)) + wi
f (k) ≤ ǫn
where wif (k) is noise. Hence, it is assumed that each bee has an internal yardstick for profitability
assessment and the noise can represent either inaccuracies in this yardstick or assessment errors.Here, we let wi
f (k) be uniformly distributed on (−wf , wf ) with wf = 0.1 (to represent up to a ±10%error in profitability assessment). The value ǫn > 0 sets a lower threshold on site profitability. Here,ǫn = 0.1. For mid-range above-threshold profitabilities the bees will on average have an accurateprofitability assessment since the expected value with respect to k of wi
f (k), E[wif (k)] = 0. Let
N i(k) = 0 for all unemployed foragers.
We assume that the probability that a bee will die during each expedition is pd, 0 ≤ pd < 1.Each bee that dies does not return to the hive with a profitability assessment; however, we assumethat this dead bee is replaced by a novice forager and hence for it we let N i(k) = 0. Since we assumethat deaths occur independent of the quality of a site, the deaths of bees will not on average affectthe pattern of forager allocations to forage sites at a single step; however, there is an impact on thepattern of allocation over many steps since these deaths cause site abandonments that result in ahive-level flexibility for reallocation. Here, to be consistent with experiments, we let pd = 0.0017 sothat about 10% of bees going on expeditions over the 10 hours will die (Dukas and Visscher 1994;Seeley 1995). We let the number of bees that die at step k be Bd(k) and the total number that dieby step k, including at step k, be Bdt(k).
Next, we must specify the locations where all the bees will go on their expeditions. Unemployedforagers stay at the hive. The locations for the Be(k) forage explorers to forage at the next step aredefined by placing them randomly on the foraging landscape with a uniform distribution. Employedforagers do not go back to the precise spot in the forage site that they visited during the last foragingexpedition. To represent this, if the employed forager was at θi(k) for expedition k, then for itsnext expedition we let
θi(k + 1) = θi(k) + [e1, e2]⊤
where e1 and e2 are zero mean Gaussian random variables with variance σ2e . Each time a recruit
tries to find the dance-indicated forage location it frequently makes errors (Seeley 1995). Hence, ifrecruiter bee i indicates to go to its last position θi(k), the jth recruited forager will go to
θj(k + 1) = θi(k) + [r1, r2]⊤
where r1 and r2 are zero mean Gaussian random variables with variance σ2r . Here, we let σ2
e = 0.001and σ2
r = 0.002, representing that it is more difficult for the recruit to find the site than the beethat has already visited the site.
2.3 Dance Strength Determination
Let Lif (k) be the number of waggle runs of bee i at step k, what is called “dance strength.” The
Bu(k) unemployed foragers have Lif (k) = 0. All employed foragers and forage explorers that have
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N i(k) = 0 will have Lif (k) = 0 since they did not find a location above the profitability threshold
ǫn so they will not seek to be unloaded and will not dance; these bees will become unemployedforagers. The novice foragers we introduce to replace dead bees have Li
f (k) = 0.
2.3.1 Wait Time to Get Unloaded
Next, we will explain dance strength decisions for the employed foragers and forage explorers withN i(k) > ǫn. To do this, we first model wait times to get unloaded and how they influence the “dancethreshold.” We will ignore the effect of tremble dances on recruitment to nectar processing since itjust produces a transient in the unload wait-times that are proportional to nectar influx. By ignoringthis we are essentially assuming that via waggle and tremble dances the colony is consistentlymaintaining a balance between its rates of nectar collecting and processing, a characteristic thathas been validated experimentally (Seeley 1995).
Define Nt(k) =∑B
i=1Ni(k) as the total nectar profitability assessment at step k for the hive.
Foragers at profitable sites tend to gather a greater quantity of nectar than at low profitabilitysites. Let N i
q(k) be the quantity of nectar (load size) gathered for a profitability assessment N i(k).We assume that N i
q(k) = αN i(k) where α > 0 is a proportionality constant. We choose α = 1 sothat N i
q(k) ∈ [0, 1], with N iq(k) = 1 representing the largest nectar load size. Notice that if we let
Ntq(k) be the total quantity of nectar influx to the hive at step k,
Ntq(k) =B∑
i=1
N iq(k) = α
B∑
i=1
N i(k) = αNt(k)
so the total hive nectar influx is proportional to the total nectar profitability assessment. Also,Ntq(k) ∈ [0, αB] since each successful forager contributes to the total nectar influx.
The average wait time to be unloaded for each bee with N i(k) > ǫn is proportional to the totalnectar influx. Suppose that the number of food-storer bees is sufficiently large so the wait timeW i(k) that bee i experiences is given by
W i(k) = ψmax{
Ntq(k) + wiw(k), 0
}
= ψmax{
αNt(k) + wiw(k), 0
}
(2)
where ψ > 0 is a scale factor and wiw(k) is a random variable that represents variations in the
wait time a bee experiences. We assume that wiw(k) is uniformly distributed on (−ww, ww). Since
Ntq(k) ∈ [0, αB], ψ(αB +ww) is the maximum value of the wait time which is achieved when totalnectar influx is maximum. For the experiments in (Seeley and Tovey 1994) (July 12 and 14 data)the maximum wait time is about 30 sec. (and we know that it must be under this value or bees willtend to perform a tremble dance rather than a waggle dance to recruit unloaders (Seeley 1995));hence, we choose ψ(αB + ww) = 30. Note that ±ψww seconds is the variation in the number ofseconds in wait time due to the noise and ww should be set accordingly. We let ψww = 5 to geta variation of ±5 seconds. If B = 200 is known, we have two equations and two unknowns, socombining these we have ψB + ψww = 30, which gives ψ = 25/200 and ww = 40.
That there is a linear relationship between wait times and total nectar influx for sufficientlyhigh nectar influxes is justified via experiments described in (Seeley and Tovey 1994) and (Seeley1995), p. 112. Deviations from linearity come from two sources, the wi
w(k) noise and the “max” inEquation (2). Each successful forager has a different and inaccurate individual assessment of thetotal nectar influx since each individual bee experiences different wait times in the unloading area.
