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Stigmergy is a class of mechanisms that mediate animal-animal interactions. Its introduction in 1959 by Pierre-Paul Grassé made it possible to explain what had been until then considered paradoxical observations: In an insect society individuals work as if they were alone while their collective activities appear to be coordinated. In this article we describe the history of stigmergy in the context of social insects and discuss the general properties of two distinct stigmergic mechanisms: quantitative stigmergy and qualitative stigmergy.
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A Brief History of Stigmergy
Guy Theraulaz
CNRS - UMR 5550
Laboratoire d’
et psychologie animale
e Paul Sabatier
118 route de Narbonne
31062 Toulouse
Eric Bonabeau
Santa Fe Institute
1399 Hyde Park Road
Santa Fe, NM 87501
stigmergy, coordination, social
insects, self-organization, self-
Abstract Stigmergy is a class of mechanisms that mediate
animal-animal interactions. Its introduction in 1959 by
Pierre-Paul Grass
e made it possible to explain what had been
until then considered paradoxical observations: In an insect
society individuals work as if they were alone while their
collective activities appear to be coordinated. In this article
we describe the history of stigmergy in the context of social
insects and discuss the general properties of two distinct
stigmergic mechanisms: quantitative stigmergy and qualitative
1 Introduction
When the concept of stigmergy was first introduced in 1959 by French zoologist Pierre-
Paul Grass
e, an important step toward understanding the mechanisms underlying the
emergence, regulation, and control of collective activities in social insects was made
[26]. Until then two antagonistic theories had been attempting to deal with how in-
sects coordinate their activities in a social insect colony. Some scientists considered
that entirely new properties appeared at the level of the society. This level of biolog-
ical organization, they believed, possesses its own laws and requires its own causal
system to be understood: From their perspective, it is the “whole” that explains the
behavior of the parts. At the other extreme, others considered that each individual in
an insect society behaves as if it were alone: Any collective behavior or division of
labor that would appear at the colony level was thought to exist only in the eye of
the beholder. Stigmergy helped researchers understand the connection between the
level of the individual and the level of colony, showing that an alternative theory could
explain the “paradox” of coordination in social insects: Although the behavior of the
colony as a whole looks wonderfully organized and coordinated, it seems that every
insect is pursuing its own agenda without paying much attention to its nestmates. After
a brief historical survey of pre-stigmergic theories (Section 2), some of which are still
alive today in the biology community (see, e.g., [35]), we will show how the concept
of stigmergy provided an elegant explanation to the coordination paradox. We will
finally describe some recent advances in the application of stigmergy to understanding
social insects (Section 3); in particular, we will introduce a subdivision of stigmergy
into quantitative stigmergy and qualitative stigmergy. Although our focus will be social
insects, the scope of stigmergy is clearly immense.
2 From the Superorganism to Stigmergy
2.1 Insect Societies as Superorganisms
The collective behavior of social insects has been a puzzling problem for philosophers
and scientists for a long time. How is it possible for such simple creatures to coordinate
° 1999 Massachusetts Institute of Technology Artificial Life 5: 97–116 (1999)
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
their actions, create complex patterns, and make up such complex “republics” some-
times held up as examples for human society? For instance, how are myriads of bees
able to adjust their actions so precisely when they build their hive? How is it possible to
understand such a coordination? Everything happens for the observer as if there were
a coordinating agent “virtually” present at the center of the colony. The first authors
who tried to understand this astonishing phenomenon used an organicist metaphor. As
was put forward by Herbert Spencer during the second part of the 19th century, any
society is an organism [42]. Between what he called the social organism and any other
living system, Spencer pointed out a whole set of common characteristics: growth,
progressive and joint differentiation of structures and functions, mutual shaping of the
parts they are made of, division of labor, and finally similar properties between the
social organism and each of its constituent units, with the exception that the former has
a greater lifetime. In 1877, French scientist and philosopher Alfred Espinas was the first
author to apply this metaphor to describe the behavior of animal societies [20]. He was
convinced that whatever the level of biological organization considered, living units
are specifically charaterized by their tendency to join together into ever bigger units; in
this way, the formation of societies from individuals appeared just as an extension of
the natural trend of cells to assemble into multicellular organisms.
During the 20th century, the organicist metaphor continued to expand in science. In
1911, American entomologist William Morton Wheeler wrote a famous paper entitled
The ant colony as an organism, in which he introduced what can be considered as the
first systemic approach to the study of social phenomena in insects [46]:
An organism is a complex, definitely coordinated and therefore individualized
system of activities, which are primarily directed to obtaining and assimilating
substances from an environment, to producing other systems, known as
offspring, and to protecting the system itself and usually also its offspring from
dangers emanating from the environment. The three fundamental activities
enumerated in this definition, namely nutrition, reproduction and protection,
seem to have their inception in what we know, from exclusively subjective
experience, as feelings of hunger, affection and fear respectively.
Wheeler justified his assertions by considering the fact that both a society and a
single organism share common features. Both behave as a single unit. Both show
some idiosyncrasies that are peculiar to the species, in behavior, size, and structure,
and other idiosyncrasies that distinguish one colony from another belonging to the
same species. Both undergo growth and reproduction cycles that are clearly adaptive.
Finally, both are differentiated in the same way: Queens and males would appear to
be the equivalent of a germ plasma while the workers should be equivalent to the
cellular soma. If these features may easily be dismissed as mere analogies, similar to
those used by Spencer, they clearly had in Wheeler’s mind a deeper meaning. Indeed,
Wheeler suggested that there might exist within an insect colony, some functioning
constraints that allow the whole society to behave as if it were a single and unique
organism. The organization of a society must therefore be controlled by laws similar
to those that govern any living organism: This would be sufficient to explain how
coordination is achieved. Unfortunately, the scientific tools and concepts available at
the time were useless in identifying such laws and Wheeler ended its study with the
following observation [47]:
[W]e can only regard the organismal character of a colony as a whole, as an
expression of the fact that it is not equivalent to the sum of its individuals but
that it represents a different and at present inexplicable emergent level.
