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 [email protected] December 14, 2006

Abstract There are two main components of social foraging by honey bees. First, there is a continuous adjustment of the number of bees dedicated to exploitation of forage sites and wide-area exploration for new forage options. Second, there is a persistent reallocation of the proportions of the foraging workforce dedicated to exploiting each site in order to match unpredictably changing relative forage site profitabilities. In this paper we 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 model also represents that as additional bees forage at a site its profitability generally degrades due to exploitation competition; hence, we can study the intimate coupling between the dynamics of forage site profitability and the hive’s reallocation of foragers. To illustrate these dynamics, we first show how in spite of the fact that no bee can know the profitability of all sites, the hive achieves an approximate “ideal free distribution” (IFD) of foragers with an allocation of foragers proportional to relative site profitability. We identify the mechanisms underlying the achievement of the emergent distribution and the hive’s ability to completely ignore forage sites of relatively inferior quality. We explain how the mechanisms lead to fast reallocations when there are sudden and significant site profitability changes or new site discoveries. Next, we show that natural selection seems to have settled on values of the behavioral parameters representing the dance strength determination rule 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 errors since the foraging process effectively filters such errors at the colony level. Finally, we identify the close relationships between the dynamics of social foraging and nest-site selection by honey bees since it provides an excellent example of how changing a single individual-level behavioral rule can result in dramatically different emergent group behavior. Building on these relationships we briefly discuss how social foraging can be viewed as a group cognition process.

Keywords: Apis mellifera, honey bee, foraging, group decision making, collective decision making.

1

1

Introduction

The aggregate behavioral properties of a group of social animals are typically emergent phenomena that arise from many individual decisions that are made with local inaccurate information and limited inter-individual cues or signals (reviewed by Camazine et al. 2001). Examples where the emergent 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 a mathematical model of social foraging by honey bees and use it to investigate how local poorlyinformed decisions by bees, mediated by limited inter-bee communications, can lead to fast yet effective allocations of nectar foragers in a changing foraging environment. Several models of social foraging by honey bees have been published. A differential equation model of functional aspects of dynamic labor force allocation of honey bees is developed and validated 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 represent social foraging processes in both bees and ants. Like (Sumpter and Pratt 2003), our model of recruitment uses the idea from (Seeley et al. 1991; Camazine and Sneyd 1991) that dance strength proportioning 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 discussed by Seeley (1995)). Here we make such connections more concrete by modeling the bee recruitment process in an analogous manner to how survival of the fittest and natural selection are modeled in genetic 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 models and experiments in (Seeley and Tovey, 1994; Seeley 1995) and is consistent with the analysis in (Anderson and Ratnieks 1999) on how the random unload wait-time information is used. To determine the proportion of foragers that should explore versus exploit known forage sites, we use the experimental results in (T. Seeley 1983) to model “recruitment to exploring” as being inversely proportional to the wait time to find a recruiting dancer. The optimal proportion of explorers and resters 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 the relative 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 illustrate the emergent allocation dynamics. de Vries and Biesmeijer (1998) introduce an “individual-oriented” model of social foraging and validate it against one set of experimental conditions as was done in (Seeley et al. 1991; Camazine and Sneyd 1991). More recently, de Vries and Biesmeijer (2002) expanded and improved the model in (de Vries and Biesmeijer 1998) (e.g., taking into account the findings in (T. D. Seeley and Tovey 1994)) and studied how the number of foragers visiting two equally profitable sites can diverge (“symmetry breaking”), cross inhibition (as studied in (Seeley 1995), p. 143), and for different currencies what they call an “equal harvest rate distribution” (EHD) where patch size provides negative feedback for recruitment (i.e., their patch degradation is driven by “interference” between foraging bees (Parker and Sutherland 1986; Giraldeau and Caraco 2000)). Additional detailed models of the social foraging process are found in (Cox and Myerscough 2003; Schmickl and Crailsheim 2003). These models quantify most of the features of the other models and additional characteristics, and like others rely on the work in (T. Seeley et al. 1991; Camazine and Sneyd 1991). 2

Moreover, Cox and Myerscough (2003) include polyandry-induced individual bee differences and use simulations to show that a heterogeneous colony can reduce the variance in the average amount of nectar collected. Bartholdi et al. (1993) study the pattern of forager allocation and the optimality of it. In related work, Dukas and Edelstein-Keshet (1998) study the spatial distribution of solitary and social food provisioners under different currency assumptions. The work in (Bartholdi et al. 1993) and (Dukas and 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 conditions under which their EHD (see above) is reached and discuss the impact of currency and relations to the IFD. Experimental evidence that the honey bees seem to achieve an IFD of foragers has been found in a number of experimental scenarios (e.g., in (Seeley 1986; Seeley et al. 1991; Seeley and Towne 1992)). Of course, for such studies hive information-flow restrictions and physical bee travel 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, only approximate achievement of matching the relative numbers of employed foragers to relative site profitability is viewed as achievement of an IFD. Moreover, there is the practical matter that the theory of the IFD (Fretwell and Lucas 1970; Giraldeau and Caraco 2000) normally assumes that there is a large (theoretically infinite) number of foragers so that forager density is a continuous variable. Otherwise, achievement of equal fitness is not always possible since it depends on the relative number of sites and foragers. In experiments, even relatively few bees have been found to achieve an “IFD.” Here, consistent with experimental work, our simulations result in achievement of a distribution that is close to what the literature refers to as an IFD. We sidestep using a detailed characterization of energetics and currency (since there is not enough experimental evidence to justify which currency to use) in favor of using a generic measure of forage site profitability. While this lessens the predictive power of our model, we are still able to make conclusions about the essential underlying mechanisms that lead to an emergent IFD, and we can match the qualitative behavior found in a wide array of experiments. Unlike in (de Vries and Biesmeijer 2002) where the 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 a patch (we have also tested suitability functions that represent “interference” and get qualitatively similar results). Unfortunately, there is no experimental work to fully justify either our approach or the approach in (de Vries and Biesmeijer 2002); however, both provide plausible representations for 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 bee expeditions (exploration or foraging), unloading, dance strength decisions, explorer allocation, recruitment on the dance floor, and accounting for interactions with other hive functions. The experimental studies we rely on are summarized in (Seeley 1995). Our primary sources for constructing 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 floor and recruitment rates (Seeley et al. 1991); and explorer allocation and its relation to recruitment (Seeley 1983; Seeley and Visscher 1988).

