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Brood comb construction by the stingless bees
Tetragonula hockingsi and Tetragonula carbonaria
Rute M. Brito ·Timothy M. Schaerf ·
Mary R. Myerscough ·Tim A. Heard ·
Benjamin P. Oldroyd
Received: 1 June 2011 / Accepted: 31 March 2012
© Springer Science + Business Media, LLC 2012
Abstract Tetragonula hockingsi and T. carbonaria are two closely related species of Aus-
tralian stingless bees. The primary species-speciﬁc character is the architecture of the brood
comb. The brood comb of T. hockingsi is an open lattice comprising clumps of about ten
cells that are connected by vertical pillars. In contrast, in T. carbonaria the brood comb is
a compact spiral in which all brood cells (except on the margins) are connected by their
walls to adjacent cells at the same height. We made detailed observations of the cell con-
struction process in two colonies of each species. From these observations we formed a
species-speciﬁc hypothesis about the algorithm followed by the bees during cell construc-
tion. The two algorithms allowed us to make predictions about the locations of new cells.
Both T. hockingsi and T. carbonaria share a preference for constructing new brood cells
in the clefts formed by two or three adjacent existing brood cells, but there are differences
in detail for other components of the building process. The fundamental difference in the
cell construction process of the two species is that for T. hockingsi, when a cluster of cells
contains ten cells, the next cell added to the cluster is offset upwards by half a cell length,
Instituto de Genética e Bioquímica, Universidade Federal de Uberlândia, Av. Pará 1720, 2E sala 34,
CEP 38400-902, Uberlândia, MG, Brazil
T.M. Schaerf ()·B.P. Oldroyd
Behaviour and Genetics of Social Insects Laboratory, School of Biological Sciences,
University of Sydney, Macleay Building A12, Sydney, NSW 2006, Australia
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
CSIRO Ecosystem Sciences, Ecosciences Precinct, GPO Box 2583, Brisbane 4001, Australia
or, less often, a vertical pillar rather than a new cell is constructed. In T. carbonaria, cell
construction is continuous at the comb margin so that there are no gaps between cells. Fur-
thermore, it seems that T. hockingsi only makes use of local knowledge of the brood comb
when deciding to place new brood cells, whereas T. carbonaria could make some building
decisions based on knowledge of the total structure. We translated the species-speciﬁc algo-
rithms into agent-based lattice swarm computer simulations of the cell construction process
for the two species. These simulations produced representations of brood combs that are
similar to those seen in vivo, suggesting that our biological rules are realistic.
Keywords Brood cell ·Nest construction ·Stigmergic algorithm
The nests of social insects are often highly elaborate, complex and, to our eyes, beautiful
structures. The construction of the nest involves the activities of many, sometimes thousands
of individuals. For example, a particular worker honey bee’s lifetime contribution to nest
construction might be to work on 3–4 different cells of a comb, on different days, for a single
week (Seeley 1982; Seeley and Kolmes 1991). It is thought that in many cases individual
workers have no concept of what the overall nest architecture should be, or how to achieve
it (Grassé 1959;Karsai1999; Theraulaz and Bonabeau 1999; Camazine et al. 2001; Garnier
et al. 2007). How then are the activities of hundreds of cell-building workers coordinated so
that a symmetrical comb is constructed?
Instead of workers building to a ‘blueprint’ that is hardwired into their neurology, it
is thought that the overall nest structure emerges from the construction rules followed by
individual workers as they work on the modular units that make up the overall structure
(Camazine et al. 2001). Compellingly realistic simulations of real-life nest structures can be
generated using agent-based models that follow simple nest-building rules in silico (Ther-
aulaz and Bonabeau 1995a; Karsai and Pénzes 1998,2000). For example, paper-wasp-like
combs can emerge from a simple algorithm that speciﬁes that if two joined cells are present,
the next cell is built between them, and if three joined cells are present, the next cell should
be built in the cleft created by the three (Camazine et al. 2001).
Despite the pleasing similarity between in silico structures and real life combs, simu-
lations of a comb construction process do not (and cannot) ‘prove’ that real-life workers
use the same nest building rules as are speciﬁed in a computer algorithm (Camazine et
al. 2001). Nonetheless, support for the idea that insect workers follow simple ‘stigmergic’
(Grassé 1959) rules to produce structures like combs would be increased if we could ob-
serve nest construction in vivo, derive the rules followed by individual workers as they build
cells, make precise predictions about what the workers would do next based on our concep-
tion of the rules that they follow, and then observe the workers complete the work that we
had predicted. The veracity of our hypothesis would be further improved if we could study
two related species with different comb structures, discern the different rules followed by
workers during the comb construction process, make testable predictions about where cells
would be built based on the rules thus derived, and then produce lifelike models of the two
different comb structures in silico by specifying the species-speciﬁc rules in two versions
of an algorithm. This is what we have attempted to do here with two species of Australian
stingless bee, Tetragonula hockingsi and T. carbonaria.
Tetragonula Moure, previously considered to be a subgenus of Trigona (s.l.), is a
cosmopolitan genus of Asian and Australian stingless bees. In Australia, it is repre-
sented by 6 species: T. carbonaria Smith, T. hockingsi Cockerell, T. mellipes Friese,
T. davenporti Franck, T. clypearis Friese, and T. sapiens Cockerell (Dollin et al. 1997;
Franck et al. 2004; Rasmussen 2008). T. carbonaria,T. hockingsi,T. mellipes,andT. dav-
enporti are collectively known as the monophyletic group ‘Carbonaria’ (Dollin et al. 1997;
Franck et al. 2004).
Workers of the different species within the Carbonaria species group are morphologically
similar, but can be readily distinguished based on DNA sequence data (Franck et al. 2004).
T. carbonaria has the broadest distribution and is found from Sydney to Cooktown while
T. hockingsi is distributed from Brisbane to the tip of Cape York (Dollin et al. 1997).
Stingless bees (Hymenoptera: Apidae: Meliponini) build brood cells within a nest in a
space called the brood chamber. They do not reuse brood cells like many other Hymenoptera
but constantly build new brood cells for the next cohort of young. These cells are destroyed
after the adult bees have emerged. When the new cells are constructed at a speciﬁc place,
this area is called the advancing front. The brood chamber is often surrounded by a multilay-
ered envelope called an involucrum (Roubik 2006). Although highly eusocial, stingless bees
produce brood in the manner of solitary bees, with an egg placed on top of a food mass in a
sealed cell. Hence all brood cells are sealed, except for those yet to complete the provision-
ing and ovipositioning process (Roubik 2006). Tetragonula carbonaria, for example, builds
approximately 5 batches of new cells each day with each batch consisting of approximately
80 new cells (Yamane et al. 1995).
Despite the morphological similarity of the workers, the nests of the various species of the
Carbonaria species complex differ strongly in the morphology of their brood combs (Fig. 1,
Michener 1961, Dollin et al. 1997). T. hockingsi and T. davenporti build what is probably
the ancestral nest morphology: a loose aggregation of cells or cell clusters with numerous
spaces between adjacent cells that are built more or less randomly with respect to the hor-
izontal plane, and are joined by vertical wax pillars (Fig. 1(a)). In contrast, T. carbonaria
builds combs with no spaces between the cells of the horizontal plane, in a continuous spiral
(Figs. 1(b)–(d)). This difference is so striking that it remains the primary diagnostic feature
for distinguishing T. carbonaria from the other species (Michener 1961; Dollin et al. 1997;
Franck et al. 2004).