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Wheeler did not use the term superorganism until 1928, when his ideas on that
matter had somewhat changed. As was underlined by E. O. Wilson [49], the notion
of superorganism was initially used to describe homeostatic processes: “We have seen
that the insect colony or society may be regarded as a super-organism and hence as a
living whole bent on preserving its moving equilibrium and integrity” [48 p. 29].
In the following years, the notion of a superorganism was revived by another Amer-
ican entomologist, Alfred E. Emerson, in a series of general papers on social insects in
which he also emphasized the homeostatic properties resulting from social dynamics
[14–19]. But for Emerson the supraorganism, as he preferred to call it from 1950 on,
remained a strict analogy-based concept with the aid of which he hoped to detect the
similarities that could have appeared in the course of evolution between an insect so-
ciety and a living organism. In particular, if an insect colony represents a distinct unit
of selection, strong parallels can be drawn between the adaptive structures that can be
found within a colony and those that characterize a single organism. Such adaptive
structures must have evolved in both cases to improve the homeostasis of the whole ma-
chinery. In this way, the progressive improvements of division of labor, communication
and trophallactic exchanges between insects, the partition into a soma (the workers) and
a germen (the reproductive queens and males) are intended to increase the regulatory
ability of a colony to reach a near-optimal adaptation. Emerson’s attempt put a tempo-
rary end to the use of the organicist theory. This attractive analogy did not provide any
insight because it did not have any explanatory value. It thus remained a mere metaphor
with which it was easy to find similar global properties in a society and a single organ-
ism, yet without providing any information on the underlying mechanisms that induce
these similarities. Higher-level laws of organization can only be discovered through
analytical methods that require both a deep understanding of individual insect behav-
ior and a careful examination of the way in which interactions among and exchanges
between individuals regulate the collective behavior observed at the level of the colony.
2.2 Insect Societies as Collections of Independent Individuals
Since the 1930s the study of individual behavior within a society, which is often as-
sociated with an analytical approach, has spread rapidly, sometimes at the expense
of an oversimplification of the real nature of social phenomena. The most significant
scientist who used this approach was French biologist Etienne Rabaud, who was suspi-
cious of any holistic explanation. His fundamental postulate was that the only cause of
behavior lies within an individual. In his mind any other theory was mere metaphysical
speculation [37]. His entire work on insect societies was an attempt to demonstrate that
each individual insect in a society behaves as if it were alone, as it appears in this text
describing building behavior in paper wasps [38]:
In this way, each worker is indistinctly in charge of any part of the nest. Is this
cooperation? In any case, if cooperation occurs it is only by chance and as a
result of unexpected incidents. Are these building wasps following a general
plan? When the founding female starts building its nest she decides to place it in
a certain way. It seems that workers that come after on the nest continue the
foundress’s work in a well defined direction. In fact, they build cells without
any regularity, but they build it in the same way the foundress did, in particular
they enlarge the envelope as the number of cells in the comb increases. All this
does not involve any prerequisite plan at all. This simply means that both the
founding female and the workers are responding in the same way and with the
same reflexes to the same stimuli. ...In fact, whether they are isolated or in
group, all individuals behave in the same way; whether they carry on their own
work or that of the others, the entire environmental conditions remain exactly
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the same; in any case, everything happens as if they were doing their own work,
without paying attention to their neighbors’ work that exerts in this respect no
noticeable influence on them. Apparently, a contradiction is emerging from
these facts; in appearance only, because nothing in the work resulting from the
combined efforts by several individuals is different from the work that is done
by a solitary founding female. Only the speed at which cells are built increases
because the size of the group is large enough. Thus all these individuals are
working in the same way and each one is working as if it were alone; the
“collective” work is only the juxtaposition of individual works. Neither the
enlargement of the nest, nor the feeding of the larvae require that a whole plan
has to be executed in such a way that a task performed by an individual induces
that of his neighbor and conversely: [T]he common work is no more than a side
effect of interattraction that gather[s] individuals together. (pp. 150–151)
Rabaud considered that division of labor requires as a precondition that a common
sign code be used by the members of a society, which, according to him, was quite out
of the question in the case of insects. Such a dogmatic assertion was unacceptable at the
time it was put forward and is even less acceptable today, when numerous examples
of social coordination are clearly established. This theoretical option was motivated by
a relentless fight against mentalism and anthropomorphism.
Nevertheless, the anti-organicist controversy in which Rabaud was involved, despite
its excess, led to the introduction of two concepts that turned out to be of utmost
importance for understanding collective phenomena.
The first concept is that of interaction. Rabaud claimed that individual behavior was
the essential drive of any collective action or process. In social animals that constantly
live next to each other, one individual’s action may influence another individual, thereby
modifying its behavior. The term interaction refers to this kind of reciprocal action. Any
social mechanism might be reduced to a set of elementary interactions. This simple
idea opened the road for a research program that broke with the organicist theory by
establishing a bridge between the individual and collective levels instead of considering
them separately and independently.
The second concept introduced by Rabaud is interattraction, which he defined as a
basic social phenomenon: Interattraction refers to the fact that any animal that belongs
to a social species is attracted in a specific way by any other animal that belongs to the
same species [38]. It is a specific interaction that can be triggered by the stimuli that
individuals bear. One consequence of these interactions is a nonrandom distribution
of individuals in the environment. This idea and its main consequences were later
developed by Grass
e [27]:
Social groups are above all characterized by the fact that any individual taken
separately produces a specific stimulus upon its fellows, while the group (that
can be reduced to a single fellow) produces in turn a specific stimulus that will
influence the behavior of that animal. (p. 8)
Rabaud clearly overlooked such consequences, particularly the fact that the emer-
gence of a collective and coordinated behavior inside a society can be explained by a
set of specific interactions among individuals.