3

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 i is represented by θ i ∈ 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 (Seeley 1995). During foraging, bees sample a “foraging profitability landscape” which we think of as a spatial distribution of forage sites with encoded information on foraging profitability that quantifies distance 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 is proportional to the profitability of nectar at a location specified by θ ∈ ℜ2 . Hence, Jf (θ) = 1 represents a location with the highest possible profitability, Jf (θ) = 0 represents a location with no profitability, 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] since the distance from the hive is assumed to be represented in the landscape. We assume the hive is at [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 since the hive seems to employ foraging strategies to optimize the net rate of energy gain or efficiency of energetic gain at different times (Seeley 1995) and it is not clear how the bees switch between these two strategies (or indeed if other strategies are used, such as those based on variance to achieve risk-sensitive behavior (Giraldeau and Caraco 2000)); (ii) energetic expenditure for flight to a site when the dependency on wind and temperature conditions is taken into account; (iii) energetic expenditure to search for and find both new and existing forage sites bees have been recruited to; (iv) energetic content of the nectar of multiple simultaneously available species of flowers and energetic 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 a large region that is searched and exploited during foraging. It is for this reason that we take a different approach here. We assume that Jf (θ) represents the overall profitability at location θ and this can include many of the factors above. This assumption clearly decreases the predictive power of the model; however, it allows us to easily set numeric values for several behavioral parameters in terms of experiments, and still allows us to represent general qualitative behavior of the social foraging process and validate this behavior with respect to experiments. An example of the type of foraging profitability landscape we will consider has four forage sites centered at various positions that are initially unknown to the bees. For convenience, we will refer to 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 height is proportional to the nectar profitability. Here, we use cylinders with heights Nfj ∈ [0, 1] that are proportional to nectar profitability, and the spread of each site is defined by the radius of the cylinders which we take to all be ǫf = 0.25. It must be emphasized that all foraging profitability 4

information is “encoded” in the Jf map; hence, you cannot directly think of the domain of Jf as a square foraging area for the bees. For instance, if you scaled the θ1 and θ2 axes to kilometers, then the Jf map would have to be appropriately deformed to represent that forage sites further away from the hive are less profitable. Below, we will say that bee i, θ i = [θ1i , θ2i ]⊤ , is “at forage site 1” if q

(θ i − [0.5, 0.7]⊤ )⊤ (θ i − [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 return with their foraging profitability assessments at one time (an asynchronous model with randomly spaced arrivals and departures will behave in a qualitatively similar manner). Our convention is that at time k = 0 no expeditions have occurred (e.g., start of a foraging day), at time k = 1 one has occurred, and so on. All bees, i = 1, 2, ..., B, have θ i (0) = [0, 0]⊤ so that initially they are at the hive. We assume that the bees forage for 10 hours and make a foraging expedition every 10 min. for a total of 60 foraging trips in one day (this only represents one scenario where forage sites are not too far from the hive). We will conduct our simulations over this one day and not consider day-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 that the profitability of being at site j, which we denote by Jfj for a bee at a location in site j, decreases as the number of bees visiting that site increases, and represent this by letting, for each j, Jfj

=

Nfj

1 exp − (B(j, k))2 /σf2 2 



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 assumption in theoretical ecology (e.g., in IFD theory Jfj is called the “suitability function” (Fretwell and Lucas 1970)). We will consider cases where the profitability parameter of a site, Nfj ∈ [0, 1], will stay constant, and other cases where it will vary over time. Increases in Nfj 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) “employed foragers” (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 employed foragers 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 resters and observers. We assume that there are Be (k) “forage explorers”1 that go to random positions in the environment, bring their nectar back if they find any, and dance accordingly, but were not dedicated 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 activities they perform. We assume that a bee simply samples the foraging profitability landscape once 1

In 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 the foraging and nest-site selection processes are conceptually the same.

5

on its expedition and hence this sample represents its combined overall assessment of foraging profitability for location θ i (k). It is this value that it holds when it returns to the hive. It also brings back knowledge of the forage location which is represented with θ i (k) for the kth foraging expedition. Let the foraging profitability assessment by employed forager (or forage explorer) i be i

N (k) =

   1

J

f   0

(θ i (k))

+

if Jf (θ i (k)) + wfi (k) ≥ 1 if 1 > Jf (θ i (k)) + wfi (k) > ǫn if Jf (θ i (k)) + wfi (k) ≤ ǫn

wfi (k)

where wfi (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 wfi (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 accurate profitability assessment since the expected value with respect to k of wfi (k), E[wfi (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 assume that this dead bee is replaced by a novice forager and hence for it we let N i (k) = 0. Since we assume that deaths occur independent of the quality of a site, the deaths of bees will not on average affect the pattern of forager allocations to forage sites at a single step; however, there is an impact on the pattern of allocation over many steps since these deaths cause site abandonments that result in a hive-level flexibility for reallocation. Here, to be consistent with experiments, we let pd = 0.0017 so that 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 die by step k, including at step k, be Bdt (k). Next, we must specify the locations where all the bees will go on their expeditions. Unemployed foragers stay at the hive. The locations for the Be (k) forage explorers to forage at the next step are defined by placing them randomly on the foraging landscape with a uniform distribution. Employed foragers do not go back to the precise spot in the forage site that they visited during the last foraging expedition. To represent this, if the employed forager was at θ i (k) for expedition k, then for its next expedition we let θ i (k + 1) = θ i (k) + [e1 , e2 ]⊤ where e1 and e2 are zero mean Gaussian random variables with variance σe2 . Each time a recruit tries to find the dance-indicated forage location it frequently makes errors (Seeley 1995). Hence, if recruiter bee i indicates to go to its last position θ i (k), the j th recruited forager will go to θ j (k + 1) = θ i (k) + [r1 , r2 ]⊤ where r1 and r2 are zero mean Gaussian random variables with variance σr2 . Here, we let σe2 = 0.001 and σr2 = 0.002, representing that it is more difficult for the recruit to find the site than the bee that 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 6

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 unemployed foragers. The novice foragers we introduce to replace dead bees have Lif (k) = 0. 2.3.1

Wait Time to Get Unloaded

Next, we will explain dance strength decisions for the employed foragers and forage explorers with N i (k) > ǫn . To do this, we first model wait times to get unloaded and how they influence the “dance threshold.” We will ignore the effect of tremble dances on recruitment to nectar processing since it just produces a transient in the unload wait-times that are proportional to nectar influx. By ignoring this we are essentially assuming that via waggle and tremble dances the colony is consistently maintaining a balance between its rates of nectar collecting and processing, a characteristic that has been validated experimentally (Seeley 1995). i Define Nt (k) = B i=1 N (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 profitability sites. Let Nqi (k) be the quantity of nectar (load size) gathered for a profitability assessment N i (k). We assume that Nqi (k) = αN i (k) where α > 0 is a proportionality constant. We choose α = 1 so that Nqi (k) ∈ [0, 1], with Nqi (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,

