Intelligent decisions from the hive mind: Foragers and nectar receivers of Apis mellifera collaborate to optimise active forager numbers

Journal of Theoretical Biology 271 (2011) 64–77

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Intelligent decisions from the hive mind: Foragers and nectar receivers of Apis mellifera collaborate to optimise active forager numbers James R. Edwards , Mary R. Myerscough The School of Mathematics and Statistics, The Centre for Mathematical Biology, University of Sydney, New South Wales 2006, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 July 2010 Received in revised form 27 October 2010 Accepted 22 November 2010 Available online 30 November 2010

We present a differential equation-based mathematical model of nectar foraging by the honey bee Apis mellifera. The model focuses on two behavioural classes; nectar foragers and nectar receivers. Results generated from the model are used to demonstrate how different classes within a collective can collaborate to combine information and produce finely tuned decisions through simple interactions. In particular we show the importance of the ‘search time’ – the time a returning forager takes to find an available nectar receiver – in restricting the forager population to a level consistent with colonywide needs. Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

Keywords: Apis mellifera Collective decisions Search time Receiver Mathematical model

1. Introduction Collective decisions may result from the simple aggregation of individual preferences. Often, however, they emerge as individuals’ preferences and their interactions with others combine in complex ways. In some important situations, individuals can be categorised by their behaviour into different classes whose interactions drive the decision making process. Such scenarios occur in both human and animal societies (Conradt and List, 2009). For example, social choice theory (Arrow, 1963) and its related fields (Dryzek and List, 2002) are primarily concerned with how human societies make decisions when classes of individuals assign different utility measures to the available options. Examples in animal societies are given by Couzin et al. (2005) and Raghib et al. (2010), who divide social groups such as fish, insects and birds into classes of informed and uninformed individuals which each move according to different rules whilst interacting to ensure the group travels cohesively, and by Clutton-Brock (1998) who discusses how groups decide which members will reproduce in societies containing dominant and subordinate females. Most collective decision models focus on either the individualscale (Bonabeau, 2002; Pratt et al., 2005) or the group-scale (Nevai et al., 2009; Tachikawa, 2010), and therefore do not explicitly include interactions between classes which occur at an intermediate level of organisation. In this paper we look at the intermediate,

 Corresponding author.

E-mail address: [email protected] (J.R. Edwards).

or mesoscale, interactions between two behavioural classes of Apis mellifera (honeybees), foragers and receivers, and explore how this interaction mediates the decision ‘how many foragers should go out of the hive to gather nectar?’ A representation of these three scales, and how they relate to each other, is given in Fig. 1. Nectar foraging is a particularly important and well studied task of the honey bee colony (Lindauer, 1961; Seeley, 1995; Dyer, 2002). The gathering of energy, in the form of nectar, is one of the main tasks that a colony engages in to maintain and increase the colony population. Foragers exploit nectar sources (flowers) by collecting their nectar and returning it to the hive. The nectar is delivered to waiting receiver bees who then process and store the nectar within the hive. The forager’s main behavioural choices are resting, returning (to the nectar source) or recruiting (performing the waggle dance to encourage other resting foragers gather nectar) (Camazine and Sneyd, 1991; Dyer, 2002). Foraging is not only important, but it is dangerous and costly. There is a high risk to the forager of predation and misadventure resulting in death (Dukas, 2001; Page and Peng, 2001). A forager expends energy when she flies out on a foraging expedition, which may be unsuccessful. Both forager death and unsuccessful foraging may diminish nectar inflow and, if frequent, potentially lead to a rapid decline in the colony population. Clearly, if it is to prosper, the colony must make a wise collective decision about how best to allocate foragers for maximum nectar return. The principle motivation of this paper is to model the effect on forager recruitment of forager and receiver interactions, and thereby understand more about their impact on the forager allocation decision.

0022-5193/$ - see front matter Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.11.027

J.R. Edwards, M.R. Myerscough / Journal of Theoretical Biology 271 (2011) 64–77

Fig. 1. The level of organisation at which a collective decision is formed may occur at different scales. In some cases considering the group as a single entity is appropriate (inner circle), in other cases focusing on each of the potentially thousands or millions of individuals is more natural (outer circle). For nectar foraging by colonies of A. mellifera modelling interaction at the mesoscale (middle circle), that is between a few different behavioural classes, allows us to best understand the forces driving the decision making process.

Existing models of nectar foraging focus, unsurprisingly, on foragers (Camazine and Sneyd, 1991; de Vries and Biesmeijer, 1998; Cox and Myerscough, 2003). In particular, they focus on the recruitment of new foragers through the waggle dance and the communication of nectar quality to other foragers. All such models relate recruitment rates to nectar quality; under normal conditions high quality sites will induce extensive waggle dancing and high recruitment levels whereas low quality ones may not even lead to waggle dancing at all. Comparatively little attention has been given to receiver bees compared to foragers, possibly because experimental work with receivers is much more difficult than with foragers. Because receivers work in the hive, they are much more difficult to label and observe than foragers. Farina and co-workers have investigated the behaviour and responses of individual receivers (Tezze and Farina, 1999; Martinez and Farina, 2008) and Seeley and coworkers have looked at their interactions with foragers (Seeley and Tovey, 1994; Seeley, 1995). There is very little modelling work on receiver behaviour. Models have been created for waiting times during forager–receiver interactions (Anderson and Ratnieks, 1999) and for the effects of multiple unloadings (Gregson et al., 2003) but there is not yet any modelling that examines the effects of receivers on forager recruitment. The modelling presented here is, therefore, very straightforward and somewhat abstract and idealised but it is, at least, a first step in examining the problem theoretically. It is intended only to model the present interactions of foragers and receivers and does not consider evolutionary fitness or evolutionary adaptation. Clearly there is scope for more elaborate model once the results of the basic modelling have been analysed. Experienced foragers know about the quality of the foraging environment, which includes not only the concentration of sugar in the nectar but also all other factors that a forager evaluates when determining the number of waggle runs they will perform. Receivers

65

learn about the quality of the incoming nectar (i.e. its sucrose concentration, etc.) (Anderson and Ratnieks, 1999), but they have very little first hand knowledge of the foraging environment and nectar sources. On the other hand, receivers, because they are hive workers, may reasonably be expected to know more about the state of the colony’s reserves than foragers do (Seeley et al., 1996). We introduce a model that incorporates both foragers (knowledgeable about and working in the external environment) and receivers (who work inside the hive and so know about that environment) to show how interactions between these classes determine the size of the active forager population. The model is designed to explore how the two classes that we consider collaborate to direct the colony commitment to foraging. To this end we simplify the events that would occur in nature and restrict ourselves to an environment where the colony has no memory of previous forage sources and foraging is solely directed by current nectar quality, receiver expectations (i.e. population) and a set of standard parameters controlling base rates of recruitment, resting, nectar storage and gathering. This scenario broadly mimics the setup of classic foraging recruitment experiments (e.g. Seeley et al., 1991; Seeley, 1995). Further, the model is less concerned with which forage source is exploited than with how many foragers are exploiting sources, as this is a better indication of the colony’s commitment to nectar gathering. The basic model uses a system of three nonlinear ordinary differential equations representing the populations of active foragers, foragers unloading their nectar, and receivers available to unload nectar. We find both analytic and numerical results of the model and use these results to explore A. mellifera foraging in particular and collective decision making in general. Finally, we extend the model to incorporate multiple, dynamic foraging sources and changes in receiver population. These models are neither individual-oriented nor group-oriented. They operate at the mesoscale (Fig. 1) in which classes of individuals (in this model foragers and receivers) are the entities that are considered.