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The noise wiw(k) in Equation (2) represents this. Some foragers can get lucky and get unloaded
quickly and this will give them the impression that nectar influx is low. Other foragers may beunlucky and slow to get unloaded and this will result in an impression that there is a very highnectar influx. Since nectar influx estimates are what determine dance thresholds for decisions,there can be a wide variance on dance strength even for the same forage site (as shown in (Seeleyand Towne 1992)). The second source of deviation from linearity in Equation (2) is due to the“max.” This nonlinearity has two effects on the wait time. First, as seen in the experiments in(Seeley and Tovey 1994) it puts a “knee” in the curve representing the functional relationshipbetween Ntq(k) and W i(k) that Equation (2) implements since: (i) for low values of Ntq(k) smallincreases in nectar influx will not increase wait times significantly (since unloaders are relativelyabundant so foragers can easily find them), and (ii) for high values of Ntq(k) increases in nectarinflux will cause more significant increases in wait times (since there are relatively few unloadersand crowding in the unloading area significantly increases wait times). The second effect of the“max” in Equation (2) is that for low values of Ntq(k), even though there are plentiful unloaders,the average value of the wait time will be positive, with increasingly higher values for higher ww.This effect is the result of the “max” operation not allowing negative wait times, an effect thatonly occurs for relatively small values of Ntq(k), and results in a positive average wait time even forvery low nectar influxes. A similar effect is seen in the experimental results in (Seeley and Tovey1994) for low Ntq(k) values. In summary, the curve implemented by Equation (2) roughly has theform of an increasing exponential on average for low values of total nectar influx, but allows forsignificant variations as seen in experiments.
2.3.2 Choice of Dance Threshold and Strength: The Dance Decision Function
Next, we assume that the ith successful forager converts the wait time it experienced into a scaledversion of an estimate of the total nectar influx that we define as
N̂ itq(k) = δW i(k) (3)
So, we are assuming that each bee has an internal mechanism for relating the wait time it experiencesto its guess at how well all the other foragers are doing. The proportionality constant for this isδ > 0 and since W i(k) ∈ [0, ψ(αB + ww)] = [0, 30] sec. we have N̂ i
tq(k) ∈ [0, 30δ].
So, how does total nectar influx influence the dance strength decision, and in particular thedance threshold? This is explained in (Seeley and Towne 1992) and on p. 118 in (Seeley 1995).Here we build on this by defining a “decision function” for each bee that shows how the dancethreshold for each individual bee shifts based on the ith bee’s estimate of total nectar influx. Thedecision function is
Lif (k) = max
{
β(
N i(k) − N̂ itq(k)
)
, 0}
(4)
which is shown in Figure 1. The parameter β > 0 affects the number of dances produced for anabove-threshold profitability.
In Figure 1, −βN̂ itq(k) is the intercept on the dance strength axis. The diagonal bold line in
Figure 1 shifts based on the bee’s estimation of total nectar influx since this is proportional toN̂ i
tq(k). Now, the key is to notice that since the line’s slope is β, and since we take the maximum
with zero in Equation (4), the lowest value of nectar profitability N i(k) that the ith bee will decideto still dance for is the “dance threshold” N̂ i
tq(k) and from Equation (3), N̂ itq(k) = δW i(k), the
bee’s scaled estimate of the total nectar influx. Note that changing β does not shift the dance
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L (k)i
N (k)i
N (k)tq-
N (k)= W (k)tq
Slope=
Dancestrength,number ofwaggleruns
Nectar profitabilityfor bee i
Nectar influxincrease
Nectar influxdecrease
f
^
^ i
i Dance thresholdfor bee i
iδ
β
β
Figure 1: Dance strength function and dance threshold (δ adjusts the threshold and β affects thenumber of dances per increase in nectar profitability for above-threshold cases).
threshold. The parameter β will, however, have the effect of a gain on the rate of recruitment forsites above the dance threshold. In the case where Ntq(k) = N̂ i
tq(k) = 0 there is no nectar influxto the hive and it has been found experimentally (Seeley 1995) that in such cases, if a bee finds ahighly profitable site, she can dance with 100 or more waggle runs. Hence, we choose β = 100 soLi
f (k) = 100 waggle runs in this case. Then, Lif (k) ∈ [0, β] = [0, 100] waggle runs for all i and k.
The dance threshold in Equation (3) is defined using the parameter δ. What value would weexpect a bee to hold for δ? Since the nectar profitability N i(k) ∈ [0, 1], δ needs to be defined sothat N̂ i
tq(k) ∈ [0, 1] so that the dance threshold is within the range of possible nectar profitabilities.This means that we need
0 < δ ≤1
30(5)
To gain insight into how to pick δ in this range notice that δ is proportional to the site abandonmentrate: (i) if δ ≈ 0, then the dance threshold N̂ i
tq(k) ≈ 0 independent of wait times and so sites ofsignificantly inferior relative profitability will never be abandoned, something that does not occurin nature; and (ii) if δ ≈ 1
30 = 0.0333, then almost all sites are not danced for since the dancethreshold is so high and the foraging process fails completely, something that does not occur innature. Hence, δ must be somewhere in the middle of the range in Equation (5); in our simulationswe tuned the value of δ to match experiments and found δ = 0.02.
Next, it is useful to consider some typical foraging scenarios, how the dance threshold varies,and its effect on the pattern of forager allocations. First, suppose that no bees have found a goodsite, but then a few suddenly do. Those bees will have above-threshold profitability assessmentssince the threshold would have been around zero. The dance strength is then proportional to siteprofitability and this will set the recruitment rates and proportion allocation across the sites. Also,it will raise nectar influx, and on average the bees’ estimate of it, so that the dance threshold willgo up. Then, as more bees are recruited, forage, and get unloaded, the site recruitment rates willstay proportional to the total amount of waggle runs for the site, but with a higher threshold,so relatively poor sites will be abandoned and the bees committed to those will be proportionallyreallocated to the above-threshold sites. Now, if this process continued, it is possible that all
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sites except the best one are abandoned. If, however, the maximum value of the dance thresholdis somewhat below the maximum possible nectar profitability assessment, then a set of relativelyequal high profitability sites will be simultaneously exploited in proportions corrresponding to theirrelative profitabilities, and there will be a simultaneous abandonment of inferior sites.
2.3.3 Dance/No-Dance Choice
The set of bees that, after dance strength determination as outlined in the previous section, haveLi
f (k) > 0 are ones that consider dancing for their forage site. In nature, even bees that visit themost profitable site being considered by the entire hive will not necessarily dance for it. Corre-spondingly, bees dedicated to a relatively poor site might, or might not, dance for it. The tendencyto dance is proportional to the relative site profitability (Seeley 1995).
Here, we let pr(i, k) ∈ [0, 1] denote the probability that bee i with Lif (k) > 0 will dance for the
site it is dedicated to. We assume that
pr(i, k) =φ
βLi
f (k)
where φ ∈ [0, 1] (which ensures that pr(i, k) ∈ [0, 1]). We choose φ = 1 since it resulted in matchingthe qualitative behavior of what is found in experiments. Hence, a bee with an above-thresholdprofitability is more likely to dance the further its profitability is above the threshold. In this way,relatively high quality new discoveries will typically be danced for, but as more bees are recruitedfor that site and colony nectar influx increases, it will become less likely that bees (e.g., the recruits)will dance for it and this will limit the number of dancers for all sites. Relatively low quality sitesare not as likely to be danced for; however, bees that decide not to dance will still go back to thesite and remain an employed forager for it. Other shapes of functions also make sense for the pr
function, such as functions that are sigmoidal with respect to Lif (k), with pr = 0 when Li
f (k) = 0
and pr = 1 when Lif (k) reaches its maximum value. We have found, however, that the simple linear
relationship works well.