2.3 The Organizing Role of Interactions and Stigmergy
At the same time Rabaud was conducting his research, a growing number of scientists,
sometimes with more of a naturalist than a theoretical background, discovered that
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Figure 1. Pierre-Paul Grass´e (1895–1985) was the coordinator of a masterpiece treatise of zoology and a synthetic
work on termites. He was also at the origin of the publication of the Bulletin of the International Union for the
Study of Social Insects that later became the journal Insectes Sociaux.
specific interactions among individuals were involved in the genesis of certain collective
phenomena in insects such as gregariousness in locusts, closure of the colony in many
ant species, caste regulation in termites, bee dances, and so on. Pierre-Paul Grass
e (see
Figure 1) and his students were the first to provide a synthetic view of such collective
phenomena that combined both the specificity of interactions among social individuals
and organized collective behavior at the colony level. Grass
e’s basic idea was that
sociality is not a trivial consequence that results from interattraction, but a biological
characteristic deeply rooted in the ethological heritage of every species. This becomes
obvious if one considers the reactions of a social animal in the presence of one of its
nestmates: Even if this animal is alone its behavior is different from that of a solitary
animal. A social animal displays what Grass
e called a social appetition that drives it to
seek its nestmates. Such a social dependence may prevent this animal from behaving
in a normal way when it is isolated and it may even be unable to survive outside a
social environment, which is the case, for instance, in bees. Another consequence of
interindividual stimuli is what has been called group effect [24, 25]. It describes the fact
that when an animal is submitted in certain conditions to a critical number of specific
stimuli from its nestmates, its behavioral state is altered. Group effect not only involves
a stimulus-response sequence, it also leads to a deep change in the reactional state
of the animal and sometimes of its whole physiology. One of the most spectacular
examples of group effect is the gregarious phase of locusts resulting from stimulus
exchanges [7].
Moreover, since a critical number of interactions is reached in an animal society,
integrative and regulatory processes emerge. This idea was developed by Grass
e who
called social regulation the fact that a society is able to re-establish a population equi-
librium that has been broken or to coordinate the collective performance of a given
task through psychophysiological and psychomotor mechanisms. Such a regulation is
a social property that results from idiosyncratic features of individual behavior. These
mechanisms rely on stimulus-response sequences in which a stimulus is an action per-
formed by one individual and the response another action that has been triggered by
this previous action. Each individual is a direct source of stimuli for the other individu-
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G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 2. Stimulus-response sequence leading to the construction of the mud funnel in the nest of the Eumenid wasp
Paralastor sp. Each new building stage n is completed after a stimulus S
triggers a new ensemble of building actions
. The completion of each building stage n gives rise to a new stimulus S
that triggers new building actions R
leading to the construction of the next building stage n + 1. When the fifth stage has been completed, there exists
no more stimulus on the funnel to trigger new building actions and the construction stops.
als. This mechanism opens the way for an indirect coordination of individual activities.
The processes that regulate such interactions are not limited to the direct influence of
the stimuli produced by individuals. Indeed, each animal’s activity is organizing the
environment in such a way that stimulating structures are created; these structures can
in turn direct and trigger a specific action from any other individual from the same
species that comes into contact with them. Chemical trails that are produced by some
ants species [10, 23], muleteer trail networks, and even dirt tracks and trail systems in
man [31, 32] result from interactions of this kind.
One of the most interesting examples studied by Grass
e is the building behavior
of termites. Stigmergy (from the Greek stigma: sting and ergon: work) was initially
introduced to explain indirect task coordination and regulation in the context of nest
reconstruction in termites of the genus Bellicositermes [26, 28]. Grass
e showed that
the coordination and regulation of building activities do not depend on the workers
themselves but are mainly achieved by the nest structure: A stimulating configura-
tion triggers a building action of a termite worker, transforming the configuration into
another configuration that may trigger in turn another (possibly different) action per-
formed by the same termite or any other worker in the colony. Stigmergy offers an
elegant and stimulating framework to understand the coordination and regulation of
collective activities. The main problem is then to determine how stimuli are organized
(in space and time) to generate robust and coherent patterns: Colonies of a given
species produce qualitatively similar patterns, be they nest architectures or networks of
foraging trails and galleries.
2.4 From Sequential to Stigmergic Activity
To better understand how multiple, “independent” building actions can be coordinated
through a stigmergic behavioral algorithm, it is instructive to look at nest construc-
tion in solitary species. The experiments performed by Smith in 1978 on a solitary
wasp shed some light on the origin of coordination of building activities and on the
preadaptation (to sociality) of this behavior [41]. Nest construction in the Eumenid
wasp Paralastor sp. occurs as a stimulus-response sequence in which the completion
of one stage provides the stimulus for commencement of the next (see Figure 2). A
wasp begins with the excavation of a narrow hole, approximately 8 cm in length and
8 mm in width. When the nest hole has been completely lined with mud, the wasp
begins the construction of a large and elaborate mud funnel above its entrance. The
funnel is built in five distinct stages from a series of mud pellets that are applied in a
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Figure 3. The construction of an abnormal mud funnel in the nest of the Eumenid wasp Paralastor sp. When the
funnel is almost completed, a spherical hole (indicated by the arrow) is made. This hole is equivalent to stimulus S
which triggers funnel construction. As a consequence, the wasp builds a second funnel, over the hole and on top of
the first one already built.
highly stereotyped sequence. Stage 1 involves the building up of the funnel stem by
application of a series of mud pellets until it reaches a length of 3 cm. At Stage 2 the
wasp ceases to build uniformly upward, and by adding more mud to one side begins
the construction of a uniform curve in the stem of the funnel. Once the curve has been
completed, Stage 3 begins with the formation of a bell with the splaying of the stem to
form a uniform flange of approximately 2 cm diameter. At Stage 4, the flange is next
widened more on the side nearest to the stem than elsewhere, thus giving the bell a
characteristic asymmetry in one direction. Finally at Stage 5, the sides of the bell are
formed by building uniformly downward from the edge of the flange. At the end of
each stage of building, the stimuli for the responses that lead to the completion of the
next stage are those that the animal encounters as a consequence of its earlier behavior.