2. Simple model We consider two behavioural classes of A. mellifera, foragers and receivers, and construct a model to explore how the interaction of these classes determines the active forager population. We assume an unchanging, predator and competition free environment with constant forager round-trip times. We also assume that the foragers’ tendency to rest, to return or to recruit is determined by two factors: the quality of the nectar (Camazine and Sneyd, 1991) and the time spent searching for a receiver (Seeley and Tovey, 1994). Initially, we introduce two simplifying assumptions: the total receiver population is considered to be constant throughout the model and there is a single source containing nectar of constant quality. Later we will extend the model and drop these assumptions. The interactions in this model are illustrated in Fig. 2. Two separate sets of data are likely to be perceived by different classes of the colony, the availability of nectar by the forager class and the need of the colony for nectar which is more likely to be perceived by hive bees, including the receiver class, than by foragers. The colony nectar need to determine the population of receivers and the ‘ready to receive’ subset of this population. This population, and the population of those foragers ready to unload, determines the average time that a forager spends searching for a receiver to unload nectar, known as the ‘search time’. A forager’s tendency to choose to rest, to return or to recruit is dependent upon the nectar quality and the search time (Seeley and Tovey, 1994). The outcome of each individual decision then feeds back into the population of foragers, and hence into future search times. This system places

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J.R. Edwards, M.R. Myerscough / Journal of Theoretical Biology 271 (2011) 64–77

part of the active receiver population represented in the model. If ss models the constant time taken when approaching another bee we can calculate the expected search time SðtÞ ¼ ss iq

F ðtÞ þ RðtÞ : Q RðtÞ

ð1Þ

The expected search time S(t) is critical; it aggregates the outcomes of numerous interactions between individual foragers and receivers into a single value that strongly influences the rate of forager recruitment in the model. It is worth noting that this model, because it is a system of ordinary differential equations, can only be used to calculate the average search time for foragers from a particular nectar source when, in reality, the actual search time experienced by individual foragers will vary significantly around this average. Nevertheless, since we are modelling a colony-wide phenomenon rather than individual behaviour, using the average search time still gives a reasonable indication of the expected behaviour of the system.

2.2. Equations for forager and receiver numbers

Fig. 2. The decision making process of a colony of Apis mellifera engaged in nectar foraging. Nectar quality and colony nectar need are the two inputs, principally perceived and analysed by foragers and receivers, respectively. The interactions of foragers and receivers produce a value, search time. Nectar quality and search time are used by foragers to decide whether to rest, return or recruit.

search time at the centre of a web of interactions, and so we will describe and define it in more detail. 2.1. Search time of returning foragers for receivers Upon entering the hive after a successful foraging expedition, a returning forager seeks a receiver who will accept her load of nectar. Foragers often unload to several receivers, but, since foragers and receivers have the same crop size (Hart and Ratnieks, 2001), we assume that multiple unloadings are equivalent to a one to one interaction and assume the search time can be thought of as the sum of all the search times experienced by an individual. The expected time spent searching for a receiver can be modelled using the proportion of foragers and receivers (Seeley and Tovey, 1994). Let F be the total active forager population, R be the total receiver population, F be the population of foragers seeking to unload near the hive entrance and R be the population of receivers ready to receive. Then, assuming that the population in the hive entrance is exclusively composed of foragers seeking to unload and receivers willing to receive nectar loads, the probability that a forager’s approach to another bee in the entrance is to a receiver can be modelled by R=ðF þRÞ. Disregarding the other hive bees simplifies the equations without significantly affecting the results; note that foragers and receivers engaged in nectar transfer after a successful search are included in these ‘other hive bees’. The expected number of interactions before finding a receiver is the inverse, ðF þ RÞ=R. However, it has been demonstrated (Scheiner et al., 2004; Martinez and Farina, 2008) that the likelihood of a receiver accepting a nectar delivery is affected by nectar quality and experience and genotype of both forager and receiver, resulting in foragers having to search longer to find a receiver willing to accept low quality nectar. We will model this by scaling the population of receptive receivers by the true nectar quality divided by a high quality iq so that the expected number of interactions are ðF þ RÞ=ððQ =iq ÞRÞ. The factor iq will be chosen so that Q r iq and in a single nectar source model with high quality nectar, we can generally let iq ¼Q, since those receivers not willing to unload for this quality are effectively not

Let Q be the quality of a nectar source as defined by its sucrose concentration measured in moles per litre since A. mellifera has a preference for sucrose over other common sugars in nectar (Scheiner et al., 2004). There are other factors that affect the general quality of a nectar source – source to hive distance is implicitly considered by the increased round trip time experienced by foragers, abundance and risk of predation are also considerations (Seeley et al., 1991; Dukas, 2001) – however we will not consider these other factors since they are not communicated to receivers (Seeley et al., 1991) and can be accounted for by assuming predator-free and abundant nectar sources. Foragers exploiting a source are increasingly likely to recruit (i.e. will perform more waggle runs) as Q increases or as S(t) decreases. Conversely, their likelihood of resting increases for low Q and for high S(t). Defining fr and fs as maximal forager recruitment and resting rates respectively we construct an equation for the changes in forager population, F(t), dFðtÞ ¼ fr FðtÞ G1 ðSðtÞ,Q Þ fs FðtÞ G2 ðSðtÞ,Q Þ: |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dt recruitment

ð2Þ

resting

An extra factor of the form 1 F/FT where FT is the total number of foragers available could be included in the first term of Eq. (2) to model the effect of having a limited number of foragers. However, because of the way that dancers recruit unemployed foragers, such a term is unnecessary unless the number of active foragers is close to the number of potential foragers, so that there are very few unemployed foragers in the hive. When a dancer advertises a site she is usually followed by only about six bees at most. It is only these followers who acquire the information that she is propagating, and not the lookers on. Therefore, the rate of spread of information is linear, i.e. it depends only on the number of dancers, unless there are insufficient unemployed foragers to form the usual complement of dance followers. In Eq. (2), G1 and G2 are functions that scale the recruitment and resting rates by mapping nectar quality and search time to values between 0 and 1. G1 should increase recruitment rates as quality improves or as search time decreases. G2 should increase resting rates as quality decreases or as search time increases. We have chosen to define these functions using Hill equations since this type of equation permits a good phenomenological fit and is algebraically convenient. We define mQ and mS as the values that produce half-maximal scalings for Q and S(t), and j and k as the coefficients that control the sensitivity of the functions to changes in Q and S(t).