If bee i dances, then it uses a dance strength of Lif (k). If it does not dance, we force Li
f (k) = 0and the bee simply remains an employed forager for its last site. We let Bfd(k) denote the numberof employed foragers with above-threshold profitability that dance. The value of φ will influencethe total number of bees that decide to dance at each step; however, on average it will not changethe relative proportioning of the total dance strength pattern for the currently exploited foragesites. Choosing not to dance has the benefit of not wasting valuable time and energy recruitingwhen the bee could be foraging. There is, however, another important benefit to not having all theforgers dance: each individual dance counts for more in the percentage of dances for each exploitedsite. Hence, if there is a site profitability change, or a newly discovered site, a single bee’s dancingcan quickly result in a reorientation of a large portion of the foraging workforce. The speed ofreorientation is aided by the positive feedback due to recruiters also recruiting. So, the sensitivityof the the hive to changes in foraging environment (and hence reaction speed) is enhanced byless, not more, dancing. Of course, some minimal level of dancing is needed or there will be noproportionate allocation, or a noisy allocation. These observations agree with experimental studiesthat show that a relatively small percentage of foragers dance (Seeley 1995).
10
2.4 Explorer Allocation and Forager Recruitment
2.4.1 Resters and Observers
The bees that either were not successful on an expedition, or were successful enough to get unloadedbut judged that the profitability of their site was below the dance threshold, become unemployedforagers. Some of these bees will start to rest and other dance “observers” will actively pursuegetting involved in the foraging process by seeking a dancing bee to get recruited. Here, at eachk we let po ∈ [0, 1] denote the probability that an unemployed forager or currently resting beewill become an observer bee; hence 1 − po is the probability that an unemployed forager will restor a currently resting bee will continue to rest. It has been seen experimentally (Seeley 1995)that in times where there are no forage sites being harvested there can be about 35% of the beesperforming as forage explorers, but when there are many sites being harvested there can be as fewas 5%. Hence, we choose po = 0.35 so that when all bees are unemployed, 35% will seek dances.
2.4.2 Explorers and Recruits
Modeling the dance floor is complicated by a lack of understanding of how explorers are allocatedin the social foraging process. The problem is that while the recruitment process to forage sitesvia waggle runs is relatively well-understood, it is not precisely known why a bee chooses to berecruited rather than to explore (and vice-versa). Here, we assume that an observer bee on thedance floor searches for dances to follow and if it does not find one after some length of time, itgives up and goes exploring. That is, we assume the bee uses the wait-time to find a dancer as a cueabout whether it should explore. Full verification of this awaits experimental study. However, it iswell-known that in social foraging bees use the wait-time to get unloaded in order to estimate hivenectar influx and hence compare the profitability of the site it just visited to how other foragersare doing so that it can decide whether to dance and for how long (Seeley and Tovey 1994). Andthere is some evidence (Seeley 1983) that foragers decide to explore for a new food source, ratherthan get recruited to one, based on how long they search to find a dancer advertising a food source.A characteristic that would certainly impact the delays in finding dancers is that there are only somany bees that can observe the dancing at any one time (due to physical constraints that lead toonly a limited number of observers for each dancer). Such physical constraints lead to a limitednumber of recruits, and impose natural delays in any observer bee getting recruited. It seems logicalthat if the bee is delayed too much, it will decide simply to go find a site on its own. The resultwill be consistent with experiments in that when nectar intake is low there will be less dancing andhence more exploring in order to find new sources.
To model explorer allocation based on wait-time cues, we assume that the wait-time is inverselyproportional to the bees’ tendency to be recruited to a forage site. Considering experimentalstudies (Seeley and Towne 1992; Seeley 1994; Seeley and Tovey 1994), the wait-time is assumedto be proportional to the total number of waggle runs on the dance floor. The more bees that aredancing strongly on the dance floor, the more likely it is that an observer bee will quickly find adancer to follow. Let
Lt(k) =
Bf (k)∑
i=1
Lif (k)
be the total number of waggle runs on the dance floor at step k. We take the Bo(k) observer bees
11
and for each one, with probability pe(k) we make it an explorer. We choose
pe(k) = exp
(
−1
2
L2t (k)
σ2
)
(6)
Notice that if Lt(k) = 0, there is no dancing on the cluster so that pe(k) = 1 and all the observerbees will explore (e.g., Lt(0) = 0 so initially all observer bees will choose to explore). If Lt(k) is low,the observer bees are less likely to find a dancer and hence will not get recruited to a forage site.They will, in a sense, be “recruited to explore” by the lack of the presence of any dance. As Lt(k)increases, they become less likely to explore and, as discussed below, will be more likely to find adancer and get recruited to a forage site. Here, we choose σ = 1000 since it produces patterns offoraging behavior in our simulations that correspond to experiments.
The explorer allocation process is concurrent with the recruitment of observer bees to foragesites. Observer bees are recruited to forage sites with probability 1− pe(k) by taking any observerbee that did not go explore and have it be recruited. To model the actual forager recruitmentprocess we view Li
f (k) as the “fitness” of the forage site that the ith bee visited during expeditionk. Then, the probability that an observer bee will follow the dance of bee i is defined to be
pi(k) =Li
f (k)∑Bf (k)
i=1 Lif (k)
(7)
In this manner, bees that dance stronger will tend to recruit more foragers to their site.
3 Simulation of Allocation Dynamics
In this section we validate that the model achieves the qualitative behavior seen in experimentsand explain the mechanisms and resulting dynamics of the social foraging process. While eachsimulation run of the foraging process for one day is slightly different due to random effects, wenext discuss the general features of the dynamics for a representative case (i.e., one simulation run).
3.1 Emergent Dynamics for an Ideal Free Distribution
We consider the foraging profitability landscape in Section 2.1 with N1f = 1, N2
f = 0.9, N3f = 0.6,
and N4f = 0.3 in decreasing order of profitability if the same number of bees is at each site. First,
in Figure 2 we show the foraging profitability landscape along with the points where bees foragedon the landscape. The bees sample much of the landscape during the day, but most foragersconcentrate their activities on the profitable foraging sites, with proportionally more for moreprofitable sites.
The actual numbers of bees visiting sites 1-4 is shown in Figure 3 and the corresponding prof-itabilities are shown in Figure 4. Figure 3 shows that more bees are allocated to more profitablesites, with the number of bees allocated so that the profitabilities of the best sites are made nearlyequal as shown in Figure 4. Moveover, the bees manage to allocate almost no bees to the leastprofitable site 4 since the achieved profitability level of sites 1-3 is above that of site 4; in fact, thevisitors there are mostly ones that discover the site by random chance, not ones that are employedas foragers for it.