What happens when the stimuli that trigger the beginning of a previous building stage
are encountered by the wasp just as it finishes the end of a particular stage? Smith
examined this question in one of his experiments. A spherical hole located in the
neck of a funnel is made just after Stage 3 has been completed (see Figure 3). After
examining the damage several times, the wasp begins the construction of a second
funnel, over the hole and on top of its first funnel. This result is extremely important
for anyone who wants to understand the coordination of building activities in social
wasps and more generally in social insects. In a solitary species such as Paralastor
sp., the indirect coordination of its behavior through the previous consequences of its
building actions results in a sequential-like behavior. There are two consequences to
this behavior.
First, the order in which stimuli arise in the course of the construction must follow a
precise sequence. If by chance a stimulus triggering a set of building actions that gives
rise to a previous subelement of the architecture is present at a later stage, this will
automatically lead to a redundant structure and an abnormal nest architecture. This
observation, as we will see in the next section, has important consequences in the
coordination of building activity in social wasps.
Second, if one wasp does not distinguish the product of its own activity from that of
another wasp, the two wasps can in principle work at completing the same nest struc-
ture. One wasp could continue the work of the other at whatever stage of construction
of the nest. Such a mechanism may then in turn be a step toward indirect cooperation
between individuals. This is precisely the mechanism that Grass
e had in mind when
he introduced the concept of stigmergy.
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G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
3 Stigmergy Revisited
Stigmergy refers to a class of mechanisms that mediate animal-animal interactions.
Therefore, it has to be supplemented with an additional mechanism that makes use
of these interactions to coordinate and regulate collective building in a particular way.
At least two such mechanisms have been identified: quantitative stigmergy [1, 9, 11]
and qualitative stigmergy [43, 44]. With quantitative stigmergy, the stimulus-response
sequence comprises stimuli that do not differ qualitatively (such as pheromone fields
and gradients) and only modify the probability of response of the individuals to these
stimuli. Qualitative stigmergy differs from quantitative stigmergy in that individuals
interact through, and respond to, qualitative stimuli.
3.1 Quantitative Stigmergy and Self-Organized Dynamics
One of the best examples of quantitative stigmergy is the construction of pillars in ter-
mites, studied by Grass
e [26, 28]. Workers use soil pellets impregnated with pheromone
to build pillars. Two successive phases take place [26]. First, the noncoordinated phase
is characterized by a random deposition of pellets. This phase lasts until one of the
deposits reaches a critical size. Then, the coordination phase starts if the group of
builders is sufficiently large: Pillars or strips emerge. The existence of an initial de-
posit of soil pellets stimulates workers to accumulate more material through a positive
feedback mechanism, since the accumulation of material reinforces the attractiveness
of deposits through the diffusing pheromone emitted by the pellets ([5]; see Figure 4).
This autocatalytic, snowball effect leads to the coordinated phase. If the density of
builders is too small, the pheromone disappears between two successive passages by
the workers, and the amplification mechanism cannot work, which leads to a nonco-
ordinated phase. The system undergoes a bifurcation at this critical density: No pillar
emerges below it, but pillars can emerge above it.
This example illustrates three important properties or signatures of the self-organized
dynamics associated with quantitative stigmergy [1]: (a) the emergence of spatiotem-
poral structures in an initially homogeneous medium, that is, a random spatial dis-
tribution of soil pellets. The basic mechanism that leads to the emergence of these
structures is positive feedback (the snowball effect); once the structures are created,
they are stabilized through negative feedback, mainly pheromone decay and compe-
tition among neighboring pillars. (b) the possible coexistence of several stable states
(multistability): Structures emerge by amplification of random deviations, and any such
deviation can be amplified, so that the system converges to one among several possible
stable states, depending on initial conditions (path dependency). (c) the existence of
(parameter-driven) bifurcations, where the behavior of a self-organized system changes
dramatically. In 1977, Deneubourg ([8], see also [2]) designed a chemotaxis-based
reaction-diffusion model of the emergence of regularly spaced pillars in termites that
exhibits the desired properties for appropriate parameter values. Figure 5 shows the
one-dimensional spatial distribution of pillars obtained with his model. In this example,
coordination between workers and the regularity of the pillars’ distribution emerges out
of indirect interactions among termites without being coded in an explicit way at the
level of each insect.
3.2 Qualitative Stigmergy and Self-Assembling Dynamics
3.2.1 Qualitative Stigmergy
Qualitative stigmergy differs from self-organization-based stigmergy in that individuals
interact through, and respond to, qualitative stimuli. The resulting dynamics is similar
to that resulting from a self-assembling process. Qualitative stigmergy is based on a
discrete set of stimulus types: For example, an insect responds to a type-1 stimulus with
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Figure 4. Schematic representation of quantitative stigmergy illustrated by the successive stages leading to the emer-
gence of pillars in termite nests. To build their nest termite workers use soil pellets impregnated with pheromone.
The existence of an initial deposit of soil pellets stimulates workers to accumulate more material through a positive
feedback mechanism, since the accumulation of material reinforces the attractiveness of deposits through the diffus-
ing pheromone emitted by the pellets. Assume that the architecture reaches state S
, which triggers response R
from worker I. S
is modified by the action of I (e.g., I drops a new soil pellet) and transformed into a new stimulating
configuration S
, which may in turn trigger a new response R
from I or any other worker I
, and so forth. The
successive responses R
, R
, R
may be produced by any worker carrying a soil pellet. Each worker creates new
stimuli in response to existing stimulating configurations. In this example, the successive stimuli differ from one
another only in the quantity of pheromone that is present on the pillar. This example therefore illustrates positive
feedback in quantitative stigmergy.
action A and responds to a type-2 stimulus with action B, in the same way that the mud
wasp Paralastor builds the funnel of its nest. In other words, qualitatively different
stimuli result in different responses: For example, a type-1 stimulus triggers action A by
individual I
; action A transforms the type-1 stimulus into a type-2 stimulus that triggers
action B by individual I
. At first sight, it is unclear how coordination and regulation
can be achieved with qualitative stigmergy. Building in social wasps provides a good
illustration (Figure 6) of how this can work.
