A model of collective nectar source selection by honey bees: Self-organization through simple rules

A model of collective nectar source selection by honey bees: Self-organization through simple rules

J. theor. Biol. (1991) 149, 547-571 A Model of Collective Nectar Source Selection by Honey Bees: Self-organization Through Simple Rules SCOTT CAMAZIN...

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J. theor. Biol. (1991) 149, 547-571

A Model of Collective Nectar Source Selection by Honey Bees: Self-organization Through Simple Rules SCOTT CAMAZINEt AND JAMES SNEYD•

t Corneil University, Section of Neurobiology and Behavior, Mudd Hall, Ithaca, N Y 14853 and ~ Department of Biomathematics, UCLA School of Medicine, 10833 Le Conte Ave, Los Angeles, CA 90024, U.S.A. (Received on 25 May 1990, Accepted in revised form on 24 September 1990) The honey bee colony chooses among different nectar sources available in the field, selectively foraging from those which are most profitable. We present a model that describes the colony's decision-making process. The model consists of a system of non-linear differential equations describing the activity of the foraging bees. Parameter estimates are based on previously published data. Numerical solutions of the equations agree closely with experimental observations. Selective exploitation of the most profitable nectar sources occurs through an autocatalytic, self-organizing process.

Introduction Understanding how collective, colony-level patterns emerge from the behaviors of the individual colony members is a major challenge in the study of insect societies. Among honey bees, the colony chooses among different flower patches, selectively exploiting the most profitable nectar sources. Previous studies have documented that the colony precisely and rapidly adjusts its pattern of foragers in time and space, tracking the environment's changing mosaic of nectar sources (Visscher & Seeley, 1982; Seeley, 1986, 1989; Seeley et al., 1991). This raises the question of how such an organized pattern of collective foraging emerges from the activities of thousands of individual bees. Our approach is to consider the colony as a system of interacting components--the foraging bees. We first examine the relevant behaviors of the individuals and the information that is shared among them. We then develop a model which demonstrates how the properties of the system (the colony-level pattern of selective foraging) emerge automatically from the dynamic interactions among the constituent components (the individual foragers). We show that the pattern of collective foraging is an autocatalytic, self-organizing process resulting from positive feedback in the interactions of many bees following a few simple behavioral rules. Previous models of foraging in ants have emphasized the crucial role of self-organization in colonylevel processes (Pasteels et al., 1987; Deneubourg et al., 1989, 1990; Deneubourg & Goss, 1989; Goss & Deneubourg, 1989; Goss et al., 1989). The remarkable feature of these models is that the "collective intelligence" of the colony is an emergent property of the system, requiring only limited and local knowledge by the participants. 547 0022-5193/91/080547 + 25 $03.00/0

© 1991 Academic Press Limited

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A Model of Colony-level Foraging THE

BIOLOGICAL

BASIS

We begin with a pool of foragers lacking knowledge of the potential nectar sources available in the field. Each o f these bees (which we call "follower bees") reaches a nectar source by following the dance o f a nestmate who has already discovered a patch o f flowers. Consider the behavior o f one such bee as she begins her day o f foraging (Fig. 1). The dance floor area of the hive contains bees dancing for different nectar sources. The follower bee selects a dancer for nectar source A. After following the bee's dances, she flies to that nectar source. Upon arrival, the forager gathers a load of nectar and returns to the hive. After relinquishing her nectar to a food storer bee, the forager may do one of three things, as indicated by the branch points (diamonds) in Fig. 1. First, she may abandon the food source and return to the pool of uncommitted followers. Alternatively, if she decides to continue to forage from the nectar source, she may either perform recruitment dances before returning to her patch of flowers or continue to forage at the food source without recruiting nestmates. Many factors affect the probability that an individual bee dances for or abandons the food source (Seeley, 1986; Seeley et al., 1991): nectar sweetness, distance to the food source, ease o f nectar collection, colony intake rate. In this model, for simplicity, we consider nectar sources that differ in quality only with respect to sugar concentration, all other factors being equal. THE

MATHEMATICAL

STRUCTURE

OF THE

MODEL

These essential features of the colony-level decision-making process can be incorporated into a mathematical model of foragers choosing between two nectar sources. First, we assume that, at any moment, each foraging bee is in one of the seven places (compartments) shown in Fig. 2. These compartments are: Ha : unloading nectar from food source A, Hb: unloading nectar from food source B, Do : dancing for food source A, Db : dancing for food source B, A: feeding at food source A, B: feeding at food source B, F: following a dancer. The dance floor (shaded area in Fig. 2) contains three separate compartments: those bees dancing for A, those bees dancing for B, and those bees following a dancer. In contrast to Fig. 1, note that Fig. 2 consists of two separate cycles, one for each food source, with the follower compartment, F, the only one shared by the two cycles. Thus, bees from one feeder can switch over to the other feeder only by passing through the dance floor and following a dancer for the other food source. The figure suggests that the dance area plays the central role in the decision-making process. Whatever information is transferred among the bees is assumed to take place here.

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FIG. 1. The behavioral cycle of a honey bee foraging for nectar. She begins foraging by following a nestmate performing a recruitment dance. She then arrives at the food source, gathers a load of nectar and returns to the hive to unload the nectar. The cycle is repeated, though now she has the option of continuing to forage from her current nectar source (and perhaps to dance for this source), or of abandoning the nectar source and following a dance for a new source. The two black diamonds represent these decision points. The grey circle represents the dance floor area inside the hive.

Two factors affect the proportion of bees in each of these seven compartments: (1) the rate at which a bee moves from one location to another, and (2) the probability that a bee takes one or the other fork at the five branch points (diamonds) of Fig. 2. We assume that the bees leave each c o m p a r t m e n t at a given rate. For each o f the seven c o m p a r t m e n t s we specifiy a rate constant p~_~, in units min -j. Thus,

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TABLE 1

Parameter values used in the model of nectar foraging. Based upon nectar source A with 2.5 M and nectar source B'with 0-75 M sucrose. Data from Seeley et al. (1991). For a complete definition of each parameter see text and Fig. 2 Parameter: definition Ti: time from start of unloading to start of following, dancing or foraging (A foragers) T2: time from start of dancing to start of foraging (A foragers) T3: time from start of foraging to start of unloading (A foragers) 7"4: time from start of following dancers to start of foraging (A and B foragers) T5 : time from start of unloading to start of following, dancing or foraging (B foragers) Tr: time from start of dancing to start of foraging (B foragers) /'7 : time from start of foraging to start of unloading (B foragers) f~ : probability of abandoning A, per foraging trip f~ : probability of abandoning B, per foraging trip f,~ : probability of dancing for A f~: probability of dancing for B ra: proportion of time in D a spent dancing rb: proportion of time in D b spent dancing