12
θ1
θ 2
Profitability Jf (contour) and bee sample points (dots)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 2: Foraging profitability landscape (contour) with each black dot representing a locationwhere a bee sampled the landscape at some time during the day. Site 1 is in the upper-right corner,site 2 is in the upper-left corner, site 3 is in the lower-right corner, and site 4 is in the lower-leftcorner.
0 2 4 6 8 100
20
40
60
B(1
,k),
#be
es s
ite 1
0 2 4 6 8 100
20
40
60
B(2
,k),
#be
es s
ite 2
0 2 4 6 8 100
20
40
60
Time, hours
B(3
,k),
#be
es s
ite 3
0 2 4 6 8 100
20
40
60
Time, hours
B(4
,k),
#be
es s
ite 4
Figure 3: Number of bees B(j, k) visiting site j at k.
To further illustrate the distribution of employed foragers see Figure 5. Here, we see that inspite of the fact that no bee knows the profitability of all the sites, the foragers are allocated so as to
13
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
J f1 , pro
fitab
ility
, site
1
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
J f2 , pro
fitab
ility
, site
2
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Time, hours
J f3 , pro
fitab
ility
, site
3
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Time, hoursJ f4 , p
rofit
abili
ty, s
ite 4
Figure 4: Profitability Jjf for site j at k.
achieve what is close to an IFD. The numbers allocated seek to make the average profitability of allvisited sites the same, and this is illustrated by how close the three horizontal lines correspondingto sites 1-3 are to each other. The noise in the process, and the lack of perfect information flow,causes the deviations from achieving perfectly equal profitabilities at sites 1-3. But, these deviationscan be viewed as useful since the distribution does not “lock-in” and stagnate; the hive maintainsflexibility via a noisy process where there is always a small amount of abandonment and recruitment.The abandonment results in resting and sometimes exploration so that the entire environment ispersistently monitored even as exploitation of excellent sites proceeds.
3.2 Underlying Mechanisms and Dynamics
In this section we study the dynamics of a number of underlying variables to explain how the IFDis achieved. We pay particular attention to how the bees manage to ignore the least profitable site,even though relatively few bees visit it.
First, the number of bees in various roles during the day is plotted in Figure 6. Here, as thenumber of employed foragers increases, the number of explorers decreases, but some explorationis done over the entire day to monitor the foraging environment for changes. The number of newrecruits is initially large, but decreases as the distribution of foragers reaches an equilibrium. Thenumber of dancers increases quickly initially as the sites are discovered, and then the dancing iscontinued all day in order to maintain the distribution of foragers. Finally, note that the numberof unemployed foragers, and obervers and resters, decreases as the number of employed foragersincreases but never becomes zero. Having a nonzero number of unemployed bees is useful as a“pool” of ready recruits in case a new highly profitable site appears and in case there is a quickloss in profitability of all sites since then explorers can be quickly deployed.
14
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of bees
Sui
tabi
lity
func
tions
, site
s 1−
4
Profitability avg (horizontal −−), avg no. of bees (vertical −−)
Figure 5: Suitability functions for sites 1-4 (solid lines, Gaussian shapes, in order from top tobottom), average profitability levels for sites 1-4 for last 5 hrs. (dashed horizontal lines), averagenumber of bees visiting sites 1-4 for last 5 hrs. (dashed vertical lines).
0 1 2 3 4 5 6 7 8 9 100
50
100
150
200
Bf(−
), B
dt(−
−),
Bfd
(−.)
Numbers of bees in different roles
0 1 2 3 4 5 6 7 8 9 100
50
100
150
200
Be(−
), B
rec(−
−)
0 1 2 3 4 5 6 7 8 9 100
50
100
150
200
Bu(−
), B
o(−−
), B
r(−.)
Time, hours
Figure 6: Number of employed foragers Bf , total number of bees that die Bdt, number of bees thatdance Bfd (top plot); number of explorers Be and number of recruited bees Brec (middle plot); andnumber of unemployed foragers Bu, number of observers Bo, and number of resters Br.
15
Figure 7 shows the mean and standard deviation of the dance threshold estimate for all beesvisiting sites 1-4. For sites 1-3 the threshold is on average the same since bees are allocated tomake those sites ultimately have the same profitability. For site 4 first note that by conventionwe use zero to be the mean wait time in the plots for the degenerate case when there are no beesvisiting that site. When a bee does visit site 4 and unloads, its estimate of the dance threshold isin the same range as for bees visiting sites 1-3 since they are all getting unloaded on in the samearea and no bee is treated preferentially in unloading. So, when a bee visits site 4 the maximumaverage profitability assessment it can have is 0.3 (since the maximum of its suitability function is0.3); hence, most often the profitability assessment is not above (or only slightly above) the dancethreshold for the bee so it is unlikely to dance for the site, and even if it does dance it will notperform many waggle runs since it will not be much above the threshold. This results in therebeing very few recruits to site 4. And, whenever the profitability assessment for site 4 is below thethreshold the bee will abandon the site; since abandonment is more likely than recruitment, thesite is ignored.
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
Ntq
hat
avg
/std
(bar
s), s
ite 1
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
Ntq
hat
avg
/std
(bar
s), s
ite 2
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
Time, hours
Ntq
hat
avg
/std
(bar
s), s
ite 3
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
Time, hours
Ntq
hat
avg
/std
(bar
s), s
ite 4
Figure 7: N̂ itq mean and standard deviation (bars) dance threshold estimates for bees visiting sites
1-4. If no bees visit a site, then we use the convention that the estimate is zero (e.g., for site 4).
These points are further clarified by considering the number of waggle runs for each of the foursites in Figure 8. Notice that in this case there is initially a significant amount of dancing and hencerecruitment, but eventually the amount of dancing for sites 1-3 decreases as shown in Figure 4. Site4 has almost no dancing for it, but once in a while a bee gets lucky and gets unloaded quickly sothat its relative profitability is above the threshold so it dances. Next, notice that the dynamics ofthe variation of the dance strengths for sites 1-3 correspond to the number of bees that appear atsites 1-3 as seen in Figure 3. In particular, notice that bees are quickly recruited to site 1, and thisinhibits somewhat (slows down) recruitment to site 2 which is nearly as profitable, and significantlyinhibits the rate of recruitment to site 3. This coupling between rates of allocation to the sitesoccurs since the pool of unemployed foragers is depleted by the recruitments to site 1, so that the
16
recruitment rates for sites 2 and 3 go down. Moreover, the initial recruits to site 1 cause the wait-time to be unloaded to go up and hence the dance threshold to go up so that there are relativelyfewer dances for the other sites since the amount of dancing is set by how much the profitability isabove the threshold. Overall, we see that the bees try to quickly deploy foragers to the best sites,but also can achieve a long-term equalization of profitabilities (as illustrated in Figure 5). Boththese features help the hive optimize their foraging performance. Quick deployment ensures returnsin spite of a quickly time-varying environment. Long-term IFD achievement is advantageous sinceeven small variations off the equalization of profitabilities can ultimately have a large negativeimpact on foraging performance since foraging losses due to such variations off the best foragerallocation accumulate over time.
