Behavioral Ecology Vol. 11 No. 4: 387-395
© 2000 International Society for Behavioral Ecology
Optimality of cell arrangement and rules of thumb of cell initiation in Polistes dominulus: a modeling approach
a Department of Biology, University of Missouri, St. Louis, 8001 Natural Bridge Road, St. Louis, MO 63121-4499, USA b Department of Evolutionary Zoology, Kossuth Lajos University, PO Box 3, Debrecen, H-4010, Hungary
Address correspondence to I. Karsai. E-mail : bioikars@jinx.umsl.edu. Z. Pénzes is now at the Department of Ecology, József Attila University, Szeged H-6720, Hungary.
Received 27 May 1999; revised 30 September 1999; accepted 30 October 1999.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Possible adaptivity and mechanisms of nest construction of a paper wasp, Polistes dominulus, were studied by analyses of nest structures and modeling. Results suggest that nests are not built in agreement with the currently accepted "economy material usage" hypothesis because (1) the number of natural forms is much less than expected under this criterion, and (2) there are non-optimal structures. Maximization of nest compactness is a new hypothesis that better predicts natural structures. By examining the predictions of different building rules and comparing model-generated structures to natural nests, we found that the nest structure provides sufficient (quantitative) information for governing the building process on (or very near) the optimal path. We assume that non-optimal natural forms are the consequence of rules of thumb being used by wasps during construction. A family of rules based on information on the age of cells was able to account for all natural forms, including the assumed optimal and non-optimal forms.
Key words: construction behavior, Polistes, social wasps, stigmergy..
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Nests of social insects are one of the most sophisticated artifacts animals can create (Hansell, 1984
; von Frisch,
1975), and as Franks et al.
(1992) pointed out, the
building behavior of insect societies can provide a good framework for
studying the fundamental problem of biological pattern formation (see
Murray, 1989). Students of
artificial life commonly use the construction behavior of social insects for
inspiration and as a model system to study decentralized decision making and
control (Connell, 1990,
Beckers et al., 1994,
Bonabeau, 1997,
Karsai, 1999). The majority of
these studies have focused on how different structures emerge from a
homogenous medium through self-organized processes
(Camazine et al., 2000 ;
Deneubourg, 1977 ;
Franks and Denebourg, 1997 ;
Franks et al., 1992 ;
Karsai and
Pénzes, 1993 ;
Skarka et al., 1990 ;
Theraulaz and Bonabeau, 1995).
Studying architectural variation also has proved useful in understanding
aspects of the evolution of such diverse groups as termites
(Emerson, 1938) and paper
wasps (Ducke, 1914 ;
Karsai and
Pénzes, 1998 ;
Wenzel, 1991). Using building
behavior to connect productivity, colony size, and behavioral flexibility
provided interesting insights into the organization of colonies and the role
of parallel processing (Karsai and Wenzel,
1998).
The nests of social insects are generally much larger than an individual
builder and have a coherent structure formed through the repetition of few
construction units. The main unit of the nest of social wasps and some bees
(but not in other social insects such as ants and termites) is the hexagonal
cell, which is the modular basis of the comb. This regular unit generally
harbors only one offspring at a time. Combs are attached to a substrate
directly or by a petiole, and in larger nests combs can be a unit of
construction themselves in forming stacked comb nests
(Wenzel, 1991). Social wasp
nests seem to illustrate Darwinian adaptation through both specific function
(Jeanne, 1975) and widespread
convergence (Wenzel, 1991) of
many architectural details. Arguments about adaptivity in social wasps' nests
include the relationships between phylogeny and adaptation in cryptic forms
(Wenzel and Carpenter, 1994),
heat insulation (Seeley and Heinrich,
1981 ; Yamane,
1988, economy of material usage, and brood protection against ants
(Jeanne, 1975,
1979) or flying parasites
(London and Jeanne, 1999 ;
West-Eberhard, 1969), although
the majority of these concepts have not yet been tested or examined in
detailed.