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G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 5. Spatial distribution of soil pellets obtained from Deneubourg’s [8] model in one dimension [2]; x: time, y:
space, z: quantity of building materials.
3.2.2 Building in Polistes
One example is nest building in the primitively eusocial wasp Polistes. The article by
Karsai in this special issue gives a detailed overview of stigmergic procedures used by
paper wasps to build their nests. Polistes use long wood fibers and plant hairs that
are mixed with salivary secretions. The resulting carton is easily shaped and has a
strong robustness though it is extremely fine and light [29]. A nest consists of a sin-
gle comb connected to the substrate by a rodlike pedicel [45]. The comb is round
shaped and mature nests can contain about 150 cells [39]. New cells are added to the
comb during nest growth and the number of potential sites where a new cell can be
added increases as construction proceeds. Several building actions may take place in
parallel. However, this raises the question of where new cells are to be added to get
a nest architecture that is species-specific. Due to the possibility of parallel activities,
another problem arises: Construction may become messy as conflicting actions may be
performed simultaneously.
These two issues are connected and the architecture itself provides enough con-
straints to canalize the building activity and prevent its deorganization. Let us consider
the comb shown in Figure 7, with six cells already built; twelve potential building sites
can be found on this comb: seven sites with one wall already present (labeled S
four sites with two adjacent walls (S
), and one other site with 3 adjacent walls (S
Figure 8 shows how the mean number of potential building sites (with either 1, 2, or
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G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 6. Schematic representation of qualitative stigmergy illustrated by the successive stages leading to the con-
struction of a comb in wasps. The wasps use wood pulp to build the various elements of their nest. Here, each
building stage corresponds to the addition of a new cell to the pre-existing comb. At the beginning (top of the
figure) all potential building sites are equivalent. Each one of them (S
) has two adjacent walls. No construction is
observed on the six other sites that have only one wall. The building of the new cell (R
) by wasp I results in the
creation of a new potential building site with two walls, but all building sites (S
) remain equivalent from a qualitative
point of view. The addition of a second cell to the comb (R
) produces building site (S
) that has three adjacent
walls and is qualitatively different from the others (S
). Each wasp I
creates new qualitative stimuli in response
to existing stimulating configurations. Successive stimuli differ from one another in their spatial configurations of
external walls.
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G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 7. Representation of the potential building sites that have one (S
), two (S
), or three (S
) walls in common
with the new cell added to the comb.
3 adjacent walls) varies as a function of the number of cells already built in the comb.
Since it appears that the growth of the comb is isotropic, that is, it occurs with equal
probability in all directions of space, the question that arises is: How are cells added
to ensure that the comb will grow fairly evenly in all directions from the pedicel? Pre-
vious studies showed that wasps are obviously influenced by previous construction,
and that building decisions are made locally on the basis of perceived configurations
in a way that possibly constrains the building dynamics (see also the article by Karsai
in this issue). Cells are not added randomly to the existing structure and wasps have
a greater probability to add new cells to a corner area where three adjacent walls are
present than to initiate a new row by adding a cell on the side of an existing row [12,
34]. One building rule is that wasps tend to finish a row of cells before initiating a new
row. Figure 9 shows that the probability to add a cell to a three-wall site is about 10
times higher than in the case of a two-wall site. What are the consequences of this
probabilistic stigmergic behavior on the development of the comb structure?
3.2.3 Lattice Swarms
In a series of previous articles we introduced a class of models to explore the potential
of qualitative stigmergy as a model of nest construction [43, 44]. The models are
based on asynchronous automata that perform random walks in a three-dimensional
discrete space, have access to local space and time information, and act on a pure
stimulus-response basis: A set of builders (automata, agents or artificial wasps) move
in three dimensions and respond to particular stimuli present on the developing comb
structure. The three-dimensional space in which the automata move is a discrete cubic
or hexagonal lattice—hence the name lattice swarms. The deposit of an elementary
building block, or brick, by an agent depends on the local configuration of bricks in
the cells surrounding the cell occupied by the agent (see Figure 10). Several types
of brick can be deposited, and no brick can be removed once it has been deposited.
Increasing the number of brick types increases the repertoire of local configurations
that are perceived by the agents. In the simulations reported below two types of bricks
were used, which is the minimal number of types with which “complex” patterns can
be generated. All simulations start with a single initial brick. A microrule is defined
as the association of a particular stimulating configuration and a brick to be deposited.
108 Artificial Life Volume 5, Number 2
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 8. Mean number of potential building sites with 1, 2, and 3 walls adjacent to the new cell as a function of
the number of cells already built in the comb in Polistes dominulus. The curve is based on 155 mesures made on 13
colonies reared in laboratory conditions and observed at various stages of development.
Two microrules are incompatible if they have the same stimulating configuration but
command different actions, here the deposits of different bricks. An algorithm is defined
as any collection of compatible microrules: It consists of a set of if-then rules, in which
the “if” statement tests for a particular configuration of bricks in the structure, and the
“then” statement results in the deposition at that site of one of the two types of bricks.
An algorithm can be characterized by its microrule table, a lookup table comprising
all its microrules, that is, all stimulating configurations and associated actions. Finally,
an algorithm can be deterministic or probabilistic: If it is deterministic any applicable
Figure 9. Probability of adding a new cell at sites that have 1, 2, and 3 walls adjacent to the potential new cell.
Artificial Life Volume 5, Number 2 109
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 10. Schematic representation of an agent’s perception range in a hexagonal lattice. Each one of the 20
neighboring cells can either be empty or contain either a type-1 brick or a type-2 brick. Each local configuration can
be associated with a building action, that is, a brick deposit. There exist 3
possible elementary microrules, and
the building behavior of an agent relies on several microrules.
microrule is applied with probability 1; in the second case, a probability between 0 and
1 is assigned to each microrule.