J.R. Edwards, M.R. Myerscough / Journal of Theoretical Biology 271 (2011) 64–77

This leads to definitions for G1 and G2 ,

G1 ðSðtÞ,Q Þ ¼

G2 ðSðtÞ,Q Þ ¼

mkS k

Q

SðtÞk

mjQ

j

SðtÞ þ mkS Q j þmjQ

k

SðtÞ

þ mkS

Q j þmjQ

,

ð3Þ

:

ð4Þ

In Fig. 3a we show how different search times affect the searchrelated part of G1 for a range of coefficient values. Compared to foragers, there is a paucity of information about receivers in the existing literature and so we rely extensively on observations recorded by Seeley (1992, 1995). We find that the closest comparison to the data given in Seeley (1992, Fig. 7) is with k¼4 and mS ¼10. Similarly, we plot the quality term of G1 for a range of coefficients in Fig. 3b and determine that the best fit to the data in Fig. 5.31 of Seeley (1995) is j¼4 and mQ ¼ 1.5. In the absence of similar data for resting rates we used these same coefficients for the equivalent terms in G2 . This also keeps the number of parameters in the model to a minimum. Eq. (2) emphasises nectar quality and search time, it does not consider other factors such as the tendency

1

1

0.8

k=2 k=4

0.6 0.4 0.2 0

0

50

100

Qj / (mQj + Qj)

mkS / (mkS + S (t)k)

k=1

150

of foragers to resume foraging at sources they were exploiting previously. Next we define the changes in the population of foragers that are seeking to unload nectar within the hive. These foragers, denoted F , are a subset of the total foragers F. The number of foragers not seeking to unload, i.e. those currently unloading or outside the hive gathering nectar, is FðtÞF ðtÞ. These foragers arrive at the hive with nectar at a rate determined by the average foraging trip time fa. They then take an average of S(t) seconds to find a receiver and commence unloading. The equation for this population is dF ðtÞ ¼ dt

F ðtÞ : SðtÞ |ffl{zffl}



ð5Þ

rate of finding a receiver

For this simple model we assume that R(t), the total number of available receivers, does not change and so R(t) ¼ R0, the initial receiver population. Each receiver can be either waiting to receive nectar or engaged in nectar storage tasks. The population of receivers engaged in storing is given by RðtÞRðtÞ. They complete this task at a rate determined by the average time for nectar storage, denoted rs. Meanwhile, as foragers find receivers available for unloading, these receivers begin trophallaxis, dropping out of the available receivers population and joining those engaged in storing nectar. The equation for the population of available receivers is R0 RðtÞ rs |fflfflfflfflffl{zfflfflfflffl ffl}

rate of returning from storing

0.6



F ðtÞ : SðtÞ |ffl{zffl}

ð6Þ

rate of receiving nectar

The complete system of ODEs from Eqs. (2) to (6) is

0.4

mjQ mkS dFðtÞ Qj SðtÞk ¼ fr FðtÞ f FðtÞ , s dt SðtÞk þ mkS Q j þmjQ SðtÞk þ mkS Q j þmjQ

j=2

0.2 0

FðtÞF ðtÞ fa |fflfflfflfflfflffl{zfflfflfflfflfflffl}

rate of arriving in entrance

dRðtÞ ¼ dt

0.8

67

j=4 j=6

0

1

S (t)

2

3

Q

Fig. 3. Plots of the Hill equation terms in G1 with different coefficient values. Halfmaximal values are derived by comparison with experimental data (Seeley, 1992, 1995) with mS ¼ 10,mQ ¼ 1:5. Search time ranges from 0 to 100 s, quality from 0 to 3 mol/L and simulated values of j and k are 1,2,4 and 2,4,6, respectively. Coefficient values are chosen by comparing these curves to this experimental data. (a) Shows how increases in search time decrease recruitment. For very low search times, recruitment (effectively the number of waggle runs the forager performs due to quality) is essentially unchanged. For very high search times waggle dancing may effectively cease. Three different curves are shown representing different values of the coefficient k. (b) Shows how increases in quality lead to foragers recruiting more. For low quality nectar recruitment is negligible. As the quality approaches 1.5, recruitment is half the maximal rate. As quality approaches 3 the rate of recruitment plateaus. Three different curves are shown representing different values of the coefficient j.

dF ðtÞ FðtÞF ðtÞ F ðtÞ ¼ ,  dt fa SðtÞ dRðtÞ R0 RðtÞ F ðtÞ ¼ :  dt rs SðtÞ

ð7Þ

Unless otherwise stated, we will use the parameters given in Table 1. In many cases a single value has been chosen from a range of appropriate values (e.g. the average forager trip time varies greatly). 2.3. Steady states We seek information about the structure and stability of the   system and so consider the equilibrium points ðF  ,F ,R Þ of the simple model (Eq. (7)). This system has three equilibrium points

Table 1 Standard parameter values. Parameter

Symbol

Value

Source

Single interaction time

ss

Camazine and Sneyd (1991)

High quality (for scaling receiver response) Forager recruitment rate Forager resting rate Half-maximal search time Search time coefficient Half-maximal forage quality Forage quality coefficient Forager round-trip time

iq fr fs mS k mQ j fa

Receiver storage time

rs

5s (range is 2 s-7 s) 3 mol=L 0:0010 s1 0:0002 s1 10 s 4 1:5 mol=L 4 15 min (range is 1=2 min-20 min) 20 min (range is 1 min-20 min)

Seeley (1995); Scheiner et al. (2004) Seeley and Towne (1992) Camazine and Sneyd (1991) Seeley (1992) Seeley (1992) Seeley (1986) Seeley (1986) Seeley (1994) Seeley (1995)

J.R. Edwards, M.R. Myerscough / Journal of Theoretical Biology 271 (2011) 64–77

which are found by assuming that dF=dt ¼ dF =dt ¼ dR=dt ¼ 0 and then solving the resulting equations to find the conditions that make this situation, in which populations do not change, true. The first corresponds to the state observed at the start of foraging when no foragers are active, the second to the population mix that the colony approaches over time in a normal foraging environment and the third to a biologically impossible situation where the population of foragers seeking to unload is negative. Here we will restrict our attention to the first two equilibria, the one near which the colony typically begins and the other which it typically approaches over time. Further details on the derivation of these results is included in Appendices A and B. The first equilibrium point is 0 1 0 1 0 F B C B 0 C @F A¼@ A R



R0

in which there are no foragers active and no receivers storing nectar. To find when the forager population increases away from this point we determine when it is unstable. We find that this steady state is unstable whenever Q 4 mQ ðfs =fr Þ1=j ðss =mS Þk=j . For the parameter values of Table 1 this corresponds to Q \ 0:6, so for all except very low quality environments (de Vries and Biesmeijer, 2002, for example, consider 1:25 mol=L to be poor quality) the forager populations will increase from zero. The second equilibrium point is 0 1 0 1 ðg þfa Þðgss Þ F R0 B C B gðgs Þ C s @F A¼ @ A ðrs þ ss Þgrs ss  gss R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  k for g ¼ k ðfr =fs ÞQ j mj Q mS ¼ S . This shows that the final populations are linearly related to the total receiver population, and are only slightly dependent upon nectar quality (through g). In other words, the population of receivers determines how many foragers should be allocated. Calculating the stability of this point is analytically intractable, but numerical simulations indicate it is an attracting stable point and an analytic analysis using an approximation (Appendix B) also demonstrates this. This analysis of the system dynamics indicates that in a foraging environment where recruitment to a new source is taking place, the forager population will increase from zero towards a single stable value. It is surprising that this value is principally dependent on forager arrival rate, receiver storage time and receiver population. In some cases, the stable equilibrium point in the model is approached but not reached within the limited time that a colony has available in which to forage. We note that for low search times, ðd=dtÞFðtÞ  ððfr Q j fs mjQ Þ=ðQ j þmjQ ÞÞFðtÞ, so that FðtÞ  F0 expðððfr Q j  fs mjQ Þ=ðQ j þmjQ ÞÞtÞ, where F0 is the initial population. Clearly, the population grows faster for higher values of Q than for low values of Q. So Q strongly determines the rate that the colony approaches the steady state (in the model) but only weakly affects the value of that steady state. This is illustrated for the parameters of Table 1 in Fig. 4. This suggests that the forager population is more likely to attain (and even, temporarily, exceed) the stable equilibrium point in a high quality environment than in a low quality one. To understand the details of how this works in practice, we solve the system numerically.