Here we present detailed observations of the comb building process of two contrasting
species: the spiral comb of T. carbonaria, and the semi-comb of T. hockingsi (Fig. 1). By
time-lapse photography of the comb construction process (detailed in Sect. 2), we gener-
ated a hypothesis concerning the simple rules followed by comb building workers for each
species, and identiﬁed how these rules differed between the two species (summarized in
Sects. 3and 4). We then tested the ability of the rules derived from our observations to
make predictions about where new cells will be constructed (see Sect. 5). In conjunction
with these predictions, we further quantiﬁed the preferences of each species to build in the
presence of two common conﬁgurations of existing brood cells. Finally, we incorporated our
observations into realistic lattice swarm models (Theraulaz and Bonabeau 1995a,1995b)in
an attempt to reproduce the complex structures produced by the bees in real life and to ex-
plore if those structures could be reproduced using stigmergic rules. Details of our models
are provided in Sect. 6and results from our simulations appear in Sect. 7. In Sect. 8,wedis-
cuss the implications of our observations and simulation results and make some concluding
Fig. 1 Brood nest structure of the two Tetragonula species studied: (a) semi-comb structure of T. hockingsi
(plan view); (b) spiral shaped brood of T. carbonaria (plan view); (c) the three dimensional structure of a
T. carbonaria nest; (d) the upper portion of a T. carbonaria spiral stretched upwards
2 Observations on the cell construction process
Two colonies of each species, T. carbonaria and T. hockingsi, were obtained from suburban
Brisbane and transferred to Sydney for observations. The colonies were housed in standard
artiﬁcial nest boxes (Heard 1988; Heard and Dollin 2000) ﬁtted with a transparent Perspex
observation window. The colonies were placed in a controlled temperature room with access
to the outside via plastic tubing.
Like all stingless bee species, T. hockingsi and T. carbonaria construct brood combs as
part of the Provisioning and Oviposition Process, or POP. In the POP, workers ﬁrst construct
a new brood cell, and then provision the cell with brood food. The queen then lays in the
freshly provisioned cell, and the workers then cap it. The details of the POP are species-
speciﬁc, and have been described in detail for numerous species (reviewed in Drumond et
al. 2000), including T. carbonaria (Yamane et al. 1995), but not T. hockingsi.
For all colonies we observed comb construction by direct observations and through video
recording with a Digital Webcam Logitech 9800 Pro. Observations were made from January
to February 2009 on a daily basis. We observed multiple POP batches for one hive from
each species in the ﬁrst month, which we used to derive the rules for brood construction. All
observations were made on well established colonies.
By noting the locations of newly-constructed cells with respect to existing cells, we
formed hypotheses about the cell-construction process for T. carbonaria and T. hockingsi.
In particular, we inferred the local rules used by workers concerning the locations of new
After deriving the rules, we followed at least three POP batches for each of four colonies,
which occur at 4 to 5 hour intervals in both species. High quality images taken before and
after each POP were compared. Positions of potential sites for new cells before each POP,
and where new cells were actually built after each POP, were mapped. We counted the
number of potential building sites adjacent to the walls of two or three existing brood cells on
a fraction of the brood comb before each POP and determined the number of these building
sites that were ﬁlled after each POP. The average proportions of available two- and three-
wall sites ﬁlled during each POP batch were used as the cell-building probabilities within
3 The rules of construction of brood cells in T. hockingsi
Our observations of the building process of T. hockingsi are summarized below and in Fig. 2.
In Fig. 2, pre-existing cells are represented by light grey cylinders and newly-constructed
cells are drawn with a dark grey top. A ‘layer’ is a horizontal plane of clusters of cells.
A ‘pillar’ is a vertical wax structure built to support a brood layer and provides a connector
to allow movement of bees between layers. In spite of brood cells being cylindrical, bees
build a new cell using sides or angles of existing cells, as if they were hexagonal.
(i) A new layer is initiated near the centre of the brood chamber and/or on the perimeter.
(ii) If cells are to be constructed on a cluster where there is only one existing cell, or a
vertical pillar, a new cell is always constructed beside the existing one (Fig. 2(a)).
(iii) If on a layer there is a cluster of two adjacent cells and there are no three-cell cell-
clusters present on the layer, a new cell is constructed in the cleft formed between the
two adjacent cells (Fig. 2(b)).
(iv) If there are any three-cell clusters present on the layer, new cells are constructed in
the clefts formed by the existing cells (Fig. 2(c)).
(v) Construction proceeds following rules (ii) to (iv) until a cluster reaches about 10 cells.
(vi) If there are two 10-cell clusters one cell diameter apart, a horizontal connection is built
between the two 10-cell clusters. Immediately following construction of the connec-
tor, a new cell is constructed supported by the connector in the cleft formed by the
two nearest cells (Fig. 2(d)).
(vii) When there is a cluster of approximately 10 cells, the next new cell is constructed
50 % higher than the existing layer (Fig. 2(e)). This offset cell initiates the new layer
to be constructed above the existing layer. Cell construction proceeds on the, new,
higher layer following rules (i) to (vii).
(viii) If a 10-cell cluster is at the edge of the brood chamber, instead of building an offset
cell as in (vii) (Fig. 2(e)), a vertical pillar is constructed on top of the cell at the far
edge near the chamber wall.
(ix) If there is a vertical pillar at the edge of the brood chamber the next cell is constructed
adjacent to the pillar (Fig. 2(f)). Subsequent building on the new layer proceeds fol-
lowing rules (ii) to (vii).
(x) Cell clusters are constructed following rules (ii) to (ix) until layers on the edge of the
brood comb reach the top of the chamber.
Fig. 2 Diagrams of rules derived for brood construction by Tetragonula hockingsi. Pre-existing cells are
represented by light grey cylinders and newly-constructed cells are drawn with a dark grey top.Hexagons
represent cells viewed from above: (a) new cells built beside a single cell or beside a pillar; (b) new cell built
in the cleft formed by two pre-existing cells; (c) new cell built in between three pre-existing cells; (d)new
cell attached to a connector between two groups of cells; (e) new cell built at a higher level in a three-wall
cleft on the edge of a group of cells; (f) a new cell built beside a pillar previously built on top of a group of
4 The rules of construction of brood cells in T. carbonaria
Our observations of the building process of T. carbonaria are summarized below and in
Fig. 3 Diagrams of rules derived for brood construction by Tetragonula carbonaria. Pre-existing cells are
represented by light grey cylinders and newly constructed cells are drawn with a dark grey top.Hexagons
represent cells viewed from above: (a) new cells built beside a single cell; (b) new cell built in the cleft of two
pre-existing cells; (c) sequential building of new cells from a single previous cell, following the preference to
build next to two pre-existing walls in (b); (d) new cell built in between three pre-existing cells; (e) new cell
built at a higher level in two or three-wall clefts
(i) Cell construction is initiated from the centre of an existing layer.
(ii) The second cell is constructed adjacent to the existing cell (Fig. 3(a)).
(iii) The third cell is built in the cleft formed by the ﬁrst two cells (Fig. 3(b)).
