swarm-intelligent foraging in honeybees: beneﬁts and costs of...

SWAM INTELLIGENCE

Swarm-intelligent foraging in honeybees: benefits and costsof task-partitioning and environmental fluctuations

Thomas Schmickl • Ronald Thenius •

Karl Crailsheim

Received: 30 May 2009 / Accepted: 18 March 2010

� Springer-Verlag London Limited 2010

Abstract For honeybee colonies, it is crucial to collect

nectar in an efficient way. Empiric experiments showed

that the process of decision making, which allows the

colony to select the optimal nectar source, is based on

individual decisions. These decisions are made by return-

ing nectar foragers, which alter their dancing behaviours

based on the nectar source’s quality and based on the

experienced search time for a receiver bee. Nectar receiv-

ers, which represent a shared limited resource for foragers,

can modulate the foraging decisions performed by the

colony. We investigated the interplay between foragers and

receivers by using a multi-agent simulation. Therefore, we

implemented agents which are capable of a limited set of

behaviours and which spend energy according to their

behaviour. In simulation experiments, we tested colonies

with various receiver-to-forager ratios and measured col-

ony-level results like the emerging foraging patterns and

the colony’s net honey gain. We show that the number of

receivers prominently regulates the foraging workforce.

All tested environmental fluctuations are predicted to cause

energetic costs for the colony. Task-partitioning addition-

ally influences the colony’s decision-making concerning

the question whether or not the colony sticks to a nectar

source after environmental fluctuations.

Keywords Swarm intelligence � Honey bees �Task partitioning � Foraging � Equal foraging distribution �Cross inhibition � Choice � Nectar economics

1 Introduction

Honeybee colonies have to collect huge amounts of nectar

in their environment to accumulate enough honey for

periods of dearth. Colonies that perform the nectar col-

lection process in a less efficient way will have a lowered

chance to survive (e.g., throughout winter), thus natural

selection favours those colonies that efficiently forage for

nectar. The collective foraging of social insects (ants, bees,

wasps, termites) was studied intensively, revealing many

insights into the proximate mechanisms that are used by

animals to decide upon their foraging behaviour. Inside of

a colony, these individual foraging behaviours (recruit-

ment, abandonment, navigation) sum up to colony-level

foraging strategies, which can differ significantly between

species or even within one species in differing environ-

mental situations.

Mathematical models are used to analyse and interpret

these foraging decisions of social insect colonies. Colony-

level decisions that arise from known mechanisms of

individual decision-making were studied with top-down

models, which predict dynamics of forager and receiver

group sizes with sets of linked differential equations.

A comparative study [1] showed the similarities and dif-

ferences between ant foraging and honeybee foraging.

Detailed studies of recruitment mechanisms and their effect

on forager group sizes, foraging decisions, and resource

throughput were performed by means of mathematical top-

down modeling [2–4]. In addition, bottom-up (individual-

based, agent-based) models were frequently used to predict

T. Schmickl (&) � R. Thenius � K. Crailsheim

Artificial Life Lab of the Department of Zoology,

University of Graz, Universitatsplatz 2, 8010 Graz, Austria

e-mail: [email protected]

R. Thenius


K. Crailsheim


123

Neural Comput & Applic

DOI 10.1007/s00521-010-0357-9

how well local cues can serve as a source of information

for individual foragers [5–8]. In honeybees, foraging

decisions are based on such local cues, which are exploited

by individual forager and receiver bees. Based on these

cues, and on communication transferred through bee dan-

ces (see for eample [9, 10]), one worker can affect the

behaviour of other nearby workers. This way ‘behavioural

feedback loops’ emerge, which lead to a self-organised

regulation of workforce and foraging decisions [11].

Honeybee foraging is a very good example of natural

‘swarm intelligence’, because the group of foraging bees is

not controlled by a central decision-making unit and

because the collective decisions of colonies in varying

environments were found to act in an intelligent way.

1.1 Honeybees select foraging targets by swarm

intelligence

At first sight, foraging of honeybees looks similar to the

foraging of other mass-recruiting social insects, which

were shown to forage for food (and also for new nests) in a

swarm-intelligent way. When looking closer onto details,

there are significant differences between the foraging per-

formed by ants (or termites) and honeybees:

1. Honeybees perform waggle dances, which communi-

cate vectors (distance, direction) pointing to the

foraging targets. For further details, see [9, 12]. The

decision-making upon which foraging target is

exploited the most is based on the likelihood of naive

bees to encounter such waggle dances, and not on

direct quality information passed from one bee to

another [13].

2. Honeybees do not simply select the nearest nectar

source; they collectively take also into consideration

the quality of the source, what is the sugar concentra-

tion of the offered nectar and/or the flow of nectar at

the source [13]. This multi-factorial information pro-

cessing is an impressive ability, because it is achieved

without a central unit of decision-making, and it was

found to be very robust and flexible [3].

3. The foraging of honeybees involves task partitioning

between forager bees and receiver bees. Foragers

collect the nectar in the environment and bring it back

to the hive. There, receiver bees take over the nectar

load from the forager bees, transport it to the storage

area of the hive and store it there. As this sequential

task partitioning involves nectar transfer from forager

bees to receiver bees, this generates the need for

swarm-intelligent coordination of these two groups of

co-workers, which is regulated by the ‘tremble dance’

behaviour. This dance is (sometimes) performed by

forager bees and was found to trigger unemployed bees

to engage in the nectar-receiving task [10].

These characteristics of the honeybees’ way to forage

for food in a swarm-intelligent way is of high significance

also for the community of engineers and mathematicians

who develop new bio-inspired optimisation algorithms or

novel swarm robotic systems. The foraging decision-mak-

ing is performed collectively by the foraging bees and it

involves also individual information processing that mod-

ulates the behaviour of single bees. The decisions made by

individual bees can be modelled by rather (mathematically)

simple equations as their behavioural modulations are

usually not very complex (stimulus–response thresholds

and correlations). Frequent interactions among bees and the

exploitation of behavioural modulations of other bees as

cues for their own behavioural decisions make this forag-

ing a clear example of swarm intelligence: it was shown

that individual bees assess the quality of a visited food

source (=net energy efficiency) in a way that is expressed

best by

qualitysource ¼gain� cost

cost; ð1Þ

as was shown in Seeley [13, 14]. In Eq. 1, the variable gain

represents the energetic value of the collected nectar and

the variable costs represents the metabolic (energetic) costs

of the foraging trip, both variables are measured in Joules.

Based on this dimensionless quality index, the foragers

decide how many dancing rounds they perform. As the

ultimate effect, this leads to the ability to collectively select

the optimal nectar source. For further details, see [3, 13].