3. Numerical results Fig. 5 displays the results of numerical simulations of Eq. (7) with the parameters of Table 1. All simulations were generated using Matlab (MATLAB, 2009) with a Runge–Kutta solver. The simulated period was 8 h and in all cases each nectar source

140

4h 8h 12 h equilibrium

120 100 80 F

68

60 40 20 0 0

0.5

1

1.5

2

2.5

3

Q Fig. 4. The population of foragers after 4, 8 and 12 h for simulations with different values of Q and otherwise identical parameters corresponding to the values given in Table 1. The receiver population is 100 and the initial forager population is 1. Actual populations are shown as stars (after 4 h), squares (after 8 h) and diamonds (after 12 h), the equilibrium population as a circle. The equilibrium point changes only slightly as quality increases as it is principally dependent upon R. For low values of Q ði:e:Q o 1Þ, the population does not grow close to equilibrium, even after 12 h. For moderate values of Q (i.e. Q A ½1,2), the population approaches equilibrium quickly and smoothly. For high values of Q (i.e. Q 4 2) the population rapidly exceeds the equilibrium population and then, upon entering a stabilising phase dominated by inhibition of recruitment, slowly decreases towards it.

initially has a single forager. This period was chosen because it conforms to the periods used in existing models (Camazine and Sneyd, 1991; de Vries and Biesmeijer, 1998; Cox and Myerscough, 2003). The four plots (Fig. 5a–d) show the population timecourses for simulations using a different combination of quality and receiver numbers. Fig. 5a and b are generated from simulations that use the same high quality environment (Q¼ 3) and vary only in receiver population. This gives the simulation results in Fig. 5b, that has half the receiver population of the simulations displayed in Fig. 5a, producing half the forager population. The shape of the population timecourse is preserved and the search time is identical. The simulations used to produce Fig. 5c and d also differ only in receiver populations, however they both have a low quality environment (Q¼1.3). Very different results are produced to those in Fig. 5a and b; the populations are never near equilibrium and, most significantly, there are no differences between their population timecourses. In other words, the size of the receiver population does not affect the forager population in a low quality environment but strongly affects it in a high quality environment. These results suggest that, in a high quality foraging environment, the population of foragers rapidly increases, as discussed in Section 2.3, until the number of foragers seeking to unload becomes large relative to the number of receivers available for unloading. Once this occurs the search time increases and recruitment is inhibited, leading to much slower changes in population in accordance with Eq. (2). The transition between these ‘rapid growth’ and ‘stabilising’ periods can be quite sudden. In Fig. 5a it can be observed to occur at a cusp in the forager population timecourse just before t¼2. This cusp typically occurs near or above the equilibrium population of F. In a low quality environment the rate of increase in the forager population is far slower and so the number of foragers remains small and search time never becomes large. This suggests that in a low quality environment the population of foragers is highly sensitive to Q, which controls its rate of growth.

J.R. Edwards, M.R. Myerscough / Journal of Theoretical Biology 271 (2011) 64–77

1000 200

500

100

0

0

2

4

6

8

0

1000

500

100

0

0 0

2

population (bees)

time (hours)

4

6

8

time (hours)

20

10

20

10

10

5

10

5

0

0

0

0

2

4

search time (s)

F F¯ S

6

8

time (hours)

search time (s)

population (bees)

200

69

0 0

2

4

6

8

time (hours)

Fig. 5. Timecourses using the model defined by Eq. (7) with varying Q and R and other parameters as defined in Table 1. (a) and (b) have high quality (Q¼ 3.0) environments and (c) and (d) have moderately low quality (Q¼ 1.3) environments. All have iq ¼ 3.0 and are initialised with one forager at the nectar source. (a) and (c) have a large number of receivers (R¼ 100) and (b) and (d) have a smaller number of receivers (R¼ 50). The legend is defined in (a). Changing R dramatically affects the results of simulations using high values of Q, however this has no effect on simulations using low values of Q.

In a high quality environment the rate of growth is so fast that the onset of high search times is most important in determining the forager population. We find then that the quality, perceived by foragers, determines the population in low quality environments. When the quality is high however, it is receiver numbers that determine the forager population, since by Eq. (1) the search time is dependent on the receiver population. That is, we can consider the quality of the foraging environment as providing positive feedback to foragers and the receiver population as providing negative feedback to foragers. The positive feedback dominates when the forager population is small and the negative feedback dominates when the forager population is large. We offer a biological interpretation for this. A colony with large nectar reserves may require only a small number of foragers to engage in source exploitation to maintain food supplies. An efficient and simple way to limit the number of foragers is to limit the population of receivers, which will increase search time once this small number of foragers is active, preventing further recruitment. When the colony has depleted its reserves, for example after an extended period of rain, it can make many receivers available and foraging numbers will become constrained only by forager population size. Foragers, however, need not respond to high receiver numbers by recruiting to the maximum level if the nectar quality is low. Thus, the two behavioural classes are potentially able to assess different sets of relevant information i.e. foraging environment quality or the state of nectar reserves inside the colony, and using these different sets of information to collaboratively reach a decision about the appropriate population of foragers. The information is able to be transferred via the

search time cue, which arises through the interactions of foragers and receivers. A graphical representation of this is given in Fig. 6. This figure shows the population of foragers 8 h after a single forager commences exploitation of a nectar source against a range of values for R and Q. It is clear that, for most values, the population is sensitive to either R or Q but not to both. That is, for low qualities it is the quality of the nectar that determines the forager population and for high qualities it is the number of receivers that determines the active forager population. For Q o 1, the forager population is typically still growing after 8 h. For some values of R there is a ridge at Q  1:4 because the population in the simulation with a slightly higher quality has already commenced its stabilising phase and its forager population is already decreasing (cf. Fig. 5a). For Q A ½1:4,2 the plot is flat along the Q-axis since the simulations have reached equilibrium within 8 h. For higher qualities the plot’s steepness increases, reflecting those simulations whose forager populations have peaked well above equilibrium and not settled back equilibrium after 8 h (cf. Fig. 4).