(iv) New cells are added in the cleft of every two cells in clockwise or anti-clockwise
direction until a small layer of 7 cells is complete (Fig. 3(c)).
(v) If there are any three-cell clusters present on the layer, new cells are constructed in
the clefts formed by the existing cells (Fig. 3(d)).
(vi) The subsequent cells are built in the three cell or two cell clefts formed on the perime-
ter of the emerging comb. The spiral shape is achieved because each cell is built
slightly higher than the existing cell in either a clockwise or an anticlockwise direc-
(vii) Cell construction proceeds following rules (i) to (vi) until a layer reaches 2/3ofthe
chamber width. At this point a new layer is initiated by offsetting cells at the centre of
the comb vertically by 50 % (Fig. 3(e)).
(viii) Cell construction stops on a layer when it reaches the chamber wall.
Fig. 4 Brood comb of the Tetragonula species studied showing the predicted sites for new cell construction
in the clefts formed between existing cells (yellow circles), sites where cells were actually constructed (green
circles) and the sites where we incorrectly predicted a cell would or would not appear (red circles)before
and after the provisioning and oviposition process: (a)T. hockingsi before the POP; (b)T. hockingsi after the
POP; (c)T. carbonaria before the POP; (d)T. carbonaria after the POP (Color ﬁgure online)
5 Testing the predictions of our rules
Using the hypothesized building rules derived from our observations in Sects. 3and 4,we
mapped the predicted locations of new cells on still images from video of the brood comb of
two colonies from both species obtained during three inter-POP periods. We then compared
our predictions against images taken after the next POP.
As an example, Fig. 4(a) shows a comb of T. hockingsi before a POP was initiated. The
predicted locations for new cells, based on the rules above, are indicated by the yellow cir-
cles. In Fig. 4(b), we show the actual locations of newly constructed cells following the POP
(green circles). Similarly, we show the predicted (Fig. 4(c)) and observed (Fig. 4(d)) loca-
tions of new cells for T. carbonaria. For instance, we predicted the positions of 23 new cells
for T. carbonaria during one POP period. Of these predicted sites, the bees completed 17
during the next batch of POP (Fig. 4(d)). Thus we incorrectly predicted the location of 6 new
cells. However, it is important to highlight that bees completed 20 cells after another POP
batch, including many of those that we had predicted. Our overall success in predicting the
sites of new cells during a single POP was 90 % for T. carbonaria and 65 % for T. hockingsi.
Tab l e 1 The number of available and used building sites bounded by two or three existing cells
Available Used (*) Proportion used
2 cells 3 cells 2 cells 3 cells 2 cells 3 cells
Colony 1–POP 1 23 11 3 4 0.130 0.364
Colony 1–POP 2 38 24 6 +213+10.211 0.583
Colony 1–POP 3 36 21 9 12 0.250 0.571
Colony 2–POP 1 46 19 8 +211+50.217 0.842
Colony 2–POP 2 44 23 11 +113+40.273 0.739
Colony 2–POP 3 53 14 9 8 +30.259 0.786
Mean 0.223 0.648
Colony 1–POP 1 39 22 13 19 +20.333 0.955
Colony 1–POP 2 41 15 11 +18+40.293 0.800
Colony 1–POP 3 23 12 6 6 +30.261 0.750
Colony 2–POP 1 17 23 3 22 +10.118 1.000
Colony 2–POP 2 33 13 7 11 0.212 0.846
Colony 2–POP 3 28 15 9 15 0.321 1.000
Mean 0.256 0.892
(*) numbers to the right of the plus sign indicate partially complete cells
Tabl e 1shows the breakdown of our observations over six POP periods for both species,
listing the number of available building sites adjacent to two or three existing cells and
the number of those sites subsequently built in by the end of each POP period. The mean
proportion of sites bounded by two or three cells that were ﬁlled during a POP period were
translated to building probabilities for use in our model (described in detail in Sect. 6.3).
6 Standard stigmergic lattice swarm model and extensions
We used the stigmergic lattice swarm model developed by Theraulaz and Bonabeau
(1995a,1995b) for simulating the building of complex nest architectures. The purpose of
using a standard stigmergic lattice swarm model was twofold. First, we sought to determine
if the local rules of construction that were observed for T. hockingsi and T. carbonaria were
sufﬁcient to reproduce structures through the use of simulation that were similar to real
nests. Second, the results of our simulations add to observational evidence as to whether
realistic nests could be constructed reliably either with or without some global knowledge
of the overall structure. The building rules for T. hockingsi seemed to be based on purely
local knowledge of small parts of the larger structure. Implicit in some of the building
rules for T. carbonaria was apparent knowledge of the larger structure at a more global
level, particularly the ability to identify the central part of the comb when building up-
Throughout our experimentation with the lattice swarm model, we found that our at-
tempts to reproduce the complex architecture of a T. carbonaria brood comb were hampered
by some of the model’s restrictions, particularly on the vertical placement of brood cells. To
address this limitation, we modiﬁed the standard lattice swarm model to allow for building
of brood cells at any vertical position, rather than only allowing building at a set of discrete
vertical positions. The necessary modiﬁcations to the model are discussed in Sect. 6.3 and
Appendix A. We adopt a modiﬁed version of the ODD protocol (Grimm et al. 2006)inthe
description of our implementation of the model of Theraulaz and Bonabeau (1995a,1995b).
6.1 Model overview
Details of the lattice swarm model are complex, but the ideas behind the model are straight-
forward. A number of construction agents representing comb-building workers are randomly
distributed throughout a three-dimensional domain. The domain comprises discrete ﬂat hor-
izontal layers of ﬁnite thickness. Each layer is made up of a lattice of regular hexagonal
prisms; we refer to these hexagonal prisms as elements. Each agent examines its local sur-
roundings, deﬁned as the set of elements that share an edge with the element where an
agent is currently located, including elements in the layers immediately above and below an
agent. All the agents then compare their local surroundings with a list of local building rules
referred to as stimuli-to-build. If the local surroundings match a stimulus-to-build identi-
cally, then an agent will construct a new cell at its current location with a given probability.
Once any building action takes place, agents are randomly re-distributed throughout the do-
main at the beginning of the next time step before they examine their local surroundings
We now elaborate on our implementation of the above model.
Details of each horizontal layer of the domain were stored as regular, rectangular matri-
ces. The numerical values of each of the elements of the matrices represented the presence
or absence of building structures. Elements representing individual brood comb cells were
set to 1, elements representing vertical wax pillars were set to 2, and elements representing
empty space were set to 0. To construct the hexagonal geometry, where each element has 6
adjacent elements, we offset every even numbered row of the matrix, as illustrated in Fig. 5.
Generally, the indices of the 6 elements adjacent to the element in row jcolumn iwere (row,
column): (j −1,i−1),(j −1,i),(j , i +1),(j +1,i),(j +1,i −1)and (j , i −1)when j
was odd, and (j −1,i),(j −1,i+1),(j , i +1),(j +1,i+1),(j +1,i) and (j , i −1)when
jwas even. There were 13 special cases where the element in row j, column i,hadfewer
than six nearest neighbors. These degenerate cases occurred when the element of interest
was located either in one of the corners of the hexagonal array, or anywhere in the top row,
or anywhere in the bottom row (different rules needed to be applied if the bottom row was
oddly or evenly numbered), or in the ﬁrst column (dependent on if the element was in an
odd or even row) or in the last column (again dependent on if the element was in an odd or
Construction agents were characterized by the row, j, column, i, and layer, z,wherethey
were located at a particular time.