0 2 4 6 8 100
20
40
60
80
100
L f avg
/std
(bar
s), s
ite 1
0 2 4 6 8 100
20
40
60
80
100
L f avg
/std
(bar
s), s
ite 2
0 2 4 6 8 100
20
40
60
80
100
Time, hours
L f avg
/std
(bar
s), s
ite 3
0 2 4 6 8 100
20
40
60
80
100
Time, hours
L f avg
/std
(bar
s), s
ite 4
Figure 8: Ljf mean and standard deviation (bars) of number of waggle runs for bees visiting sites
1-4. If no bees visit a site, then we use the convention that the average is zero (e.g., for site 4).
Finally, the number of employed foragers that dance is shown in Figure 9. Notice that thenumber of dancers is proportioned per the relative site profitabilities, and that very few bees dancefor site 4 due to its relative inferiority. This proportional allocation of dancers per profitabilitymakes it so that better estimates of the dance threshold will be obtained by the group of beesvisiting the best sites (simply since the average is determined by more bees); this helps the beesavoid abandonment of relatively profitable sites, and at the same time allows for a flexibility thatresults in abandonment of relatively weak sites. Moreover, the proportioned amount of dancingper profitability ensures that the proportional allocation of the number of foragers is maintainedas seen in experiments (Seeley 1995).
3.3 Ability to Ignore Inferior Sites
Next, we further consider the ability of the hive to abandon an inferior site as it is a key propertycontributing to foraging success. Consider the case where there are only two forage sites with the
17
0 2 4 6 8 100
5
10
15
20
25
Bfd
(1,k
), #
danc
ers
site
1
0 2 4 6 8 100
5
10
15
20
25
Bfd
(2,k
), #
danc
ers
site
2
0 2 4 6 8 100
5
10
15
20
25
Time, hours
Bfd
(3,k
), #
danc
ers
site
3
0 2 4 6 8 100
5
10
15
20
25
Time, hoursB
fd(4
,k),
#da
ncer
s si
te 4
Figure 9: Number of employed foragers that dance Bfd(j, k) for site j (solid line) and the meanover the entire day (dashed).
same suitability functions used in the last section, but now with N1f = 1 (good site) and we vary
N2f ∈ [0, 1]. For each value of N2
f we run the simulation 100 times and compute the average andstandard deviation of the profitability of each site. The results are in Figure 10. Notice that forvalues of N2
f > 0.6 the average site profitability of the two sites is equalized due to the achievement
of the IFD. For N2f = 0 site 2 is ignored entirely and the average profitability level of site 1 is
about 0.25. This value represents the profitability level below which the hive should ignore anyother site besides site 1. Indeed, this is what happens. Notice that the average profitability of site2 essentially increases linearly with a linear increase in N2
f in the range of N2f ∈ [0, 0.25] and this
shows that there were almost no visitors to site 2 for this profitability range. If there were morevisitors then the profitability level of site 2 would have decreased. This is what we see a slight effectfrom right at N2
f = 0.25 where there is a slight decrease in profitability from what a line fit would
predict. Now, for values of N2f > 0.25 the hive nearly achieves an IFD, with better achievement as
N2f increases.
3.4 Effect of Significant Profitability Variations
Next, we discuss how the colony reacts to fast and significant changes in profitability of foragesites. We use the scenario described in (Seeley et al. 1991; Camazine and Sneyd 1991) that issummarized in (Seeley 1995). There are two sites, one much more profitable than the other, thatthe bees forage at in the morning and as expected the colony allocates foragers so that there aremany more bees at the more profitable site. Then, in the afternoon the profitability of the twosites are swapped and the colony reallocates so that number of bees at each site corresponds totheir relative profitability. We represent this here by having two nonzero profitability sites with thesame suitability functions used in the last section, but now with N1
f = 1 (good site) and N2f = 0.5
18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Pro
fitai
lity
avg/
std,
site
s 1
(−),
2(−
−)
Nf2
Figure 10: Average and standard deviations (bars) of profitabilities for sites 1 (solid line) and 2(dashed line) when N1
f = 1 and N2f ∈ [0, 1].
(mediocre site) in the morning, and then these values are swapped at t = 5 hrs. For our model,the results for this case are shown in Figure 11 and we see that indeed the qualitative behavior ofthe reallocation pattern matches what is found in experiments (Seeley et al. 1991; Camazine andSneyd 1991). The number of bees at site 1 by t = 5 hrs. is roughly the same as the number ofbees at site 2 by t = 10 hrs. showing that the colony dynamically reallocates bees under dynamicchanges in environmental conditions. This allocation is achieved via simultaneous abandonment ofsite 1 and recruitment to site 2 since on average, after the profitability swap, bees visiting site 2 willbe further above the dance threshold so that they will be more successful in recruiting. This resultsin a “cross-inhibition” effect (Seeley 1995) where the recruitment for one site inhibits recruitmentto another site. Moreover, the number of bees dancing for each site realigns (see Figure 12) inorder to achieve the reallocation. In Figure 12 the mean number of dances over the entire day foreach site is the same since the swap occurred half way through the day, but in the morning thereare more dancers for site 1 and in the afternoon there are more for site 2. The hive’s ability toshift the proportion of dancers to keep it aligned to the relative profitability of forage sites is a keymechanism underlying the dynamic reallocation of foragers in response to changes in the foragingenvironment. The mechanisms underlying this ability are the relatively low number of dancingbees, the negative feedback due to the dance threshold’s effect on abandonment, and the positivefeedback that results from recruitment.
Finally, we note that similar reallocation dynamics are found for other scenarios (in the interestof brevity we omit plots). First, if there are three equal profitability sites and one suddenly appearsthat is much more profitable, then we see cross-inhibition (Seeley 1995) where the more profitablesite receives more foragers and simultaneously inhibits foraging at the other relatively inferior sitesdue to an increase in wait times. Second, when there are two equally profitable sites, then oneincreases in profitability while the other one keeps the same profitability reallocation does not
19
0 2 4 6 8 100
20
40
60
80
B(1
,k),
#be
es s
ite 1
0 2 4 6 8 100
20
40
60
80
B(2
,k),
#be
es s
ite 2
0 2 4 6 8 100
20
40
60
80
Time, hours
B(3
,k),
#be
es s
ite 3
0 2 4 6 8 100
20
40
60
80
Time, hoursB
(4,k
), #
bees
site
4
Figure 11: Number of bees visiting sites during forage site profitability change (bottom plots showthat sites 3 and 4 are rarely visited).