The sequential task partitioning between forager bees

and receiver bees adds interesting features to the honeybee

foraging system: It was shown that nectar receivers play an

important role, because they represent a limited resource

shared by the whole cohort of foragers. If a forager sear-

ches too long for a receiver bee that unloads the collected

nectar, it does not perform a waggle dance, thus this for-

ager does not recruit additional foragers. It was pointed out

in Johnson [15] that nectar receivers work also in pro-

cessing the food, thus they need to have a specific physi-

ological constitution. This physiological status is usually

found in ’middle-aged bees’ (11–20 days old), which are

usually present in high numbers (several thousands) in a

colony. However, this age-class performs not only nectar-

receiving, it also performs a high variety of other tasks:

e.g., guarding, wax-building, or undertaking [16, 17]. In

recent studies, we showed that the age-structure of a hon-

eybee colony is affected by external (e.g., weather) and

internal (e.g., cannibalism) factors, thus the size of age-

groups can change quickly in a colony [18, 19]. In con-

sequence a limitation in the number of available receiver


123

bees, which are mid-aged bees located at the nectar transfer

places and that actually engage in the food handling task,

can significantly decrease the recruitment rate of new for-

agers bees. It was shown by Seeley [20] that honeybees use

a dynamic system to regulate the number of active receiver

bees: when the searching (queuing) delay of a loaded for-

ager exceeds a certain period, this forager starts to perform

a tremble dance, which recruits additional nectar receivers

instead of performing a waggle dance. This way, a

decentralised auto-balancing mechanism is established that

regulates the size of the two groups of workers (foragers

and receivers) in a way that keeps them at an optimal ratio.

Remembering the fact that the two groups split their task

sequentially, such a regulation system could be interpreted

as an adaptation to optimise the overall performance of the

colony, which is the unit of natural selection in honeybees.

For further details on this regulation regime, see [21].

Figure 1 shows how colony saturation (amount of stored

honey in the hive) and the distance to the exploited nectar

source influence the ratio of receivers to foragers in the

colony: upon interaction, foragers get unloaded and

receivers get loaded. The fuller the colony is, the longer it

takes for receivers to get unloaded again, as they have to

search longer for empty storage cells. The further away the

foraging target is, the lower is the frequency of returning

loaded foragers.

1.2 Hypotheses and motivation

The task-partitioning between receivers and foragers is a

remarkable characteristic of honeybees. Our goal was to

study the ‘swarm intelligence’ of honeybee foraging and

how the group size of receivers affects the collective

decision-making process. We hypothesise that the receiver-

to-forager ratio is ‘shaping’ the collective foraging in the

following ways:

H1: Changing the receiver-to-forager ratio affects the

collective foraging decision made by the simulated hon-

eybee colony significantly.

H2: How these foraging decisions are affected by the

receiver-to-forager ratio additionally depends on the envi-

ronmental situation or the intensity of an environmental

fluctuation.

H3: Every foraging decision induced by our simulated

environmental fluctuations—characterised by a sudden

change in location of the most profitable food source

without change in mean quality of all available nectar

sources—causes an energetic loss for the foraging honey-

bee colony due to the (physiological) costs of the decision-

making process that is involved. The receiver-to-forager

ratio also affects the nectar economy of the colony sig-

nificantly in these cases.

To allow such an analysis, we extended an existing

multi-agent simulation platform (HoFoSim, honeybee for-

aging simulator), which is described in detail in Schmickl

et al. [22–24]. As we wanted to investigate the role of

receiver bees, we extended our model by implementing

also receiver bees as autonomous agents, which make their

own behavioural decisions and which have their own

metabolic expenditures. This new model is called

HoFoReSim (honeybee forager and receiver simulator) and

has the following key characteristics:

1. We model the foraging purely based on individual

behaviours, no global communication, no global

supervising unit is implemented. For example, waggle

and tremble dances affect only local neighbouring

agents.

2. The platform HoFoReSim allows a variety of interac-

tive simulation experiments to address many questions

that arise during the study of honeybee foraging.

3. HoFoReSim implements all relevant factors that

represent a realistic honeybee foraging scenario, e.g.,

errors in communication and navigation, metabolic

costs of behaviours, differences among individuals

(dance–response curves, weights, speeds, crop sizes,

etc.).

Using this multi-agent model, we performed a series of

simulation experiments in which we predicted the number

of foragers on different nectar sources, as well as the col-

ony’s predicted honey accumulation. We exposed the

modeled colony to a variety of environmental fluctuations

and observed how the colony was able to react to these

disturbances. By evaluating the honey accumulation, we

were able to assess the energetic costs induced by these

Fig. 1 Regulation of the number of active receiver bees and of active

forager bees according to the distance of the nectar source, according

to the quality of the nectar source and according to the colony’s nectar

need


123

environmental fluctuations. By varying the receiver-to-

forager ratio, we could investigate the resulting changes in

foraging strategies and also how this ratio affects the col-

ony’s energetic efficiency.

2 Methods

Our multi-agent simulation platform is able to simulate up

to 5,000 foraging bees and up to 5,000 receiver bees

simultaneously. In contrast to the model described in

Schmickl et al. [22, 23], we extended our model in a way

that also receiver bees are now simulated individually.

They can be loaded by returning foragers and they are then

occupied for the time it takes to store the nectar. For

details, see [24]. Our model focuses on the tasks ‘nectar

foraging’, ‘nectar receiving’, ‘idle’ (=work reserve) and

‘scouting’. We modelled the three signals that affect this

process prominently that are ‘waggle dancing’, ‘tremble

dancing’, and ‘nectar offering’. In addition, we modelled

physiological costs of all tasks in much detail, to allow us

to predict colony-wide economics. Other signals (e.g. the

‘shaking signal’) were omitted so far, to keep the com-

plexity of the model in reasonable range.

2.1 Environment

Our simulation platform is able to perform a variety of

experiments focusing on the foraging of honeybees. The

bees (foragers and receivers) move randomly around in the

hive area and they interact (e.g., nectar transfer) when they

meet on the same patch, which represents an area of

7.5 mm 9 7.5 mm of comb space. The foragers leave the

hive through a special ‘entrance patch’. In the outside

environment, one patch represents 2.7 m 9 2.7 m of

space. The outside environment holds up to three nectar

sources, which represent artificial feeders that can be freely

positioned in the environment. The sugar concentration of

the offered nectar (sucrose) is adjustable during runtime

between 0.0 Mol/l (feeder removed) and 2.5 Mol/l. For the

spatial layout of our model, see Fig. 2.

2.2 Agent behaviours

We implemented two types of agents: ‘foragers’ and

‘receivers’. All agents are modeled as finite state automata

(Fig. 3). They are modeled as ‘reflex agents with inner

status’ (see [25]). Each agent is situated in one behavioural

state, which determines its behaviour. This behaviour is

performed every time step unless a behavioural transition is

triggered. Such transitions can be caused by one of the

following reasons: a fixed time delay has expired, an

external event has happened, an internal event has

happened, a communication event has happened or an

event was triggered with a fixed probability. For details,

see [22–24]. The ‘inner state’ of the agent is characterised

not only by its ‘behavioural state’ as described above, but

also by its crop load (nectar load) and sometimes by its

experienced queuing delay when waiting for unloading. In

addition, it possesses a memory to recognise one foraging

target, which is the vector towards a food source and the

distance of this source from the hive.

In our model, most of the transitions from one behav-

ioural state to another are implemented on a threshold

based system following the suggestions of Bonabeau et al.

[26], which we already compared to empiric data in

Schmickl and Crailsheim [23]. The probability to change

behavioural states depends on the intensity of local stimuli

(cues) in a sigmoid-shaped relationship. For example, the

time period tsearch(i, t) of a forager i in time step t that

searches for a receiver bee is exploited as a cue (stimulus).