Value 1.0 rain 1-5 min 2.5 min 60 min 3.0 min 2.0 min 3.5 rain 0-0 0-04 1.0 0.15 0.38 0.02

f.~, i n d i c a t e s the p r o b a b i l i t y t h a t a bee, u p o n l e a v i n g Ha, will a b a n d o n n e c t a r s o u r c e A to b e c o m e a f o l l o w e r . T h e m o d e l a s s u m e s t h a t a b e e w h i c h a b a n d o n s a p a r t i c u l a r f o o d s o u r c e d o e s n o t s u b s e q u e n t l y a v o i d f o l l o w i n g a d a n c e r for t h a t f o o d source; thus, in t h e m o d e l , it is p o s s i b l e for a b e e to a b a n d o n f o o d s o u r c e A, b e c o m e a f o l l o w e r , f o l l o w a b e e d a n c i n g for A, a n d e v e n t u a l l y e n d u p b a c k at A. A b a n d o n m e n t d i m i n i s h e s the n u m b e r o f b e e s c o m m i t t e d to a f o o d s o u r c e a n d p r o v i d e s a p o o l o f u n c o m m i t t e d b e e s w h i c h f o l l o w d a n c e r s f o r o n e n e c t a r s o u r c e o r t h e other. T h e s e c o n d b r a n c h p o i n t d e t e r m i n e s the p r o p o r t i o n o f the c o m m i t t e d bees that d a n c e for t h e n e c t a r s o u r c e t h e y have j u s t visited. A l t h o u g h at t h e s e c o n d b r a n c h p o i n t t h e r e is n o filtering o f bees a w a y f r o m the f o o d s o u r c e to w h i c h t h e y are c o m m i t t e d , this b r a n c h p o i n t affects the p r o b a b i l i t y with w h i c h an u n c o m m i t t e d f o r a g e r f o l l o w s a d a n c e r f o r o n e o r t h e o t h e r f o o d source, as d e s c r i b e d b e l o w . T h e p r o b a b i l i t y t h a t a b e e b e c o m e s a d a n c e r for h e r f o o d s o u r c e is d e n o t e d b y the f u n c t i o n fd, t h e d a n c i n g f u n c t i o n . As for t h e a b a n d o n m e n t f u n c t i o n , its v a l u e d e p e n d s on the q u a l i t y o f t h e f o o d s o u r c e , with fad i n d i c a t i n g the p r o b a b i l i t y that a b e e f o r a g i n g at n e c t a r s o u r c e A p e r f o r m s r e c r u i t m e n t d a n c e s . T h e t h i r d b r a n c h p o i n t o c c u r s on the d a n c e floor w h e n b e e s f o l l o w d a n c e r s for o n e o r a n o t h e r n e c t a r s o u r c e . T h e f r a c t i o n o f the f o l l o w e r b e e s l e a v i n g the d a n c e floor to go to f o o d s o u r c e A is d e n o t e d b y the f u n c t i o n f ~ , the f o l l o w i n g function. A l t h o u g h c r u c i a l to t h e m o d e l , the m e c h a n i s m w h e r e b y a bee selects a p a r t i c u l a r d a n c e r h a s n o t y e t b e e n e l u c i d a t e d e x p e r i m e n t a l l y . T h e r e f o r e , we p r e s e n t two alternative hypotheses.

Hypothesis 1 A b e e e n t e r i n g the d a n c e a r e a r a n d o m l y e n c o u n t e r s d a n c e r s , a n d f o l l o w s the first d a n c e r s h e e n c o u n t e r s . I n t h e s i t u a t i o n o f j u s t t w o n e c t a r s o u r c e s , A a n d B, the

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probability o f a follower encountering dancers for A can be roughly estimated by D J ( D a + D b ) where D~ and Db are the number of bees in each of the dance compartments A and B, respectively. However, since only a portion o f a bee's time in the dance area is actually spent dancing, it is necessary to multiply D , and Db in the above expression by the proportion of time that the foragers actually dance. These fractions are denoted by r~ and zb. Thus, f ~ = D a r J ( D j a + D b r b ) . This fraction takes into account the number of dancers for each food source as well as the time spent dancing, and thus indicates the proportion o f the total dancing for each nectar source.

Hypothesis 2 The alternative hypothesis presents a different behavioral mechanism for selecting a dancer---comparison and choice o f the best quality nectar source. If the dancers were able to communicate not only the location, but also the quality of the food source, then followers might benefit by sampling a number of dances and selecting the best nectar source being advertized. This would result in a higher proportion of followers heading for the most profitable food source. We assume, as in Hypothesis 1, that the probability o f encountering a dancer for food source B is DbT"b/(Daz a + Dbrb), where B is the poorer quality food source. If n different dancers were sampled randomly, then the chance that a follower encounters only dancers for nectar source B is [Dbrb/(DJa + Dbrb)] ~. The probability diminishes rapidly as n increases with the result that nearly all the follower bees [1 - ( D b r b / ( D ~ r a + Dbrb))"] will head for A, the better food source. The implications o f the two alternative hypotheses are considered in the Results section. For simplicity in the model, we make several further assumptions: (1) all the foragers go to either one o f the two nectar sources. (2) A bee is considered to be in one compartment until it reaches the next compartment. Thus, the travel time to the next compartment is included in the time spent in that compartment. (3) The total number of foraging bees is fixed. However, all of the bees do not begin foraging simultaneously at the start o f the day. An arbitrarily chosen portion of this pool of foragers initially begins to forage, and thereafter, new bees begin foraging at a rate proportional to the difference between the eventual total and the number presently foraging. This assumption has the effect o f initially recruiting bees into the forager pool rapidly, but at a rate which gradually decreases. In the Appendix we derive a system of differential equations that describes this model o f honey bee foraging. The reader is referred to the Appendix for the mathematical details of the model.

DETERMINATION

OF THE PARAMETERS

Based on the time required to get from one compartment to another, we can calculate each rate constant p~, i = 1-7, measured in units of min -~. Each rate constant, p~, is equal to 1/T~, where each T~ is the time required to get from one compartment to the next. Values of T,. for two food sources are given in Table 1

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(based upon Seeley et al., 1991). Nectar source A is 2-5 M and nectar source B is 0.75 M sucrose.

Results This model predicts the distribution over time of numbers of foragers in each of the seven compartments. In terms of colony efficiency, the recruitment rate to the two nectar sources is the crucial variable to be considered. This recruitment rate can also be considered a discovery rate, as it provides a measure o f how quickly the colony finds and exploits a newly-discovered food source. It is calculated as the increase in total number of bees utilizing a food source per unit time. This rate varies over time, with the maximum rate occurring early in the recruitment process. Another result obtained from the model is the long-term, steady-state distribution o f bees in each of the compartments. This distribution develops over the course of 4-6 hr, after the initial rate of recruitment levels off. The data presented here are obtained by numerical solution of the model equations of the Appendix. (A) CORRESPONDENCE OF THE MODEL WITH EMPIRICAL OBSERVATIONS AND COMPARISON OF HYPOTHESES t AND 2 Using the parameter values in Table 1 (obtained from Seeley et al., 1991), we can determine how well the model's predictions correspond with results o f actual field experiments. Figure 3 shows the results of an actual experiment (Seeley et al., 1991)

I00 ~

80 south feeder bee

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j\/(f

~ 4o 2O

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I0

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(hr) FIG. 3. Results o f an experiment (see Seeley el al., 1991) showing the recruitment o f nectar forager bees to feeders containing sugar solutions. The first 4 hr (08.00-12.00 hours) shows the time course o f buildup o f recruits to the s o u t h feeder containing a 2.5 M sugar solution vs. the north feeder provided with a 1-0 M sugar solution. D u r i n g the next 4 hr (12.00-16.00 hours) the sugar solution at the south feeder was switched to 0.75 M a n d that at the north feeder to 2.5 M. During this period there is a decline in the n u m b e r of bees at the south feeder and a corresponding increase in the n u m b e r of bees at the north feeder, The forager g r o u p size indicates the n u m b e r of different individual bees visiting the feeder at least once during the half-hour observation period.