Animals are likely to use "rules of thumb" that approximate
optimal solutions (Clark, 1991
; Janetos and Cole, 1981 ;
Houston and McNamara, 1984).
As Roughgarden (1991 ;
104-105) emphasized, "There is a need to explore how simple behavioral
decision rules (`rules of thumb') predict behavior that converges on an
optimal solution that we mere humans can only deduce with an expensive
computer." In social insects the ability to build at more than one
location on a nest simultaneously was an important evolutionary step for the
development of complex nest architecture (Karsai and Wenzel, 1998 ;
Michener, 1964).
Grassé
(1959,
1984) proposed stigmergy
theory to explain how stimuli and work are organized when parallel
construction is performed by several individuals. He envisioned the nest as
the result of a succession of stimulus response steps. The workers modify
their environment, produce a structure that provides new stimuli, and so
induce new responses. In this theory, no direct interactions are required
between the builders. The workers interact with each other through the
by-product of their previous activity. Stigmergy can be contrasted with the
use of recipes, where a set of predefined instructions specifies the sequence
of behavior. This results in a rigid behavioral program without feedback from
the structure being built, and it can be found in some solitary builders
(Smith, 1978). However, this
approach would make coordination difficult in large colonies
(Camazine et al., 2000).
In the case of a stigmergic type of construction, the key problem is to
understand how stimuli are organized in space and time to ensure coherent
building (Bonabeau et al., 1997
; Karsai, 1999). Theraulaz and
Bonabeau (1995) modeled
collective building with lattice swarms and showed that coherent structures
can be generated only by coordinated stigmergic algorithms. By modeling the
nest excavation of ants with a simple stigmergic mechanism, Denebourg and
Franks (1995 ;
Franks and Denebourg, 1997)
generated adaptive collective responses and showed how physical constraints
can be exploited by the behavioral program to produce these global patterns.
Karsai and Pénzes
(1993),
1998 ;
Karsai, 1999) showed that
stigmergy is a key mechanism to understanding both the diversity and the
development of nest structures in paper wasps.
Although several cues are known to affect the building behavior of social
wasps (Downing, 1994 ;
Downing and Jeanne 1990 ;
Karsai, 1997 ;
Karsai and Wenzel, 1995 ;
Karsai et al., 1996 ;
Wenzel, 1989), understanding
is incomplete with regard to what kind of stimuli ensure coherent building and
how they are organized in space and time. Our aim in this study, beyond
examining the optimality of round one-comb nests in regard to material economy
and structural compactness, was to explore the predictive power of simple
stigmergic rules of thumb that may be used by builders in Polistes
dominulus Christ colonies.
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METHODS |
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Nests
We mapped and analyzed 74 nests from different wasps in different laboratory cages (i.e., every nest was independent). Using a hexagonal grid, every cell was mapped as a regular hexagon, and their relative positions were aligned to the grid. This is a commonly used method that makes computerized analyses possible (Karsai and Pénzes, 1996
). No irregular
displacement or nonhexagonal cell was observed (see
Wenzel, 1989, for occurences
of pentagons and heptagons in larger nests of Polistes annularis).
Although very small cells have some rounded walls, for simplicity they were
also assumed to possess hexagonal shape. Different numerical variables were
derived to characterize the nests (Figure
1). The number of walls (W) makes it possible to estimate
the material used for construction. In cases of the same cell number
(N), the nests have the same reproductive output (both are able to
breed the same number of brood). However, if the number of shared walls is
higher (i.e., W smaller), then less material is necessary to build
the nest. Compactness (C) is a sensitive index to compare different
cell arrangements of same-sized nests
(Karsai and
Pénzes, 1996). Compactness is the
summed distance of every cell center from the two-dimensional geometric center
of the nests (Figure 1). Every
nest form was characterized by its cell number and relative compactness value.
For example, form F9a consists of nine cells and it is maximally compact,
whereas F9b has the same cell number but has less relative compactness. If
there is no letter after the cell number (e.g., F8), then only the most
compact form was found (Table
1).