It influences the chance to perform a waggle dance or a

tremble dance (see Fig. 4) according to the following two

equations:

To model an agent’s probability to dance a tremble

dance, we modeled

ptrembleði; tÞ ¼tsearchði; tÞn

tsearchði; tÞn þHn; ð2Þ

where we found best agreement to empiric literature data

(see [23]) with n = 5 and H ¼ 40. The probability of an

agent to perform a waggle dance was modeled by

pwaggleði; tÞ ¼ 1� tsearchði; tÞn

tsearchði; tÞn þHn; ð3Þ

where we set n = 2.5 and H ¼ 10 according to literature

(see [23]).

Bees within a distance of rhearing (6 patches = 4.5 cm)

are able to perceive a dance and are able to react according

Fig. 2 Screenshot of the simulation. a The hive: inside the hive

foragers and receivers interact. Waggle dances and tremble dances

take place there. b The hive entrance: foragers leave and enter the

hive here. Most interactions between foragers and receivers take place

near the hive’s entrance. c, d The outside: in the environment outside

the hive nectar sources are placed. e, f Experienced foragers fly to the

nectar sources. g Some scout bees do not collect nectar from the

known nectar sources but fly randomly in the environment to find new

nectar sources


123

to Fig. 3. The duration of the waggle dance is modeled

according to Seeley [13], see also Fig. 5. For further

details, please see [27].

For modelling the tremble dance, we followed a similar

scheme, but had to consider the topology of a bee hive and

the spatial distributions of dance-searching naive forager

bees and of inactive potential receiver bees. In our

simulation, we did not model all combs of a full-sized bee

hive. The modelled colony area is 58 9 58 patches wide,

representing 1892 cm2 of comb surface. This area repre-

sents approximately the two comb sides of the ‘dance floor’

area on which foragers perform waggle dances and on

which the majority of nectar transfers take place. As

waggle dancers and potential foraging recruits preferen-

tially locate in this area, we could take over the dance

length and probabilities of dances as they are reported in

empiric studies. In contrast to foraging recruits, inactive

mid-aged bees, which can be activated by tremble dances,

are located in all remote areas of the hive. Based on the

data given in Seeley [10], we calculated that a tremble

Fig. 3 Two linked finite state automata, that we used to model the

behaviour of bees inside and outside the colony. The two behaviours

‘foraging’ and ‘scouting’ are associated with high metabolic costs

(Eq. 11), all other behavioural states are associated with lower

metabolic costs (Eq. 12). a Forager–forager interaction: one forager

can recruit another forager to a nectar source by performing a waggle

dance. b Forager–receiver interaction: a forager can recruit an idle

receiver to start receiving nectar by performing a tremble dance.

c Nectar transfer: during the unloading-process, nectar is transferred

from a forager to a receiver bee

0 50 100 1500

0.2

0.4

0.6

0.8

1

searching time for receiver bee [s]

prob

abili

tyof

danc

e

waggle dancetremble danceno dance

Fig. 4 Sigmoid-shaped stimulus–response curves that model the

dancing behaviour of returning forager bees depending on the

forager’s searching-time for an available nectar receiver. For details,

please see Eq. 2 and equation. These stimulus–response curves are

based on empiric observations reported in Seeley [10]

20 30 40 50 60 70 800

10

20

30

40

50

60

source quality

danc

ero

unds

perf

orm

edbee #1bee #2bee #3bee #4bee #5bee #6bee #7

Fig. 5 Duration of dances performed by foragers depending on the

source quality, as it is described by Eq. 1. Redrawn from [13]


123

dance can cover 2–5% of hive space. We assumed the same

communication distance for waggle dances and tremble

dances. We parametrized our model that one tremble dance

can reach 4% of all receiver bees in the modelled hive,

which corresponds to a tremble dance length of 1 time step.

Compared to a comparable recent model [5], where every

tremble dance recruits always one additional receiver, our

modelling approach is closer to empiric data, as more

inactive mid-aged bees can be recruited by one dance.

In contrast to the sigmoid-shaped curves that model the

chances of dancing, the duration of waggle dances is lin-

early scaling with the quality of the nectar source visited by

the dancing forager bee, as it is expressed by Eq. 1. These

stimulus–response curves for the calculation of the dance

duration are implemented according to Seeley [13]. The

duration dwaggle of a dance depends on the quality index

qualitysource of a source (see Fig. 5) and is calculated

according to the Eqs. 4, 5 and 6:

kwaggle ¼ 0:51 � gð2Þ: ð4Þ

owaggle ¼ 6 � cnnþ 22

1:032e8:798kwaggle� 25 ð5Þ

dwaggle ¼ qualitysource � kwaggle þ owaggle ð6Þ

In these equations, g(x) indicates a uniformly distributed

floating-point random number, which is:

0 � gðxÞ\x: ð7Þ

The variable cnn indicates the actual ‘colony nectar

need’ according to Seeley [13], who showed that the global

status of honey stores (=accumulated recent honey income

rate) modulates the offset of the linear dance–response

curves of individual foragers. The more nectar gets

accumulated in the colony, the less they dance. The

variable kwaggle indicates the slope of the individual

stimulus–response curve, and owaggle indicates the curve’s

offset from the origin. qualitysource is modeled according to

Eq. 1 and expresses a dimensionless quality index (net

energetic gain) of the foraging trip.

The quality index qualitysource also influences the

probability of a bee i in time step t to abandon a source, as

is expressed by the following equation

pabandonði; tÞ ¼ðc� qualitysourceÞ

n

ðc� qualitysourceÞn þ cn

; ð8Þ

which was parametrised with c = 100 and n = 4 to reflect

empiric data of abandonment given in Seeley [11].

All modeled behaviours of forager bees and receiver

bees are associated with metabolic costs, which also

depend on the current weight of the bee. This metabolism

forces the agents to consume a portion of their own crop-

load each time step. How much nectar an agent actually

consumes depends on the sugar concentration of its nectar

load. After the crop-load falls below a certain threshold, an

agent refills by decreasing the honey stores of the colony.

An agent that runs out of nectar outside of the colony dies

immediately in our model, what just happens with extreme

settings (parametrisations) that are not shown in this article.

After a successful foraging flight, nectar is transferred to

receiver bees. Receiver bees increase the colony’s honey

stores when they deposit the nectar into the stores. Thus,

we can interpret the optimality of all emerging foraging

patterns by observing the dynamics of the colony’s honey

stores. Finally, we state that the nectar sources in the

environment, the nectar in the bees’ crops and the honey

stores differ significantly concerning their sugar concen-

trations. Thus, the concentrations of nectar in the crop of a

bee could vary due to the possibility of mixture of nectar of

different sources. Due to this, we had to model all dilution

and concentration processes in detail to retrieve robust

results from our model. Nectar sources offer sucrose in the

empiric experiments described in Seeley [28]. In contrast,

concentrations of sugars in the bees’ crops are tracked in

energetic equivalents of glucose. Bees have different nectar

concentrations in their crops resulting from their past

activities.

2.3 Metabolism

The metabolic rate of a honeybee in our model depends on

the actual weight of an agent i, which is expressed by

wagent(i, t). The metabolic rate depends also on the activity

level of the bee (for details see [24]). The empty weight of

a bee is modeled as wempty(i) = 83.32 ± 6.89 mg [29].

The current crop-load volume is expressed as Vload(i, t).