The model uses equally spaced, discrete time steps. At the beginning of each time
step, all construction agents were each moved simultaneously to a random location in
the domain. For T. hockingsi and T. carbonaria simulations, each of the row, column
and layer coordinates that each agent was moved to were determined by selecting a
uniformly distributed random number between 0.5 and ns+0.5(wherenswas the to-
tal number of rows, columns or layers in the domain, depending on the coordinate be-
ing calculated), and then rounding the random number to the nearest integer. The ran-
dom distribution of agents throughout the domain differed slightly from the standard ap-
proach used by Theraulaz and Bonabeau (1995a,1995b), where agents were moved to
Fig. 5 A matrix can be used to store elements that are interpreted as being part of a hexagonal lattice. In the
left hand matrix, which is drawn in the usual rectangular format, the element represented by the ﬁlled circle
(•) has eight nearest neighbors, illustrated by the square () symbols. If even numbered rows of the matrix
are treated as being offset by a half-step to the right, as in the right hand matrix, then the element represented
by the ﬁlled circle only has six nearest neighbors, represented by diamonds (♦), which is equivalent to the
cells of a hexagonal lattice. The indices of the nearest neighbors to a particular element of the hexagonal array
are a function of whether the row containing the cell is odd or even numbered, as detailed in the main text
a random element adjacent to their previous location at the beginning of each time step.
There is considerable time lag between the construction of sets of cells for T. hockingsi
and T. carbonaria (of the order of 4 to 5 hours), and it would not be realistic for our
agents to only move a very small distance over that time frame. Instead, our agents es-
sentially sampled a random portion of the domain for potential building sites at each time
After all the agents had moved, they examined their local environments to determine
the conﬁguration and type of structures nearby. The local environment was deﬁned as the
set of elements that shared an edge with an agent located in row j, column i, layer z.The
local environment of an agent consisted of cells in the agent’s current layer (z), the layer
immediately bellow the agent (layer z−1) and the layer immediately above the agent (layer
z+1). Each agent would then compare its surroundings to each item on a list of stimuli-to-
build. Graphical representations of the stimuli-to-build available to our agents are provided
in Figs. 6and 7. For each stimulus-to-build the focal agent is located in the central cell of
layer z(marked with a cross). Empty cells are illustrated in white, cells with brood comb
present are ﬁlled in gray, and cells containing components of vertical wax pillars are ﬁlled
in black. Each row of the diagrams in Figs. 6and 7represents a different stimulus-to-build.
The leftmost column uniquely identiﬁes each stimulus for each species with a number of the
form n.m,wherenis the type of cell to be constructed at the agent’s current location and mis
the mth stimulus for construction of cell type n. Stimuli for building brood comb have n=1,
stimuli for building vertical pillars have n=2. The numerical identiﬁers for each stimulus-
to-build are preceded by either an h, for stimuli for T. hockingsi,orac,forT. carbonaria.
Probabilities of building in the presence of each stimulus are denoted by pbuild.Wherever
possible, these probabilities were determined empirically from our observations, or inferred
using a combination of the empirically-determined probabilities and complementary obser-
vations (detailed in Sect. 6.3). Once all agents had examined their surroundings and it was
determined if they were to build at their current location (row j, column i, layer z), all
new cells were constructed simultaneously. The building of new cells concluded each time
6.2 Additional implementation details
In practice, comparisons between an agent’s local environment and stimuli-to-build were
made numerically rather than visually (see, for example, Pilat 2004). The ﬁrst step in making
Fig. 6 Graphical representation of stimuli-to-build for T. hockingsi.Each row of the diagram corresponds
to a different stimulus-to-build. Columns,from left to right, correspond to the layer below the agent (z−1),
the layer containing the agent (z), and the layer above the agent (z+1). The agent is always located in the
central cell of layer z. Numbering to the left of each stimulus-to-build is of the form n.m where nindicates
the type of block to be deposited by the agent in the central cell of layer zand mis a reference number
used to identify each stimulus for each block type uniquely. n=1 corresponds to rules where brood cells
are to be constructed. n=2 corresponds to construction of wax pillars. Existing brood cells are illustrated in
gray, existing wax pillars are illustrated in black, and empty cells are white. Probabilities of depositing a new
block in the presence of a stimulus-to-build are given to the right of each stimulus. The location of the agent
examining its surroundings is marked with a cross. Stimuli-to-build h2.1 and h2.2 relate to the construction
of wax pillars near the involucrum and are only applied within four rows or four columns of the edge of the
the comparison was to ‘unwind’ the elements surrounding an agent’s location to construct
a row vector representation of the local environment in a particular layer; see Fig. 8.Foran
agent located in layer z, row vectors representing the surrounding cells in layers z−1, z
and z+1 were required. A ‘surroundings’ matrix was then constructed using all three row
vectors, with the row vector representing layer z−1 becoming the ﬁrst row of the matrix, the
row vector representing layer zbecoming the second row and the row vector representing
surroundings in layer z+1 becoming the third row. Within our programme, stimuli-to-build
were stored in an identical format to a surroundings matrix.
Our observations indicated that building preferences of the real bees were based on the
number of adjacent cell walls and not on building on a particular side of existing cells. There-
fore, all six rotations about the central cell of each listed building stimulus were included as
All construction rules were applied stochastically. Whenever an agent encountered a rec-
ognized stimulus-to-build, a random number uniformly distributed between 0 and 1 would
be generated. If the random number was less than the probability of building when faced
with such a conﬁguration, pbuild, then the appropriate structure would be placed in the cell
currently occupied by the agent.
Fig. 7 Graphical representations of stimuli-to-build for T. carbonaria used by agents in the standard lattice
swarm model. Stimulus-to-build c1.6 can only be applied in one of the central 4 cells of a given layer. Refer
to Fig. 6for further explanation of this diagram
6.3 Parameter estimation and initial conditions
T. hockingsi and T. carbonaria colonies consist of approximately 10,000 workers of which
a small fraction are involved in cell construction. For our simulations we assumed that 2000
workers (agents) were involved in cell construction.
The standard artiﬁcial nest boxes where colonies were housed for observation had in-
ternal dimensions of approximately 150 mm ×230 mm ×200 mm. The median diameter
and height of brood cells for T. hockingsi are 3.35 mm and 4.0 mm, respectively (Dollin
Fig. 8 The elements surrounding and including the central elements, g,nand u, in three consecutive layers,
z−1, zand z+1, are re-written as row vectors. The row vectors are then combined to form a surroundings
matrix. The surroundings matrix is used to compare an agent’s surroundings with each stimulus-to-build,
each of which is stored in the same format as the surroundings matrix
et al. 1997). The median diameter and height of brood cells for T. carbonaria are 2.5 mm
and 3.5 mm, respectively. If we assume that a layer of cells is constructed so that cells are
packed tightly in approximately the same conﬁguration as a hexagonal lattice, then a sin-
gle horizontal layer in the observation box would comprise an approximate maximum of
44 ×78 cells for T. hockingsi and 60 ×105 cells for T. carbonaria. Approximately 50 lay-
ers of T. hockingsi brood cells and 57 layers of T. carbonaria cells could be packed tightly
into one of the observation boxes. Brood comb usually only covers approximately one third
of the available ﬂoor space for both species (personal observations). In terms of number of
cells, one third of the ﬂoor space would be covered by approximately 34 ×34 brood cells
for T. hockingsi and 46 ×46 brood cells for T. carbonaria. Therefore, we chose domains of
simulation of size 34 ×34 ×50 elements for T. hockingsi and 46 ×46 ×57 elements for
We used the observed mean probabilities of building next to two or three existing walls
(see Table 1) as the basis for determining the values of pbuild associated with each stimulus-
to-build for both species. Stimuli-to-build for T. hockingsi and T. carbonaria are illustrated
in Figs. 6and 7; see Sect. 6.1 for details of the naming convention for all stimuli.