Lattice swarms can be used to study the dynamics of comb enlargement in Polistes
wasps. Comb structures produced with deterministic and probabilistic algorithms can
easily be compared. We shall consider only the two following rules (see Figure 11):
Rule 1: Add a new cell at a two-wall site location.
Rule 2: Add a new cell at a three-wall site location.
In the first simulation, both rules are applied every time an agent encounters the right
stimulating configuration. In the second case, rules are applied with the experimental
probabilities found in natural Polistes colonies: respectively 0.057 for Rule 1 and 0.55
for Rule 2. Figure 12a shows a comb that has been obtained using deterministic rules.
The comb is indented in many places, with several lobes. On the other hand, using
probabilistic rules leads to the construction of a round-shaped comb similar to what
is found in natural Polistes nests (Figure 12b). Similar results have been obtained by
Karsai and P
enzes [33]. These differences can be simply explained by the fact that
in the probabilistic rule, three-wall sites trigger cell building with a higher probability;
this process results in building closely packed parallel rows of cells and the nest grows
fairly evenly in all directions.
Using lattice swarm models, it is possible to build much more complex architectures
[43, 44]. Figure 13 shows a few architectures that have been obtained using determin-
istic rules in cubic and hexagonal environments. According to their neighborhoods
and lookup tables, agents may put down two types of brick (type 1 or type 2). Al-
110 Artificial Life Volume 5, Number 2
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 11. Two local microrules used to build the comb of a Polistes-like nest.
though the underlying behavioral principle is quite simple, complex architectures can
form, some of which closely match those found in nature. Future work will be aimed
at verifying that such complex architectures can still be produced with probabilistic
4 Conclusions
The basic principle of stigmergy is extremely simple: Traces left and modifications made
by individuals in their environment may feed back on them. The colony records its
activity in part in the physical environment and uses this record to organize collective
behavior. Various forms of storage are used: gradients of pheromones, material struc-
tures (impregnated or not by chemical compounds), or spatial distribution of colony
elements. Such structures materialize the dynamics of the colony’s collective behavior
and constrain the behavior of individuals through a feedback loop. Stigmergy also
solves the coordination paradox: Individuals do interact to achieve coordination but
they interact indirectly, so that each insect taken separately does not seem to be in-
volved in a coordinated, collective behavior.
As a consequence of the medium used, physical and geometrical constraints influ-
ence subsequent choices of the colony. It is well known that in ant species using
mass recruitment, when several food sources are discovered independently, the closest
source to the nest is selected by the colony [23]. Indeed when the distance between
a food source and the nest is long, the time interval between the trips of two foragers
may exceed the evaporation latency of the pheromone and the trail disappears. If the
path is shorter, the traffic is sufficiently intense for the pheromonal trace to remain. This
dynamic improves the exploitation of the environment by the colony. An interesting
property of this process is that various organizations may be produced with the same
individual behavioral algorithms: Several foraging patterns can be produced according
Artificial Life Volume 5, Number 2 111
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 12. (a) Comb structure obtained when deterministic rules are used. (b) Comb structure obtained when
probabilistic rules are used.
to the distribution of resources, as illustrated by army ants [10]. Similar coupling be-
tween environmental and social organization may be found in the self-organization of
brood patterns in bees and ants. In a bee hive (Apis mellifica), the comb is organized in
three concentric regions of cells: a central brood area, a rim of pollen, and a peripheral
zone of honey. It has been shown using computer simulation that such patterns may
arise spontaneously from the dynamics of interactions, according to behavioral rules
bearing only upon local cues [6]. In lepthoracin ants, eggs, larvae, and cocoons are
sorted out by workers according to their developmental stage. A model worked out
by Deneubourg et al. [11] showed that this structuration may arise from a tendency
of workers to deposit brood elements on heaps of the same category, in a positive
feedback loop similar to those reported in termites.
Marking the environment with glandular secretions, urine, or feces has been re-
ported in a number of mammals, the scent-mark of an individual frequently inducing
remarking by others [4]. Besides the function of advertising conspecifics, marks may
help individuals orient themselves in their range [see 3, 22, 30, 40]. Though scent-marks
can be used as reminders by individuals, they should not be viewed only as landmarks:
They are actually part of the environmental structure and contribute to its definition.
In species forming social groups, marking is a collective affair; chemical signals are de-
posited throughout the range, their distribution being denser at much-visited sites such
as junctions, dens, or zones where individuals from other groups may be encountered
[4]. Group members continuously come across olfactory signals; they obtain informa-
112 Artificial Life Volume 5, Number 2
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
Figure 13. Simulations of collective building on a 3D cubic (a) and hexagonal (b to i) lattice. Simulations were run
with a 40 × 40 × 40 lattice for architectures a and b; with a 20 × 20 × 20 lattice for architecture c; and with a
16 × 16 × 16 lattice for architectures d to i. These architectures are reminiscent of natural wasp nests and exhibit
a similar design. For each architecture we give the name of the corresponding natural wasp nest species and in
brackets the total number of microrules used to build it. (a) Agelaia nest-like architecture (13). (b) Parapolybia
nest-like architecture (12). (c) Parachartergus nest-like architecture (21). (d) Vespa nest-like architecture (13). (e)
Same architecture as (d) shown in front section. (f, g) Stelopolybia nest-like architectures (12). (h) Chartergus nest-like
architecture (39). (i) Same architecture as (h) but a portion of the external envelope has been partly removed to
show the internal structure of the nest.
tion about each other’s movements that may improve foraging efficiency and maintain
group cohesion [3, 13, 21].