4. Extending the model with receiver recruitment via the tremble dance The tremble dance (Seeley et al., 1996; Dyer, 2002) is performed by foragers experiencing high search times, and is followed by increase in the receiver population. It can also discourage other foragers from dancing for the same source. We seek to understand

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Fig. 6. Forager populations after 8 h against a range of likely Q and R values. The initial forager population is 1 and the receiver population is 100. All parameters otherwise correspond to the definitions of Table 1. At a threshold of Q  1 the population switches from being sensitive to Q to being sensitive to R. The populations may exceed the stable equilibrium point temporarily. This is apparent around Q¼ 1.4 where the graph displays a dip as Q increases since simulations with quality Q \ 1 have decreased further from their peak after 8 h than those with a lower quality. This corresponds to the cusp that occurs as the forager numbers change in the transition from the ‘rapid growth’ to the ‘stabilising’ phase. A similar effect is visible for Q 42, where there is considerable over-shoot beyond the equilibrium population and the populations are yet to decrease to equilibrium.

how receiver recruitment affects the collective decision about forager numbers, and so we extend the simple model (Eq. (7)) to include recruitment of receivers by tremble dancing. Let U be the unit step function and o, m and mT be, respectively, the maximal recruitment rate due to tremble dancing, the ‘steepness’ of the response to tremble dancing and the minimum search time for which tremble dancing occurs. The equation that we use to model receiver population change is dRðtÞ oSðtÞm ¼ UðSðtÞmT Þ : dt SðtÞm þ mm T

0.6

ð8Þ

Here there is no increase in receiver population unless the search time is greater than mT and no possibility of the receiver population decreasing. While recruited receivers may abandon their new task and revert to other hive duties (Seeley et al., 1996), the process is not well understood and – at least over the timescale of a day – it appears that the abandonment rate is fairly low (perhaps 10% for high quality, high nectar influx environments) and so we have disregarded it. The effect of this on the model is that the receiver population can increase but it cannot decrease. We have however suggested a possible mathematical expression that incorporates this in the discussion. When SðtÞ 4 mT the receiver population increases at an increasing rate as search time increases. For o ¼ 0 the system reduces to the simple model of Eq. (7). The values of mT and m were determined in a similar manner to mS, mQ, j and k, by plotting for a range of values (Fig. 7) and comparing to Seeley (1992, Fig. 7); we find that m¼5 and mT ¼30 are suitable values. We begin our analysis of the new system by calculating the effect that the introduction of Eq. (8) has on the steady state. F(t) is unchanging whenever F(t)¼0 (trivially, though this may be unstable when the population is perturbed) or whenever the search time is stable such that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u fr mkS k  SðtÞ ¼ S ¼ tQ j : ð9Þ fs mjQ When S omT , R(t) eventually stabilises and then the rest of the system also stabilises. If, however, S ZmT then the search time

0

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S (t) Fig. 7. The effect of different values of m on the trembling response. m takes values 3,5,9 and mT ¼30. The best fitting median and coefficient values were chosen by comparing with results in Seeley (1992).

steady state S (Eq. (9)) continually induces as receiver population increases and the entire system loses stability. By rearranging Eq. (9) we can see that the system will be stable whenever Q omQ  k 1=j ðmT =mkS Þðfs =fr Þ . For the standard parameters used in the simulations and defined in Table 1 this implies that for Q t 3 the forager population should reach a stable value, for higher values of Q it will continually increase. This stability criterion is independent of R(t), so although high search times will increase the receiver population, this increase does not resolve the instability. Hence for high quality nectar, the increase in both receivers and forager populations will continue until the colony has exhausted its populations of potential foragers or potential receivers. This is observed in the numerical simulations presented in Fig. 8. Fig. 8a is generated using an identical configuration to Fig. 5a, including that the initial receiver population is 100 and the initial forager population is one, with the extra inclusion of Eq. (8) with o ¼ 5. The forager and receiver populations become unstable, and continually increase. This would extend the model into situations where it is not valid. In reality the colony would soon exhaust its population of potential foragers and so a new and more complicated model which includes the finite size of the forager population would be needed. A series of jagged oscillations occur for the search time when it exceeds the threshold mT defined in Eq. (9). When the search time exceeds this threshold there is a sudden influx of receivers which then causes the forager population to increase until the search time threshold is reached again. In Fig. 8b we have a similar set of parameters, however we have reduced the foraging quality so that Q¼ 2.0, which is below the stability threshold. In this simulation, as predicted, the forager population trends towards a single value. During the rapid growth phase, receiver recruitment occurs due to the high amplitude of the search time oscillations, but this ceases as the system stabilises. A simulation (not shown) generated over the same parameter set except with Q¼1.3 produces results identical to that in Fig. 5c. We find here that the inclusion of Eq. (8) has no effect in this low quality environment, since S(t) is always less than mT. This is consistent with the previous conclusions concerning low quality environments in that receiver numbers have no influence over the forager population. Fig. 8c and d were generated with parameters identical to those used for Fig. 8a and b, respectively, except that o was set to 0.1

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Fig. 8. Population timecourses for a model with parameters identical to Fig. 5a except that Eq. (8) is included in the model. Parameters are as defined by Table 1 with mT ¼ 30 and m¼ 5. For (a) and (b) o ¼ 5, for (c) and (d) o ¼ 0:1. In all simulations there are initially 100 receivers and 1 forager. The dynamics are identical up to t  2, at which point tremble dancing commences and receiver numbers begin to increase. This point coincides with the end of the rapid growth phase and commencement of the stabilising phase, discussed in Section 3. In (a) Q ¼ 3 and S 4 mT , and so the receiver population increases indefinitely. The search time undergoes oscillations due to repeated cycles of receiver increase followed by forager increase causing the populations to vary widely. In (b) Q ¼ 2 and S o mT , and so the receiver and forager populations eventually stabilise. (c) and (d) are included to display the difference in the dynamics due to different values of o, they are otherwise identical to (a) and (b), respectively. The smaller o value leads to greater oscillations in search time and somewhat higher forager and receiver populations. For each simulation iq ¼ 3.0.

rather than 5. The purpose of these figures is to demonstrate the differences in the dynamics with different values of this parameter. We find that the amplitude of oscillations in the search time increases with a smaller o, because the slower rate of receiver increase causes greater search times to temporarily be attained, and that the forager and receiver populations increase. These results describe a mechanism by which the equilibrium population of foragers may increase. The stimuli thresholds at which individuals initiate action vary (Myerscough and Oldroyd, 2004), due to, amongst other factors, the high degree of genetic diversity in A. mellifera colonies, this is no doubt true for the tremble dancing response amongst the pool of potential receiver bees. The stable state populations would increase smoothly as the nectar quality increases so the effect of a mechanism similar to those described by Fuchs and Moritz (1999) and Jones et al. (2004) would be beneficial. When there is a range of response thresholds, receivers play a similar role to that described in this simple model. Fundamentally, however, this model states that receivers can adapt to exceptionally high quality environments in such a way that forager numbers can continue to increase and thereby take full advantage of foraging conditions.