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in which a c o l o n y o f bees was given a choice b e t w e e n two feeders ( l a b e l e d n o r t h a n d south). F o r c o m p a r i s o n , Fig. 4 shows the c o m p u t e d s o l u t i o n s o f H y p o t h e s i s 1 (no choice a m o n g d a n c e r s ) a n d H y p o t h e s i s 2 o f the m o d e l ( c o m p a r i s o n a n d choice a m o n g f o u r dancers). T h e initial c o n d i t i o n s for the m o d e l were c h o s e n to m a t c h the e x p e r i m e n t a l d a t a o f Seeley e t a l . (1991) where there were a p p r o x i m a t e l y 12 bees c o m m i t t e d to each feeder, a n d d u r i n g the course o f the d a y a total o f approxim a t e l y 125 different bees visited the two feeders. T h u s , A a n d B = 11, Do a n d Db = 1, Ha a n d Ha = 0, a n d F = 101. A c o m p a r i s o n o f the two figures shows that in the c o m p u t e d s o l u t i o n s o f the m o d e l , as in reality, the c o l o n y always exploits the most profitable n e c t a r source, a n d r a p i d l y r e s p o n d s to c h a n g e s in the l o c a t i o n o f the richer feeder. I n the e x p e r i m e n t , the s o u t h feeder s h o w e d a r a p i d b u i l d u p o f bees b e t w e e n 08.00 a n d 12.00 h o u r s w h e n it was l o a d e d with 2.5 M sugar s o l u t i o n f o l l o w e d by a d e c l i n e in the n u m b e r o f bees u s i n g the feeder over the next 4 h r w h e n the feeder was switched to 0-75 M. T h e c o m p u t e d s o l u t i o n s o f the m o d e l show a s i m i l a r p a t t e r n o f r a p i d u t i l i z a t i o n of feeder A ( l o a d e d with a 2.5 M sugar s o l u t i o n ) , a n d a d e c l i n e 4 hr later w h e n the feeder was switched to 0.75 M. Over the initial 4-hr p e r i o d , the

120 I00

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2o 0

Jl I 2

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(hr)

FIG. 4. The computed solutions of the mathematical model based upon the conditions of the experiment shown in Fig. 3. The first 4 hr show the buildup of bees using a 2.5 M sucrose feeder (offered along with a 0.75 M sucrose feeder), and the second 4 hr show the decline in the number of bees using the feeder when the nectar sources were switched. Note that the predictions of the model are similar to the actual experimental results. Note also that the results of Hypothesis I (a follower bee randomly selecting a dancer, i.e. no choice among dancers) and Hypothesis 2 (a follower bee following four different dancers and choosing the one advertizing the best nectar source) are nearly the same. The initial conditions were chosen to match the experimental data of Fig. 3 where there were approximately 12 bees committed to each feeder, and during the course of the day a total of approximately 125 different bees visited the two feeders. Thus, A = B = 11, D a = D b = 1, H a = H t, = 0 , and F = 101. Ntot= 125, and N, the number of bees that initially start foraging, was arbitrarily set to 50. The number of bees using the feeder is defined as the sum of number of bees at the feeder, the number of bees at the hive unloading from that feeder, and the number of bees dancing for that feeder. (..... ), Hypothesis 1; ( - - - ) , Hypothesis 2.

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number of bees using feeder A increased to 119, compared with 91 for the south feeder. When the sugar solutions were switched, the number of bees at feeder A declined to 30, compared to ten for the south feeder. In the actual experiment, the north feeder initially contained a 1 M sugar solution between 08.00 and 12.00 hours, rather than 0-75 M, in order to prevent total abandonment of this feeder. It therefore showed a slight buildup of bees. In the simulation, feeder B was loaded with a 0.75 M sugar solution and showed a slight decline in the number of bees. When the feeders were switched to 2.5 M at noon, the number o f bees reported at the north feeder increased to 122, compared with the computed result of 80 at feeder B. Figure 4 also demonstrates that the results of Hypothesis 1 and 2 are nearly the same. Despite the intuitive expectation that the most efficient exploitation of the best food source would occur when follower bees make comparisons (Hypothesis 2), this is not the case. Even with as many as four comparisons, the maximum recruitment rate to the feeder is hardly increased. Thus, little is gained over the mechanism o f Hypothesis 1 in which a bee follows only one dancer, the first that she randomly encounters in the dance area. In fact, since the model of Hypothesis 2 neglects the additional time required to make comparisons among four dancers (rather than following the first dancer encountered), it actually overestimates the rate at which bees would discover the food source. The model also predicts that over a longer period of time all the bees will abandon the poorer nectar source in favor of the better one (Fig. 5, also see Appendix). However, considering the long time required to reach this steady state (about 5 hr), it may not, in general, be achieved prior to the depletion of the food source, or before increasing numbers of bees at the nectar source leads to sufficient competition

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FIG. 5. The long-term (steady-state) behavior predicted by the model showing the time course of the number of bees using feeder A, the number of bees using feeder B and the number of bees in the follower compartment. As in Fig. 4, the graphs for feeder A or feeder B show the number of bees using the feeder, and thus is the sum of the number of bees at the feeder, unloading from the feeder and dancing for the feeder, at a particular moment. Initial conditions as in Fig. 4. ( ), Bees using feeder A; (- - -), bees using feeder B; (- - -), follower bees.

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to disrupt this steady-state. Therefore, this result of the model may be of minor ecological significance. O f greater importance is that the initial exploitation (discovery) o f a good nectar source or the desertion of a poor nectar source proceeds rapidly. (B) E F F E C T S