For T. hockingsi we set pbuild =0.223 for stimulus h1.2 (the stimulus derived from the
real bees’ tendency to build in the cleft of two adjacent, existing cells) and pbuild =0.648 for
stimulus h1.3 (derived from the tendency to build in the cleft of three adjacent cells). T. hock-
ingsi tends to build clusters of approximately ten cells. Stimuli h1.4 and h1.5 represented
the starting point for new clusters of cells initiated by building upwards in the cleft formed
by two or three adjacent cells, respectively. We set pbuild =0.0223 and pbuild =0.0648 for
stimuli h1.4 and h1.5, respectively; these probabilities are 1/10th the probability of building
in the same layer next to two or three existing brood cells. We set the probabilities of form-
ing a connector to start a new cluster in the same layer (stimulus h1.1), connecting existing
clusters (stimulus h1.6) or starting a new cluster connected to a wax pillar (stimulus h1.7)
as one tenth the sum of the probabilities of building next to either two or three existing cells
(i.e. pbuild =0.0871). Stimuli h1.8 and h1.9 were an extension to the process illustrated in
Fig. 2(f), where a second brood cell is placed next to a wax pillar and an existing brood cell.
We set pbuild =0.223 for stimuli h1.8 and h1.9 due to their close association with stimulus
h1.2. We had no quantiﬁed observational data for the frequency at which wax pillars were
built on top of brood cells, so we approximated pbuild as 0.1 for stimuli h2.1 and h2.2, close
to the probabilities associated with starting a new cluster of cells. In reality, wax pillars are
most commonly constructed near the involucrum (at the edges of the domain for simulation
purposes), so stimuli-to-build h2.1 and h2.2 were restricted to working within four rows or
columns of the edge of the domain.
For T. carbonaria we set pbuild =0.256 for stimulus c1.1 (the stimulus for building in a
cleft formed by two adjacent existing cells) and pbuild =0.892 for stimulus c1.2 (the stimu-
lus for building in the cleft formed by three adjacent cells). We used observation (viii) from
Sect. 4as the basis for stimuli to build upwards. First, we allowed for a central vertical con-
necting cell to be placed on any of the four brood cells located at the centre of a horizontal
layer (stimulus c1.6). Our observations indicated that T. carbonaria tend to build upwards
from the centre when a circular layer of cells ﬁlls approximately two thirds of the width
of the domain. In our simulations, such a circular layer would comprise a little over 1000
brood cells. To match observations, we regulated the tendency to build upward by setting
pbuild =0 for stimulus c1.6 initially, and then increasing pbuild by increments of 1/1000 for
every new cell that was constructed. Each time a central connector was built using stimu-
lus c1.6, we reset pbuild to zero for this stimulus. Stimuli c1.7 and c1.8 relate to initiating
building in the layer above the central connector following Fig. 3(a); we set pbuild =1for
both these stimuli. pbuild was set to 0.256 for stimuli c1.9 and c1.10, and to 0.892 for stim-
uli c1.11, c1.12 and c1.13, which are rules for building adjacent to existing two and three
cell groupings close to the central connector. During preliminary calculations, we employed
stimuli-to-build based on Fig. 3(e) (and identical to stimuli h1.4 and h1.5), which reﬂect the
tendency of T. carbonaria bees to build upwards as they are building outwards. We found
that implementing the rules based on Fig. 3(e) resulted in structures that did not resemble
T. carbonaria brood combs. Two of the key reasons for the discrepancy between real and
simulated structures were that our agents built upwards irrespective of whether they were
building towards or away from the centre of the structure, and that the large discrete jumps
in building height imposed by the model did not reﬂect the subtle incline that T. carbonaria
brood combs possess. We abandoned stimuli based on Fig. 3(e) when using the standard lat-
tice swarm model, but revisited them when we modiﬁed the model to allow for continuous
variation in building heights (see Appendix A). Throughout the preliminary calculations, we
also tested whether or not it was necessary to conﬁne the vertical connector to the centre of
the domain; we found that it was necessary to do so as in some instances multiple connectors
could be built with a gap of a single element between them, and close building of connec-
tors would hinder building on the layer above because the local surroundings did not match
a stimulus to build. Another difﬁculty in reproducing the brood comb of T. carbonaria was
that we found that ﬁssures were forming in each of the layers of the comb. We explored
this problem in detail, see Appendix B, and ultimately determined that it was necessary to
add some stimuli that ﬁll in any gaps that could not be dealt with by stimuli c1.1 and c1.2.
The additional gap-ﬁlling rules are listed as stimuli c1.3 to c1.5 and stimuli c1.14 to c1.23.
pbuild =1 for all gap-ﬁlling stimuli. It is rare that even the early stages of ﬁssures are evi-
dent in real T. carbonaria nests. Interestingly, after performing the calculations described in
Appendix B, we noticed that gaps such as those dealt with by stimulus c1.3 also appear in
real T. carbonaria combs, but they are quickly ﬁlled.
Tab l e 2 Correspondence between the observations of Sects. 3and 4and stimuli-to-build that appear in
Figs. 6and 7
Species Observation Stimulus-to-build Notes
T. hockingsi (ii) (Fig. 2(a)) h1.1
(iii) (Fig. 2(b)) h1.2
(iv) (Fig. 2(c)) h1.3
(v) – Implicit in probabilities associated
with starting new clusters.
(vi) (Fig. 2(d)) h1.6
(vii) (Fig. 2(e)) h1.4, h1.5
(viii) h2.1, h2.2 Only used within four rows or
columns of the edge of the domain
(ix) (Fig. 2(f)) h1.7, h1.8, h1.9
(ii) (Fig. 3(a)) – Forms part of initial condition
(iii) (Fig. 3(b)) c1.1
(iv) (Fig. 3(c)) – Implicit in stimuli c1.1 and c1.2.
There is no control on building
clockwise or anti-clockwise. Basis
for initial condition.
(v) (Fig. 3(d)) c1.2
(vi) – No stimulus.
(vii) (Fig. 3(e)) – Implicit in the variable probability
associated with building upwards
for c1.6. Stimuli c1.7 to c1.23
model early stages of new layer
– c1.3, c1.4, c1.5 Rules included to ﬁll-in gaps in
the brood comb.
Tabl e 2summarizes the connections between the observations in Sects. 3and 4and the
stimuli-to-build in Figs. 6and 7for both species.