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Assumptions of the models
We used models to describe the early development of the round nests characteristic of P. dominulus and several other species that possess the same nest type (Downing and Jeanne, 1986
; Karsai and
Pénzes, 1998). Nests were
generated by different algorithms, and these generated nests were compared to
the nest collection and the supposed optimal shapes. The models are based on
different construction algorithms but common assumptions :
- The growth of the nest is simplified into consecutive cell additions. Other
types of construction behaviors such as petiole strengthening are rare or, as
in case of both petiole strengthening and cell elongation, their stimuli seem
to be independent of cell initiation
(Downing, 1994 ;
Karsai and
Pénzes, 1996 ;
Karsai and Theraulaz, 1995).
Thus, the models are specified into a two-dimensional lattice, where only the
position of the next new cell is predicted. In this lattice the cells are
represented as regular hexagons, and the nest structures are defined as nest
forms.
- Analyzing a great number of wasp nests of different species (Karsai and
Pénzes,
1996,
1998) and the actual behavior
of the builder (Downing and Jeanne,
1990 ; Karsai and Theraulaz,
1995) has shown that new cells (except the first and second cell)
are always built so that at least two older cells support the new one. The new
cell will share common walls with those neighbors onto which the wasp builds
the given new cell. We refer to this as a structural constraint.
- Polistes initiate cells individually, and only a small number of
cells are initiated per day owing to the low fecundity of the queen
(Gervet, 1964 ;
Karsai, 1997). Because cell
initiation is rare, it is assumed that every cell is initiated independently
in turn. No rearrangement of the already built structure is allowed. Every new
cell is integrated into the previous structure governed by the structural
constraint described by assumption 2 and the construction rule being
analyzed.
- The embryonic stage in P. dominulus (from construction of the
first cell until the first larvae reach fourth instar) corresponds to a nest
size of 1-15 cells (Karsai and
Pénzes, 1996). In this stage both
the nest structure and the social milieu of the colony are the simplest. Large
larvae and pupae, which have considerable effect on building behavior
(Downing, 1994 ;
Karsai, 1997), and workers,
which generally take part in the construction, have not yet emerged. In this
stage only the queen, or mainly the queen in the case of a polygynous nest,
initiates the cells (Pratte,
1989).
- The largest extent of the embryonic comb generally is never greater than
two times the body size of the wasp. P. fuscatus, on average, spends
38 s and checks 18 positions before making a final building-site decision
(Downing, 1994). This leads us
to assume that in these small nests the wasps have the opportunity to check
every position ; thus the probability of the occurrence of the builder in any
buildable position is the same.
- We assumed that information collected for the decision is local. The wasp
can examine only the cell below its head and the next cell neighbors (those
with which the given cell shares walls) for decisions necessary for
construction. The collected information stems only from the nest structure ;
neither any external information (e.g., from brood or properties of substrate)
nor information flow between nest mates are presumed to affect cell
initiation. Because we assume that the collected information is local,
isometrical structures (mirror images) were considered to be the same nest
forms.
- Although the buildable positions are determined by the existence of two
older cells (structural constraint ; assumption 2), these positions provide
different stimulus values that are measured locally (assumption 6). We assumed
that the probability of initiation in a given position is determined
exclusively by this stimulus value (i.e., the probabilities of initiation of
positions with equal stimulus value are the same). Given the assumption that
searching behavior precedes a construction decision (assumption 5), it was
assumed that cell initiation happens where the stimulus is maximal, and in
case of equal stimulus values one position is chosen randomly.
- The nest size of a given colony is dependent on the egglaying capacity of
the egg layer, the larval development times, and the magnitude of oophagy
(Karsai et al., 1996). That is
why mature nest sizes vary considerably
(Karsai and
Pénzes, 1996). Because a final or
optimal nest size and form cannot be derived or assumed in advance,
prospective approach is used, and only the interplay between the current nest
form and the building algorithm determine the position of the next cell (and,
therefore, the subsequent nest form).