AND

RELATIVE

IMPORTANCE

OF

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COMPONENT

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MODEL

One advantage of a model is that it permits the simulation of experiments that are difficult or impossible to perform in the field. Each o f the branch points (Fig. 2) affects the distribution of foragers exploiting the two nectar sources. In the field, a change in nectar source quality affects the partitioning of bees at all these branch points. In order to understand better the process o f colony-level selection, it is useful to assess the relative importance of each branch point independently. Therefore, we compared the recruitment rate predicted by the full model with the predictions o f modified versions o f the model. Figure 6 demonstrates the effect o f eliminating the differential filtering of the bees at each of the branch points of Fig. 2. As in Fig. 5, the graphs show the change in the number of bees over time using feeder A (2-5 M sucrose), using feeder B (0-75 M sucrose), and in the following compartment. In the first simulated experiment, we set the probability of abandonment per foraging trip, fx, to 0-04 for both feeders A and B, keeping all other parameters fixed. Intuitively one might expect that increasing the probability of abandonment o f the better feeder would have a significant effect on the recruitment rate to that feeder. However, the initial recruitment rate to feeder A over the first half-hour [Fig. 6(a)] was nearly the same as in the complete model. Figure 6(b) shows that there was no increase in the number of bees at feeder B. The most significant effect o f increasing the abandonment of feeder A was a decrease in the long-term (steadystate) buildup of bees using feeder A, and a corresponding increase in the number o f bees in the follower compartment [Fig. 6(c)]. Increasing the abandonment rate at both feeders does not result in an increase in the utilization of the poorer food source (B), but only results in a decrease in the steady-state number of bees at the better food source (A). This is largely due to the length o f time spent in the follower compartment (60min, Table 1). The time required to find a food source after following a dancer is the bottleneck in the foraging process. An individual bee often follows a dancer yet fails to find the indicated nectar source, repeatedly returning to the dance floor to follow another dancer before discovering the nectar source (Seeley & Vissher, 1988). Next we examined the effect o f eliminating different rates o f dancing for the two nectar sources by setting fd = 1"0 for both A and B, and ra = zb = 0"38. This resulted in a decrease in the maximum recruitment rate to feeder A, and a temporary increase in the n u m b e r o f bees using the poorer nectar source, B. However, the long-term (steady-state) distribution of bees at nectar source B is still zero. Finally, we simulated the situation in which bees had an equal probability of choosing to go either nectar source after visiting the dance area by setting f~ = 0.5 for both A and B. This change removes the non-linearity from the model. The result

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. i .......................

60

40

20 I

I

I

(b) loo

-

80-

B d

z

40--

20-

.. +.................................... ,:S .............................................

0 ~" ,,,,,,; ,: ,;

,~' '1'

1

I

.....................

(c) I00 ~ , ~

60

"-. "-.

I

1

I

2

3

4

(hr)

FIG. 6. Computed solutions comparing the Full model with various modifications of the model, as described in the text. Initial conditions as follows: A = B = Da = Dh = 1, H,, = H;, = 0 , and F = 113. Ntot = 117 a n d N = 117. (a) N u m b e r of bees using feeder A. (b) N u m b e r of bees using feeder B. (c) N u m b e r of follower bees. (- - -), equal probability of following dancer A or B; (- - -), equal probability of d a n c i n g (all bees dance); ( . . . . ), equal probability of a b a n d o n m e n t (.f~ = 0.04); ( ), full model.

558

S. C A M A Z I N E

AND

J. SNEYD

is in an even greater decrease in the initial recruitment o f new bees to the better nectar source, A, and a larger increase in the recruitment rate to B. Unlike the previous cases in which the increase in the n u m b e r of bees using feeder B was temporary, now the final steady-state n u m b e r of bees using feeder B is non-zero. The results of these simulated experiments can be understood in a qualitative sense by considering the a b a n d o n m e n t as a process that creates an opportunity for bees to switch from one food source to another, while the differential following is the process that acts upon these bees and selectively filters them to the best food source. The differential dancing for one food source or another determines the degree o f selectivity that occurs in this filtering process. In other words, a b a n d o n m e n t provides a pool of u n c o m m i t t e d bees, u p o n which the dancing and following rules act to selectively filter foragers to the best food source. (C) EXTENSION

OF THE

MODEL:

COMPETITION

AT THE

NECTAR

SOURCES

Thus far we have used the model to examine the initial recruitment phase to a new nectar source. However, once a substantial n u m b e r of bees have discovered the food source, competition may occur. This can readily be incorporated into the differential equations (see Appendix) by assuming that a buildup of bees at the nectar source will degrade the bee's evaluation o f the quality of the nectar quality. Thus, food source quality can be considered a function not only of sugar concentration, but also the n u m b e r o f bees feeding from the nectar source. Various factors o f the food source will determine the nature o f the competition. These include the size of the flower patch and the type o f flowers. At one extreme, an enormous patch o f widely spaced flowers can provide nectar for m a n y bees without competition a m o n g the bees. A patch o f a few, clustered flowers will a c c o m o d a t e only a few bees before competitive interactions occur. As we show in the Appendix, if there is little competition at the food sources, then the behavior of the model with competition is very similar to the behavior of the model without competition. The recruitment rate to the better food source is similar to that discussed previously and the final steady-state is one in which there are no bees at the poorer food source. However, as competition increases, the poorer food source increases in relative value, until (for large enough competition) the final steady-state is one in which there are bees at both food sources. In this steady-state o f the model with competition, the food sources become equal in value. Discussion (A) SELF-ORGANIZED

PATTERN

OF

FORAGING

IN

HONEY

BEE COLONIES

Nectar and pollen foraging show a continually changing mosaic pattern in time and space as the colony tracks the waxing and waning o f flowers. Efficient colony functioning depends u p o n the rapid discovery and efficient utilization o f the best o f these ephemeral food sources as they a p p e a r throughout the foraging season. What is remarkable about the nectar foraging process is that colony-level exploitation o f nectar sources emerges automatically as each bee follows a few simple behavioral

COLLECTIVE

DECISION-MAKING

BY

HONEY

BEES

559

rules, each based on limited knowledge of the array of available nectar sources in the field (Seeley et al., 1991). Each bee need only have knowledge of the nectar source at which she is presently feeding. Efficient exploitation of the food sources can occur without any bees comparing the relative profitabilities of the nectar sources. Simulations based on Hypothesis 1 of the model demonstrate how differential rates of dancing and a b a n d o n m e n t based upon nectar source quality create a positive feedback system that rapidly filters the majority of uncommitted bees to the best food source. This hypothesis invokes a simpler behavioral mechanism than Hypothesis 2 which requires that a bee follow more than one dancer and make comparisons a m o n g them. The a p p r o a c h advocated here is to consider the honey bee colony as a single functional unit consisting of thousands of interacting individuals, each with a limited behavioral repertoire. The model shows how the interactions a m o n g a large number of individuals, each following the same few rules, and each having access only to local information, are sufficient to generate an efficient, self-organizing, colony-level pattern of nectar foraging. Why have we taken this atomistic a p p r o a c h to colony processes? Since colony-level processes can be extraordinarily complex, involving thousands of individuals, subtle changes in the hive, or in the field can have reverberating effects throughout the colony. Our intuition concerning the global behavior of such multicomponent systems can be very misleading. However, by incorporating the behavioral rules of the individual bees into a mathematical model we can readily appreciate their effect on colony function. (B) C O M P A R I S O N S