P

Ntq (k) =

B X

Nqi (k) = α

i=1

B X

N i (k) = αNt (k)

i=1

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 total nectar influx. Suppose that the number of food-storer bees is sufficiently large so the wait time W i (k) that bee i experiences is given by n

o

n

o

i i W i (k) = ψ max Ntq (k) + ww (k), 0 = ψ max αNt (k) + ww (k), 0

(2)

i (k) is a random variable that represents variations in the where ψ > 0 is a scale factor and ww i (k) is uniformly distributed on (−w , w ). Since wait time a bee experiences. We assume that ww w w Ntq (k) ∈ [0, αB], ψ(αB + ww ) is the maximum value of the wait time which is achieved when total nectar 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 will tend 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 of seconds in wait time due to the noise and ww should be set accordingly. We let ψww = 5 to get a variation of ±5 seconds. If B = 200 is known, we have two equations and two unknowns, so combining 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 sufficiently high nectar influxes is justified via experiments described in (Seeley and Tovey 1994) and (Seeley i (k) noise and the “max” in 1995), p. 112. Deviations from linearity come from two sources, the ww Equation (2). Each successful forager has a different and inaccurate individual assessment of the total nectar influx since each individual bee experiences different wait times in the unloading area. 7

i (k) in Equation (2) represents this. Some foragers can get lucky and get unloaded The noise ww quickly and this will give them the impression that nectar influx is low. Other foragers may be unlucky and slow to get unloaded and this will result in an impression that there is a very high nectar 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 (Seeley and 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 relationship between Ntq (k) and W i (k) that Equation (2) implements since: (i) for low values of Ntq (k) small increases in nectar influx will not increase wait times significantly (since unloaders are relatively abundant so foragers can easily find them), and (ii) for high values of Ntq (k) increases in nectar influx will cause more significant increases in wait times (since there are relatively few unloaders and 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 that only occurs for relatively small values of Ntq (k), and results in a positive average wait time even for very low nectar influxes. A similar effect is seen in the experimental results in (Seeley and Tovey 1994) for low Ntq (k) values. In summary, the curve implemented by Equation (2) roughly has the form of an increasing exponential on average for low values of total nectar influx, but allows for significant 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 scaled version of an estimate of the total nectar influx that we define as i ˆtq N (k) = δW i (k)

(3)

So, we are assuming that each bee has an internal mechanism for relating the wait time it experiences to its guess at how well all the other foragers are doing. The proportionality constant for this is i (k) ∈ [0, 30δ]. ˆtq δ > 0 and since W i (k) ∈ [0, ψ(αB + ww )] = [0, 30] sec. we have N So, how does total nectar influx influence the dance strength decision, and in particular the dance 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 dance threshold for each individual bee shifts based on the ith bee’s estimate of total nectar influx. The decision function is n   o i ˆtq Lif (k) = max β N i (k) − N (k) , 0 (4) which is shown in Figure 1. The parameter β > 0 affects the number of dances produced for an above-threshold profitability.

i (k) is the intercept on the dance strength axis. The diagonal bold line in ˆtq In Figure 1, −β N Figure 1 shifts based on the bee’s estimation of total nectar influx since this is proportional to i (k). Now, the key is to notice that since the line’s slope is β, and since we take the maximum ˆtq N with zero in Equation (4), the lowest value of nectar profitability N i (k) that the ith bee will decide ˆ i (k) = δW i (k), the ˆ i (k) and from Equation (3), N to still dance for is the “dance threshold” N tq tq bee’s scaled estimate of the total nectar influx. Note that changing β does not shift the dance

8

i

L (k) f

Nectar influx decrease

Dance strength, number of waggle runs

Nectar influx increase Slope= β

^i Ntq (k)= δW i(k) ^i - β Ntq (k)

Dance threshold for bee i

i

N (k) Nectar profitability for bee i

Figure 1: Dance strength function and dance threshold (δ adjusts the threshold and β affects the number 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 for ˆ i (k) = 0 there is no nectar influx sites above the dance threshold. In the case where Ntq (k) = N tq to the hive and it has been found experimentally (Seeley 1995) that in such cases, if a bee finds a highly profitable site, she can dance with 100 or more waggle runs. Hence, we choose β = 100 so Lif (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 we expect a bee to hold for δ? Since the nectar profitability N i (k) ∈ [0, 1], δ needs to be defined so ˆ i (k) ∈ [0, 1] so that the dance threshold is within the range of possible nectar profitabilities. that N tq This means that we need 1 (5) 0 0 are ones that consider dancing for their forage site. In nature, even bees that visit the most profitable site being considered by the entire hive will not necessarily dance for it. Correspondingly, bees dedicated to a relatively poor site might, or might not, dance for it. The tendency to 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 i L (k) β f

where φ ∈ [0, 1] (which ensures that pr (i, k) ∈ [0, 1]). We choose φ = 1 since it resulted in matching the qualitative behavior of what is found in experiments. Hence, a bee with an above-threshold profitability 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 recruited for 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 sites are not as likely to be danced for; however, bees that decide not to dance will still go back to the site 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 Lif (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 Lif (k) = 0 and the bee simply remains an employed forager for its last site. We let Bf d (k) denote the number of employed foragers with above-threshold profitability that dance. The value of φ will influence the total number of bees that decide to dance at each step; however, on average it will not change the relative proportioning of the total dance strength pattern for the currently exploited forage sites. Choosing not to dance has the benefit of not wasting valuable time and energy recruiting when the bee could be foraging. There is, however, another important benefit to not having all the forgers dance: each individual dance counts for more in the percentage of dances for each exploited site. Hence, if there is a site profitability change, or a newly discovered site, a single bee’s dancing can quickly result in a reorientation of a large portion of the foraging workforce. The speed of reorientation is aided by the positive feedback due to recruiters also recruiting. So, the sensitivity of the the hive to changes in foraging environment (and hence reaction speed) is enhanced by less, not more, dancing. Of course, some minimal level of dancing is needed or there will be no proportionate allocation, or a noisy allocation. These observations agree with experimental studies that show that a relatively small percentage of foragers dance (Seeley 1995).

10

2.4 2.4.1

Explorer Allocation and Forager Recruitment Resters and Observers

The bees that either were not successful on an expedition, or were successful enough to get unloaded but judged that the profitability of their site was below the dance threshold, become unemployed foragers. Some of these bees will start to rest and other dance “observers” will actively pursue getting involved in the foraging process by seeking a dancing bee to get recruited. Here, at each k we let po ∈ [0, 1] denote the probability that an unemployed forager or currently resting bee will become an observer bee; hence 1 − po is the probability that an unemployed forager will rest or 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 bees performing as forage explorers, but when there are many sites being harvested there can be as few as 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 allocated in the social foraging process. The problem is that while the recruitment process to forage sites via waggle runs is relatively well-understood, it is not precisely known why a bee chooses to be recruited rather than to explore (and vice-versa). Here, we assume that an observer bee on the dance floor searches for dances to follow and if it does not find one after some length of time, it gives up and goes exploring. That is, we assume the bee uses the wait-time to find a dancer as a cue about whether it should explore. Full verification of this awaits experimental study. However, it is well-known that in social foraging bees use the wait-time to get unloaded in order to estimate hive nectar influx and hence compare the profitability of the site it just visited to how other foragers are doing so that it can decide whether to dance and for how long (Seeley and Tovey 1994). And there is some evidence (Seeley 1983) that foragers decide to explore for a new food source, rather than 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 so many bees that can observe the dancing at any one time (due to physical constraints that lead to only a limited number of observers for each dancer). Such physical constraints lead to a limited number of recruits, and impose natural delays in any observer bee getting recruited. It seems logical that if the bee is delayed too much, it will decide simply to go find a site on its own. The result will be consistent with experiments in that when nectar intake is low there will be less dancing and hence more exploring in order to find new sources. To model explorer allocation based on wait-time cues, we assume that the wait-time is inversely proportional to the bees’ tendency to be recruited to a forage site. Considering experimental studies (Seeley and Towne 1992; Seeley 1994; Seeley and Tovey 1994), the wait-time is assumed to be proportional to the total number of waggle runs on the dance floor. The more bees that are dancing strongly on the dance floor, the more likely it is that an observer bee will quickly find a dancer to follow. Let Bf (k)

Lt (k) =

X

Lif (k)

i=1

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 1 L2t (k) pe (k) = exp − 2 σ2

!