5. Extending the model with multiple forage sources In the field an A. mellifera colony will simultaneously exploit multiple sources, each with different quality. We therefore extend the model defined by Eq. (7) to include more than one nectar source. We denote the quality of n sources as Q1 ,Q2 , . . . ,Qi , . . . ,Qn , and add new forager equations for each source so that Eq. (7) is replaced by dF i ðtÞ ¼ fr Fi ðtÞG1 ðSi ðtÞ,Qi Þfs Fi ðtÞG2 ðSi ðtÞ,Qi Þ, dt dFi ðtÞ F ðtÞF i ðtÞ F i ðtÞ ¼ i ,  dt fa Si ðtÞ n dRðtÞ RðtÞRðtÞ X Fi ðtÞ ¼  dt rs S ðtÞ i¼1 i

and search time (Eq. (1)) is given for each nectar source by !, n X Si ðtÞ ¼ ss ðF i ðtÞÞ þRðtÞ ðRðtÞQi =iq Þ: i¼1

ð10Þ

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population at each source is equal to that of the population of foragers in Fig. 5c, which presents results from the corresponding simulation of a single source environment with Q¼1.3, and so both nectar sources initially have one forager and hence the total forager population in this plot is extremely close to twice that of the single source simulation. This is consistent with the results of the model with a single source of low quality nectar, in that foragers determine their own population on the basis of nectar quality without being influenced by receivers and therefore if there are twice as many initial foragers there will be very close to twice as many foragers at all times up until the search times become large. The distribution of foragers is determined by the quality, and since Q1 and Q2 are nearly identical so are the populations foraging at these sources. The numerical results of the multiple source simulations are consistent with those of the single source simulations. This suggests that the analytic results for the simple model (Section 2) are also relevant to this multiple source model. As such, we can consider the single source model as aggregating multiple sources into one general term by thinking of Q as representative of the average quality of the entire environment rather than just a single nectar source.

We use this extended model to investigate how the inhibitory effect of high search times affects the population and the allocation of foragers among different sources, and to explore whether the results of the single source model (Eq. (7)) also apply to multiple source models. The results of simulations with two nectar sources, with qualities denoted Q1 and Q2, are presented in Fig. 9. In all cases both nectar sources initially have a single forager exploiting them and iq ¼3.0. Fig. 9a displays the population timecourses from a simulation of a high quality environment, in which Q1 ¼3 and Q2 ¼2. The total population of foragers in this plot is approximately equal to the population of foragers in Fig. 5a, which also presents results from a simulation of a high quality, though single source, environment with Q¼3. The equilibrium populations of the two simulations are identical. The similarity between the total populations is consistent with the idea, explored previously, that the receiver population imposes an upper limit on the forager population. The simulation suggests that receivers achieve this by limiting the total population of foragers rather than by limiting the population at each nectar source. Further, the time when the population dynamics move from a ‘rapid growth’ phase to a ‘stabilising’ phase is the same. Other characteristics of the figures are very similar, such as search times, available receiver population and the shape of the timecourses. By a variety of measures then, the single source model of Eq. (7) is equivalent to aggregating the population of foragers at multiple sources in high quality environments. The distribution of foragers between the two sources changes during the rapid growth phase, with the higher nectar quality source acquiring a larger percentage of the foragers until the stabilising phase is entered, when the distribution becomes nearly static. Within the model receivers affect the distribution of foragers between sources by preferentially unloading higher quality nectar—there are less receivers willing to unload lower quality nectar which increases the search time for foragers of the low quality nectar. This is the only way in which receivers direct the foragers to one particular source, and it is done within the context of specifying a colony-wide nectar goal via the receiver population size. Foragers, conversely, determine their allocation to each source based solely on each source’s nectar quality and its unloading time. The model predicts that, as usual, bees will display a decentralised and self-organised approach to colony management. Fig. 9b presents the population timecourses from a simulation of a moderately low quality environment, in which Q1 ¼1.3 and Q2 ¼1.31. (We have chosen slightly different qualities so that the lines on the graph do not line on top of one another.) The forager

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6. Extending the model with dynamic forage sources The quality of nectar at a source may change over the course of a day (Ginsberg, 1983), due to changes in weather, inter-species competition levels or plant characteristics. How a colony adapts to this is crucial to its success, and so we seek to understand how search time affects forager populations when a source’s nectar quality changes. We follow the scenario of Camazine and Sneyd (1991) in which there are two nectar sources, one initially having low quality nectar and the other initially having high quality nectar, that switch nectar qualities after a few hours (denoted Ts) so that ( ( 2:5 if t o Ts 1:5 if t oTs , Q1 ¼ and Q2 ¼ 1:5 otherwise 2:5 otherwise: Using the multi-source model of Eq. (10), we generate simulations of two different scenarios with this type of nectar quality switching. Each simulation initially has 100 receivers, one forager at each nectar source and other parameters as defined in Table 1. Population timecourses of the forager populations and search times are presented in Fig. 10. In Fig. 10a where Ts ¼ 1, the foraging population rapidly changes its allocations to exploit the better source, as with

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Fig. 9. Population timecourses for a model with the parameter definitions of Table 1 and with other factors also identical to those in Fig. 5, except that there are two forage sources, each initially with 1 forager. In both plots there are 100 receivers, so they correspond to Fig. 5a and c: (a) has two high quality nectar sources (Q1 ¼ 3,Q2 ¼ 2) and (b) has two low quality ones (Q1 ¼ 1:3,Q2 ¼ 1:31).

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Fig. 10. Population timecourses for simulations of sources that switch nectar qualities. Sources switch quality between 2.5 and 1.5 with iq ¼2.5. Parameter values are from Table 1, tremble dancing is defined with mT ¼30, m¼ 5 and o ¼ 5, and R¼ 100 with each source having a single forager at t ¼0. In (a) nectar sources switch quality at t ¼1 and there is no tremble dancing. The colony adapts to the changed environment. In (b) nectar sources switch quality at t ¼ 2 and there is no tremble dancing. The colony takes a long time to adapt to the changed environment. In (c) nectar sources switch quality at t¼ 1 and there is moderate tremble dancing. The colony rapidly adapts to the changed environment. In (d) nectar sources switch quality at t¼ 2 and there is moderate tremble dancing. The rate and extent of adaption is improved as compared with (b). Search time is that experienced by the foragers of the better quality nectar and is proportionally greater for the other foragers.

Camazine and Sneyd (1991). In Fig. 10b the parameters are identical but Ts ¼2. The better source is not exploited in this scenario, suggesting a failure in the decision making process. The colony’s inability to adapt to the quality change in Fig. 10b is surprising given its successful adaption to the apparently similar scenario recorded in Fig. 10a. The reason for the failure is that during the first simulation the switch occurred during the ‘rapid growth’ phase, during which search times are small, and the second simulation’s switch occurred during the ‘stabilising’ phase, during which search times are high. High search times inhibit recruitment, regardless of nectar quality, and so during the second simulation there simply was not enough recruitment occurring to increase the forager population at source two. By contrast, in the first simulation there were low search times for more than 2 h after the nectar quality change. Previous results (e.g. Fig. 5) consistently show that the rapid growth phase is associated with fast changes in population whereas the stabilising phase is associated with slow population changes. We note, further, that by Eq. (2) when F1 ðtÞ b F2 ðtÞ and Q1 oQ2 (as occurs at t ¼Ts) it may still be true that dF 1 =dt 4 dF 2 =dt so that there is still more recruitment to the poor quality site than to the high quality one. This can be observed in the timecourses of the populations in Fig. 10b around t ¼ 4. Nevertheless, the better quality site will increase its forager population at a greater proportional rate and so eventually, over a period of time much longer than a day, have the greater population as well.