0 2 4 6 8 100
5
10
15
20
25
Bfd
(1,k
), #
danc
ers
site
1
0 2 4 6 8 100
5
10
15
20
25
Bfd
(2,k
), #
danc
ers
site
2
0 2 4 6 8 100
5
10
15
20
25
Time, hours
Bfd
(3,k
), #
danc
ers
site
3
0 2 4 6 8 100
5
10
15
20
25
Time, hours
Bfd
(4,k
), #
danc
ers
site
4
Figure 12: Number of employed foragers that dance Bfd(j, k) for site j (solid line) and the meanover the entire day (dashed). Bottom plots show that no bee dances for sites 3 and 4.
occur primarily by abandonment of the relatively inferior site, but by recruiting from a relativelylarge pool of unemployed foragers (the pool is large in this case since there are only two sites being
20
exploited so there is still a significant pool of bees that are exploring or resting since there has notbeen a significant amount of dancing). This shows that reallocation can occur in two ways: viaredirection of foragers from one site to a more profitable one, or by deployment of more foragers. Ineither case, the effect is that the relative number of foragers at each site matches the site’s relativeprofitabilities. Third, if there are many relatively inferior sites and one excellent one suddenlyappears, then the rate at which the hive can reorient will degrade over the case where there is, forinstance, just one inferior site being exploited by the colony. The reason for this is that it takeslonger to reallocate employed foragers than to simply draw on a pool of unemployed foragers.
4 Adaptation: Effects of Behavioral Parameters and Assessment
Noise
The behavioral parameters are ones that characterize the decision rules of the individual bees.Here, we consider the effects of the two behavioral parameters δ and β that characterize the dancestrength decision, and the profitability assessment noise magnitude wf , on the average total quantityof nectar obtained in one day and the average total number of waggle runs over a day needed to getthat quantity of nectar. Moreover, we count the number of bees that are employed foragers on eachexpedition and that dance after an expedition, and then sum these totals across all expeditionsin a day to obtain totals. Then, we compute the averages and standard deviations of these. Ourstatistics for each case are computed using 100 simulations of the foraging process over a day, foreach parameter value considered. For all cases, we use the foraging landscape in Section 2.1 withN1
f = N2f = N3
f = N4f = 0.9. Qualitatively similar results were found for other landscapes. We
vary only one parameter at a time and keep all others at the values used in the last section.
4.1 Effect of Dance Threshold Parameter
We first vary δ over the range given in Equation (5). The top-left plot in Figure 13 shows thatthe average total amount of nectar profitability peaks at a point near the value of δ = 0.02 thatwe used in the last section. The average total amount of dancing goes down as δ increases (seebottom-left plot) since fewer dancers are deployed (see bottom-right plot) and also since with ahigher δ the difference between the dance threshold and profitability is lower. The top-right plotconfirms that if the threshold is set too high the foraging process can fail. The main conclusion,however, is reached by simultaneously considering all four plots. The value of δ that maximizesforaging return for the investment lies in a range around 0.015–0.025. This further confirms ourmodel’s validity, and shows that optimizing foraging performance involves complex trade-offs thatemerge from individual-level behavioral rules.
4.2 Effect of Dance Strength Function Slope
We vary β over the range [50, 150]. Figure 14 shows that nectar quantity gathered increasesas β increases but only slightly above β = 100 (top-left plot). The increased nectar quantitygathered costs on average more dancing (bottom-left plot) by more dancers (bottom-right plot).Fewer foragers are deployed for low values of β (top-right plot), but there is little difference inthe number deployed in the range of β ∈ [100, 150]. The main conclusion, however, comes fromconsidering all four plots at once. Simultaneous maximization of the average total profitability
21
0 0.01 0.02 0.03
5600
5800
6000
6200
6400
Ntq
tota
l avg
/std
0 0.01 0.02 0.030
1
2
3
x 105
L t tota
l avg
/std
δ
0 0.01 0.02 0.03
9000
9500
10000
Bf to
tal a
vg/s
td0 0.01 0.02 0.03
2000
3000
4000
5000
6000
Bfd
tota
l avg
/std
δ
Figure 13: Foraging performance as a function of δ. Average (solid lines) and standard deviation(bars) of total nectar profitability gathered in one day (top-left), average and standard deviation oftotal amount of dancing in one day (bottom-left), average and standard deviation of total numberof employed foragers (top-right), average and standard deviation of total number of foragers thatdance in one day (bottom-right).
returned, minimization of the number of dances, maximization of the number of employed foragers,and minimization of the number of dancers comes at an intermediate value of β around 80–120,which agrees with the the value of 100 found experimentally in (Seeley 1995).
4.3 Effect of Profitability Assessment Accuracy
We vary wf ∈ [0, 0.5] and get the results in Figure 15 which show that the average total nectarprofitability return is maximized for any value of wf ∈ [0, 0.3]. Similarly, the average total numberof deployed foragers is relatively constant over the range of wf ∈ [0, 0.2]. There is little effect onthe average total number of dances and the number of dancers until wf > 0.1, but as there is moreassessment noise there is a need for more dances to maintain a profitable distribution. A value ofwf = 0 is physically unrealistic, but we see that our chosen value of wf = 0.1 results little adverseimpact on foraging performance. The effect of individual-level bee assessment noise is filtered outat the group level so that an effective allocation can emerge.
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Figure 14: Foraging performance as a function of β. See Figure 13 caption for a description ofvariables.
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Figure 15: Foraging performance as a function of wf . See caption of Figure 13 for a description ofvariables.
5 Discussion
5.1 Mechanisms and Dynamics
The key mechanisms unique to social foraging that enable the dynamical achievement of an effectiveemergent forager allocation primarily revolve around dance strength determination (as represented23
in the experimentally validated dance decision function in Equation (4)). First, via the wait-timeto be unloaded experienced by each successful forager, a bee can produce a scaled estimate ofthe current total nectar influx of the colony each time it returns to the hive. This estimate isproportional to the current average site profitability of all employed foragers in the colony. Anindividual bee can compare this estimate with its last site profitability assessment and decidewhether the forage site it is currently exploiting is above or below the average profitability of theother sites currently being exploited by the colony. Then, if it is exploiting what is currently arelatively inferior site it can abandon it. And, for increasingly relatively profitable sites it can bothraise its dance strength and its tendency to dance. This results in a proportionate distribution offoragers across the current relatively profitable sites.