WITH

MODELS OF ANT

OF SELF-ORGANIZING

PATTERNS

FORAGING

It is enlightening to c o m p a r e our model of collective foraging with a similar model of collective decision making in ants (Pasteels et al., 1987; Beckers et al., 1990). The ant foraging model also presents a system of non-linear differential equations describing the self-organizing pattern of foraging at two food sources. However, the ant system and the honey bee system differ in several crucial aspects. First, the non-linearity is based upon a different aspect of the foraging mechanism. In the ant model, the branch point that determines the fraction of ants which follow a trail to a food source vs. the n u m b e r of ants that become lost is a function of the n u m b e r of ants using the food source. This occurs because ants lay down a p h e r o m o n e trail as they return to the nest after successful foraging. In contrast, the positive feedback in the bee system occurs at the branch point where an uncommitted follower bee decides which dancer to follow. In this case, Since the amount (per capita) of dancing is a function of food source quality, and since a follower bee selects a dancer for one source or another based upon the amount of dancing for each source, the follower bee is more likely to be recruited to the better food source. Thus, recruitment a m o n g bees is always a function of the quality of the food source, whereas in the ant model, food source quality plays a secondary role. Recruitment is primarily a function of the n u m b e r of ants already utilizing the food source. As

560

s. CAMAZ1NE AND J, SNEYD

a result, it is possible in the ant model for a rich food source to be neglected if a poorer source is discovered earlier (Beckers et aL, 1990). This detail is also responsible for the observed bifurcation which results when random fluctuations in foraging cause one of two identical food sources to become preferentially exploited. Among honey bees, the differential dancing as a function of food source quality always assures that the better food source will be preferentially exploited. A second difference between the two models concerns the fate of lost ants or uncommitted follower bees. These are individuals seeking a food source. In the case o f the lost ants, there is an equal probability that such an ant will discover one food source or the other. However, in the case of the followers, bees are preferentially filtered to the better food source through differential dance following. When presented with two identical feeders, this randomizing influence and the nature of the trail following results in a bifurcation in the numbers of ants at the feeders as one source becomes preferentially exploited. In the case of the bees, however, the model predicts that there is an infinity of stationary states associated with the exploitation of two identical food sources. Other interesting behavioral difference are seen among the various species of ants discussed by Beckers et al. (1990), and these have been incorporated into variations o f their model. But what is most important to emphasize here is that seemingly minor biological differences concerning the behavior and ecology o f a species may result in major differences in the pattern of foraging, and the nature o f the autocatalytic process. A carefully formulated model should be able to embody these differences and generate the observed experimental observations. (C) S E L F - O R G A N I Z I N G

PROCESSES

IN INSECT

SOCIETIES

A common feature of the advanced social insects is their highly co-ordinated collective endeavors. For example, army ant raids, numbering up to 200000 individuals, follow a precise compass bearing each day as they methodically sweep the forest floor for prey. The army ant nest, with its half a million inhabitants, can accurately regulate its temperature to within I°C (Franks & Fletcher, 1983; Franks, 1989; Deneubourg et aL, 1989). Likewise, honey bee colonies demonstrate remarkable thermoregulatory skill throughout the long temperate winter (Heinrich, 1981, 1985). Termites build complex, colossal structures (GrassY, 1959, 1967; Deneubourg, 1977) thousands of times larger in size than the individuals who take part in the construction. A colony of thousands of individuals faces the same challenges to survival that confront single organisms--foraging, defense and protection against environmental extremes. The colony's collective solutions to these problems of survival have therefore prompted a view of the society as a single functional entity, a "superorganism," capable of adaptive decision-making and co-ordinated colony-level activity (Maeterlinck, 1927; Wheeler, 1911; Wilson & Sober, 1989; Seeley, 1989; Franks, 1989). But paradoxically, the colonial superorganism is only a loose assemblage of separate individuals, and it lacks mechanisms such as a nervous or circulatory system that physically integrate the organism. While it is clear that analogous integrating

COLLECTIVE DECISION-MAKING

BY H O N E Y BEES

561

mechanisms must exist to co-ordinate the collective activities o f social insects, they have been difficult to elucidate. To unravel this mystery we must understand how the interactions a m o n g individuals generate complex collective behaviors. Wilson (1971) remarked that the reconstruction of mass behavior from a knowledge of the behavior of single colony members is the central problem of insect sociology. In a broader context, this question is one of understanding how the global properties of a system emerge from the dynamic interactions of its constituent subunits. This question has plagued developmental biology (Maynard Smith, 1986); how can we understand morphogenesis, the emergence of an orderly complex adult structure, in terms of the properties and interrelations a m o n g the cellular subunits of the developing embryo ? The question arises once again in neurobiology where patterns of neutral activity (locomotion or thought) emerge from the connections and dynamic interactions of neurons (Sch6ner & Kelso, 1988). It appears, at least in part, that diverse biological p h e n o m e n a such as these can be understood as self-organizing processes in which order emerges spontaneously from the dynamic interactions of the subunits (Rosen, 1981; Murray, 1989). With this approach, it is often possible to characterize the overall properties of a system based upon a detailed knowledge of the properties of its constituent components. Much of this work was conducted at the Centre for Mathematical Biology, Mathematical Institute, Oxford University, U.K. and was supported, in part, by grants from the U.S. National Science Foundation (grant BNS-8916006) and the U.S. Department of Agriculture, through Hatch grant NY(C)-191407 to Tom Seeley, and a National Science Foundation Graduate Fellowship to S.C.J.S. acknowledges the support of the University of California, Los Angeles. We are grateful to T. Seetey for providing experimental data (used in Table 1 and Fig. 3), for his careful reading of the manuscript, and for numerous enlightening discussions. We also wish to thank two of the reviewers, J. L. Deneubourg and S. Goss, for their helpful comments. REFERENCES BECKERS, R., DENEUBOURG, J. L., GOSS, S. & PASTEELS, J. M. (1990). Collective decision making through food recruitment, lnsectes Sociaux 37, 258-267. DENEUBOURG, J. L (1977). Application de l'ordre par fluctuations ~i la description de certaines &apes de la construction du nid chez les termites, lnsectes Sociaux 24, t 17-130. DENEUBOURG, J. L., ARON, S., GOSS, S. & PASTEELS, J. M. (1990L The self-organizing exploratory pattern of the Argentine ant. Z Insect Behav. 3, 159-168. DENEUBOURG, J. L. & GOSS, S. (1989). Collective patterns and decision-making. Ethol. EcoL E~ot. 1, 295-311. DENEUBOURG, J. L., GOSS, S., FRANKS, N. & PASTEELS, J. M. (1989). The blind leading the blind: modelling chemically mediated army ant raid patterns. J. Insect Behao. 2, 719-725. FRANKS, N. R. (1989). Army ants: a collective intelligence. Am. Sci. 77, 139-145. FRANKS, N. R. & FLETCHER, C. R. (1983). Spatial patterns in army ant foraging and migration: Eciton burchelli on Barro Colorado Island, Panama. Behav. Ecol. SociobioL 12, 261-270. Goss, S. & DENEUBOURG, J. L. (1989). The self-organising clock pattern of Messor pergandei (Formicidae, Myrmicinae). Insectes Sociaux 36, 339-347. Goss, S., ARON, S., DENEUBOURG, J. L, & PASTEELS, J. M. (1989). Self-organized shortcuts :.n the Argentine ant. Naturwissenschaften 76, 579-581. GRASSY, P. (1959). La reconstruction du nidet les coordinations interindividuelles chez Bellicositermes natalensis et Cubitermes sp. La th6orie de la stigmergie: Essai d'interpretation du comportement des termites constructeurs, lnsectes Sociaux 6, 41-83.