The maximum crop volume a bee can load with nectar

is modeled as Vmaxcrop ¼ 45:9� 8:7 ll [29]. The weight

wload(i, t) of the actual nectar load having a sugar con-

centration of cload(i, t) is calculated by Eq. 9, based on own

empiric measurements (data not shown), we modelled

wloadði; tÞ ¼ Vloadði; tÞ � ð/ � cloadði; tÞ þ nÞ: ð9Þ

We found best agreement to our own empiric

measurements (data not shown) by setting / =

0.065 mg/Mol and n = 1.0026 mg/ll. According to this

equation, the actual weight of a bee wagent(i, t) is calculated

by

wagentði; tÞ ¼ wemptyðiÞ þ wloadði; tÞ: ð10Þ

Based on data presented in Seeley [13], the metabolic

rate of a bee l(i, t) [ {lhigh(i, t), llow(i, t)} is calculated

either by

lhighði; tÞ ¼�high � wagentði; tÞqhigh

r � cloadði; tÞ� Dt; ð11Þ


123

or by

llowði; tÞ ¼�low � wagentði; tÞqlow

r � cloadði; tÞ� Dt; ð12Þ

whereby ehigh = 0.00287, elow = 0.00248, qhigh = 0.629,

qlow = 0.492, r = 2.827 (according to Seeley [13]),

depending on the activity level of the agent i in time step t.

The metabolic rate is the volume of nectar which an agent

consumes per time step.

Receiver bees add the nectar that was transferred to them

to the honey stores after a period of search. The nectar

deposition process is modeled as follows: receivers stay in

state ‘storing’ for a period of time dstore, which depends on

the current colony’s nectar need [21], which we express by

the parameter cnn (=‘colony nectar need’) in our model. In

Seeley [21], the author refers to an ‘empty’ hive and to a

‘full’ hive, which we parametrise in our model as cnn = 0.0

and 1.0, respectively. The author notes that the receivers

spend 10 min (smin = 600 s) to store the nectar in an empty

hive and 28 min (smax = 1,680 s) in a full hive. We assumed

a linear negative regression of the storing duration between

the two extremes of cnn, thus we modeled the storage cycle

time (in time steps of Dt ¼ 0:5 s) as

dstore ¼smin þ ðsmax � sminÞ � ð1� cnnÞ

Dt: ð13Þ

Receiver bees consume energy (and thus nectar)

according to their current weight, also during their storage

trips. The remaining transported nectar is then added to the

colony’s honey stores at the point in time a receiver switches

from the ‘storing’ state to the ‘receiving’ state again, see

Fig. 3.

2.4 Parametrisation and individuality of bees

Our simulation experiments shown in this article were

performed with time step size Dt = 0.5 s. Other important

parameters of the Simulation are the spatial distances in

and outside the hive in the simulation environment: the

edge length of one patch inside the colony corresponds to

0.75 cm, outside the colony 2.7 m. The speed s of a bee

was implemented with s = 5 m/s outside the colony (flight

speed), within the colony s = 1.5 cm/s (running speed).

The sugar concentration of honey in the honey stores is

set to 6.25 Mol/l in our model (own measurements). To

produce honey, nectar receivers have to evaporate water

from the nectar and to alter sugars in an enzymatic way.

The water evaporation leads to a decrease of volume when

nectar is stored as honey in the colony’s honey stores, what

is also part of the material flow, which is modelled in detail

in our model. The sugar concentration of honey is of big

importance for the ‘refilling’ processes of honeybees inside

the colony. If bees run out of nectar inside the colony they

refill their crop directly from the honey stores of the col-

ony. Foragers upload 1 ± 0.2 ll of honey, receivers upload

9 ± 1.8 ll of honey. This data is estimated, based on

Huang and Seeley [29]. At the beginning of a simulation

run 15 foragers in the hive have the information about

heading and distance of each nectar source, and 50% of the

receivers are in the state ‘‘receiving’’.

The individuality of foragers regarding the duration of

the waggle dance (see Fig. 5) is implemented by a indi-

vidual value, represented by g(2) in Eq. 4, as described in

detail in Thenius et al. [27]. Other parameters, like the

probability to leave the colony for scouting, the probability

to forget a source or the probability to start foraging were

(uniformly) distributed randomly around ±30% around the

mean values. Other parameters of the simulation were

parametrized using empiric data from literature as it is

mentioned in the previous sections.

2.5 Statistics

We repeated all analysis multiple times, as we applied ran-

dom uniform noise symmetrically to the bees communica-

tion and to individual agent properties based on the

parametrised mean values. The exact repetition number is

given with every experiment in the text and in the figure

legends. To analyse differences between two datasets, we

used two-sided Mann–Whitney U-tests. To analyse the effect

of multiple factors in datasets, we used ANOVA (two-way

randomised blocks), whereby experiment repetitions with

the same settings were interpreted as blocks. In these anal-

yses, means were analysed by Student–Newman–Keuls

tests. For correlation analysis, we performed Pearson cor-

relation tests. In all analyses, the level of significance (a) was

set to a = 0.01.

3 Results

In the simulation experiments described in this article, we

focus on the role of receiver bees in the decision-making

process. To test the reliability of our model, we performed a

set of simulation experiments that reflect empiric experiments

performed with real honeybees. Then, we performed a series

of simulation runs in fluctuating environments and observed

the energetic optimality of the emerging foraging patterns.

Therefore, we investigated the nectar-economic effects that

arise from the intensity of the simulated environmental fluc-

tuations and from the receiver-to-forager ratio.

3.1 Experiment 1: equal foraging distribution

In an experiment described in Seeley [13], the dependency of

waggle dance duration on the net energy efficiency


123

(see Eq. 1) was shown. Two nectar sources were present in

the environment, one in a distance of 250 m (source A) and

one in a distance of 550 m (source B) from the hive. Source

B offered always a sucrose solution of 2.5 Mol/l. By

manipulating the concentration of the near feeder (source A),

the scientists searched for the concentration which stimu-

lated the bees to perform exactly the same dance length for

both nectar sources. We modeled this set-up with our sim-

ulator by simulating 500 foraging bees and 400 receiver

bees. To demonstrate the reliability of our model, we para-

metrised our model to the details given in Seeley [13]. We

observed the number of bees carrying information about one

of the two sources. This correlates with the waggle dance

durations. Additionally, we observed the size of the forager

groups that were visiting both nectar sources.

We found an equal foraging distribution (=EFD) with a

sugar concentration of 1.34 Mol/l at the near feeder, which is

within the range of concentrations reported in Seeley [13] for

real bee experiments (1.32–1.46 Mol/l). As Fig. 6 demon-

strates, there are some difficulties in searching the point of

EFD: while the number of bees which hold information

about either one of the two sources stays almost in a ratio of

1:1 (Fig. 6a), the number of bees actually visiting the near

source is always higher than those visiting the far source

(Fig. 6b). This difference is due to the fact that our model

incorporates communication error in the communication of

nectar sources (see for example [12]). Also, the flight of the

bees was modeled to be erroneous, according to the data

given in von Frisch [9]. In addition to that the flight cycle of

those bees that forage on the more distant source is longer,

thus more bees are—on average—in the flying state, in turn

decreasing the average number of bees found drinking on the

distant source per time interval. This effect can be seen by

comparing Fig. 6a with Fig. 6b. As Fig. 6c shows, the

quality assessments of the bees for their nectar sources are

similar at EFD conditions.