The initial building conditions for both species had the central element in the bottom layer
(layer 2 in practice) of the domain and the six elements in the bottom layer adjacent to the
central element ﬁlled with brood comb initially; all other potential building sites were empty
at time t=0 for each simulation. This initial condition was equivalent to the early stages
of a layer being formed for T. carbonaria (see Figs. 3(a)–(c)), or the partial completion of a
cluster of cells for T. hockingsi.
The T. carbonaria-like brood comb produced by a standard lattice swarm model lacked
some of the deﬁning features of a real brood comb. These deﬁning features included the
details of the spiral and the tendency to build incrementally upwards moving away from
the centre of the structure. Both of these features required more ﬂexibility in the vertical
placement of cells than is allowed by the discrete layers of a standard lattice swarm model.
We extended the standard lattice swarm model to allow for building at any height within the
domain to determine if we could more accurately reproduce the structure of a T. carbonaria
nest. We also modeled the tendency of T. carbonaria to preferentially build upwards when
Fig. 9 Brood comb constructed
by our model T. hockingsi agents.
Brood cells are represented by
red hexagonal prisms and wax
pillars constructed near the edge
of the domain are represented by
blue hexagonal prisms.The
domain of the simulation
consisted of 50 horizontal layers
of brood comb, each comprising
34 ×34 hexagonal cells. 2000
agents were involved in the
construction of the brood comb.
The illustrated structure
comprises 13134 brood cells and
706 wax pillar components
(Color ﬁgure online)
building in a particular sense (clockwise or anti-clockwise) to try the reproduce the conical-
helix seen in a real T. carbonaria nest. Extensions to the model are detailed in Appendix A.
7 Simulation results
The stigmergic lattice swarm model was capable of producing a complex nest architecture
that resembles the brood comb of T. hockingsi. Example output from the model is given by
a three dimensional perspective plot provided in Fig. 9, and cross-sections of some of the
layers of the comb are provided in Fig. 10. In both ﬁgures, brood comb cells are represented
by red hexagonal prisms, and wax pillars are illustrated as blue hexagonal prisms. In Fig. 9,
a light source (and shadows) have been included to better illustrate the three dimensional
structure. Distinct clusters of groups of approximately ten cells are visible throughout the
structure, similar to a real brood comb. At the edges of the brood comb, the structure has
adopted the shape of the domain boundaries. In reality, the involucrum surrounding T. hock-
ingsi’s brood comb is round, so real combs comprise circular, rather than square horizontal
cross-sections. The similarities between real and simulated brood combs where only local
rules for building were employed suggests that our observed building rules were realistic,
and it is plausible that a T. hockingsi brood comb is constructed via a similar stigmergic
The standard lattice swarm model with the addition of the non-stigmergic building rule
for the construction of a vertical connector at the centre of the domain (c1.6) was capable
of constructing a brood comb that bears many of the features of a real T. carbonaria comb.
A three dimensional plot of output from the model is provided in Fig. 11, with horizontal
Fig. 10 Plan view of selected horizontal layers of the brood comb in Fig. 9. The pictured layers correspond
to z=8, 16, 24, 32, 40 and 48 (Color ﬁgure online)
Fig. 11 Brood comb constructed
by our model T. carbonaria
agents using the standard lattice
swarm model. Brood cells are
represented by red hexagonal
prisms. The domain of the
simulation consisted of 57
horizontal layers of brood comb,
each comprising 46 ×46
hexagonal cells. 2000 agents
were involved in the construction
of the brood comb. The
illustrated structure comprises
31799 brood cells
Fig. 12 Plan view of selected horizontal layers of the brood comb in Fig. 11. The pictured layers correspond
to z=24, 28, 32, 36, 40 and 44
cross sections of the comb provided in Fig. 12. At the upper levels of the simulated comb
the circular layering that is evident in real combs is present. The squarer shape of the comb
in lower layers is due to the shape of the domain’s boundary.
The least convincing nest architecture came from simulations performed using the ex-
tended version of the lattice swarm model described in Appendix A, where building of cells
could occur at any height. Within this model, we attempted to replicate the spiral seen in
T. carbonaria’s brood comb by using an initial condition that would be conducive to spi-
ral formation, and making our agents build incrementally upwards when cells were built
upwards in a clockwise direction relative to the heights of existing brood cells. Figure 13
provides a plot of a structure produced by the extended lattice swarm model. The general
upward curve of brood cells moving outwards from the centre of the comb that is seen in
real T. carbonaria brood combs is evident, but there are ﬁssures in the comb in spite of
stimuli being used to ﬁll in any gaps. The particular simulation illustrated stalled because no
buildable sites could be found at the centre of the domain during the phase where cells were
to be built upwards from that region. Hence, many fewer cells are present than in the ﬁgures
for standard lattice swarm simulations of T. hockingsi and T. carbonaria.
Our behavioral observations allowed us to postulate a set of rules for the construction of
brood combs of two closely related Australian Tetragonula stingless bee species. Once de-
termined, these rules could be translated into stimuli-to-build for use in the stigmergic lattice
swarm model developed by Theraulaz and Bonabeau (1995a,1995b). The resulting simu-
Fig. 13 Brood comb constructed
by our model T. carbonaria
agents using the extended version
of the lattice swarm model that
allows for greater ﬂexibility in
vertical brood cell placement.
2000 agents were involved in the
construction process. The
illustrated structure comprises
3185 brood cells
lations produced representations of comb structures that are similar to the morphologies of
the natural combs of the two species.
Our simulations successfully generated the overall structure of a T. hockingsi brood nest:
the resulting structure was made up of many interconnected clusters of brood comb cells—
similar to those observed in nature. We had less success in reproducing a structure that
resembled a T. carbonaria brood nest. Over short distances, the cells were connected in a
continuous manner, similarly to the natural comb. At a larger scale, the simulations repli-
cated the layered appearance of a T. carbonaria brood comb, although the spiral structure
did not emerge. We sought to overcome some of the discrepancies between real and sim-
ulated T. carbonaria nests by making several extensions to the lattice swarm model. The
ﬁrst major extension removed the restriction that only allowed brood cells to be placed at
particular vertical heights. The other major extension made agents build upwards relative to
nearby cells in a clockwise direction. Even with these extensions, our simulation was unable
to reproduce the T. carbonaria spiral, although it did generate some additional features of
the natural brood comb such as the upward curving of the structure moving outwards from
Our behavioral observations show that there are two fundamental similarities in the
choice of site to build a new brood cell by both T. hockingsi and T. carbonaria. These sim-
ilarities are the preferences of both species to build adjacent to existing clusters of two or
three brood cells. In addition to the two species of stingless bee studied here, other species
such as Polistes wasps exhibit similar preferences when constructing new cells (Downing
and Jeanne 1990; Karsai and Theraulaz 1995; Karsai and Pénzes 1996,1998,2000).
The details of the building processes employed by T. hockingsi and T. carbonaria differ in
detail beyond the two and three-walled building site preferences. T. hockingsi tends to build
clusters of approximately ten brood cells; often new clusters of cells are initiated by placing
a new brood cell signiﬁcantly higher than other cells in an existing cluster, usually of the
order of half a cell height. In contrast, subtle height differences in the placement of adjacent
brood cells in a T. carbonaria nest contribute to the brood comb’s concavity. The preference
to build upwards in either a clockwise or anti-clockwise sense results in the characteristic
spiral of T. carbonaria.