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GRASSY,P. (1967). Nouvelles exp6riences sur le termite de Miiller (Macrotermes mailed) et consid6rations sur la th6ode de la stigmergie, lnsectes Sociaux 14, 73-102. HEINRICH, B. (1981). The mechanisms and energetics of honeybee swarm temperature regulation. J. expl. Biol. 91, 25-55. HEINRICH, B. (1985). The gocial physiology of temperature regulation in honeybees. In: Experimental Behavioral Ecology and Sociobiology (HSlldobler, B. & LINDAUER,M., eds) pp. 393-406. Sunderland, MA: Sinauer. MAETERLINCK, M. (1927). The Life o f the White Ant. London: George Allen and Unwin. MAYNARD SMITH, J. (1986). The Problems o f Biology. Oxford: Oxford University Press. MURRAY, J. D. (1989). Mathematical Biology. Berlin: Springer Verlag. PASTEELS, J. M., DENEUBOURG, J. L. & Goss, S. (1987). Self-organization mechanisms in ant societies (I): Trail recruitment to newly discovered food sources. In: From Individual to Collective Behavior in Social Insects (Pasteels, J. M. & Deneubourg, J. L., eds) pp. 155-176. Basel: Birkh/iuser. ROSEN, R. (1981). Pattern generation in networks. Pro& theor. Biol. 6, 161-209. SCHONER, G. & KELSO, J. A. S. (1988). Dynamic pattern generation in behavioral and neural systems. Science 239, 1513-1520. SEELEY, T. D. (1986). Social foraging by honeybees: how colonies allocate foragers among patches of flowers. Behav. Ecol. Sociobiol. 19, 343-354. SEELEY, T. D. (1989). The honey bee colony as a superorganism Am. Sci. 77, 546-553. SEELEY, T. D., CAMAZINE, S. ~g SNEYD, J . (1991). Collective decision-making in honey bees: how colonies choose among nectar sources. Behav. Ecol. Sociobiol., in press. SEELEY, T. D. & VISSCHER, P. K. (1988). Assessing the benefits of co-operation in honeybee foraging: search costs, forage quality, and competitive ability. Behav. Ecol. Sociobiol. 22, 229-237. VISSCHER, P. K. & SEELEY, T. D. (1982). Foraging strategy of honeybee colonies in a temperate deciduous forest. Ecology 63(6), 1790-1801. WHEELER, W. M. (1911). The ant colony as an organism. J. Morphol. 22, 307-325. WILSON, D. S. & SOBER, E. (1989). Reviving the superorganism. J. theor. Biol. 136, 337-356. WILSON, E. O. (1971). The Insect Societies. Harvard: Harvard University Press. APPENDIX

Construction of the M o d e l I n ' t h i s A p p e n d i x we c o n s t r u c t a simple m a t h e m a t i c a l m o d e l o f h o n e y bee f o r a g i n g b a s e d o n Fig. 2. A l t h o u g h the m o v e m e n t tff each bee from o n e c o m p a r t m e n t to a n o t h e r c a n best be d e s c r i b e d in terms o f p r o b a b i l i t i e s , the b e h a v i o r o f a large n u m b e r o f bees can be w e l l - d e s c r i b e d in terms o f the c h a n g e in the m e a n n u m b e r o f bees in each c o m p a r t m e n t . We m a y thus c o n s t r u c t a d e t e r m i n i s t i c m o d e l ( i n v o l v i n g a system o f o r d i n a r y differential e q u a t i o n s ) that p r o v i d e s a c o n t i n u o u s d e s c r i p t i o n o f the b e h a v i o r o f a large n u m b e r o f bees. Let A d e n o t e the n u m b e r o f bees in c o m p a r t m e n t A a n d s i m i l a r l y for the o t h e r c o m p a r t m e n t s . T h e n , f r o m Fig. 2, we c a n write d o w n the f o l l o w i n g system o f differential e q u a t i o n s for the n u m b e r o f bees i n each c o m p a r t m e n t : dA

dt dD~ dt

d~ dt

=

(1 -~)(1 -f])plH~ +p2Da +p4f~F-p3A

- ~ d ( 1 - f ~ ) p~ H~ - p 2 D ~

- p 3 A - pI H~

dF ~ b d--t = f ~ P ' Ha + f ~ P 5 l i b -- p 4 F

(A.1)

COLLECTIVE

dB ~ ( 1

dt

DECISION-MAKING

BY

HONEY

BEES

563

b - f a)PsHb b -fx°)(1 + p6Db + p , ( 1 - f ~ ) F - p T B

d D b _ f ~ ( 1 - f~)psHb - p6Db

dt

d~ dt

- p 7 B -psHb.

Note that we have used the identity f~ = 1 - f ~ . In the model we shall denote the quality of a food source by a : a~ and ab denote the quality o f A and B respectively. In the initial model we shall assume that a is constant for each food source, although in later extensions of the model this assumption is relaxed. In general, fx (the abandonment function) and fd (the dancing function) are functions of a i.e. fx = f x ( a ) andfd =fd (Or). [In the above system we use the notation f,~ =fd(ot~), f x~ =fx(a~) and similarly for f~ and f b . ] However, it is important to note that, since we are assuming that a~ and ab are both constant, we do not need to know the functional forms of fx and fd in order to perform computations with this initial model: the experimentally obtained values o f f x and fa for two different food sources are sufficient for this purpose. Further, the rate constants are also functions o f the quality o f the food source. However, the above argument holds for them as well, and thus they may be effectively treated as constants. The function f~' (which we shall call the following function), is assumed to be a function o f D~ and Db. Since it is not known precisely how an individual bee selects a particular dancer to follow, we shall study the behavior of the model for several choices of f~(D~, Db). Values for the rate constants can be obtained from experimental data. If the rate at which bees leave compartment X is Px then the average time a bee stays in X is 1/px. Thus, by observing how long (on average) a bee stays in a particular compartment, it is possible to determine the rate constants. Experimentally obtained values for the rate constants are given in Table 1. It can be seen from eqn (A.1) that d

- ~ ( A + B+ D~+ Db + H~+ Hb + F) =0. Thus, we can write down the conservation equation

A+B+D~+Db+H~+Hb+F=N, where N is the total number o f bees in the system. Using the conservation equation to eliminate one o f the variables in (A.1), F say, we get the new system

dA = (1 -f,~)(1 -f~)p,H~ +p2Da +p4f~F-p3A dt

dD~ -~d(1-f~)plHa-p2D~ dt

564

S. C A M A Z I N E

AND

J.

SNEYD

dHo dt

= paA-plHa

(M) d__B= ( 1 - f ~ ) ( 1 -- fbd)p5 l i b + p 6 D b + dt

d Db dt

d/-/b dt

b b -- f d ( 1 -- f ~ ) p s n b

p4(1 -- f ~ ) F - p 7 B

-- p 6 D b

- p T B -- p s H b .

where F = N - A - B - D o

-

D b -- H a - H b .

We shall call this system of equations, M.