Markov Chains were used to generate all possible forms according to the general assumptions (1-8) and the specific construction rules (see later). This approach made it easy to calculate the absolute probabilities of the different nest forms (Appendix). The aptness of the models (predictability) was measured by simple indices :
PredComp = N° possible maximally compact forms/ (N° possible maximally compact forms + N° generated non maximally compact forms + N° non-generated maximally compact forms) ;
PredNat = N° found natural forms/ (N° found natural forms + N° generated non-natural forms + N° non-generated natural forms) ;
RelOccNat = 
(relative occurrence of natural forms) /N° natural
forms.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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The natural and optimal forms
In Polistes dominulus 18 nest forms were found : one form at every nest size except in the cases of 6, 7, and 9 cell nests where 1 additional form in each case was observed (Table 1). Although the collected nests were independent structures, these forms can be arranged according to their size and compactness value to form a series for possible derivation of these forms from each other (Figure 2). In this series, there is no missing form ; every form can be deduced from a smaller structure, and every structure can be conceived as the progenitor of a following larger form. The chain of the most compact forms is continuous, and returning to this chain from less compact forms is always possible in the next step (except from F6b, where two steps are necessary).
Using Delaurance's (1957)
scheme of the order of cell initiation in P. dominulus, we could not
derive any form not found in our collection. Morimoto
(1953) published a series of
nest maps of another species (P. chinensis antennalis Perez) that
builds round nests. These nests were marked as built in a "typical
way" (Morimoto, 1953),
but again this data series provided no new nest forms
(Table 1). The small number of
observed forms and their strict linkage suggest that the construction of nest
forms is canalized, and none or only a small number of natural forms remained
undetected. Without any canalization (i.e., when every buildable position has
the same stimulus value), 2994 nest forms could emerge in the same nest-size
range (Table 1).
The material economy hypothesis (proposed by
Jeanne, 1975) predicts that
nests possess a minimal number of walls, due to being built in such a way that
the number of shared walls is maximized. Our analyses showed that all natural
forms have the minimal possible wall number for each nest size with the
exception of the F7b structure, which possesses 31 walls instead of the
minimal 30 (Table 1).
Twenty-seven "economical" nest forms exist from the set of all
possible forms. We found only 17 of these 27 forms, and we also found
"non-economical" natural nests. This either means that not all
structures with minimal wall number are built in nature or that the material
economy hypothesis is insufficient to explain the nest shapes.
Our compactness hypothesis predicts that nests are built in such a way that the arrangement of cells ensures the most compact and condensed structure (i.e., C minimal). All optimal structures were found in nature. All natural forms had a minimal compactness value with three exceptions. These exceptions caused the bifurcations of the putative development path of natural structures (Figure 2). Although the relative compactness value of the F7b structure was low, the other two exceptional forms (F6b, F9b) were closer to the maximum (Table 1). All most compact structures posessed the minimal possible wall number (i.e., building in a compact way also solved the minimization of building material). From this finding we suppose that the current nest forms are the consequence of constructing in a compact way. Although the system may gain evolutionary benefit from this global property (see Discussion), the questions of how these forms are actually built and why non-optimal forms can be found are addressed in the next section.
Rules of thumb
By experimenting with a simple model, we can examine different rules of
thumb that are able to produce structure according to the basic common
assumptions and a specific algorithm. Our goal is to find algorithms that
generate both the supposed optimal (minimal compactness) and natural (optimal
+ errors due to the rules of thumb) forms.
Wall number
In our two-dimensional model, cell initiation can be simplified into wall
additions to peripheral cells, where these new walls and one outer wall of
each of two or more adjacent existing cells form the new cell. One older cell
can provide only one wall for this new cell ; thus, the number of walls needed
to create a new cell in position i is determined by the number of
i's neighbors. Assume that the stimulus related to the number of
neighbors, rule Max W, was established : a new cell is added to a
position where the number of new walls necessary to form the new cell is the
minimum (i.e., the number of neighbors of a buildable position is the maximum
; Figure 3 and Appendix).
