Self-organizing nest construction in ants: individual worker behaviour and the nest s dynamics

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1 Anim. Behav., 1997, 54, elf-organizing nest construction in ants: individual worker behaviour and the nest s dynamics NIGEL R. FRANK* & JEAN-LOUI DENEUBOURG *Centre for Mathematical Biology, University of Bath Centre for Non-linear Phenomena and Complex ystems, Université Libre de Bruxelles (Received 24 June 1996; initial acceptance 1 August 1996; final acceptance 25 October 1996; M. number: 5274R) Abstract. We examine nest construction in the ant Leptothorax tuberointerruptus at two levels: (1) the building behaviour of individual workers and (2) the collective properties (temporal and spatial) of the structures they create. We also explore, for the first time explicitly, the linkage between these two levels. Leptothorax tuberointerruptus nests occur in flat cavities which provide the roof and the floor of their dwelling places. Hence, they construct only a peripheral encircling wall, breached by one or more entrance passageways. The wall is constructed brick by brick. This facilitates experimental estimation of the probabilities of individual workers picking up and depositing building material in response to different stimuli. We incorporate both the qualitative and quantitative behavioural rules that workers employ during building into a mathematical model. This model confirms that a surprisingly small and simple set of behavioural rules are not only sufficient for wall construction but also for the formation of one or more nest entrances. In addition, this model predicts that the nests of these ants are likely to exhibit interesting dynamics, in which, for example, the tendency to build a new larger nest may lag behind growth of the population that the nest has to house. We present experimental evidence that suggests that this prediction is valid The Association for the tudy of Animal Behaviour The nests of social insects are amongst the most complex and sophisticated of all animal artefacts (von Frisch 1975; Hansell 1984). As Franks et al. (1992) pointed out, the building behaviour of social insects is also a classic example of the fundamental problem of biological pattern formation (see Murray 1989). In an earlier paper on building by the ant Leptothorax, Franks et al. (1992) described the overall dynamics of building, the relationship between colony size and nest size and how some simple rules of thumb employed by individual workers might generate both the pattern and the process of such nest construction. In that earlier paper the question was posed how do social insects collectively build sophisticated nests in the apparent absence of central planners, architects, blueprints or quality control overseers? An alternative to such centralized control is Correspondence: N. R. Franks, Centre for Mathematical Biology, chool of Biology and Biochemistry, University of Bath, Bath BA2 7AY, U.K. J.-L. Deneubourg is at the Centre for Non-Linear Phenomena and Complex ystems, CP231, ULB, Bd du Triomphe, 15 Bruxelles, Belgium. self-organization in which larger scale patterns and processes emerge in a system through large numbers of (smaller scale) local interactions among the components of the system. In selforganization the components interact locally without an overview of the pattern or the process they are helping to create. Furthermore, there are no administrators organizing the system by monitoring and communicating information on the status of the global pattern (see, for example, Deneubourg et al. 1989). In this paper, we further examine building behaviour in the ant Leptothorax (Myrafant) tuberointerruptus. We focus first on the behaviour of individual building workers and estimate their probabilities of picking up and depositing building blocks. We then incorporate these quantitative building rules into a mathematical model which makes testable predictions about the properties of this system. Leptothorax tuberointerruptus colonies are unusually good subjects for studies of building behaviour in ants. This can be attributed, in part, to the simplicity of the structures they create. The /97/ $25.//ar The Association for the tudy of Animal Behaviour 779