Both our observations and simulations suggest that an important difference between the
cell construction processes of T. hockingsi and T. carbonaria is whether or not each species
build their brood comb through stigmergy alone. T. hockingsi seems to rely on purely local
stimuli to decide on the placement of new cells, and our simulation results suggest that
aT. hockingsi-like brood comb can be reproduced using a stigmergic algorithm. Implicit
in our observations of T. carbonaria is that the species could make use of some of the
global properties of its brood comb to decide on cell placement. The use of global properties
is evident in the tendency to start building from the centre of the domain once the brood
comb ﬁlls approximately two thirds of the brood chamber’s width and in the preferential
construction in either a clockwise or anticlockwise direction. Following our observations
and preliminary simulations, we found that the use of a non-stigmergic building rule, based
around conﬁning the placement of a central connector cell to the centre of the domain, was
necessary to produce a brood comb that resembled a T. carbonaria nest.
It is still plausible that T. carbonaria relies only on local stimuli to decide on cell place-
ment, but not in the restricted sense in which only local stimuli are applied in the standard
lattice swarm model used here. For example, the centre of a layer of cells could be identiﬁed
as a local minimum in cell height by bees walking across the brood comb. Bees working
on cell construction might not only take into account the placement of existing cells im-
mediately adjacent to a potential building site, but might take note of cells a little further
away that could still reasonably be regarded as part of a local stimulus. The complexity of
identifying building rules in reality and applying such rules within a simulation increases
dramatically as a function of the number of cells considered, but it might still be tractable
to extract and analyze such information. It is possible that another important consideration
in constructing a T. carbonaria brood comb, particularly the spiral, is that the workers may
explicitly choose not to build in particular locations. Such a decision not to build could be a
function of the age of a cell, or a bi-product of a strong preference to build next to younger
cells. The effect that cell age might have on construction work was examined in detail in the
theoretical study of Karsai and Pénzes (2000) where it was found that biologically realistic
cell arrangements like those made by Polistes dominulus could be achieved by simulation
by preferentially connecting the youngest existing cell to its next youngest nearest neighbor.
It should be noted that the building rules used by Karsai and Pénzes (2000) were not strictly
stigmergic because they relied on evaluation and comparison of multiple potential building
sites. Beyond the choice of building sites for new brood cells, future modeling work might
make use of more realistic movement rules for agents. Such movement rules could include
exploration of elements close to the location of each agent via a short random walk, or rules
that only allow agents to ‘walk’ across solid surfaces, such as the involucrum, or existing
Roubik (2006) observed that individual colonies of stingless bees from a variety of gen-
era (Melipona,Plebia,Plebina,Nannotrigona,Trigona and Tetragona) would at different
times produce brood combs that were either a series of connected, stacked, ﬂat, circular
components or a continuous spiral. It is still uncertain what triggers these changes in archi-
tecture, but possible inﬂuences include: low titers of queen pheromone due to aging, food
availability, infections and changing seasons. In the case of Tetragonula,T. carbonaria and
T. hockingsi colonies are often found in nature side by side, and the characteristic shape of
each species’ brood comb remains constant throughout the entire year (personal observa-
tions). This suggests that the differences in brood comb structure are likely due to genetic
differences, rather than environmental inﬂuences. Given the genetic similarities between
T. hockingsi and T. carbonaria it is extraordinary to note that the fundamental differences in
the building activity of these two species could be plausibly encoded by a single gene, or a
small group of genes.
Acknowledgements We thank the anonymous reviewers of our work whose suggestions and comments
contributed to a greatly improved manuscript.
RMB and TMS contributed equally to this work; RMB made the behavioral observations and derived the
algorithms; TMS conducted the computer simulations; MRM provided assistance with technical aspects of
the modeling and advice on the manuscript; TAH provided the hives and discussed behavioral data; BPO per-
formed the literature review. RMB, TMS and BPO wrote the paper. This work was supported by an Australian
Research Council grant to Madeleine Beekman and MRM. RMB was supported by an Endeavour Fellowship
and a CNPq Fellowship (PDE 201470/2008-0).
Appendix A: Model extensions
Here we describe extensions to the standard lattice swarm model that allowed for cells to
be built at any height within the domain, and an in-built tendency for all agents to build
upwards in a clockwise or anti-clockwise direction. Implementation of these extensions re-
quired some alteration to stimuli-to-build for T. carbonaria and the initial building condition,
which are also detailed below.
The random row and column that each agent was moved to at the beginning of each
time step was calculated using the same method described in Sect. 6.1. The random height
that each agent was moved to was a uniformly distributed random number between 0.5 and
nl+0.5, where nlwas the vertical extent of the domain measured in cell heights. Domain
dimensions used in our simulations were 46 ×46 ×57 brood cells.
Each agent examined its local surroundings in turn; if an agent determined that it was
at a buildable site, it constructed a new cell immediately. This is different to the standard
model, where building did not take place until all agents examined their surroundings. For
the purposes of comparing local surroundings with stimuli-to-build, an agent located at a
vertical height of zcidentiﬁed all cells with the zcoordinate of the top of each cell, zt,inthe
c+has belonging to layer z,wherehwas half the height of a brood
comb cell. Similarly, existing brood cells with zc−3h≤zt<z
c−hwere identiﬁed as
belonging to layer z−1, and existing brood cells with zc+h≤zt<z
as belonging to layer z+1. Existing cells were no longer recorded as part of a matrix for each
horizontal layer, but rather stored in vectors identifying the row, column and the location of
the top of each cell. All cells were assumed to be of the same height (2h=1 cell height
for our simulations). The coordinates in these vectors were used to construct a surroundings
matrix for each agent.
We built in a preference for building upwards when moving clockwise relative to existing
cells. To determine relative clockwise/anti-clockwise placement of cells we drew vectors
from the centre of the domain to the coordinates of a focal agent, f, and to the coordinates of
the centre of adjacent existing cells, b. We formed the vector cross product b×f,andthen
examined the sign of the vertical component of the resulting vector. If the vertical component
was positive then the placement of an existing cell was anti-clockwise relative to an agent’s
location. If the vertical component was negative then the placement of an existing cell was
clockwise relative to an agent’s location. The building height of a new cell was set as the
maximum height of any cells anti-clockwise of an agent’s location, plus a small increment
of 0.1 cell heights. If there were no cells anti-clockwise of an agent, then building height
was set as the minimum height of any cells that were clockwise relative to the agent.
Fig. 14 Graphical representations of stimuli-to-build for T. carbonaria used by agents in the extended lattice
swarm model. Stimuli preceded by c1are available to agents during stage 1 of the two alternating stages of
construction. Stimuli preceded by c2are available during stage 2, and can only be applied in one of the central
4 cells of a horizontal cross-section of the domain. Refer to Fig. 6for further explanation of this ﬁgure
We explicitly applied the extended lattice swarm model in two alternating stages; such
a multi-stage or modular construction algorithm is referred to as a coordinated building
algorithm (see Theraulaz and Bonabeau 1995b for details). Stage 1 related to the ﬁlling of a
layer with 1058 brood cells (the approximate number of cells required to construct a circular
region that ﬁlled two thirds of the width of the domain). Stage 2 related to building 2 cells
near the centre of a layer to initiate a new layer before returning to stage 1. The stimuli-
to-build for these two stages are provided in Fig. 14, and are closely related to the stimuli
used for T. carbonaria in the standard model. We set pbuild =0.256 for stimuli c11.1and
c11.6 in stage 1—the stimuli for building in the cleft formed by two adjacent existing cells.