We must emphasize that the dance threshold value is unique to each bee since it is basedon its individually experienced wait time and the site profitability assessment for the last site itvisited. There will then be many “mistakes” made by individual bees. Due to the noise on theexperienced wait time, a bee may mistakenly abandon a site, or choose a dance strength thatcorresponds to it seeking more recruits than are currently warranted for the site. But, the key tothe hive’s success is based on the combined actions of all the bees, not solely a single individual’serror-prone actions. First, note that the average profitability assessment and average wait timeexperienced by a sufficiently large group of bees visiting any site are quite accurate representationsof the site profitability and total nectar influx. Hence, the total set of abandonment/recruitmentactions take by a group of bees visiting a site can be quite accurate. In fact, as more bees visita site, the group will make more accurate decisions; this ensures that the hive will rarely makemistakes in abandoning a great site, and will not over-recruit when two sites of close profitabilityare being simultaneously exploited, so an effective proportionate allocation is achieved. However,it does mean that the hive will be more likely to sometimes make allocation mistakes for relativelypoor sites. These “mistakes,” however, amount to the hive being flexible in its view of a currentlyrelatively inferior site: if the site profitability changes it can then more quickly abandon or recruitto it. The hive continually explores, but pays increased attention to relatively superior sites. Theaggregate effect is a simultaneous proportioned monitoring of the entire foraging environment thatenables the forager allocation.
To confirm these ideas, the simulations shown in Figures 2, 3, 4, and 5 illustrate how the hiveproduced an effective allocation of foragers by nearly achieving an IFD. Also, Figure 7 illustratesthat the group of bees visiting each of the four sites does indeed on average have an accurate estimateof the scaled average nectar influx (e.g., for sites 1-3 it is the same as it should be). Figure 8 showsthat average dance strengths of the groups visiting each site (and hence recruitment rates) arehigher for relatively more profitable sites. The variances Figure 8 show, however, that the dancedetermination by individuals is relatively error-prone. Figure 10 shows that the hive manages tolargely ignore an inferior site, but that as its profitability increases, it receives increasing attention.The “flexibility” discussed above results in the continual monitoring of the environment and itseffect is illustrated by the discrepancy between the average profitabilities for the two sites for therange of 0.25 < N2
f < 0.6 in Figure 10. In this range, the hive “over-samples” the relatively inferiorsite in order to monitor it, a characteristic found in other IFD studies (Tregenza 1995).
Next, Figures 11 and 12 show that the hive can quickly reallocate foragers if there is a signif-icant change in site profitability. The key to this reallocation is how the hive manages the poolof unemployed foragers (see discussion below), and the fact that bees decide not to dance. Whenrelatively few bees dance (as illustrated in Figures 6 and 12), but the dance strengths are pro-portioned according to currently exploited site profitabilities, each dance has a larger impact on
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the change in the allocation than if all employed foragers danced. So, by saving time-energy costsof dancing the hive also gains allocation speed. Note that experiments suggest that keeping theamount of dancing relatively low is also a characteristic of pollen and water foraging (Seeley 1995).Since dancers for nectar, pollen, and water share the same dance floor, this ensures that dances foreach resource can have a relatively large impact on recruitment (e.g., if suddenly there is a strongdemand for water, dancers for water sources will be able to effectively recruit to supply it). Hence,the multitude of individual choices not to dance helps (i) decouple foraging processes for differentresources and (ii)ensure that the hive can simultaneously and quickly respond to make all resourcesupplies match all hive demands.
It is important to highlight how the resting, observing, exploring, and recruitment occurs in away to contribute to the effectiveness of the foraging. First, the hive essentially manages the size ofthe pool of unemployed foragers and explorers. When there is little available forage, there are moreexplorers dispatched. As sites are found and the number of recruits increases, the number of unem-ployed foragers decreases and the hive gradually switches from a search mode to a forage exploitationmode, but there are always some explorers. This is illustrated in Figure 6. Other simulations showthat if there are relatively few good sites, then there is more exploration and the pool of resters islarger and this contributes to the ability of the hive to quickly exploit a newly discovered highlyprofitable site. Hence, depending on conditions, forager reallocation can occur via the deploymentof unemployed foragers and/or the abandonment-recruitment process. Finally, we note that a recenttheoretical study on the optimal proportion of forage searchers versus recruited foragers shows animportant dependency on food source availability duration (Dechaume-Moncharmont et al. 2005).It would be of interest of experimentally validate the mechanisms underlying scout/recruit alloca-tion (see discussion in Section 2.4), then use the principles in (Dechaume-Moncharmont et al. 2005)to study the adaptation of the relevant behavioral parameters (this would require, however, a num-ber of modifications to our model along with new simulation test scenarios for time-varying foragesite profitabilities).
5.2 Behavioral Parameter Adaptation
As further confirmation of the above ideas, in Figures 13 and 14 we show that natural selectionseems to have settled on values of the dance threshold parameter and dance function slope thatrepresent a compromise between maximizing nectar intake and minimizing the number of dances toachieve that intake. Notably, the value of the dance function slope parameter β is approximatelythe one found experimentally in (Seeley 1995). Moreover, the dance threshold parameter δ isnear the higher end of the feasible range in Equation (5). This value enables the hive to quicklyabandon inferior sites, yet still achieve and maintain a proportional allocation since the average siteprofitability is above this value. Figure 13 also confirms the model formulation by showing that ifδ is at either extreme in the range given by Equation (5) nectar influx suffers. If δ is at the highend of the range, then the foraging process fails since all sites are abandoned. If δ is at the low endof the range significant time-energy costs are incurred due to dancing.
Next, we illustrated that foraging performance suffers little if the individual profitability as-sessment noise magnitude wf ∈ [0, 0.1], but that if wf > 0.1, then to maintain an effective foragerallocation it costs more dances, and if the noise is too high average nectar influx can also degrade.This shows that individual level assessment errors of sufficiently low magnitude do not propa-gate to adversely impact colony-level foraging performance (i.e., the social foraging process filtersindividual-level errors). Finally, note that in other simulations (that we omit to in the interest
25
of brevity) we have shown that there are similar trade-offs to what we found in Section 4 for thetendency to explore σ and tendency to dance φ.
5.3 Relations to Nest-site Selection by Honey Bees: Dynamics and the Speed
and Accuracy Trade-off
Unlike any of the other models discussed in the Introduction, ours is formulated to uncover thecommon features of social foraging and nest-site selection by honey bees. There are only twodifferences between the model here and the one in (Passino and Seeley 2006b). First, we use a“forage profitability landscape” rather than a “nest-site quality landscape” since the task hereis foraging. Second, we change how the individual bees choose dance strength. The remainingparts of the model, particularly recruitment, explorer allocation, resting, and dancing observingare structurally the same as in (Passino and Seeley 2006b). The change in dance strength choiceresults in very different emergent decisions: choice of one site vs. proportionate allocation of beesacross many sites. However, the resulting dynamics of the two distributed decision-making processesshare a number of features that provide insights into generic features of the functionality of socialdecision making by honey bees.