(6)

Notice that if Lt (k) = 0, there is no dancing on the cluster so that pe (k) = 1 and all the observer bees 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 a dancer and get recruited to a forage site. Here, we choose σ = 1000 since it produces patterns of foraging behavior in our simulations that correspond to experiments. The explorer allocation process is concurrent with the recruitment of observer bees to forage sites. Observer bees are recruited to forage sites with probability 1 − pe (k) by taking any observer bee that did not go explore and have it be recruited. To model the actual forager recruitment process we view Lif (k) as the “fitness” of the forage site that the ith bee visited during expedition k. Then, the probability that an observer bee will follow the dance of bee i is defined to be Lif (k) pi (k) = PB (k) f i i=1 Lf (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 experiments and explain the mechanisms and resulting dynamics of the social foraging process. While each simulation run of the foraging process for one day is slightly different due to random effects, we next 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 Nf1 = 1, Nf2 = 0.9, Nf3 = 0.6, and Nf4 = 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 foraged on the landscape. The bees sample much of the landscape during the day, but most foragers concentrate their activities on the profitable foraging sites, with proportionally more for more profitable sites. The actual numbers of bees visiting sites 1-4 is shown in Figure 3 and the corresponding profitabilities are shown in Figure 4. Figure 3 shows that more bees are allocated to more profitable sites, with the number of bees allocated so that the profitabilities of the best sites are made nearly equal as shown in Figure 4. Moveover, the bees manage to allocate almost no bees to the least profitable site 4 since the achieved profitability level of sites 1-3 is above that of site 4; in fact, the visitors there are mostly ones that discover the site by random chance, not ones that are employed as foragers for it. 12

Profitability J (contour) and bee sample points (dots) f

1 0.8 0.6 0.4

θ2

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 θ1

0.2

0.4

0.6

0.8

1

40

20

0

B(3,k), #bees site 3

B(2,k), #bees site 2

60

0

2

4

6

8

60

40

20

0

0

2

4 6 Time, hours

8

60

40

20

0

10

B(4,k), #bees site 4

B(1,k), #bees site 1

Figure 2: Foraging profitability landscape (contour) with each black dot representing a location where 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-left corner.

2

0

2

4

6

8

10

8

10

60

40

20

0

10

0

4 6 Time, hours

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 in spite of the fact that no bee knows the profitability of all the sites, the foragers are allocated so as to 13

1 J2f , profitability, site 2

f

J1, profitability, site 1

1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

0.4 0.2

0

2

0

2

4

6

8

10

8

10

1 J4f , profitability, site 4

f

J3, profitability, site 3

0.6

0

10

1 0.8 0.6 0.4 0.2 0

0.8

0

2

4 6 Time, hours

8

0.8 0.6 0.4 0.2 0

10

4 6 Time, hours

Figure 4: Profitability Jfj for site j at k. achieve what is close to an IFD. The numbers allocated seek to make the average profitability of all visited sites the same, and this is illustrated by how close the three horizontal lines corresponding to 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 deviations can be viewed as useful since the distribution does not “lock-in” and stagnate; the hive maintains flexibility 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 is persistently 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 IFD is 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 the number of employed foragers increases, the number of explorers decreases, but some exploration is done over the entire day to monitor the foraging environment for changes. The number of new recruits is initially large, but decreases as the distribution of foragers reaches an equilibrium. The number of dancers increases quickly initially as the sites are discovered, and then the dancing is continued all day in order to maintain the distribution of foragers. Finally, note that the number of unemployed foragers, and obervers and resters, decreases as the number of employed foragers increases 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 quick loss in profitability of all sites since then explorers can be quickly deployed.

14

Profitability avg (horizontal −−), avg no. of bees (vertical −−) 1 0.9

Suitability functions, sites 1−4

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

20

40

60

80

100 120 Number of bees

140

160

180

200

Figure 5: Suitability functions for sites 1-4 (solid lines, Gaussian shapes, in order from top to bottom), average profitability levels for sites 1-4 for last 5 hrs. (dashed horizontal lines), average number of bees visiting sites 1-4 for last 5 hrs. (dashed vertical lines).

Bf(−), Bdt(−−), Bfd(−.)

Numbers of bees in different roles 200 150 100 50 0

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 Time, hours

6

7

8

9

10

Be(−), Brec(−−)

200 150 100 50

Bu(−), Bo(−−), Br(−.)

0 200 150 100 50 0

Figure 6: Number of employed foragers Bf , total number of bees that die Bdt , number of bees that dance Bf d (top plot); number of explorers Be and number of recruited bees Brec (middle plot); and number 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 bees visiting sites 1-4. For sites 1-3 the threshold is on average the same since bees are allocated to make those sites ultimately have the same profitability. For site 4 first note that by convention we use zero to be the mean wait time in the plots for the degenerate case when there are no bees visiting that site. When a bee does visit site 4 and unloads, its estimate of the dance threshold is in the same range as for bees visiting sites 1-3 since they are all getting unloaded on in the same area and no bee is treated preferentially in unloading. So, when a bee visits site 4 the maximum average profitability assessment it can have is 0.3 (since the maximum of its suitability function is 0.3); hence, most often the profitability assessment is not above (or only slightly above) the dance threshold for the bee so it is unlikely to dance for the site, and even if it does dance it will not perform many waggle runs since it will not be much above the threshold. This results in there being very few recruits to site 4. And, whenever the profitability assessment for site 4 is below the threshold the bee will abandon the site; since abandonment is more likely than recruitment, the site is ignored.

0.3 Ntq hat avg/std(bars), site 2

Ntq hat avg/std(bars), site 1

0.3 0.25 0.2 0.15 0.1 0.05 0

0

2

4

6

8

0.15 0.1 0.05 0

2

0

2

4

6

8

10

8

10

0.3 Ntq hat avg/std(bars), site 4

Ntq hat avg/std(bars), site 3

0.2

0

10

0.3 0.25 0.2 0.15 0.1 0.05 0

0.25

0

2

4 6 Time, hours

8

0.25 0.2 0.15 0.1 0.05 0

10

4 6 Time, hours

i mean and standard deviation (bars) dance threshold estimates for bees visiting sites ˆtq Figure 7: N 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 four sites in Figure 8. Notice that in this case there is initially a significant amount of dancing and hence recruitment, but eventually the amount of dancing for sites 1-3 decreases as shown in Figure 4. Site 4 has almost no dancing for it, but once in a while a bee gets lucky and gets unloaded quickly so that its relative profitability is above the threshold so it dances. Next, notice that the dynamics of the variation of the dance strengths for sites 1-3 correspond to the number of bees that appear at sites 1-3 as seen in Figure 3. In particular, notice that bees are quickly recruited to site 1, and this inhibits somewhat (slows down) recruitment to site 2 which is nearly as profitable, and significantly inhibits the rate of recruitment to site 3. This coupling between rates of allocation to the sites occurs 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 waittime to be unloaded to go up and hence the dance threshold to go up so that there are relatively fewer dances for the other sites since the amount of dancing is set by how much the profitability is above 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). Both these features help the hive optimize their foraging performance. Quick deployment ensures returns in spite of a quickly time-varying environment. Long-term IFD achievement is advantageous since even small variations off the equalization of profitabilities can ultimately have a large negative impact on foraging performance since foraging losses due to such variations off the best forager allocation accumulate over time. 100 Lf avg/std(bars), site 2