With such a poor result observed in Fig. 10b, it is natural to ask if there is something missing from the model that could improve the outcome and make it more realistic. We will apply the same values as used to generate Fig. 10a and b into a model extended to include tremble dancing (Eq. (8)) with o ¼ 0:1. Tremble dancing is only performed by foragers of the high quality nectar, there is no mention in the literature of tremble dancing occurring for low quality nectar. The resulting population timecourses are displayed in Fig. 10c and d. The results of Fig. 10c are similar to those in Fig. 10a, though with the increased populations expected due to the inclusion of tremble dancing. The results of Fig. 10d, however, improve considerably upon those in Fig. 10b. The improvement occurs because tremble dancing relieves some inhibitory pressure on forager recruitment, as the increased receiver numbers temporarily decrease the search time. Increasing numbers of these newly recruited foragers exploit the higher quality nectar source. Even with the modifications, the change in forager allocation when the quality switch occurs at Ts ¼ 2 still appears to be slow. There are two possible reasons for this. Firstly, the model could be showing a result that is qualitatively accurate, yet its outcome is much better in practice due to behaviours that are not modelled. Secondly, the model could be quantitatively correct and this slow response is useful in some way to the colony, representing perhaps a collective memory effect that highly values nectar sources that have previously proven to contain good quality nectar. It could also

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be seen as a part of the colony’s risk-reduction strategy, as the colony allocates its foragers sub-optimally (in terms of maximising nectar influx) so that it can robustly protect against another sudden loss of the better source.

7. Discussion We have presented a class-oriented model of A. mellifera nectar foraging whose principal novelty is that it incorporates the receiver population. Numerical and analytic results derived from the model all suggest that the size of the receiver population determines the equilibrium population of active foragers. In general, foragers exploiting a high quality nectar source rapidly recruit more new foragers, but only while there are receivers available to quickly unload them. Once the demand for receivers significantly exceeds supply, forager recruitment is inhibited and their population stabilises. Under the simple model defined and analysed in Sections 2 and 3, receiver numbers are fixed. Once this fixed population is reached no more bees will become receivers. The model is modified in Section 4 so that new receivers are recruited by tremble dancing. The results of the simulations produced by a model that includes tremble dancing suggest that when a source with extremely high quality nectar is found the colony may indefinitely recruit new receivers, at least until the worker population is exhausted. Under standard conditions, when the colony is not recovering from an extended inability to forage, and with mid-range nectar concentrations of up to 2.5 mol/L (de Vries and Biesmeijer, 1998) tremble dancing allows the colony to increase its population of receivers to a greater, but still limited, size if this is justified by the discovery of a source with high quality nectar. Another system is also plausible; some worker bees are dedicated receivers, others are available to receive under certain criteria and others are never available. Such a system allows a staggered response to tremble dancing, so that individual bees may have different thresholds (Jones et al., 2004) at which they will switch to receiving; some will switch as soon as they perceive a tremble dance, others only when there are many foragers who are tremble dancing. The actual scheme is of great importance in determining the role of receivers. If workers always switch roles to become receivers upon noticing a tremble dance, then they are only passive participants in the decision making process. If, however, they are influenced by their environment as well as the tremble dance then they have an important role to play in deciding how many foragers should be active (as we have modelled here). The effect on the decision making process is immense, and warrants further study. Under such a scheme, the tremble dance recruits non-receivers to act as receivers but this will only occur amongst those bees whose recruitment threshold, perhaps influenced by colony need, is sufficiently low. Once again, this makes biological sense; bees that are available to receive should be engaged in other housekeeping roles until such time as there is demand for their labour. The results reported in Section 5 suggest that receivers may affect the foraging decision in two ways. Firstly, their preferential unloading of foragers carrying higher quality nectar (Scheiner et al., 2004; Martinez and Farina, 2008) tends to affect the distribution of foragers by encouraging a greater proportion at sources of higher quality nectar. Secondly, their population implicitly specifies a target size for the population of foragers, who in turn arrange their own distribution amongst different sources. These observations enable us to propose a possible sequence of events that determine forager population size and distribution in the idealised environment envisaged by the present model: 1. The receiver population is determined by the colony’s size and possibly nectar need.

2. Foragers commence exploiting nectar sources. Those with higher quality nectar provoke more intense recruitment and thereby gain larger forager populations. 3. The active forager population increases to the point where search times become greater than normal. This inhibits recruitment and forager populations begin to stabilise. 4. As receivers preferentially unload foragers with higher quality nectar, foragers from these nectar sources experience relatively lower search times and the distribution of foragers begins to favour sources of higher quality nectar. Section 6 models a colony adapting to a dramatic change in nectar quality. Interpreted qualitatively, the colony response depends on whether the change in quality occurs before or after the start of the ‘stabilising’ phase—i.e. when the demand for receivers outstrips the supply. The colony smoothly adapts to a change before this phase, but takes much longer to respond correctly when the change occurs afterwards, when recruitment is inhibited by high search times. The inclusion of tremble dancing improves the response significantly. This is interesting because it highlights how flexibility within the colony, in this case in the receiver population, leads to improved outcomes. 7.1. Experimental implications Nectar receivers have received less attention than foragers. However, our results suggest that the nectar receiver population may have a significant influence on the colony’s exploitation of forage sources. For instance, the model suggests that the daily foraging cycle may consist of periods in which the nectar receiver population may inhibit forager population change—even as conditions themselves change. It would be interesting to see whether this inhibition does in fact occur in real hives. In addition, the population of active nectar receivers may fluctuate due to the level of tremble dancing and other factors (Seeley et al., 1996). The model results suggest that receivers should respond to tremble dancing only when the foraging environment is excellent and/or the colony has low nectar stores. It is possible that the population of receivers, especially that at the start of a foraging cycle, reflects the internal state of the colony – that is the level of the nectar reserves and its consumption rate – which then, in turn, affects the forager population. 7.2. Model limitations and future extensions A number of extensions are suggested by the biology and by the model results. It is possible that the colony’s combs will become nearly full over time (Seeley, 1995), making it harder for a receiver to find a storage location. Until new honeycomb is constructed this may have the effect of increasing storage time; storage time is represented in this model by the constant parameter rs. We see from Eq. (2.3) that as rs increases, the steady state populations decrease. The model could be extended to make rs vary with the level of nectar stored within the hive. This may result in greater nectar stores leading to smaller foraging levels, and exploring this in the model may elucidate the system further. In addition, including the tendency of workers to construct new honeycomb as the hive fills and being able to visualise its effect on the forager and receiver populations would be useful and interesting. The tremble dancing equation we use (Eq. (8)) does not cater for abandonment of receiving by hive bees. An alternative form that does is   dRðtÞ oSðtÞm RðtÞ ¼ UðSðtÞmT Þ þ oa 1 , ð11Þ m m dt R0 SðtÞ þ mT where oa is the rate of abandonment and is generally much less than o. The extra term assumes there is a natural size for the receiver

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population that will always be approached in the absence of recruitment. We have not included it here since the actual mechanism is poorly understood, numerical simulations using the new term indicate that it is sensitive to the value for oa but broadly the augmented model will predict that receiver population will temporarily increase as the forager population increases and then decrease towards R0 after accommodating the initial surge in nectar influx. One further extension would be to include inspector bees (foragers who repeatedly check a previously good source of high quality nectar to ascertain whether it has returned to providing good quality nectar) and different subclasses of foragers that respond differently to different values of Q and S(t). This should be particularly relevant when studying task allocation, where models of which often rely on different individual response thresholds (Beshers and Fewell, 2001; Myerscough and Oldroyd, 2004).