The Incorporation of a Variable Number of Foragers

The above equations were derived assuming that all the foragers start foraging simultaneously. However, this assumption can be easily relaxed. We can instead model the dynamics of the foraging pool by including in the system M the equation dN = k,,(Ntot- N), dt

k, a constant,

which allows the number of foraging bees to increase with time. The number of foraging bees, N, will increase quickly at first, rising in an exponential curve to a maximum of Ntot. This phenomenologicai equation qualitatively describes the dynamics of the forager pool. Incorporation of this equation into the model will make no difference to the following steady-state analysis (except that N is then replaced everywhere by Ntot); in the Appendix we thus restrict our attention to the system M.

Analysis of the Model

A steady-state of the system M is found by solution of the equations (1 -f.~) (1 - f ~ ) p , H ~

+ p2D. + p4f'[ F-p3A f~(1 -f~)p,

= 0

n~ - p2D. = 0 p3A-p1H.,

(1 -f~)(1 -- fba) p s H h + p 6 D b +p4(l - f ' ; ) F - p 7 B

=0 = 0

f~(1 -- f ~ ) p s H b -- p 6 D b = 0 p T B - p s H ~ = O.

(A.2)

COLLECTIVE

DECISION-MAKING

BY H O N E Y

BEES

565

Thus

Ha = p3A Pl Da "-=f~(1 -f~)p3A P2 p7 B lib= P5

(A.3)

Db f ~ ( 1 -f~)p7B P6

and using (A.3) to eliminate D~, Ha, Db and Hb in (A.2), gives two equations for A and B in terms o f f ~ and the parameters:

f~p3A = P4f~ F

(A.4)

fbxp7B = p4(1 - f ~ ) F.

In general, f~ is a function of D~ and Db and so the solution of (A.4) will depend on the functional form chosen for f~. However, eliminating f~ in (A.4) gives

f~p3A = p4F -- fb p7B = P4(N - A - B - D~ - Db - Ha - Hb) -f~pTB =P4{ N-A-B-P3API

p7Bp5 f~(1-J~)p3Ap2

(A.5)

fbd(1--fb":)p7Bt7 J -f~p7B

(A.6)

and thus

q~tA + dp2B = p4N,

(A.7a)

where

4',={f:P~+P~[I+p3~JT'(~1-J~)p~j}

(A.7b)

P5

(A.7c)

and "~6

_lJ"

This relationship holds, irrespective of the functional form offer (Da, Db). Similarly we can eliminate F in eqn (A.4) to get

p7f~B =p3(1 - f f )

A.

(A.8)

For a given f'i(D,, Db), solution of eqns (A.7) and (A.8) will now give the steadystates of M. [Although, for a given f~, we could have solved eqn (A.4) directly for A and B, it is easier to solve using eqns (A.7) and (A.8) instead.]

566

S. C A M A Z I N E

AND

J. S N E Y D

In the text, we considered two cases in particular for f ~ : Hypotheses 1 and 2. However, since the mathematical analysis is the same for both hypotheses, we shall consider them together here. H y p o t h e s e s 1 and 2

Without loss of generality, we may assume that food source A is better than food source B, i.e. ao > ab. Then, let f ~ = 1 - [ ( % D b ) " / ( r a D o + %Db)"], where % and % are constants that describe the amount of time each dancer spends dancing for its respective food source and n is the n u m b e r o f dancers that a follower bee follows (see text). Since food source A is assumed to be better than food source B it follows that To > %. It is important to note that, although the analysis is the same for all n -> 1, the case n = 1 is fundamentally different (biologically speaking) than the case n > 1. This is discussed in detail in the text. Let us now consider the solutions o f eqn (A.8) with this choice for f ~ i.e. (%Db).

]

pTf~B 1 (%/~+%---l)b)"J

( %Db)~

-(roDo+%Db)"p3f~A.

(A.9)

Clearly, A = 0 and B = 0 are solutions of eqn (A.9) [Note that A = 0 does not imply that B = 0: for, if A = 0, then DQ = 0 also, which means that f ~ = 0. Thus, A = 0 is a solution o f (A.9) for any B. Similarly for B - - 0 ] . However, are there any other solutions o f (A.9) such that neither A nor B are zero? The answer is no, and we show this as follows. Without loss of generality, assume that B # 0 and rewrite (A.9) as

pvf~[ ( %Do + %Db )" - ( rbDb)'~] = ( %Db )"p3f: ( A). Since B # 0 by assumption we may divide through by (rbDb)" to get

Now,

PTf~[(l+r"D'~y'-l]=P'f:(B) , let x = A/B. Then, substituting in for Do and Db gives pTf~[(1 +/3x)" - 1] =

p3f°xx,

where p6p3rof~(1 - f : )

- P2P7rbf~ ( 1 -f~)" Hence, we are looking for solutions to the equation

f,(x) = f 2 ( x ) ,

(A.IO)

where

f,(x) = (1 + fix)", and :2(x)

=

\

b

PTf ~]



COLLECTIVE DECISION-MAKING BY HONEY BEES

567

We note two things about f~ and f2- Firstly, f2 is a straight line with f 2 ( 0 ) = 1. Secondly, x >>.0 ~ f ~ " ) ( x ) >-0 for all m - 0, and f~(0) = 1. Hence, since f~ is always increasing and concave up, it can intersect the straight line, f2 in at most two places. One o f these places is clearly x = 0, which corresponds to A = 0 and is therefore disregarded. It follows that f~ will intersect f2 at some x0 > 0 if and only iff'l(0)
< P3f~x But, since ~ > f ~ , 1 - f ~ > 1 - f ~ , P6 > P2 (food source A is better than food source B and the bees will dance longer for a better food source cf. Table 1) and To > 7b it follows that nfl >P3f~/(PTf~) for all n - 1. Hence, fl can intersect f2 only at x = 0 and the desired result follows. We have not shown that for this choice o f f , ( f o r any n >- 1) there are exactly two critical points: A=0

p4N

B=fbxp7-~I+ P__Z.+fb(I --fb)p7]' P4[

L

P5

P6

(A.II)

J

and

p4N

A= f a p 3 "1t" p411

L

_tp3 ~f~(1--fa)p3]" Pl

~

_1

(A.12)

B=0. Computations show that the stable critical point is the one in which there are no bees at the food source of lesser value, i.e. the steady-state given by eqn (A.12) is stable if food source A is better than food source B.