Lf avg/std(bars), site 1

100 80 60 40 20 0

0

2

4

6

8

40 20

0

2

0

2

4

6

8

10

8

10

100 Lf avg/std(bars), site 4

Lf avg/std(bars), site 3

60

0

10

100 80 60 40 20 0

80

0

2

4 6 Time, hours

8

80 60 40 20 0

10

4 6 Time, hours

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 the number of dancers is proportioned per the relative site profitabilities, and that very few bees dance for site 4 due to its relative inferiority. This proportional allocation of dancers per profitability makes it so that better estimates of the dance threshold will be obtained by the group of bees visiting the best sites (simply since the average is determined by more bees); this helps the bees avoid abandonment of relatively profitable sites, and at the same time allows for a flexibility that results in abandonment of relatively weak sites. Moreover, the proportioned amount of dancing per profitability ensures that the proportional allocation of the number of foragers is maintained as 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 property contributing to foraging success. Consider the case where there are only two forage sites with the 17

25 Bfd(2,k), #dancers site 2

Bfd(1,k), #dancers site 1

25 20 15 10 5 0

0

2

4

6

8

10 5

0

2

0

2

4

6

8

10

8

10

25 Bfd(4,k), #dancers site 4

Bfd(3,k), #dancers site 3

15

0

10

25 20 15 10 5 0

20

0

2

4 6 Time, hours

8

20 15 10 5 0

10

4 6 Time, hours

Figure 9: Number of employed foragers that dance Bf d (j, k) for site j (solid line) and the mean over the entire day (dashed). same suitability functions used in the last section, but now with Nf1 = 1 (good site) and we vary Nf2 ∈ [0, 1]. For each value of Nf2 we run the simulation 100 times and compute the average and standard deviation of the profitability of each site. The results are in Figure 10. Notice that for values of Nf2 > 0.6 the average site profitability of the two sites is equalized due to the achievement of the IFD. For Nf2 = 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 any other site besides site 1. Indeed, this is what happens. Notice that the average profitability of site 2 essentially increases linearly with a linear increase in Nf2 in the range of Nf2 ∈ [0, 0.25] and this shows that there were almost no visitors to site 2 for this profitability range. If there were more visitors then the profitability level of site 2 would have decreased. This is what we see a slight effect from right at Nf2 = 0.25 where there is a slight decrease in profitability from what a line fit would predict. Now, for values of Nf2 > 0.25 the hive nearly achieves an IFD, with better achievement as Nf2 increases.

3.4

Effect of Significant Profitability Variations

Next, we discuss how the colony reacts to fast and significant changes in profitability of forage sites. We use the scenario described in (Seeley et al. 1991; Camazine and Sneyd 1991) that is summarized in (Seeley 1995). There are two sites, one much more profitable than the other, that the bees forage at in the morning and as expected the colony allocates foragers so that there are many more bees at the more profitable site. Then, in the afternoon the profitability of the two sites are swapped and the colony reallocates so that number of bees at each site corresponds to their relative profitability. We represent this here by having two nonzero profitability sites with the same suitability functions used in the last section, but now with Nf1 = 1 (good site) and Nf2 = 0.5 18

0.35

Profitaility avg/std, sites 1 (−), 2(−−)

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5 N2f

0.6

0.7

0.8

0.9

1

Figure 10: Average and standard deviations (bars) of profitabilities for sites 1 (solid line) and 2 (dashed line) when Nf1 = 1 and Nf2 ∈ [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 of the reallocation pattern matches what is found in experiments (Seeley et al. 1991; Camazine and Sneyd 1991). The number of bees at site 1 by t = 5 hrs. is roughly the same as the number of bees at site 2 by t = 10 hrs. showing that the colony dynamically reallocates bees under dynamic changes in environmental conditions. This allocation is achieved via simultaneous abandonment of site 1 and recruitment to site 2 since on average, after the profitability swap, bees visiting site 2 will be further above the dance threshold so that they will be more successful in recruiting. This results in a “cross-inhibition” effect (Seeley 1995) where the recruitment for one site inhibits recruitment to another site. Moreover, the number of bees dancing for each site realigns (see Figure 12) in order to achieve the reallocation. In Figure 12 the mean number of dances over the entire day for each site is the same since the swap occurred half way through the day, but in the morning there are more dancers for site 1 and in the afternoon there are more for site 2. The hive’s ability to shift the proportion of dancers to keep it aligned to the relative profitability of forage sites is a key mechanism underlying the dynamic reallocation of foragers in response to changes in the foraging environment. The mechanisms underlying this ability are the relatively low number of dancing bees, the negative feedback due to the dance threshold’s effect on abandonment, and the positive feedback that results from recruitment. Finally, we note that similar reallocation dynamics are found for other scenarios (in the interest of brevity we omit plots). First, if there are three equal profitability sites and one suddenly appears that is much more profitable, then we see cross-inhibition (Seeley 1995) where the more profitable site receives more foragers and simultaneously inhibits foraging at the other relatively inferior sites due to an increase in wait times. Second, when there are two equally profitable sites, then one increases in profitability while the other one keeps the same profitability reallocation does not 19

80 B(2,k), #bees site 2

B(1,k), #bees site 1

80 60 40 20 0

0

2

4

6

8

20

0

2

0

2

4

6

8

10

8

10

80 B(4,k), #bees site 4

B(3,k), #bees site 3

40

0

10

80 60 40 20 0

60

0

2

4 6 Time, hours

8

60 40 20 0

10

4 6 Time, hours

Figure 11: Number of bees visiting sites during forage site profitability change (bottom plots show that sites 3 and 4 are rarely visited).

25 Bfd(2,k), #dancers site 2

Bfd(1,k), #dancers site 1

25 20 15 10 5 0

0

2

4

6

8

10 5

0

2

0

2

4

6

8

10

8

10

25 Bfd(4,k), #dancers site 4

Bfd(3,k), #dancers site 3

15

0

10

25 20 15 10 5 0

20

0

2

4 6 Time, hours

8

20 15 10 5 0

10

4 6 Time, hours

Figure 12: Number of employed foragers that dance Bf d (j, k) for site j (solid line) and the mean over 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 relatively large 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 not been a significant amount of dancing). This shows that reallocation can occur in two ways: via redirection of foragers from one site to a more profitable one, or by deployment of more foragers. In either case, the effect is that the relative number of foragers at each site matches the site’s relative profitabilities. Third, if there are many relatively inferior sites and one excellent one suddenly appears, then the rate at which the hive can reorient will degrade over the case where there is, for instance, just one inferior site being exploited by the colony. The reason for this is that it takes longer 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 dance strength decision, and the profitability assessment noise magnitude wf , on the average total quantity of nectar obtained in one day and the average total number of waggle runs over a day needed to get that quantity of nectar. Moreover, we count the number of bees that are employed foragers on each expedition and that dance after an expedition, and then sum these totals across all expeditions in a day to obtain totals. Then, we compute the averages and standard deviations of these. Our statistics for each case are computed using 100 simulations of the foraging process over a day, for each parameter value considered. For all cases, we use the foraging landscape in Section 2.1 with Nf1 = Nf2 = Nf3 = Nf4 = 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 that the average total amount of nectar profitability peaks at a point near the value of δ = 0.02 that we used in the last section. The average total amount of dancing goes down as δ increases (see bottom-left plot) since fewer dancers are deployed (see bottom-right plot) and also since with a higher δ the difference between the dance threshold and profitability is lower. The top-right plot confirms 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 maximizes foraging return for the investment lies in a range around 0.015–0.025. This further confirms our model’s validity, and shows that optimizing foraging performance involves complex trade-offs that emerge 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 increases as β increases but only slightly above β = 100 (top-left plot). The increased nectar quantity gathered 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 in the number deployed in the range of β ∈ [100, 150]. The main conclusion, however, comes from considering all four plots at once. Simultaneous maximization of the average total profitability 21