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A.2. Second (non-trivial) equilibrium point We have used Mathematica to help solve these points. For dFðtÞ=dt ¼ 0 and F  a0, we have iq ss g Qr s g þss iq gss rs iq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k where g ¼ k ðfr =fs ÞQ j mj Q mS . We can use this and solve for dRðtÞ= 

R ¼ R0

dt ¼ 0 to find     Qg F ¼R 1 : iq ss Finally, these results can be used when solving for dF ðtÞ=dt ¼ 0 to find ðfa þ gÞðQ giq ss Þ ðQr s þ iq ss Þgiq rs ss

7.3. Foraging as a collective decision

F  ¼ R0

Our model treats foraging as a problem solved by a collective group intelligence; the colony analyses information and makes a decision about forager allocation. In the model two classes of bee, each with access to different, yet critical, pieces of data interact while unloading nectar in such a way that a self-organised value emerges in the form of the search time. This value provides both positive and negative feedbacks, depending on the current situation, and so the colony is able to make a wise choice when it decides on its forager allocation. The inclusion of receivers in this model provides an inhibition mechanism that is common in decision making models (Sumpter, 2006) and which counters the activation provided by forager recruitment.

which completes the definition of the second equilibrium point. A.3. Third (biologically unobtainable) equilibrium point As with the first point, we begin with F  ¼ 0. Then we solve for dF ðtÞ=dt ¼ 0 such that F ðtÞ a0. Note that this is implausible, in a biological environment there cannot be zero foragers and yet more (or less) than zero foragers attempting to unload, so that this is a purely ‘mathematical’ equilibrium point and as such is not discussed in the main text. We find, again using Mathematica, that fa ss iq fa rs Q þ iq ss ðfa þ rs Þ   fa Q fa ss iq iq ss þ fa Q   F ¼ R 1 þ ¼ R0 iq ss fa rs Q þ iq ss ðfa þ rs Þ iq ss fa ðiq ss þfa Q Þ ¼ R0 fa rs Q þiq ss ðfa þrs Þ 

R ¼ R0 Acknowledgements J.R.E. acknowledges the support of an Australian Postgraduate Award. The authors thank two anonymous referees for helpful comments.



Appendices We present some details of the derivation of the mathematical results included in the main text.

Appendix A. Steady states (equilibrium points)

which for positive parameter values implies that R o0. We have then the three equilibria.    ðfa þ gÞðQ giq ss Þ    Qg ðF  ,F ,R Þ ¼ ð0,0,R0 Þ ¼ R0 ,R 1 ðQr s þiq ss Þgiq rs ss iq ss  iq ss g R0 Qr s g þ ss iq gss rs iq   fa f a ss i q ðiq ss þfa Q ÞR0 : ¼ 0,R0 fa rs Q þ iq ss ðfa þrsÞ fa rs Q þiq ss ðfa þ rsÞ

Steady state populations are denoted with an asterisk, e.g. F  is the steady state population of F(t). Appendix B. Stability A.1. First (trivial) equilibrium point For mjQ mkS dFðtÞ Qj SðtÞk ¼ FðtÞ fr f s dt SðtÞk þ mkS Q j þ mjQ SðtÞk þ mkS Q j þmjQ

! ¼ 0,

one solution is F  ¼ 0. Using this and solving for dF =dt ¼ 0 gives ! F ðtÞ F ðtÞ 1 1 Q RðtÞ ¼ 0 ) F ðtÞ  þ ¼0 fa SðtÞ fa iq ss RðtÞ þ F ðtÞ 

which also has an obvious solution of F ¼ 0. Solving for dR=dt ¼ 0  with F  ¼ F ¼ 0 gives 1 ðR0 RðtÞ ¼ 0 ) RðtÞ ¼ R0 rs a



and so there is one equilibrium point at ðF  ,F st,R Þ ¼ ð0,0,R0 Þ.

We are interested in the nature of the two possible equilibrium points, especially their stability, as it helps us to understand how the decision will trend over time. We will do this by finding the determinants of the Jacobian matrices at the equilibrium points. B.1. Jacobian matrix The Jacobian matrix for Eq. (7) is    dFuðtÞ dF uðtÞ dRuðtÞ      dF dF   dF    dFuðtÞ dF uðtÞ dRuðtÞ  : J ¼  dF dF   dF    dFuðtÞ dF uðtÞ dRuðtÞ     dR dR dR 

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B.2. First equilibrium point

Appendix C. Supplementary data

The matrix for the first critical point corresponds to the initial point, with no active foragers. It is relatively simple and can be   solved properly. After substituting the associated F  ,F and R values in the matrix, we have    fr mk Q j fs mj ðiq ss Þk    S Q Q 0 0  j  i k q k  ðm þQ j Þðm þ ð ss Þ Þ   Q  S Q       Q J1 ¼  fa  fa þ 0 ,  i s q s     Q 1     0   iq ss rs 

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jtbi.2010.11.027.

a diagonal matrix for which the eigenvalues are the diagonal elements. The lower two eigenvalues, ðfa þ Q =iq ss Þ and 1=rs , are always negative since the constant parameters are all greater than zero. The first eigenvalue, k iq ss Q  k ! iq j k ss ðmQ þ Q j Þ mS þ Q fr mkS Q j fs mjQ



is positive for fr mkS Q j 4fs mjQ i:e: Q j 4 mjQ i:e: Q 4



fs fr



k



k

iq ss Q

iq ss QmS

,

fs j k k k m m i s fr Q S q s

,

1=ðj þ kÞ

and so for the parameter values of Table 1 this point is only stable for Q t 58. For all greater values of Q this point is unstable and so forager numbers will tend to increase. This conforms well to the biological situation in which nectar with 58 mol=L of sucrose would be considered of poor quality and foraging would be minimal. The eigenvector corresponding to the positive eigenvalue indicates that the unstable manifold is along a positive growth for F and F and negative growth for R, as we would expect for colony that was increasing its population of active foragers. B.3. Second equilibrium point The second critical point is the point typically approached when the colony is in a good quality foraging environment. The full system is analytically intractable, but if we assume that S ¼ S þ d, for d being a small constant, the Jacobian has three eigenvalues, l1 , l2 , l3 , such that fr mkS Q j fs mjQ ðg þ dÞk



, mjQ þQ j mkS þ ðg þ dÞk

l1 ¼

l2 ¼ fa 

l3 ¼ 

1

gþd

,

1 rs

for which l2 and l3 are obviously negative (since all parameters are positive). l1 is only negative for low values of Q , indicating a saddle, but l2 is the dominant eigenvalue (due to the large value of fa and that the upper bound of l1 ¼ ðfr mkS Q j fs mjQ ðg þ dÞk Þ=ðmjQ þ Q j Þ ðmkS þ ðg þ dÞk Þ ofr 5 fa ) and so we have a stable system.

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