If there is an open row on the comb, it means one or two positions provide three ready walls for the next new cell initiation (i.e., three neighbor cells would support the new one). Cell initiation will happen in this position (or in one of the possible two) because this position requires adding the smallest number of new walls. When consecutive initiations have completed the cell row, the only positions available will require building four new walls for the next cell. Given that all positions have the same stimulus value, any of these positions will accept the next new cell with the same probability. Although the forms generated by MaxW contain both all natural and all most compact forms, the number of total forms generated by this rule was much higher (155) than we observed in nature, decreasing the predictability of this model considerably (Table 1).
Age models
Because the number of walls seems unable to provide enough information for
the observed canalized construction, a new stimulus type was defined in which
the stimulus was related to the age of the cells. Because cells are added to
the nest individually, there is an order of emergence of the consecutive cells
that we can use to define the age of the cells. Counting the number of cell
initiations since a given cell has existed and assigning this value for every
cell gives us a tractable value for this type of stimulus. Mapping these
structures in the same way as for natural nests (nest forms based on cell
arrangement), we can compare the prediction of these models to both natural
structures and the prediction of other models
(Table 1).
In the first two algorithms the new cell is built next to the youngest cell. The two possibilities are that the new cell is built in such a way that it connects the youngest cell to its older (MinAo) or to its younger (MinAy) neighbor (Figure 3). Both algorithms generated 15 structures. The MinAo structures were long, two cells wide, and after the fifth cell stage they showed neither resemblance to any compact form nor to any nests built with minimal wall number (Table 1). On the other hand, minAy forms were maximally compact except in the 9-and 15-cell stages, where MinAy predicted less compact structures that possessed minimal wall number.
In the next two algorithms the new cell was initiated next to the oldest cell. When initiations were restricted only to that side of the oldest peripheral cell where its neighbor is younger (MaxAy), the result was the same spiral growth and the same structures as predicted by minAy. The two algorithms are the same because the new cell links the oldest and youngest peripheral cells (i.e., initiation happens where the difference between the ages of cells is the largest). When a new cell is initiated to that side of the oldest cell where its neighbor is older (MaxAo), then only the most compact forms are generated. This rule of thumb, using only local stimulus value, is able to generate globally optimal forms, but it cannot explain the bifurcations in the natural set of forms.
The next two algorithms assume that the wasp need not initiate the new cell next to the youngest or the oldest one, but rather it should consider the age of all neighbors of the given position. When the wasp initiates the new cell where the summed age of all neighbors is minimal (SumAmin), the predicted structures and also the construction itself are the same as was predicted by MinAo algorithm (Figure 3). When the wasp initiates new cells in a position where the summed age of the neighbors is maximal (SumAmax), then several predicted routes of construction emerged. There is a compact route (Figure 4), where only the most compact structures are built ; however, there is a bifurcation at two points because in five-cell and nine-cell nests there are two positions with the same maximal stimulus value. When the wasp initiates in the alternative position (not leading to a compact structure) on a five-cell nest (Figure 4), it will lead to F6b and F7b forms. When the alternative route is chosen on a nine-cell nest (Figure 4), a new form (F10b, which was not found in the natural nest series) emerges.
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The SumAmax predicts all compact and natural stuctures except F9b and predicted only one additional form (Table 1). The emergence of F9b can be conceived as a result of shifting the initiation a few millimeters (i.e., the new cell was initiated at the same oldest peripheral cell that the model predicted but on the other side of it ; Figure 5). If we allow this misplacement to happen, then two alternative routes emerge (depending on which route was started after the bifurcation at five cells). If the construction goes through F6b and F7b, then after the emergence of F9b only optimal forms emerge (Figure 4). On the other hand, if the initiation goes through the most compact route initially, then the shift that generates F9b will lead to one more extra structure (F14b) later (Figure 4).