Properties of the Steady-state The first thing we notice is that the steady state given by eqn (A.12), (which we shall call Po), does not depend on n. Thus, in terms of the steady state distribution, Hypotheses 1 and 2 are identical. Further, P0 does not depend on f~ or 3~a: the steady-state is independent of the abandonment rate and the dancing rate from the lesser food source. Rewriting A gives

A-

p4N

f p3( 1

p,( 1 +P3~ +ffaP3P4" Pl/

P2

568

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AND

J. S N E Y D

and thus we see that P2 > f ~ P 4 ~ A decreases as f x~ increases P2 <~dP4 :=~A increases as f~x increases. Hence, an increase in the a b a n d o n m e n t rate from A will result in a decrease in the n u m b e r o f bees at A only if the rate at which bees find a food source after following a dancer is small enough. From our experimentally measured values for the rate constants it can be seen that this process o f finding a food source is indeed very slow, and so for all our simulations an increase in the a b a n d o n m e n t rate results in a decrease in the n u m b e r of bees at A. From eqn (A.3) it follows that a decrease in A (at P0) results in a decrease in D~ and Ho and therefore an increase in F (since F = N - A - D o Ho when B = 0). Thus, as discussed in the text, as the a b a n d o n m e n t rate from A increases, the n u m b e r of bees following a dancer at the steady-state Po increases. For a very good food source with f~ = 0, there are no follower bees at PoAt first sight it is a puzzling feature of this model that even if two food sources are of very similar quality, the bees will eventually neglect the lesser one completely. However, computations show that the bees initially forage from both sources almost equally and the final steady-state (where one o f the sources is neglected) is not reached for a considerable period of time. Thus, over reasonable periods of time, it would a p p e a r as if the bees were foraging equally from two sources o f similar quality: it is unlikely that the steady-state could ever be reached before other factors (such as depletion of the food source) disrupted the process. The Case of Identical Food Sources I f the food sources are of identical value (aa = O~b), and if n = 1, there is a line of steady-states, as in this case eqn (A.9) is identically satisfied. For, if ao = tzb, then ~'o = zb and P3 =P7, and thus eqn (A.9) becomes f~BDo =f~,ADb. Substituting in the expressions for Do and Db from eqn (A.3) then shows us that (A.9) is identically satisfied as required. Thus, every point along the line ~blA + ~bzB = p3N is a steadystate for the system. In this case, the starting conditions determine the final steadystate. However, this could never be observed in the field, as even the smallest deviation from equality results in dramatically different behavior. Such inherent biological instability renders this case insignificant. The Elimination of Differential Following Let us now consider the case f~ = constant, i.e. a constant fraction of the bees always goes to each food source. In this case, the model M is a linear nonh o m o g e n o u s system and has a unique stable steady-state, in which there are more bees at the better food source, but a non-zero n u m b e r of bees at the lesser source. Thus, the behavior of the model under this assumption is qualitatively different from the behavior of the non-linear version of the model. Some results from the linear model are presented in the text.

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Competition at the Food Source Let us now investigate what happens when competition at the food source is incorporated into the model. Our discussion is not meant to be quantitatively accurate, but should give some idea of the wide range of possible behaviors of more general models of honey bee foraging. Assume that the quality o f the food source is a decreasing function o f the n u m b e r of bees at that food source, as crowding at the food source means that some bees do not get adequate access to the food. In particular, let us assume that ~ ao

kaO

a~-l+A/k

k+A'

(A.13)

and similarly for ab. Here, a ° is the worth of the food source A when there are no bees at A (and thus there is no crowding), and k is some competition factor, chosen arbitrarily. Clearly, large values o f k mean that there is little competition at the feeder while small values of k mean that there is a large amount of competition at the feeder: actual values for k will depend on the geometry of the particular feeders that are being used. Here we are assuming that food sources A and B have identical competition factors. The above functional form for an is chosen arbitrarily. In general, this function will also depend on the particular feeders being used. Since the quality o f the food source will now change over time, it is necessary to specify the functional forms offd and fx in order to study the equations. We shall specify an arbitrary form for fx such that it is of a biologically reasonable shape, and for simplicity, we shall just let fa be a constant function. Also for simplicity, let us assume that each pi, i = 1, 7 is constant and that p~ = Ps, p2 P6 and P3 P7- Let ----

1

--, a f'c(a)=l+

=

(A.14)

and suppose that f~ = fb = fa for some constant fd. Finally, let f7 = D,J ( D,~+ 196) (a simplified Hypothesis 1). Let us denote this new model, with the inclusion of competition in a simple way, as model MC. Although some of the assumptions in this version o f the model with competition are biologically unrealistic, an analysis o f this model will provide us with some insight into the behavior o f more realistic models. We shall show here that, if there is little competition at the feeders, the steady-state is one in which all the bees are at the better food source (as in Model M, Hypothesis 1 above), but that an additional (stable) critical point appears when N / k reaches a certain value, where N is the total n u m b e r o f bees. Thus, if the competition at each o f the food sources is great enough, a steady-state distribution o f bees will result, in which there are a non-zero n u m b e r of bees at each feeder. Without loss of generality, assume that a ao > a bo. For the above choice o f f T , eqn (A.8) becomes

AB[

f~

]=ABr

fa(1 -f~)_[

f~

1

Lfa(1 _ f b ) J,

(n.15)

570

s.

CAMAZINE

AND

J.

SNEYD

and substituting eqns (A.13) and (A.14) into (A.15) we get

AB

(oo)

,oo,

= AB~---~,].

(A.16)

It thus follows that the steady-state solution for A and B for this simple extension o f the model is given by solution of eqns (A.16) and (A.7). We see immediately that A = 0 and B = 0 are two solutions o f (A.16): when A = 0, it is easy to show that there is exactly one positive solution for B, and similarly for B = 0. These two critical points, in which all the bees are at one of the food sources, are analogous to the critical points o f model M, given by eqns (A.11) and (A.12). Let us look for other steady-states of model MC. Suppose that A # 0 and B # 0. Then, dividing (A.16) by A B we find that the other steady-states of model MC are given by solution of the equations 0

o

a a _ ab k+A k+B

(A.17)

c~lA + qb2B = p4N,

(A.18)

where ~bl and ~b2 are as in eqns (A.7b) and (A.7c). Notice that this steady-state is obtained when the two sources are of identical quality. We see immediately from (A.17) that

o

A

= ~....._~a

o~ B + k

o

-1

,

and thus A > B. Substituting this into (A.18), simplifying, and collecting powers of B gives

c~ B 2 + c2B + c3 = 0, where

Cl

=(1+--~b)(plp2P3 o + p2P3P,* + plp3p4)

C2 = k

Oto

- 1

I'\+

PlP2P3 +P2P3P4+PlP3P4) o

+ k / -1- r -a/aE p , p ~ p ~ ( I + ~ o~) + p , p ~ p 4 f ~o +p~p~p,(1 + o ) + p,p~p,] \

orb /

- Nplp2p3, and

c3= k2( ~Ob--1)[ plP2P3( I + aO) + plP3P4fdct ~ + P2P3p4( I + a O) + plP2P4] -- Nkplp2P3(1 + ct°).

(A.19)

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It is possible to show (the details are omitted) that (A.19) has exactly one positive root if and only if -->

k

-1

1+

pz(l+ot °) p, p 3 ( l + a ° )

'

and that this root bifurcates from the stable steady state B = 0. C o m p u t a t i o n s show that the bifurcating non-zero critical point is stable and the zero critical point becomes unstable: computations also show that if N / k is small enough, the stable critical point is the one in which all the bees are at the better food source, as in model M, Hypotheses 1 and 2 above. In conclusion we can say that when the competition at each feeder becomes large enough (either as a function of the total n u m b e r of bees or of the geometry of the feeder) a new stable critical point is generated, and the final steady-state distribution is one in which there are bees at both food sources,with more bees at the better food source.