We performed additional simulation runs to test how

much the ability to reach an EFD depends on the environ-

mental conditions and on the receiver-to-forager ratio. We

placed two nectar sources in equal distance to the hive

(400 m) and offered an equal sugar concentration CAsource ¼

CBsource on both sources. This experiment was performed with

two sugar concentrations (fCAsource ¼ CB

source ¼ 1:0 Mol=l;

CAsource ¼ CB

source ¼ 2:5 Mol=lg). We simulated several

ratios of receivers to foragers (foragers = 500 bees,

receivers [ {200, 400, 600, 800, 1000, 1200} bees). The

foraging distribution was calculated according to the fol-

lowing symmetry index

SA;Binfo ¼

maxðf Ainfo; f

BinfoÞ

f Ainfo þ f B

info

; ð14Þ

where a value of SA;Binfo ¼ 0:5 indicates an EFD. The vari-

ables f Ainfo and f B

info represent the number of foragers that

have information about one of the nectar sources A or B as

it is depicted for one single run in Fig. 6a. As Fig. 7 shows,

our model predicts distributions closer to EFD with higher

ratios of receiver-to-foragers. This effect is stronger with a

low quality CAsource ¼ CB

source ¼ 1:0 Mol=l of the available

nectar sources.

3.2 Experiment 2: cross-inhibition between groups

of foragers

In an empiric experiment [28], two nectar sources were

placed 400 m away from the hive. Both sources offered a

sugar concentration of 1.25 Mol/l. After some time, there

was an EFD on both feeding sites, comparable to the sit-

uation demonstrated in experiment 1. At this point in time,

the sugar concentration at one of the nectar sources was

increased suddenly to 2.5 Mol/l. The number of foragers

recruited to this feeding site increased very fast and

(a) (b) (c)

Fig. 6 One exemplary run showing equilibrium foraging on two

nectar sources. Source A 250 m, 1.34 Mol/l, source B 550 m,

2.50 Mol/l. a Number of foragers having information in their memory

about one of the two sources. b Number of individual foragers found

on each source within 30-min time slots. c Distribution of quality

assessments (according to Eq. 1) within the foragers (cumulative sum

of 5 snapshots during runtime, taken in intervals of 30 min). This

correlates with dance length of waggle dances. Upper graph shows

data for source A, lower graph for source B


123

simultaneously lowered the number of foragers visiting the

other feeding site, which remained unchanged in quality.

We resembled this experimental scenario and simulated it

with several realistic receiver-to-forager ratios. In all runs,

we used 500 foragers, the number of receivers were 300,

600 and 900 bees.

To investigate the effect of the quality increase of one

nectar source onto the foraging on the other source, we

performed the following series of simulation experiments.

In the first part of the experiment (before the quality

increase), we waited for two simulated hours to establish an

EFD. Then, we stored this point in time on hard disk. One

interesting feature of our simulation platform is the fact

that we are able to save the state of all agents and of the

environment to the computer hard disk and reload this

point in time later again. This allows us to perform ‘time-

machine’-like experiments, like rewinding the tape and

running the simulation from the same starting point again.

Taking this saved state, we continued the simulations

with a sudden quality increase of source A for 3 h. This

experiment was repeated 25 times. After that, again starting

with the initial state saved on disk, we simulated a scenario

without the quality increase of source A for 25 times. By

comparing the foraging patterns at the never-changed

source B in both scenarios, we investigated whether a

cross-inhibition has occurred or not. These ‘save-and-

reload’ experiments allowed us to interpret the cross-

inhibitory effect by only looking at the number of foragers

on the unchanged source after the disturbance occurred,

while other studies (e.g., [28]) had to compare the periods

before and after the occurrence of the disturbance.

In all simulation runs we observed cross-inhibition

between the two groups of forager bees. In all three set-ups

(Fig. 8a, d, g) the quality increase of source A led to an

immediate increase of the number of bees foraging on source

A. The more receivers were present in each simulation run,

the higher was the number of foragers on source A at

t = 14:00. Between 30 min and 1 h after the quality change

of source A the difference between the number of foragers on

the two nectar sources was found to be statistically signifi-

cant. This indicates that the simulated colony made a for-

aging decision according to the sudden environmental

fluctuation (two-tailed Mann–Whitney U-test, p \ 0.01,

N1 = N2 = 25 per setting). In contrast to that, in the 25 runs

without quality changes, the number of foragers on both

sources stayed the same (except for the first half hour in two

cases, two-tailed Mann–Whitney U-test, p [ 0.01, N1 =

N2 = 25 per setting; see Fig. 8b, e, h). When comparing the

number of foragers on unchanged source B in the stable

environment and in the unstable environment, we found the

following pattern. The model predicts a significant decrease

in the number of foragers on source B for the time period

30 min-1 h after the environmental fluctuation (two-tailed

Mann–Whitney U-test, p \ 0.01, N1 = N2 = 25 per setting).

Thus, we observed clear cross-inhibition in all of these

simulation runs. The number of foragers on the unchanged

nectar source at t = 14:00 correlated positively (Pearson

correlation, r = 0.67, N1 = N2 = 25 per setting) with the

receiver-to-forager ratio. This demonstrates the importance

of receiver bees as shared limited resource in explaining the

cross-inhibitory effect between forager groups.

3.3 Experiment 3: optimal source selection

Empiric experiments showed that a honeybee colony col-

lectively selects the optimal nectar source and that it is able to

reverse this decision after a severe change (=fluctuation) in

the environment [3]. Two nectar sources were placed at the

same distance from the hive (400 m), one offering a high

sugar concentration (2.5 Mol/l), the other one offering a low

sugar concentration (1.0 Mol/l). After 4 h, the concentra-

tions were altered, the former high-quality source offered

now a very poor sugar concentration (0.75 Mol/l), while the

former poor-quality source now offered a high sugar con-

centration (2.5 Mol/l). This configuration was kept for

additional 4 h. We took over this experimental set-up but

extended it by adding the following variations and

modifications:

1. We performed symmetrical fluctuations so that the

quality (=sugar concentration) of the poor source was

the same in the first and in the second half of the

Fig. 7 Distribution of foragers on two equal-quality and equidistant

nectar sources with varying ratios of receivers to foragers (SA;Binfo , see

text). Two different sugar concentrations were tested on both nectar

sources. N = 6 per setting. Bars indicate medians, whiskers indicate

quartiles


123

experiment. This is different to the study described in

Seeley et al. [3], in which the experimenters offered a

lower sugar concentration on the poor source in the

afternoon compared to the poor source in the morning.

2. We simulated different amplitudes of environmental

fluctuations. The following pairs of sugar concentra-

tions were tested (given in Mol/l): 2.5:2.0, 2.5:1.5,

2.5:1.0.

3. We performed simulations with different receiver-to-

forager ratios. We kept the number of foragers

constant at 500 bees and used the following sizes for

the receiver group: 250, 500, 1,000 and 1,500 bees. In

total, each of these combinations (number of receivers

in combination with fluctuation amplitudes) was

repeated six times.

We found both, an impact of the fluctuation amplitude

and of the receiver-to-forager ratio onto the emerging

foraging patterns. Figure 9 shows that higher numbers of

receivers lead to higher numbers of active foraging bees.