Stigmergy also has consequences with respect to the evolution of social life, and the
route from solitary to social life might not be as complex as one may think. Numerous
examples of solitary species that use stigmergy are known: Many solitary animals, such
as spiders [36], are not able to see the differences that exist between the products of their
own activities and those of conspecifics. Occasional cooperation between conspecifics
may spontaneously emerge in these “preadapted-to-social-life” species if, by accident,
they find themselves in the same location. Of course stigmergy alone is not a sufficient
condition to develop a truly social life and two additional features are required that are
closely related, yet not exactly identical: a “preadapted” species must be able to reach
locally high population density, that is, possess some kind of “homing” mechanism
or interattraction allowing cluster formation. The second condition is that once many
individuals have joined to form a group, they must be able to maintain its cohesion
long enough to benefit from cooperative effects.
In conclusion, the potential of stigmergy is still largely untapped in the biology
community, in which it originated. Perhaps, as the other articles in this special issue
indicate, the sciences of the artificial will be more open to this potential.
Artificial Life Volume 5, Number 2 113
G. Theraulaz and E. Bonabeau A Brief History of Stigmergy
E. Bonabeau is supported by the Interval Research Fellowship in Adaptive Computa-
tion at the Santa Fe Institute. This work was supported in part by a grant from the GIS
(Groupe d’Int
et Scientifique) Sciences de la Cognition to E. Bonabeau and G. Ther-
aulaz and a grant from the Conseil R
egional Midi-Pyr
ees to G. Theraulaz.
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... We speculate that this kind of dynamic could be an instance of stigmergy, a term introduced by Grassé to describe a form of indirect communication mediated by modification of the environment that he observed in some species of termites (Theraulaz and Bonabeau, 1999) and since widely suggested to be a pillar of optimization tasks performed by ant colonies and other social insects (Dorigo et al., 2000;Theraulaz, 2014). In this perspective, the L and R equilibrium could express a kind of intelligence (in a mathematical meaning of the term) known as "swarm intelligence." ...
... The theory of stigmergy, initiated in entomology, at the level of resolution individuals/colonies, presents, like the "d "coordination without coordinator", thus resol coordination: "Individuals indirectly, so that each insect collective behavior" (THERAULAZ and BONAVEU, 1999, Today, if you look up the word "stigmergy" in an Internet search engine, the vast majority of entries lead to simulation algorithms and interacting robot assemblies. What started with the humble termites Stigmergy in social insects, considered here, leads from the coordination of actions to the construction of structures in the environment: termite mounds, nests, ...
Full-text available
Abstract: This paper is a review of the place of teleology in its relationships with biological sciences, taken not in isolation but associated with the ideas of communication and normativity, the former coming from the semiotic sciences and the latter from the human sciences. The first part puts forward a classification of the sciences into natural, semiotic, and human sciences, which allows the introduction in the second part of a context for a discussion about teleology, not only in the biological sciences, but also on the borders connecting these with the semiotic and human sciences. In the first place, an account is given of the attempts to naturalize teleology in biological thought. In the second, it shows how mathematical and physical ideas are used to model living systems and processes, as well as to unify living beings under the umbrella of the idea of communication. Finally, in the third place, it addresses the idea of normativity, associated with human actions in their possible relationships with biological sciences in the matters of niche construction, stigmergy, and the directed (teleological) character of organic processes and actions.
... In this framework, the concept of emergence keeps the ideas of multiscale and holism together. As Ladyman et al. (2010, p. 40) remark, "A strong, perhaps the strongest, notion of emergence is that emergent objects, properties or processes exhibit something called 'downwards causation,'" also called "stigmergy" (Grassé 1959;Théraulaz and Bonabeau 1999) or "immergence" (Chavalarias et al. 2009). These terms refer to the idea that the whole has somehow a form of causality over the parts that compose it: The individuals make up the whole in a bottom-up relation and the whole in its turn influences and regulates its components. ...
... The builder or rather the builders' action is an element of the stigmergic system. The builder responds to the partially complete structure by monitoring the appropriate environmental variable, like the pheromone in the termite moundcalled the trace of the action in the stigmergy literature (see Heylighen, 2016;Theraulaz & Bonabeau, 1999). Being experienced, our builder is sensitive to exactly those aspects of the building-in-progress that track completion of tasks and trigger new task performances. ...
The tricky question in the plant cognition debate is what theory of cognition should be used to fix the reference of cognitive concepts without skewing the debate too much one way or the other. After all, plants are rather different to animals in many respects: they are not motile, do not possess central nervous systems or even neurons, do not exhibit an invariant morphology, interact with the world in a distributed multi-centred manner, and behave through changes in their physiology. Nonetheless, there is a significant strand in the debate that asserts that plants are indeed cognitive. But what theory of cognition makes sense of this claim without baking in prior zoological assumptions? The aim of this paper is to try out a theory of minimal cognition that makes the claim of plant cognition plausible. It is primarily inspired by the distributed cognition literature and the sensorimotor coordination theory of cognition proposed by van Duijn et al. (2006) . We take a cognitive system to be a coordinated set of semi-autonomous processes running over the organism and items in its environment. Coordination is characterised in terms of two functional conditions that ensure that the system generates goal-directed action in the world. The system is stigmergic in the sense that the material results of its actions in the environment are a crucial part of the processes that coordinate further actions. The account possesses a degree of scale invariance and helps unify cognitive explanation across microorganisms, plants and animals.
... Researchers have been inspired by many phenomena at the physical, chemical, biological, or social level to study the emergence of swarm intelligence [1] [2][3] [4][5] [6] [7]. Many models, mechanisms, and techniques for swarm intelligence have been proposed, such as the Particle Swarm Optimization (PSO) [8], Gravitational Search Algorithm (GSA) [9], Intelligent Water Drop (IWD) [10], Ant Colony Optimization (ACO) [11], Artificial Bee Colony (ABC) [12], and Holonic system model [13]. ...