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DISCUSSION |
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Studying the early construction (up to two cells) of P. fuscatus, Downing and Jeanne (1988
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concluded that the building process cannot be explained by stigmergy theory
(Grassé,
1959 ;
Wilson, 1971) because,
although the nest provides cues for the subsequent construction, the wasps use
additional types of cues and evaluate some of the same nest features at each
step of construction. Karsai and Theraulaz
(1995) revealed some of these
cues for the same early period in P. dominulus, showing the mutual
relationships between the structural constitution of the incipient nest and
the postures, movement, and working possibilities of the builder. On the basis
of a theoretical framework suggested by Dawkins
(1976), Downing and Jeanne
(1990) proposed that the
construction can be represented by a non-overlapping, branching hierarchy of
choice levels, where at each level multiple cues are evaluated either
simultaneously or in a hierarchical manner. However, simulations on the basis
of a simple stigmergic algorithm (Karsai,
1999 ; Karsai and
Pénzes, 1993 ;
Pénzes and
Karsai, 1993) showed that lifelike nests can emerge, using only
immediate local cues, from a simple algorithm that equalizes local
irregularities. The algorithm does not invoke any learning or different
subroutines, nor does it require systematic information collection,
information storage, processing, or evaluation. Moreover, different nest
shapes characteristic of Polistes can be produced by the same
fundamental building algorithm (Karsai and
Pénzes, 1998). The result of this
kind of construction should be adaptive (or at least sufficiently good) global
structures ; thus, it is reasonable to ask questions about how and why at the
same time (Houston and McNamara,
1984).
The adaptivity of the nests
We have shown in P. dominulus that the economy of nest material
hypothesis predicts numerous extra forms we did not observe in nature.
Moreover, building by giving preference to positions where the number of new
walls necessary to add to the nest is minimal (i.e., the number of ready walls
is maximal ; Max W) generated lifelike structures with only moderate
success. This conclusion would not be clear from the study of natural forms
only. We also showed that the arrangement of the cells (i.e., the shape of the
nest) is strongly constrained. The majority of observed natural forms
represent the most compact structures that it is possible to construct in this
system, and those that were not maximally compact possessed the second-most
compact cell arrangement possible. With this analysis we did not provide a
causal explanation of why this compact arrangement emerges (i.e., whether it
is adaptive for thermal biology, crypsis, or physical stability), but we
substituted the economy of nesting material hypothesis with a more
parsimonious one. We also showed that compactness is a strict, global, and
presumably adaptive property of the nests. This may support or may be
supported by some of the adaptive hypotheses listed above. For example,
physically testing the sturdiness of nests with different compactness probably
will show the benefit of compact cell arrangement, even if this is not the
real evolutionary reason for this type of structure. This compact property may
also be the consequence of the rules of thumb used for construction
behavior.
In a different study we showed that the main nest shapes characteristic of
genus Polistes can be derived from the same simple building algorithm
(Karsai and
Pénzes, 1998). The scale of that
model was much larger than the one in the current study (i.e., initiation of
cell rows was studied), but our earlier study showed that interplay between a
stigmergic building mechanism and an external parameter are sufficient to
reproduce the variability of nests in this genus. We showed that the
differences in nest shapes do not necessarily result from gross differences at
a behavioral level, but rather they arise from a quantitative modulation of
the building rule (e.g., by changing response thresholds). The adaptive
radiation of social wasps may have relied on a flexible general construction
algorithm. In this case, nest shape early in the evolution with a small
modification of the basic rule could easily be adapted to new conditions (see
adaptivity hypotheses above). In response to sustained selection forces, the
tuning parameter of the building program of different populations may have
become fixed, resulting in the current characteristic nest shapes.
Rules of thumb
Constraints from the structures seem to canalize nest shape in the early
development stage. Before the nest possesses two cells, the possible postures
the wasp can perform on this small structure are limited, and this leads to
limited variability in the early structures
(Karsai and Theraulaz, 1995).