Stimuli c11.6 applied to brood cells built above (but not usually directly connected to) cells
considered to be part of layer z−1. We set pbuild =0.892 for stimuli c11.2andc
stage 1 (stimuli for building next to a cleft formed by three cells). Stimuli c11.3toc
11.10 in stage 1 were necessary gap ﬁlling rules equivalent to those used in
the standard lattice swarm model. Stimuli c21.1andc
21.2 during stage 2 represented the
tendency of T. carbonaria to build upwards from the centre of the domain. Building heights
for the top of any cells built during stage 2 were selected as 1 cell height above the top of
any cells identiﬁed as being part of layer z−1 by an agent. Application of both stage 2
stimuli was restricted to the centre of the domain in exactly the same way as stimulus c1.6
was restricted for the standard lattice swarm model.
Tabl e 3summarizes the connections between the observations in Sect. 4and the stimuli-
to-build in Fig. 14 for T. carbonaria.
We used a similar initial condition for building with the extended model to that used for
the standard model. The central element in the bottom layer of the domain was ﬁlled with
a brood comb cell. The top of this brood comb cell was located at z=2. Six additional
brood comb cells were connected to each of the six walls of the initial hexagonal cell. These
Tab l e 3 Correspondence between Sect. 4observations and stimuli-to-build in Fig. 14
Species Observation Stimulus-to-build Notes
(ii) (Fig. 3(a)) c21.2
(iii) (Fig. 3(b)) c11.1, c11.6
(iv) (Fig. 3(c)) – Basis for initial condition.
Tendency to build upwards in
clockwise sense is in-built.
(v) (Fig. 3(d)) c11.2, c11.7
(vi) – Clockwise building increments.
(vii) (Fig. 3(e)) c21.1, c21.2 Active once approximately 2/3
the width of the domain is ﬁlled.
11.3, c11.4, c11.5, c11.8, c11.9,
Rules included to ﬁll-in gaps in
the brood comb.
brood cells were placed with their tops at z=2.1, 2.2, 2.3, 2.4, 2.5 and 2.6, respectively,
arranged so that the height of each cell increased when moving clockwise around the edge
of the central cell. This initial condition was intended to help facilitate the construction of
the spiral shape of a T. carbonaria-like nest.
Appendix B: Modiﬁcations to T. carbonaria stimuli-to-build
As with T. hockingsi,T. carbonaria was observed to have a preference for constructing new
brood cells next to the walls of pairs or trios of existing cells; see observations (iii) and (iv) in
Sect. 3and observations (iii) and (v) in Sect. 4. Such a preference for the positioning of new
structures is not unique to the two Tetragonula species that we observed. It is also known
that some species of wasp, such as those belonging to the genus Polistes,havethesame
preference (see Fig. 19.19 in Camazine et al. 2001). When only rules for building based on
the presence of two or three adjacent, existing cells are applied deterministically, the agents
of a stigmergic algorithm tend to produce structures that initially have holes in them that
can lead to the construction of distinct, disconnected lobes; (see, for example, Fig. 12(a) in
Theraulaz and Bonabeau 1999). The reason that the holes lead to the formation of lobes is
that they correspond to conﬁgurations of cells that are not included on the list of stimuli-to-
build. One example is an empty cell that has had four of its adjacent cells ﬁlled with brood
comb; in such a case the central cell will never be ﬁlled. At best the agents can build near
the unﬁllable hole, but the likely result is the formation of a ﬁssure separating sections of
brood comb. Holes and lobes are undesirable when trying to reproduce the well-connected
geometry of a T. carbonaria brood comb. To limit the production of lobes, Theraulaz and
Bonabeau (1999) applied their rules for construction probabilistically with the probability
of building when an agent encountered two existing adjacent cells set to 0.057 and the prob-
ability of construction when an agent encountered three existing adjacent cells set to 0.55.
The probabilities were derived from observations of Polistes wasps and the result was a
much rounder grouping of cells (see Theraulaz and Bonabeau 1999, Fig. 12(b)).
Preliminary simulations suggested that even the application of building rules using the
probabilities noted by Theraulaz and Bonabeau (1999) was insufﬁcient to completely pre-
vent the appearance of unbuildable sites, and then ﬁssures, within our model structures.
This was partially due to the horizontal extent of the structures produced by our model. The
structures illustrated in Fig. 12 of Theraulaz and Bonabeau (1999) lie on a region of approx-
imately 20 ×20 hexagonal cells, and even at that extent, unbuildable sites are visible on the
edges of the rounder structure in Fig. 12(b); for example, one unbuildable site appears near
the top right hand corner where an empty cell is surrounded by four ﬁlled neighboring cells.
To quantify the effectiveness of applying the probabilities of building used by Theraulaz
and Bonabeau (1999) in preventing the formation of unbuildable sites within a domain we
performed two sets of simulations where the only available stimuli-to-build were stimuli
c1.1 and c1.2 from the T. carbonaria building rules. In the ﬁrst set of simulations, both rules
were applied with certainty (the probability of building when encountering either stimulus
was set to 1). In the second set of simulations, stimulus c1.1 was applied with probabil-
ity 0.057 and stimulus c1.2 was applied with probability 0.55. Each set of simulations was
comprised of 1000 replicates. Simulations were performed in a domain with 20 horizontal
layers and each horizontal layer was made up of 20 ×20 hexagonal cells. We used 20 agents
for each simulation which were free to move anywhere in the domain, but all construction
was limited to the second layer due to the initial condition where the cell in row 10, column
10, layer 2 and the six cells adjacent to that cell were ﬁlled with brood comb. Simulations
were run until agents constructed 120 additional brood comb cells. After all simulations
were complete we identiﬁed all empty cells in the buildable layer (layer 2). We then char-
acterized each of the empty cells as being surrounded by zeros, as being a buildable site
if an empty cell and its surrounding six cells corresponded to a stimuli-to-build, or being
unbuildable if there were ﬁlled cells next to the empty cell but the entire conﬁguration did
not correspond to a stimulus-to-build. We found that the application of stimuli c1.1 and c1.2
using the probabilities derived by Theraulaz and Bonabeau (1999) was effective in reducing
the number of unbuildable sites. The mean number of unbuildable sites that formed across
1000 simulations when the stimuli were applied deterministically was 120.7 with a standard
deviation of 5.3. The mean number of unbuildable sites that formed across 1000 simula-
tions when the stimuli were applied stochastically was 26.7 with a standard deviation of 4.8.
However, even with the substantial reduction in the number of unbuildable sites in a domain
with layers with 20 ×20 cells, the presence of even a few unbuildable sites in the smaller
domain would ultimately lead to the formation of lobes in a larger structure.
Ultimately, it was necessary for us to add rules to our stimuli-to-build to allow our agents
to ﬁll in the gaps in a structure by constructing new cells when a potential building site was
surrounded by 4, 5 or 6 existing cells (stimuli-to-build c1.3, c1.4 and c1.5 for T. carbonaria
in Fig. 7).
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