6400

Bf total avg/std

Ntq total avg/std

10000 6200 6000 5800

9500

9000

5600 0

0.01

0.02

0.03

0

0.01

0

0.01

0.02

0.03

0.02

0.03

5

x 10

3 Bfd total avg/std

Lt total avg/std

6000

2

1

0

0

0.01

δ

0.02

5000 4000 3000 2000

0.03

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 of total amount of dancing in one day (bottom-left), average and standard deviation of total number of employed foragers (top-right), average and standard deviation of total number of foragers that dance 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 nectar profitability return is maximized for any value of wf ∈ [0, 0.3]. Similarly, the average total number of deployed foragers is relatively constant over the range of wf ∈ [0, 0.2]. There is little effect on the average total number of dances and the number of dancers until wf > 0.1, but as there is more assessment noise there is a need for more dances to maintain a profitable distribution. A value of wf = 0 is physically unrealistic, but we see that our chosen value of wf = 0.1 results little adverse impact on foraging performance. The effect of individual-level bee assessment noise is filtered out at the group level so that an effective allocation can emerge.

22

10100 Bf total avg/std

Ntq total avg/std

9100 9000 8900

10000 9900 9800

8800 50

100

150

50

100

150

100 β

150

5

5200

3

5150

Bfd total avg/std

Lt total avg/std

x 10 4

2 1 0 50

100 β

150

5100 5050

50

Figure 14: Foraging performance as a function of β. See Figure 13 caption for a description of variables. 10000

6100 Bf total avg/std

Ntq total avg/std

6200

6000 5900 5800 5700 5600

9500 9000 8500 8000

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

5

x 10

4400 4200

1.5

Bfd total avg/std

Lt total avg/std

2

1 0.5 0

4000 3800 3600 3400

0

0.1

0.2

0.3 w

f

0.4

0.5

w

f

Figure 15: Foraging performance as a function of wf . See caption of Figure 13 for a description of variables.

5 5.1

Discussion Mechanisms and Dynamics

The key mechanisms unique to social foraging that enable the dynamical achievement of an effective 23 dance strength determination (as represented emergent forager allocation primarily revolve around

in the experimentally validated dance decision function in Equation (4)). First, via the wait-time to be unloaded experienced by each successful forager, a bee can produce a scaled estimate of the current total nectar influx of the colony each time it returns to the hive. This estimate is proportional to the current average site profitability of all employed foragers in the colony. An individual bee can compare this estimate with its last site profitability assessment and decide whether the forage site it is currently exploiting is above or below the average profitability of the other sites currently being exploited by the colony. Then, if it is exploiting what is currently a relatively inferior site it can abandon it. And, for increasingly relatively profitable sites it can both raise its dance strength and its tendency to dance. This results in a proportionate distribution of foragers across the current relatively profitable sites. We must emphasize that the dance threshold value is unique to each bee since it is based on its individually experienced wait time and the site profitability assessment for the last site it visited. There will then be many “mistakes” made by individual bees. Due to the noise on the experienced wait time, a bee may mistakenly abandon a site, or choose a dance strength that corresponds to it seeking more recruits than are currently warranted for the site. But, the key to the hive’s success is based on the combined actions of all the bees, not solely a single individual’s error-prone actions. First, note that the average profitability assessment and average wait time experienced by a sufficiently large group of bees visiting any site are quite accurate representations of the site profitability and total nectar influx. Hence, the total set of abandonment/recruitment actions take by a group of bees visiting a site can be quite accurate. In fact, as more bees visit a site, the group will make more accurate decisions; this ensures that the hive will rarely make mistakes in abandoning a great site, and will not over-recruit when two sites of close profitability are 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 relatively poor sites. These “mistakes,” however, amount to the hive being flexible in its view of a currently relatively inferior site: if the site profitability changes it can then more quickly abandon or recruit to it. The hive continually explores, but pays increased attention to relatively superior sites. The aggregate effect is a simultaneous proportioned monitoring of the entire foraging environment that enables the forager allocation. To confirm these ideas, the simulations shown in Figures 2, 3, 4, and 5 illustrate how the hive produced an effective allocation of foragers by nearly achieving an IFD. Also, Figure 7 illustrates that the group of bees visiting each of the four sites does indeed on average have an accurate estimate of the scaled average nectar influx (e.g., for sites 1-3 it is the same as it should be). Figure 8 shows that average dance strengths of the groups visiting each site (and hence recruitment rates) are higher for relatively more profitable sites. The variances Figure 8 show, however, that the dance determination by individuals is relatively error-prone. Figure 10 shows that the hive manages to largely 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 its effect is illustrated by the discrepancy between the average profitabilities for the two sites for the range of 0.25 < Nf2 < 0.6 in Figure 10. In this range, the hive “over-samples” the relatively inferior site 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 significant change in site profitability. The key to this reallocation is how the hive manages the pool of unemployed foragers (see discussion below), and the fact that bees decide not to dance. When relatively few bees dance (as illustrated in Figures 6 and 12), but the dance strengths are proportioned according to currently exploited site profitabilities, each dance has a larger impact on 24

the change in the allocation than if all employed foragers danced. So, by saving time-energy costs of dancing the hive also gains allocation speed. Note that experiments suggest that keeping the amount 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 for each resource can have a relatively large impact on recruitment (e.g., if suddenly there is a strong demand 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 different resources and (ii)ensure that the hive can simultaneously and quickly respond to make all resource supplies match all hive demands. It is important to highlight how the resting, observing, exploring, and recruitment occurs in a way to contribute to the effectiveness of the foraging. First, the hive essentially manages the size of the pool of unemployed foragers and explorers. When there is little available forage, there are more explorers dispatched. As sites are found and the number of recruits increases, the number of unemployed foragers decreases and the hive gradually switches from a search mode to a forage exploitation mode, but there are always some explorers. This is illustrated in Figure 6. Other simulations show that if there are relatively few good sites, then there is more exploration and the pool of resters is larger and this contributes to the ability of the hive to quickly exploit a newly discovered highly profitable site. Hence, depending on conditions, forager reallocation can occur via the deployment of unemployed foragers and/or the abandonment-recruitment process. Finally, we note that a recent theoretical study on the optimal proportion of forage searchers versus recruited foragers shows an important dependency on food source availability duration (Dechaume-Moncharmont et al. 2005). It would be of interest of experimentally validate the mechanisms underlying scout/recruit allocation (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 number of modifications to our model along with new simulation test scenarios for time-varying forage site profitabilities).