The structural constraint, defined in the current study as an existence of a
groove between cells as a necessary prerequisite for cell initiation,
determines the shape of a nest possessing five or fewer cells. However, this
constraint alone cannot be responsible for the small number of natural nest
forms because as the nest becomes larger it possesses more and more positions
where initiation of new cells is possible. If all such positions have the same
stimulus value (random choice), a great number of forms (2989) emerges in the
range from 6 to 15 cells. The small number of natural forms and their compact
properties required us to find algorithms that control the nest development
more strictly (i.e., that provide different stimulus values for possible
initiation positions).
Finishing a cell row before starting a new one (i.e., to initiate at the
positions where only three new walls are necessary to complete a cell ; rule
Max W) seemed a logical building rule supported by literature and
earlier models (Downing and Jeanne,
1990 ; Karsai and
Pénzes, 1993 ;
Pénzes and
Karsai, 1993). This rule can be translated into an immediate
benefit in construction material because three walls of the new cell already
exist ; only three new ones need be constructed, and higher walls can be
obtained using the same quantity of material. By studying the transitions of
natural forms, it was obvious that any time a form emerged with an open row,
the next initiation occurred in a position where only three new walls were
required to build the new cell. The Max W rule would be sufficient to
explain how the structures go toward the most compact forms after the
bifurcations (F7b 
F8 and F9b F10 transitions). Although this
rule predicts very well where the next initiation will be when there is an
initiation position with three walls, it behaves like the random rule when
these types of initiation position are missing, i.e., when every position has
the same stimulus value (Appendix). Hence, this rule predicts a high number of
structures that we did not find in nature ; they are neither compact nor
possess minimal wall number (for example, under this rule from F5 a six-celled
form with two rows of three cells may develop).
The comparison of natural nests with the forms generated by the Max
W rule suggested that the initiation positions that possess two ready
walls do not provide the same stimuli ; some of these positions are preferred.
For example, the middle part of cell rows is commonly a preferred position for
initiation of a new row (Downing and
Jeanne, 1990). Introducing a new type of stimulus (age of the
cells) allowed us to predict the nest structures of more rules. The best rules
predicted those phenomena that the earlier rules were unable to predict. On
the other hand, two rules (logically the two are the same) led to the extreme
two-row wide, long-comb form found in this genus only in Polistes
goeldii. This finding reinforces the results of our previous model on
nest diversity of Polistes
(Karsai and
Pénzes, 1998) that showed that all
main nest forms can be derived from the same construction rule. Using the same
type of stimulus and construction rule, a very small difference in the usage
of the same information in the decision on initiation will lead to completely
different nest forms.
Rule MaxAo predicted the most compact nests only. If nests with maximal compactness are adaptive and there is a rule of thumb that can be used to construct these structures, then why did we find nonoptimal forms in nature ? Rule MaxAo requires finding the oldest peripheral cell and its older neighbor. In real wasps this may not be easy because the perception radius of the wasp may be larger than one cell wide, and stimuli that stem from neighborhood positions may be integrated in some manner. The SumAmax and MaxAo are similar rules based on the high value of local stimuli, but SumAmax integrates (sums) the stimuli for a given position rather than seeking a mere maximum irrespective of the local configuration. The SumAmax rule is more naturalistic in this manner and also was able to predict the natural bifurcations ; we considered this rule to be the best predictive model for cell initiation. This algorithm predicted only a single extra structure we did not find in nature (F10b). This structure has two positions with three ready walls, and in this manner it is exceptional. The SumAmax algorithm generally predicted well those initiations where three walls are ready to accept the new cell (as in MaxW) because in these cases the information (age) is integrated from three cells rather than two, and this generally provides the highest stimulus value of the nest. However, in the case when F10b is built, two positions (one with three neighbors and one with two neighbors) provide the same highest stimuli (value 14) that will lead to the occurrence of F10b. The SumAmax algorithm failed to predict the bifurcation after the eight-cell stage ; however, F9b can be explained by this algorithm by assuming a small error in the decision in which the new cell was initated onto the same cell predicted by the algorithm but on the other side of it. Assuming the possibility of this error will lead to structure F14b with 50% probability. This form is missing in the natural series, and it is also similar to F10b in that it has two positions with three walls. These two structures are predicted to be rare and are expected to be found in nature. We also assume that construction is not deterministic (as our model supposed) because when the structure is bigger, probably not all possible construction positions may be checked. This also means that some forms that are not predicted by the best model may be found in nature later.