To see the close relationships, here we will think of nest-site quality and forage site profitabilityas generic measures of “goodness” of sites. First, note that while dance strength is superficiallyproportional to site quality, in nest-site selection it decays linearly after the bee’s first visit whilein foraging it can remain at a high level for a long period of time. Interestingly, in the transientthere can often be an average decay of the dance strengths in social foraging due to an increase incolony nectar influx that raises the dance threshold, but the level of dancing will not go to zerofor the superior sites (e.g., see Figure 8). And, of course such transients can occur in the oppositedirection and dance strengths can increase, then decrease, especially when the dance thresholddecreases due to a reduction in colony nectar influx. Second, positive feedback is used in bothprocesses for recruitment of more bees to a site. In social foraging it ensures that the best sitesget recruits much faster than inferior sites (see Figure 8). In nest-site selection positive feedbackis stronger for higher quality sites and this leads to quicker quorum achievement for the best site.Negative feedback causes site abandonment. In social foraging negative feedback arises due to thedynamic changes in the dance threshold, while in nest-site selection it is due to the dance decaycharacteristic. Third, there are cross-inhibition effects between sites for both processes. Fourth, inboth cases the average number of bees allocated to a site is in proportion to how good it is. Innest-site selection the numbers of site-assessor bees is allocated on average according to relativenest-site quality according to an IFD. This ensures that the best sites are fully evaluated before theyare chosen so mistakes are avoided. In social foraging the proportionate allocation achieves an IFD,but by having more bees at the best sites there will tend to be fewer recruitment/abandonmenterrors for those sites. Hence, the hive’s ability to achieve an IFD is a fundamental property of itssocial decision-making. It is achieved in quite different ways for the two processes, yet is a key tothe success of both processes.
In nest-site selection there is a fundamental trade-off between speed and accuracy of choice(Passino and Seeley 2006b). Fast decisions lead to choice errors. In social foraging there is analogoustrade-off between reallocation speed and the accuracy of the allocation with respect the IFD.Recall that in social foraging if large numbers of bees visit a site: (i) the group on average makesrelatively few mistakes, but (ii) this leads to slower reallocations since then each dance cannothave as significant of an influence on the reallocation. Hence, the amount of dancing in social
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foraging represents a trade-off between achieving fast yet effective forager allocation patterns. It is animportant future direction to conduct experiments to further explore this speed and accuracy trade-off. This would involve conducting experiments where we establish forage sites whose profitabilitycan be represented as a function of the number of site visitors, then simultaneous recording ofall forage site and hive variables when there are significant site profitability changes of varyingmagnitudes.
5.4 Relations to Group Cognition
The perspective that inter-animal sociality can achieve (i) complexity in form and function via mas-sively parallel interconnections, and (ii) cognition that is functionally equivalent to neuron-basedbrains has been recognized for some time (e.g., see (Markl 1985; Bonner 1988)). Recently, it wasshown that nest-site selection is a “group cognition process” that shares close relationships withthe structure, functionality, and properties of neural-based cognition (Passino and Seeley 2006a).Here, to provide an alternative and abstract view of the dynamics of social foraging, we brieflyexplain how to view social foraging by honey bees as a group cognition process. From such aperspective, social foraging is most similar to attentional processes (Gazzaniga et al. 1998). Theforaging environment is the “field of view” of the colony and low level sensory and cognition units(bees) provide for persistent monitoring of the environment for changes. There is a random buton average predictable pattern of inter-bee communications via dances that provides for signal-ing between cognition units. The set of bees visiting each forage site provides the colony with agroup-level memory of the current allocation of attention, and the underlying dance determinationmechanisms result in proportionally more attention (foragers) being paid to the relatively impor-tant locations in the environment. Just like in nest-site selection, the quality of the group memory(a type of “internal model” of the bees’ problem domain) is enhanced by the averaging effects ofthe entire group of bees visiting a site. The “late” processing stages (Gazzaniga et al. 1998) ofthe foraging attentional process enable for reallocation of attention. The quick reallocation abil-ity enables refocusing of attention, and the dance threshold emerges as a dynamic threshold thatenables the hive to ignore relatively unimportant sites. An important future research direction isto move beyond such analogical connections to determine if our understanding of social foragingcan offer insights into the dynamics, pathways, or functionality of neural-based attentional systems.
Acknowledgements: The research reported here was partially supported by the OSU Office ofResearch. I would like to thank Thomas D. Seeley for many helpful conversations on the socialforaging of the honey bees. I also would like to thank Jorge Finke and Nicanor Quijano for checkingthe simulation code and for some suggestions on the document.
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6 Appendix: Additional Simulations–Not for Publication
A number of other simulations were conducted to determine the impact of other parameters. Theones below are not, however, very useful as they are conducted for a certain type of landscape(four relatively good constant-quality sites that can be easily found) so that in tuning the valueswe cannot consider the effects of the need to search to discover difficult-to-find sites, and theneed to maintain flexibility (e.g., via a pool of resting foragers) in case there are quick changesin the landscape (e.g., a site that suddenly becomes excellent). To fully consider the effects ofthe parameters below other landscapes would have to be considered. The simulations below simplysupport that the values chosen in the validated model indeed make sense for the chosen performancemeasures, but modified by search/flexibility considerations.
6.1 Effect of Tendency to Explore
We vary σ over the range [500, 1500]. Figure 16 shows that the total average nectar profitabilityreturn is maximized at about σ = 800. Increasing values of σ result in an increase in the averagetotal number of dances (and number of dancers) since it then takes more dances to achieve re-cruitment since the pool of unemployed foragers is smaller since there are more explorers. A smallσ maximizes the number of deployed foragers since there is then less tendency to explore. Note,however, that we are considering only one type of landscape in making these conclusions, one wherethere are four good sites that are easy to find. For other landscapes where sites are not as plentifulor profitable there is a need for a value of σ > 800 and via our simulations we found that a valueof σ = 1000 was suitable.
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Figure 16: Foraging performance as a function of σ. See caption of Figure 13 for a description ofvariables.
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6.2 Effect of Tendency to Dance
We vary φ over the range [0, 1]. Figure 17 shows that the total average nectar profitability returnis maximized for high values of φ. As φ increases more dancing is needed and more dancers used.Moreover, more foragers are deployed. What is the best value for φ? Considering the trade-offs anyvalue of φ > 0.7 seems reasonable; however, note that this would mean that even when no foragesources are being exploited, if an explorer would find a new and most highly profitable site it wouldonly dance, for example, 70% of the time. Since this does not happen in nature, we conclude thatφ must be on the higher end of the range than a value suggested by these plots would indicate.This is why we use φ = 1.
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Figure 17: Foraging performance as a function of φ. See caption of Figure 13 for a description ofvariables.
6.3 Effect of Tendency to Observe Dances
We vary po over the range [0.1, 0.5]. Figure 18 shows that the total average nectar profitabilityreturn is maximized for high values of po and the average total number of dances used decreases.This would imply that po should be chosen as po = 0.5 (or higher) so that the size of the unemployedforagers pool (resters) will go down, perhaps very low; but this does not take into account otherlandscapes that are dynamically varying and here there is the need for a reserve of foragers thatcan be quickly deployed.
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Figure 18: Foraging performance as a function of po. See caption of Figure 13 for a description ofvariables.
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