5.2

Behavioral Parameter Adaptation

As further confirmation of the above ideas, in Figures 13 and 14 we show that natural selection seems to have settled on values of the dance threshold parameter and dance function slope that represent a compromise between maximizing nectar intake and minimizing the number of dances to achieve that intake. Notably, the value of the dance function slope parameter β is approximately the one found experimentally in (Seeley 1995). Moreover, the dance threshold parameter δ is near the higher end of the feasible range in Equation (5). This value enables the hive to quickly abandon inferior sites, yet still achieve and maintain a proportional allocation since the average site profitability is above this value. Figure 13 also confirms the model formulation by showing that if δ is at either extreme in the range given by Equation (5) nectar influx suffers. If δ is at the high end of the range, then the foraging process fails since all sites are abandoned. If δ is at the low end of the range significant time-energy costs are incurred due to dancing. Next, we illustrated that foraging performance suffers little if the individual profitability assessment noise magnitude wf ∈ [0, 0.1], but that if wf > 0.1, then to maintain an effective forager allocation 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 propagate to adversely impact colony-level foraging performance (i.e., the social foraging process filters individual-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 the tendency 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 the common features of social foraging and nest-site selection by honey bees. There are only two differences 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 here is foraging. Second, we change how the individual bees choose dance strength. The remaining parts of the model, particularly recruitment, explorer allocation, resting, and dancing observing are structurally the same as in (Passino and Seeley 2006b). The change in dance strength choice results in very different emergent decisions: choice of one site vs. proportionate allocation of bees across many sites. However, the resulting dynamics of the two distributed decision-making processes share a number of features that provide insights into generic features of the functionality of social decision making by honey bees. To see the close relationships, here we will think of nest-site quality and forage site profitability as generic measures of “goodness” of sites. First, note that while dance strength is superficially proportional to site quality, in nest-site selection it decays linearly after the bee’s first visit while in foraging it can remain at a high level for a long period of time. Interestingly, in the transient there can often be an average decay of the dance strengths in social foraging due to an increase in colony nectar influx that raises the dance threshold, but the level of dancing will not go to zero for the superior sites (e.g., see Figure 8). And, of course such transients can occur in the opposite direction and dance strengths can increase, then decrease, especially when the dance threshold decreases due to a reduction in colony nectar influx. Second, positive feedback is used in both processes for recruitment of more bees to a site. In social foraging it ensures that the best sites get recruits much faster than inferior sites (see Figure 8). In nest-site selection positive feedback is 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 the dynamic changes in the dance threshold, while in nest-site selection it is due to the dance decay characteristic. Third, there are cross-inhibition effects between sites for both processes. Fourth, in both cases the average number of bees allocated to a site is in proportion to how good it is. In nest-site selection the numbers of site-assessor bees is allocated on average according to relative nest-site quality according to an IFD. This ensures that the best sites are fully evaluated before they are 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/abandonment errors for those sites. Hence, the hive’s ability to achieve an IFD is a fundamental property of its social decision-making. It is achieved in quite different ways for the two processes, yet is a key to the 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 analogous trade-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 makes relatively few mistakes, but (ii) this leads to slower reallocations since then each dance cannot have as significant of an influence on the reallocation. Hence, the amount of dancing in social 26

foraging represents a trade-off between achieving fast yet effective forager allocation patterns. It is an important future direction to conduct experiments to further explore this speed and accuracy tradeoff. This would involve conducting experiments where we establish forage sites whose profitability can be represented as a function of the number of site visitors, then simultaneous recording of all forage site and hive variables when there are significant site profitability changes of varying magnitudes.

5.4

Relations to Group Cognition

The perspective that inter-animal sociality can achieve (i) complexity in form and function via massively parallel interconnections, and (ii) cognition that is functionally equivalent to neuron-based brains has been recognized for some time (e.g., see (Markl 1985; Bonner 1988)). Recently, it was shown that nest-site selection is a “group cognition process” that shares close relationships with the 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 briefly explain how to view social foraging by honey bees as a group cognition process. From such a perspective, social foraging is most similar to attentional processes (Gazzaniga et al. 1998). The foraging 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 but on average predictable pattern of inter-bee communications via dances that provides for signaling between cognition units. The set of bees visiting each forage site provides the colony with a group-level memory of the current allocation of attention, and the underlying dance determination mechanisms result in proportionally more attention (foragers) being paid to the relatively important 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 of the entire group of bees visiting a site. The “late” processing stages (Gazzaniga et al. 1998) of the foraging attentional process enable for reallocation of attention. The quick reallocation ability enables refocusing of attention, and the dance threshold emerges as a dynamic threshold that enables the hive to ignore relatively unimportant sites. An important future research direction is to move beyond such analogical connections to determine if our understanding of social foraging can 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 of Research. I would like to thank Thomas D. Seeley for many helpful conversations on the social foraging of the honey bees. I also would like to thank Jorge Finke and Nicanor Quijano for checking the simulation code and for some suggestions on the document.

References Anderson, C. (2001). The adaptive value of inactive foragers and the scout-recruit system in honey bee (Apis Mellifera) colonies. Behav. Ecol., 12, 111–119. Anderson, C. and Ratnieks, F. (1999). Task partitioning in foraging: general principles, efficiency and information reliability of queueing delays. In C. Detrain, J. Deneubourg, and J. Pasteels (Eds.), Information processing in social insects. Boston: Birkhauser Verlag. 27