We calculated the mean number of active foragers on both

nectar sources (f). The values for f varied between f = 74.2

active foragers found with 250 receiver bees and f = 127.9

active foragers found with 1,500 receiver bees (also

depicted in Fig. 13b). The differences observed in the

number of active foragers were significantly affected by all

tested numbers of receivers (ANOVA, N = 24 per setting,

p \ 0.01 between all settings) but it was not affected sig-

nificantly by the strength of the tested environmental

fluctuations (ANOVA, N = 24 per setting, p [ 0.01).

The most important question in this experiment is:

does the receiver-to-forager ratio influence the ability of

the colony to perform a second decision within one day,

that is, to select the better feeder in the afternoon? To

investigate this, we made the following definition: if the

number of foragers on source B exceeds the number of

foragers on source A at t = 16:00, we say that the colony

has performed a second foraging decision in the

afternoon.

Figure 9 shows that an increasing amplitude of the

environmental fluctuation favours such a second foraging

decision in the afternoon (depicted in sub-figures 9i–l). The

fewer receivers were available in a simulation run, the earlier

this decision was made. Such sharp foraging decisions are

performed also with lower amplitudes of the environmental

fluctuation (e.g., see Fig. 9e), if the number of receivers is

low (250 bees). With high numbers of receivers and with low

amplitudes of the environmental fluctuation, the emerging

foraging pattern is significantly different. The colony keeps

on to exploit source A massively (which was not worsened

very much in the afternoon in these runs) and additionally

recruits to source B in the afternoon, which was improved

in quality then. Using ANOVA analysis, we analysed

the dependence of the observables foragerssource Aðt ¼16 :00Þ; foragerssource Bðt ¼ 16 :00Þ and Dforagers B�Aðt ¼16 :00Þ , which represent the number of foragers on the

sources A and B, as well as the difference between these two

group sizes at t = 16:00. Statistical analysis showed that all

three observables were significantly affected (ANOVA,

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 8 Cross-inhibition

between two groups of foragers

in scenarios with different

receiver-to-forager ratios. All

runs: 500 foragers. Left column:

300 receivers. Middle column:

600 receivers. Right column:

900 receivers. Both sources are

of same quality for the first 2 h.

For the following 3 h, two

different scenarios were

performed. First row: the quality

of source A increased

spontaneously for the last 3 h of

the experiment. Second row: the

quality of source A stayed

unchanged during the last 3 h of

the experiment. Last row:

comparison of the number of

foragers on source B in both

scenarios. Stars: significant

differences (U-test, p \ 0.01,

N1 = N2 = 25), thus they

indicate presence of a cross-

inhibitory effect. Graph shows

mean ± SD


123

N = 24 per setting, p \ 0.01) by the hight of the environ-

mental fluctuation as well as by the number of available

receivers. However, there was no statistical difference of

means between the runs with 1,000 and 1,500 receivers, as

well as between the runs with the two strongest

environmental fluctuations (ANOVA, N = 24 per setting,

p [ 0.01). In addition, ANOVA analysis showed that also

the interaction of the type of environmental fluctuation and

the number of receivers jointly affected the means

(ANOVA, N = 24 per setting, p \ 0.01).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Fig. 9 Source selection of the simulated honeybee colony in an

environment which always contains one high-quality foraging target.

For the first 4 h, source A offered the high nectar quality, for the

second 4 h period, source B offered the higher nectar quality. Three

different amplitudes of such environmental fluctuations were per-

formed, see column headings for exact values. These scenarios were

performed with different receiver-to-forager ratios; see row headings

for exact values. An intersection of the two datasets on the right half

in each figure (afternoon) indicates that the colony performed a

second foraging decision in response to the environmental fluctuation

at noon. The values of f indicate the mean number of foragers found

on both sources, thus they indicate the emerging colony’s foraging

activity. All runs: 500 foragers were simulated with different numbers

of receiver bees. N = 6 per set-up. Graph shows mean ± SD


123

If both options in the decision-making are of poor

quality, but still one option is better than the other, a

similar pattern arises (see Fig. 10). The lower the ratio of

receivers to foragers is, the quicker a second foraging

decision is made in the afternoon.

3.4 Analysis of experiment 3: how does this colony

behaviour affect the net honey gain of the colony?

Figure 11 exemplary shows for two selected simulation

experiments of experiment 3 that these environmental fluc-

tuations significantly affect the colony’s predicted net honey

gain. During the first 2 h of the simulated 8-h period, the

colony has almost no net honey gain, as all bee agents con-

sume nectar and waggle dance recruitment is a process that

takes time. In the second half of the morning, the predicted

honey gain increases, but the switch of source qualities calls

for re-recruitment of foragers to the other food source.

Again, honey gain breaks down, and it takes some time until

it reaches the previous value of honey gain per time unit.

Figure 11 also shows that the effect of the environmental

fluctuation is observable on the honey gain of the colony

after a significant time delay, which is due to the duration of

foraging flights, queuing delays and the time receivers need

to store the nectar. To allow investigation of the energetic

costs of environmental fluctuations, we compared the pre-

dicted honey gain of undisturbed simulation runs (indicated

by the final net gain ’a’ in Fig. 11) to the predicted net honey

gain of runs with environmental fluctuations (indicated by

the final net gain ’b’ in Fig. 11). The environmental fluctu-

ation induced higher costs (vabscolony ¼ �a� �b) in the simulation

runs performed with 1,500 receiver bees compared to those

runs performed with 250 receiver bees. The variable �a rep-

resents the mean final honey store in all undisturbed runs of

the same colony size and the variable �b represnts the mean

final honey stores in disturbed runs with same colony size.

When we increased the size of the receiver group, we

increased the size of the simulated colony accordingly. Of

course, bigger colonies have more active foragers (see

values of f indicated in Figs. 9, 10), and thus these colonies

accumulate more honey per time unit. This can be seen by

comparing absolute values of a and b in Fig. 11. To allow

comparison of the costs induced by environmental fluctu-

ations of differently sized colonies, we expressed the costs

as a relative measure of the potential net honey gain

achieved in an undisturbed environment. Thus, we defined

vrelcolony ¼ 1�

�b

�a; ð15Þ

expressing the fraction of the net honey gain, which was

not achieved due to the environmental fluctuation

Fig. 10 Source selection of the simulated honeybee colony between

two poor nectar source options of differing quality. For the first 4 h,

source A offered a higher nectar quality than source B. For the second

4 h period, source B offered a higher nectar quality than source A.

These scenarios were performed with different receiver-to-forager

ratios; see row headings for exact values. An intersection of the two

data sets on the right half in each figure (afternoon) indicates that the

colony performed a second foraging decision in response to the

environmental fluctuation at noon. The values of f indicate the mean

number of foragers found on both sources, thus they indicate the

emerging colony’s foraging activity. All runs: 500 foragers were

simulated with different numbers of receiver bees. N = 6 per setup.

Graph shows mean ± SD


123

compared to undisturbed simulation runs with the same

colony size.

We compared the predicted colony’s net honey cost

(vrelcolony) for all choice experiments depicted in the Figs. 9

and 10 to see how the strength of the environmental fluc-

tuation and the receiver-to-forager ratio affects this

important variable. As shown in Fig. 12, both factors affect

the costs significantly. On the one hand, lower receiver

numbers lead to lower relative costs of environmental

fluctuations. This suggests that the quick foraging decisions

induced by low numbers of receivers help the colony to

prevent costs caused by environmental fluctuations. On the

other hand, low numbers of receivers lead to less amounts

of accumulated honey.