The modeling of emergent swarm intelligence constitutes a major challenge and it has been tacked in a number of different ways. However, existing approaches fail to capture the nature of swarm intelligence and they are either too abstract for practical application or not generic enough to describe the various types of emergence phenomena. In this paper, a contradiction-centric model for swarm intelligence is proposed, in which individuals determine their behaviors based on their internal contradictions whilst they associate and in-teract to update their contradictions. The model hypothesizes that 1) the emergence of swarm intelligence is rooted in the development of individuals’ internal contradictions and the interactions taking place between in-dividuals and the environment, and 2) swarm intelligence is essentially a combinative reflection of the configu-rations of individuals’ internal contradictions and the distributions of these contradictions across individuals. The model is formally described and five swarm intelligence systems are studied to illustrate its broad applica-bility. The studies confirm the generic character of the model and its effectiveness for describing the emergence of various kinds of swarm intelligence; and they also demonstrate that the model is straightforward to apply, without the need for complicated computations.
... In these systems, agents do not interact directly with each other, but rather share information by manipulating a shared medium called a virtual stigmergy [41]. The concept of stigmergies originates from biology, where it has been used to explain the collective behaviour of social insects such as ants, termites, and bees [47], but appears well-suited to describe a much wider range of phenomena, including the creation and curation of content on the Wikipedia collaborative encyclopedia [6], or the development of open-source software [45]. The indirect and asynchronous nature of this interaction mechanism induces vast state spaces even in modestly-sized systems, making their verification challenging [10,12]. ...
Conference Paper
Collective adaptive systems may be broadly defined as ensembles of autonomous agents, whose interaction may lead to the emergence of global features and patterns. Formal verification may provide strong guarantees about the emergence of these features, but may suffer from scalability issues caused by state space explosion. Compositional verification, whereby the state space of a system is generated by combining (an abstraction of) those of its components, has shown to be a promising countermeasure to the state space explosion problem. Therefore, in this work we apply these techniques to the problem of verifying collective adaptive systems with stigmergic interaction. Specifically, we automatically encode these systems into networks of LNT processes, apply a static value analysis to prune the state space of individual agents, and then reuse compositional verification procedures provided by the CADP toolbox. We demonstrate the effectiveness of our approach by verifying a collection of representative systems.
Collective adaptive systems may be broadly defined as ensembles of autonomous agents, whose interaction may lead to the emergence of global features and patterns. Formal verification may provide strong guarantees about the emergence of these features, but may suffer from scalability issues caused by state space explosion. Compositional verification techniques, whereby the state space of a system is generated by combining (an abstraction of) those of its components, have shown to be a promising countermeasure to the state space explosion problem. Therefore, in this work we apply these techniques to the problem of verifying collective adaptive systems with stigmergic interaction. Specifically, we automatically encode these systems into networks of LNT processes, apply a static value analysis to prune the state space of individual agents, and then reuse compositional verification procedures provided by the CADP toolbox. We demonstrate the effectiveness of our approach by verifying a collection of representative systems.
The physics of signal propagation in a collection of organisms that communicate with each other both enables and limits how active excitations at the individual level reach, recruit and lead to collective patterning. Inspired by the spatio-temporal patterns in a planar swarm of bees that use pheromones and fanning flows to recruit additional bees, we develop a theoretical framework for patterning via active flow-based recruitment. Our model generalizes the well-known Patlak–Keller–Segel model of diffusion dominated aggregation and leads to an enhanced phase space of patterns spanned by two dimensionless parameters that measure the scaled stimulus/activity and the scaled chemotactic response. Together these determine the efficacy of signal communication via fluid flow (which we dub rheomergy ) that leads to a variety of migration and aggregation patterns, consistent with observations.
The concept of “Smart cities” has been criticized for imprecise and inconsistent definitions across disciplines, potential hidden agendas of power and control, and a failure to address important social aspects of cities. Here we consider a more fundamental question of centralization versus distribution of city information, and in particular the information within the city and not only about the city—a distinction we draw by applying the concept of stigmergy. After conducting a brief examination of the deeper philosophical issues of information and city structure, we consider how the application of information within the city is a mostly distributed process that can be centralized only in limited ways. The model of stigmergy illustrates how such a process of local interactions can occur between actors within the city, and between them and the evolving forms of the city itself. Evidence suggests that this self-organizing and emergent process plays an essential role in a city’s ability to satisfy multiple interests of city residents over time. An effective Smart city strategy will need to engage and support this capacity. We conclude with potential application as well as opportunities for further research.KeywordsSmart citiesStigmergyInformation theorySymmetryActor network theory
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Self-organization was introduced originally in the context of physics and chemistry to describe how microscopic processes give rise to macroscopic stuctures in out-of-equilibrium systems, Recent research that extends this concept to ethology suggests that it provides a concise description of a wide range of collective phenomena in animals, especially in social insects. This description does not rely on individual complexity to account for complex spatiotemporal features that emerge at the colony level, but rather assumes that intractions among simple individuals can produce highly structured collective behaviours.
Cultural evolutionism ("survival of the fittest" in terms of cultural and social forms); society as organism (heavy organic analogy); evolution from homogenous state to heterogeneous state, increasing differentiation, specialization, division of labor and interdependence; society has reality beyond sum of individual parts; progress is driven by man’s innate adaptability to higher states of perfection
Publisher Summary This chapter discusses the information contained in the odorous secretions of mammals and provides a classification system based on the behavioral and chemical analyses. This classification divides social odors into two groups: identifier and emotive. Identifier odors are defined as those produced through the regular metabolic processes of the animal, without specific stimulation. The emotive odors are those produced as the result of some transient emotional state or external stimulus. The chapter categorizes nine different types of information contained in mammalian social odors: species, age, sex, colony membership, individuality, social status, reproductive state, maternal state, and stress odors. Social odors are modified by diet and hormone levels and by bacterial action. When the chemicals and bacteria responsible for producing the social odors have been identified, the responses of test animals show large individual differences. Responses to olfactory stimuli depend on hormonal and experiential factors. Theoretical models in the study of population regulation, sexual selection, kinship recognition, altruism, parental care, and territoriality infer that animals recognize particular individuals and specific relationships, and such recognition may depend to some extent on the information contained in olfactory signals.