Bartholdi, J., Seeley, T. D., Tovey, C., and VandeVate, J. (1993). The pattern and effectiveness of forager allocation among flower patches by honey bee colonies. J Theor Biol, 160, 23–40. Beckers, R., Deneubourg, J., Goss, S., and Pasteels, J. (1990). Collective decision making through food recruitment. Insectes Soc., 37, 258–367. Bonner, J. T. (1988). The evolution of complexity by means of natural selection. Princeton, NJ: Princeton University Press. Camazine, S., Deneubourg, J.-L., Franks, N., Sneyd, J., Theraulaz, G., and Bonnabeau, E. (2001). Self-organization in biological systems. NJ: Princeton Univ. Press. Camazine, S. and Sneyd, J. (1991). A model of collective nectar source selection by honey bees: self-organization through simple rules. J Theor Biol, 149, 547–571. Cox, M. and Myerscough, M. (2003). A flexible model of foraging by a honey bee colony: the effects of individual behavior on foraging success. J Theor Biol, 223, 179–197. de Vries, H. and Biesmeijer, J. C. (1998). Modeling collective foraging by means of individual behavior rules in honey-bees. Behav Ecol Sociobiol, 44, 109-124. de Vries, H. and Biesmeijer, J. C. (2002). Self-organization in collective honeybee foraging: emergence of symmetry breaking, cross inhibition, and equal harvest-rate distribution. Behav Ecol Sociobiol, 51, 557-569. Dechaume-Moncharmont, F., Dornhaus, A., Houston, A., McNamara, J., Collins, E., and Franks, N. (2005). The hidden cost of information in collective foraging. Proc. Royal Society, London, 272 (1573), 1689–1695. Deneubourg, J. and Goss, S. (1989). Collective patterns and decision-making. Ethol. Ecol. Evol., 1, 295–311. Dukas, R. and Edelstein-Keshet, L. (1998). The spatial distribution of colonial food provisioners. Journal of Theoretical Biology, 190, 121–134. Dukas, R. and Visscher, P. (1994). Lifetime learning by foraging honey bees. Animal Behavior, 48, 1007–1012. Fretwell, S. and Lucas, H. (1970). On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica, 19, 16–36. Gazzaniga, M. S., Ivry, R. B., and Mangun, G. R. (1998). Cognitive neuroscience: The biology of the mind. NY: W. W. Norton and Co. Giraldeau, L. and Caraco, T. (2000). Social foraging theory. Princeton, NJ: Princeton Univ. Press. Mallon, E., Pratt, S., and Franks, N. (2001). Individual and collective decision-making during nest site selection by the ant leptothorax albipennis. Behav Ecol Sociobiol, 50, 352–359. Markl, H. (1985). Manipulation, modulation, information, cognition: some of the riddles of communication. In B. Holldobler and M. Lindauer (Eds.), Experimental behavioral ecology and sociobiology (p. 163-194). New York: Sinauer Assoc. Mitchell, M. (1996). An introduction to genetic algorithms. Cambridge, MA: MIT Press. Parker, G. and Sutherland, W. (1986). Ideal free distributions when individuals differ in competitive ability: phenotype-limited ideal free models. Animal Behavior, 34, 1223–1242. Passino, K. M. and Seeley, T. D. (2006a). Honey bee swarm cognition: Mechanisms, behavior, and adaptation. In preparation, 1, ??–?? Passino, K. M. and Seeley, T. D. (2006b). Modeling and analysis of nest-site selection by honey bee swarms: The speed and accuracy trade-off. Behav Ecol Sociobiol, 59 (3), 427–442. Pratt, S., Mallon, E., Sumpter, D., and Franks, N. (2002). Quorum sensing, recruitment, and collective decision-making during colony emigration by the ant leptothorax albipennis. Behav Ecol Sociobiol, 52, 117–127. Schmickl, T. and Crailsheim, K. (2003). Costs of environmental fluctuations and benefits of 28

dynamic decentralized foraging decisions in honey bees. In (pp. 145–152). Atlanta, GA: Proc. of 2nd Int. Workshop on the Mathematics and Algorithms of Social Insects. Seeley, T. (1983). Division of labor between scouts and recruits in honeybee foraging. Behavioral ecology and sociobiology, 12, 253–259. Seeley, T. (1986). Social foraging by honeybees: how colonies allocate foragers among patches of flowers. Behav Ecol Sociobiol, 19, 343–354. Seeley, T. (1994). Honey bee foragers as sensory units of their colonies. Behavioral ecology and sociobiology, 34, 51–62. Seeley, T. (1995). The wisdom of the hive: The social physiology of honey bee colonies. Cambridge, Mass: Harvard University Press. Seeley, T., Camazine, S., and Sneyd, J. (1991). Collective decision-making in honey bees: how colonies choose among nectar sources. Behav Ecol Sociobio, 28, 277–290. Seeley, T. and Towne, W. (1992). Tactics of dance choice in honeybees: do foragers compare dances? Behav Ecol and Sociobio, 30, 59–69. Seeley, T. and Visscher, P. (1988). Assessing the benefits of cooperation in honey bee foraging: search costs, forage quality, and competitive ability. Behavioral Ecology and Sociobiology, 22, 229–237. Seeley, T. and Visscher, P. (2004). Group decision making in nest-site selection by honey bees. Apidologie, 35, 1–16. Seeley, T. D. and Tovey, C. (1994). Why search time to find a food-storer bee accurately indicates the relative rates of nectar collecting and nectar processing in honey bee colonies. Animal Behav, 47, 311–316. Seeley, T. D., Visscher, P. K., and Passino, K. M. (2006, May/June). Group decision making in honey bee swarms. American Scientist, 94 (3), 220–229. Sumpter, D. and Pratt, S. (2003). A modelling framework for understanding social insect foraging. Behav Ecol Sociobiol, 53, 131-144. Tregenza, T. (1995). Building on the ideal free distribution. Advances in Ecological Research, 26, 253–307.

29

6

Appendix: Additional Simulations–Not for Publication

A number of other simulations were conducted to determine the impact of other parameters. The ones 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 values we cannot consider the effects of the need to search to discover difficult-to-find sites, and the need to maintain flexibility (e.g., via a pool of resting foragers) in case there are quick changes in the landscape (e.g., a site that suddenly becomes excellent). To fully consider the effects of the parameters below other landscapes would have to be considered. The simulations below simply support that the values chosen in the validated model indeed make sense for the chosen performance measures, 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 profitability return is maximized at about σ = 800. Increasing values of σ result in an increase in the average total number of dances (and number of dancers) since it then takes more dances to achieve recruitment since the pool of unemployed foragers is smaller since there are more explorers. A small σ 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 where there are four good sites that are easy to find. For other landscapes where sites are not as plentiful or profitable there is a need for a value of σ > 800 and via our simulations we found that a value of σ = 1000 was suitable.

10100

6200

Bf total avg/std

Ntq total avg/std

6400

6000 5800

10000 9900 9800 9700 9600

5600 500

1000

9500 500

1500

1000

1500

1000 σ

1500

4

x 10

3700 3600 Bfd total avg/std

t

L total avg/std

15

10

5

0 500

3500 3400 3300 3200

1000 σ

1500

500

Figure 16: Foraging performance as a function of σ. See caption of Figure 13 for a description of variables.

30

6.2

Effect of Tendency to Dance

6200

10000

6000

9500

Bf total avg/std

Ntq total avg/std

We vary φ over the range [0, 1]. Figure 17 shows that the total average nectar profitability return is 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 any value of φ > 0.7 seems reasonable; however, note that this would mean that even when no forage sources are being exploited, if an explorer would find a new and most highly profitable site it would only dance, for example, 70% of the time. Since this does not happen in nature, we conclude that φ 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.

5800

5600

9000

8500 0.4

0.6

0.8

1

0.4

0.6

0.4

0.6

0.8

1

0.8

1

4

x 10

3500 3000

10

Bfd total avg/std

t

L total avg/std

12

8 6 4 2 0

2500 2000 1500 1000

0.4

0.6

φ

0.8

1

φ

Figure 17: Foraging performance as a function of φ. See caption of Figure 13 for a description of variables.

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 profitability return 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 unemployed foragers pool (resters) will go down, perhaps very low; but this does not take into account other landscapes that are dynamically varying and here there is the need for a reserve of foragers that can be quickly deployed.

31

6500

Bf total avg/std

Ntq total avg/std

10000 6000 5500 5000

9000 8000 7000

4500 0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3 p

0.4

0.5

4

x 10

3600 Bfd total avg/std

10

t

L total avg/std

15

5

3500 3400 3300 3200 3100

0

0.1

0.2

0.3 p

0.4

0.5

o

o

Figure 18: Foraging performance as a function of po . See caption of Figure 13 for a description of variables.

32