As Figs. 9 and 10 show, the predicted foraging patterns

are significantly influenced by the receiver-to-forager ratio.

Low numbers of receivers mainly lead to low numbers of

foragers on the worsened source in the afternoon. This can

be explained by the cross-inhibition phenomenon depicted

in Fig. 8. We compared the accumulated net honey gain

within the 8 h of our experiments to investigate how these

different foraging patterns affect the colony’s energetic

foraging success.

Figure 13a shows that the number of available receiver

bees affects the net honey gain of the simulated colony:

colonies with medium ratios of receivers to foragers

accumulate more honey than colonies with extreme ratios.

When receivers are rather scarce, massive honey accumu-

lation is prevented. With too many receivers, energetic

costs of these excess receivers decrease the net honey gain.

In contrast to that the number of active foragers is posi-

tively correlated to the receiver-to-forager ratio (Fig. 13b).

When evaluating efficiency criteria like the net honey gain

per active forager (Fig. 13c) and the net honey gain per

worker bee (forager and receivers combined, Fig. 13d), a

clear optimum for medium receiver-to-forager ratios (500

receivers and 500 foragers) was found. When analysing

how the environmental conditions affect these measures,

we found a clear picture. The better the best nectar source

in the environment is, the better is the colony’s predicted

net honey accumulation. The least affected measure is the

number of active foragers (f), which is more affected

by receiver-to-forager ratios than by environmental

conditions.

4 Discussion

With our model ‘HoFoReSim’, we simulated collective

foraging behaviour of honeybees by a multi-agent simu-

lation. Our aim was to investigate how environmental

changes affect the nectar economics of honeybee colonies

and how foraging decisions could influence these eco-

nomics. In our model, the agents’ behaviour are modelled

as state machines. Changes of state are triggered by events,

probabilities, thresholds and time-outs. Both, the state

machine and the functions that model thresholds could

alternatively be modelled as artificial neural networks [30].

In the real bees, these thresholds and behavioural states are

controlled by the bees’ nervous systems and by their

physiological status (energy and hormones).

We formulated three novel hypotheses (see Sect. 1) and

designed several experiments to test these hypotheses. In

addition, we were able to give qualitative and also quan-

titative predictions of the benefits and gains of foraging

decisions under a variety of environmental conditions and

fluctuations. To allow this study, we modeled proximate

(a)

(b)

Fig. 11 Dynamics of honey accumulation performed by the simu-

lated colony. a Predicted honey dynamics of a colony having a low

number of receiver bees. b Predicted honey dynamics of a colony

having a high number of receiver bees. To assess the energetic costs

induced by the environmental fluctuation at t = 12:00, we compared

the final net honey gain of undisturbed runs to the final net gain of

disturbed runs. For all settings. N = 6. Graph shows mean nectar

dynamics


123

(a) (b)

(c) (d)

Fig. 12 Costs of environmental

fluctuations as observed in our

simulated choice experiments.

We compared four

environmental fluctuations of

different intensity of the

disturbance, which occurred at

t = 12:00 and investigated the

resulting relative colony-level

costs compared to undisturbed

runs performed with similar

sized colonies (vrelcolony). All runs:

500 foragers were simulated

with different numbers of

receiver bees. N = 6 per set-up

(a) (b)

(c) (d)

Fig. 13 Investigations of the

net honey gain of the colony

which results from the foraging

patterns depicted in Fig. 9.

a Mean accumulated net honey

gain of the colony after 8 h

[ml]. b Mean number of

foragers active in each

simulation set-up (values of fdepicted in Fig. 9). c Mean net

honey gain per active forager in

all experimental set-ups

[ml/bee]. d Mean net honey

gain per worker bee in all

experimental set-ups [ml/bee].

All runs: 500 foragers were

simulated with different

numbers of receiver bees. N = 6

per set-up


123

behavioural rules in our agents (which represent bees), and

we compared the emerging ultimate results to empiric data.

Using the same simulator and having the same parameter

settings, our results in the experiments of equilibrium

foraging, of cross-inhibition between groups of foragers

and in the source-selection experiment match published

empiric data closely [3, 4, 13, 28] (empiric data not shown

here).

Several models of honeybee foraging have been pub-

lished. Most of them took a top–down approach and

describe the rate of changes of forager groups on different

sources by a linked set of differential equations [1, 3]. Only

few individual-based simulations of the honeybee foraging

system exist. The studies of [7, 8, 31]) investigated the

forager–receiver interactions in an individual-based model,

but did not model any nectar sources or complex behav-

ioural programs of foragers and receivers, like we did it in

our model (see Fig. 3). Other studies investigated the

occurrence of multiple unloading events in forager-receiver

interactions, without dealing with other aspects of honey-

bee foraging behaviour [32].

Two studies of de Vries and Biesmeijer [5, 6] describe a

model, which has many similarities to our model presented

here, but which differs in significant aspects that are crucial

for the analyses presented here. Our model implements the

agents’ metabolic costs in more detail, e.g. by considering

the bee’s weight and task. It models the energetic expen-

ditures of all bees, also from those bees that are ‘idle’ in the

hive. In our model, receivers have to meet forager bees

locally and dances have a limited spreading area. Our

dance–response curves differ significantly form the ones

published in de Vries and Biesmeijer [5], as can be seen in

Eqs. 4 and 5, thus our curves (Fig. 5) resemble the data

reported by Seeley [13] statistically more closely. We

implemented the dynamics of the honey stores in detail and

also the focal aspects differ significantly. Our model

focuses on the economics of foraging decisions. One sig-

nificant result of our model is the fact that the second

foraging decision is not made by the colony under several

conditions (Fig. 9). Such predictions were not made by de

Vries and Biesmeijer [5]. Without implementing the bees’

metabolism and morphological features, it is impossible to

predict an energy balance of the modeled colony, like we

did it extensively in our study.

The model ‘TaskSelSim’ described in Schmickl et al.

[33–35] has also implemented the metabolic cost of for-

aging bees and receiver bees, but it is focused on emergent

division of labour and therefore does not allow choices

between different nectar sources. An individual-based

model that implements two nectar sources and metabolism

of agents is ‘FlowerSim’ described in Thenius et al. [36].

This model abstracts the waggle dances to a rough

approximation and implements the nectar sources as a field

of many small blossoms that have to be visited multiple

times by forager bees. While this model reflects the for-

aging behaviour of bees accurately, it is not comparable to

the environmental situations investigated by Seeley [28],

who used unlimited flow feeders as artificial food sources

in empiric experiments. As we wanted to compare our

predicted foraging decisions with these data provided by

Seeley [28], we had to elaborate our model ‘HoFoSim’

[22–24] to the current model ‘HoFoReSim’, which extends

the original model mainly by its additional implementation

of receiver bees and their metabolic costs.