The stimulus
The exact nature of the proposed stimulus is yet unknown. Our models worked
with variables such as number of walls and age of cells. Using these
parameters we did not assume that the wasp actually is able to count the
number of cell walls or is able to follow the age of every cell. These were
used as tractable values to represent the yet unknown stimuli. It was assumed
that the actual stimulus correlates highly with these values. The main aim of
our theoretical experiments was to examine the feasibility of several
reasonable cues. Our models have shown that we have to search for a variable
(or a combination of several) that correlates with the age of the cells.
Downing (1994) studied cell
lengthening and suggested that in the early stage the relative cell length is
the most important stimulus in cell elongation. Cells in incipient nests
increase quickly in length ; thus cells with different ages may differ
considerably in cell size. In our earlier model
(Karsai and
Pénzes, 1993), we used similar
types of stimuli for both cell lengthening and initiation, and this resulted
in lifelike, three-dimensional round nests. However, the stimuli are not
necessarily mechanical. In larger nests wasps prefer to initiate cells near
those cells that harbor pupae (Karsai,
1997), even if these cells are not longer than those containing
large larvae. This suggests that chemicals may be involved in the decision on
where to initiate a cell. Different chemicals are applied to the nest during
building and in the course of other activities
(Cervo and Turillazzi, 1989 ;
Downing, 1991 ;
Espelie et al., 1990). If
these applications are independent of the cell positions, then it will lead to
a chemical pattern where older cells contain more chemicals than younger ones.
By touching cells with its antennae, the wasp may collect information about
the quantity of these nonvolatile or slowly volatile chemicals. Of course,
this supposed stimulus may not necessarily be a single special chemical [Ishay
and Perna (1979) proposed the
existence of a building pheromone that has not yet been discovered] ; it can
be the same chemicals as used for kin recognition
(Singer and Espelie, 1996), or
it can be a more complex stimulus that also includes mechanical, chemical, and
other components.
The number of natural forms increases when the nest goes beyond the
embryonic stage (Karsai and
Pénzes, 1996). Several positions
with three ready walls may exist, and neither the most compact nor economical
nests may emerge. This may be because the wasp does not check every position
on the nest if the nest is large (Downing,
1994 ; Karsai and Theraulaz,
1995) ; thus a wasp may not necessary build in a position with the
highest stimulus value, or errors, as we assumed in SumAmax, may be
common. We also can easily imagine that as the nest increases in size and
becomes older, the differences among the initiation positions may become
smaller (e.g., cell size or chemical concentration may reach its maximum
value), which also would lead to more random building. Wenzel
(1989) suggested that the
queen and the worker have different building rules, and this could account for
more irregular older nests. However, irregularity of larger nests can also be
the consequence of the simple rule of thumb used for nest building if the
stimulus pattern equalized. Due to the compact core of the nest, which was
built in the embryonic phase and in the absence of physical obstacles or nest
damage, the simple stigmergic rule will ensure that the nest will not diverge
far from the supposed adaptive form, even if the decision about the position
of new cells is less unequivocal in later stages.
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ACKNOWLEDGEMENTS |
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We are especially grateful to Z. Varga (Kossuth University) and J. H. Hunt (University of Missouri, St. Louis), who have constantly supported and encouraged our research. We greatefully acknowledge the comments of two anonymous reviewers. This work was supported by the Hungarian National Science Foundation (OTKA F-020572) (to I.K. and Z.P.), the Hungarian Commercial Bank "A Magyar Tudományért" Foundation (to Z.P.), by the Széchenyi István Scholarship Fund (to I.K.) and by a grant from the University of Missouri Research Board (to J. H. Hunt, Principal Investigator).
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