In experiment 1, where we searched the sugar concen-

tration for the near nectar source to achieve equilibrium

foraging, we matched the published results of empiric

experiments very closely. The fact that we always found

more bees actively foraging on the nearer source in times

when the number of bees recruited to the two sources was

in a ratio of 1:1 can be explained by the communication

error associated with the waggle dancing. These waggle

dances do not communicate coordinates of the food

source, they just encode the direction and the distance to

this source. Therefore, the error in communication mis-

guides the bees on trips to the far source more than on trips

to the near source. This leads to longer searching flights

around the far nectar source and to a lower probability of

finding the source, compared to the near source. Longer

searching flights increase the costs, what leads to the small

left-shift of the lower histogram in Fig. 1c. The higher

probability of missing the source leads to the lowered bee-

counting on the far nectar source (Fig. 6b). The fact that

honeybees are able to forage simultaneously on several

foraging targets, while in parallel scaling the forager

groups on each nectar source according to its quality, is an

interesting phenomenon. It shows, in contrast to many ant

species, that the collective foraging does not quickly

converge into just one single solution, thus the swarm

intelligence of honeybees can be described to be highly

flexible. A quick reaction to environmental changes is only

possible when the colony frequently observes and com-

pares many available solutions.

In experiment 2, we found cross-inhibition in all of our

simulation scenarios. Despite that, Fig. 8 shows that the

receiver-to-forager ratio simultaneously modulates the total

growth capacities of the two groups of foragers. The cross-

inhibitory effect is important for social insect foraging

decisions, as it is a measure of the colony’s ability to reflect

changes of environmental conditions in its internal infor-

mation processing. The parallel exploitation (shown in

experiment 1) ensures that information from various

sources flows into the colony in parallel. Only the cross-

inhibitory effect allows the foragers to react to quality

changes of food sources, which they never have visited

themselves.


123

The choice experiments which we performed in exper-

iment 3 tested whether or not the phenomena demonstrated

in experiments 1 and 2 really lead to a quick and adaptive

foraging decision in a changing environment. Our experi-

ments show the huge influence of the receiver-to-forager

ratio onto the foraging patterns that emerge in fluctuating

environments (see Figs. 9, 10). With low receiver-to-for-

ager ratios, the number of foragers that exploit the wors-

ened source in the afternoon always decreases, due to the

cross-inhibition effect shown in experiment 2. The rate of

this decrease is higher, the higher the amplitude of the

environmental fluctuation is. With high receiver-to-forager

ratios, the colony reduces the number of foragers on the

worsened source only if this worsening is strong. With low

fluctuation amplitudes, the colony even increases the

number of foragers on the worsened source and addition-

ally intensifies the foraging on the improved source.

By looking at the net honey gain of the colony in the

runs of experiment 3, we conclude that a low number of

receivers (and thus a strong cross-inhibitory effect) does

not lead to a better nectar income compared to colonies

with high numbers of receivers for the same number of

forager bees. This result was expected, as bigger colonies

naturally have more active foragers and thus more nectar

income per time unit. We found that colonies with low

receiver-to-forager ratios switch quicker from one nectar

source to another after an environmental fluctuation. This

sharpening of collective foraging decisions leads to a

decrease in energetic costs caused by environmental fluc-

tuations compared to runs of a similar sized colony in

undisturbed environmental conditions. In conclusion, we

can say that colonies with low ratios of receivers to for-

agers (0.5:1) react faster and are more picky in their for-

aging decisions. This allows such colonies to make a better

deal out of the environmental situation they are confronted

with. More receiver bees allow more nectar accumulation,

but additional receiver bees increase also the metabolic

expenditures of the colony. In our simulation runs, we

found an optimum with a 1:1 ratio of receivers to foragers

when we measured the per bee efficiency of the nectar

accumulation. As was shown by Seeley [28], this optimal

ratio depends on the duration of the average foraging trip of

forager bees and on the average duration of storage trips by

receiver bees.

As shown in Fig. 14, the availability of receivers for

foragers affects the emerging foraging equilibria that arise

on both nectar sources in multiple ways. More receiver bees

per forager favour recruitment by waggle dances, as queuing

delays are short (see Fig. 4). In addition, as waggle dancers

tend to fly again to the source they danced for, abandonment

rates in forager groups decrease in parallel. The combination

of these effects favours bigger forager groups (more active

foragers, as expressed by f) but keeps foragers longer in

their foraging loop. This slows down recruitment to alter-

native foraging targets in response to environmental fluc-

tuations. This is shown in Figs. 9 and 10.

In conclusion, we can summarize the answers to our

previously formulated hypotheses:

H1: Our model predicts that the receiver-to-forager ratio

affects foraging decisions. A high ratio leads to more active

foragers, but to a slow reacting colony after the first for-

aging decision is settled. Cross-inhibition is affected by

low receiver-to-forager ratio and equal foraging distribu-

tion (EFD) is hard to find with a low number of receivers,

thus the system is more likely to show symmetry-breaking

with two food sources of equal quality.

H2: Our model predicted that stronger environmental

fluctuations enhance the speed of colony-level reactions in

the forager distribution, especially with low numbers of

receivers per forager.

H3: All simulated environmental fluctuations caused

colony-level costs (b \ a, see Fig. 11, also shown in a

summarized view in Fig. 12). Low receiver-to-forager

ratios prevent a significant fraction of these potential costs,

but keep harvests small, even in undisturbed runs. An

excess number of receivers causes an excess of metabolic

costs, thus it is lowering the colony’s harvesting efficiency.

We found an optimal receiver-to-forager ratio at 1:1.

Fig. 14 Receivers affect the emerging foraging distributions on two

nectar sources in multiple ways. More receivers per foragers favour

recruitment by waggle dances, as queuing delays are short (see

Fig. 4). In addition, as waggle dancers tend to fly again to that source,

abandonment rates in forager groups decrease. This favours bigger

forager groups (more active foragers) but keeps foragers longer in

their foraging loop, thus slows down recruitment to other foraging

targets in response to environmental fluctuations


123

Our simulation study generated testable predictions for

honeybee scientists and investigated the role of nectar

receivers concerning their effect on colony-level foraging

strategies and nectar economics. Despite a better under-

standing of the biological system, this helps to derive novel

bio-inspired swarm-intelligent algorithms based on hon-

eybee foraging, as they were already proposed [37–40].

Comparable to the honeybee foraging case, ant foraging

has significantly inspired the field of swarm-intelligent

optimisation and routing algorithms [41–43] and swarm

robotics (e.g., [44–47]). Even simple aggregation behav-

iours, like aggregation of cockroaches, has influenced

swarm robotics [48]. Also in the field of swarm robotics,

foraging for energy is a topic of interest in recent years. For

honeybee colonies, nectar is a source (and also a storage

form) of energy, thus we think that the insights we got by

studying the economic gains and costs of honeybee for-

aging will be useful also in the domain of swarm intelli-

gence and swarm robotics.

Recently, we translated several behavioural patterns

found in honeybees to swarm-robotic algorithms: temper-

ature-induced aggregation of young honeybees [49, 50] and

honeybee trophallactic exchanges [51–53]. By further

investigating the key aspects and feedbacks in the regula-

tion of honeybees’ society and collective behaviours, we

plan to elaborate these algorithms and control programs.

Such investigations are performed by empiric experiments

and by model analyses, like we present it in this article.

Acknowledgments This article was supported by the following

grants: EU IST-FET-open project (IP) I-Swarm, no. 507006. EU IST-

FET project SYMBRION, no. 216342. EU ICT project REPLICA-

TOR, no. 216240. Austrian Science Fund (FWF) research grants

P15961-B06 and P19478-B16.

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