2 78 Animal Behaviour, 54, 4 colonies nest within flat crevices in rocks which provide the roof and floor of their dwelling place. In the field these colonies often wall themselves into their chosen crevice by encircling themselves with a border of discrete debris, mostly small fragments of stone or particles of earth. In the laboratory such colonies will readily nest between microscope slides and, given a supply of carborundum grit or coarse sand, will build a wall of an appropriate circumference to house comfortably their population which typically consists of a single queen, brood and up to 5 workers. Each piece of grit or sand, that is, each stone, is manipulated separately by the building ants. The resulting brick by brick construction process also simplifies the analysis. The great advantage of using the building behaviour of these ants as a study system is that, in the laboratory, all the ants and all their building materials are visible throughout the building process. For this reason they provide an almost unrivalled opportunity to quantify not only the global pattern of the building but also the building behaviour of individual workers. OBERVATION AND EXPERIMENT General Methods Leptothorax (Myrafant) tuberointerruptus is a provisional name for a species that has been known in Britain as Leptothorax tuberum but can be distinguished by allozyme analysis (P. Douwes, personal communication; see Douwes & tille 1991; Franks et al. 1992; Bolton 1995, page 246 for further explanation). The source in the field of the experimental colonies, the culturing methods and observational techniques are identical (unless stated otherwise) to those used and described by Franks et al. (1992). Indeed, for the detailed analysis of individual building behaviour we re-examined videotapes of three of the colonies analysed by Franks et al. (1992). For consistency, we use the same colony designations as in that paper. Colony 3 had a population of approximately 9 workers. Colonies 4 and 9 both contained between 4 and 5 workers. (The overall dynamics of building by these colonies are shown in Figure 3 in Franks et al ) In these videotaped experiments the colonies were allowed to emigrate into new nest sites constructed from two microscope slides of mm held.8 mm apart by small pieces of cardboard at each of the four corners. Building material in the form of carborundum grit approximately.5 mm in diameter was available outside each nest site. We recorded the building process in each of these experiments by using a time-lapse video-recorder (Panasonic NV-85). uch a recorder permitted continuous videotaping for 48 h by capturing 1.5 frames per s instead of the normal 25. Given the time scale of individual building behaviours little information is lost through such time-lapse recording. We made qualitative and quantitative records of the behaviour of individual workers from these videotapes of the building process. In a number of cases we analysed the movements not just of ants carrying stones, but also of unburdened ants that were free to start carrying stones, and of the stones themselves. To get accurate unbiased records we often had to play the videotapes both forwards and backwards many times, often at reduced speeds, to examine each event of interest. In some experiments we removed either half or three-quarters of the worker populations of colonies and reunited them after 5 1 days when all building by the remaining part of the colony had been completed. The half removal experiments were conducted in December 1991 and the threequarters removal experiments were conducted in February Both of these experiments, in common with all the other observations of building behaviour reported both here and in Franks et al. (1992), were conducted at a standard room temperature of about 2 C. In these experiments the procedures were slightly different. ome of the colonies were collected from Goblin Combe near Bristol (as were colonies 3, 4 and 9) and some were from Bolt Head in Devon. ix colonies (designated A F) were used. All colonies had a single queen. Colonies A F had, respectively, 175, 121, 86, 76, 56 and 77 workers. We conducted all these experiments in the laboratory using nest sites of the form shown in Fig. 1. In these particular experiments the ants had been given a cavity.8 mm deep between glass plates measuring 8 8 mm. Blue-dyed, sieved sand was substituted as the building material but, as in the case of the carborundum used earlier, the diameter of each grain was about.5 mm. We estimated the internal area of completed nests by cutting around a paper tracing of the perimeter of the inside of

3 Franks & Deneubourg: elf-organizing nest construction 781 Figure 1. A Leptothorax tuberointerruptus nest in the laboratory. The ants have been given a cavity.8 mm deep between glass plates of 8 8 mm. They have built a dense wall from sieved sand. The worker ants, each approximately 2.5 mm in length, are densely clustered around the central queen, the largest individual present, and the white brood. There is a gap or corridor between the dense brood/adult ant cluster and the inside of the wall. The single entrance to the nest is at 7 o clock. the nest and feeding the paper into a suitably calibrated leaf area meter (see Franks et al for more details). Results and Discussion One result of the earlier experimental analysis was that there can be a rather precise relationship between the population size of the colony and the area enclosed by the nest wall (Franks et al. 1992). In other words, the ants appear to build themselves a nest of just the right size. Each nest is, seemingly, neither too large nor too small: each adult worker had about 5 mm 2 of floor area in the nest (Franks et al. 1992). This relationship between the size of the colony and the size of its nest held true whether the ants had to clear a cavity of scattered building materials or bring new building materials into the nest from the outside world. In this paper, only nest building in nest cavities that were initially free from building materials was examined.

4 782 Animal Behaviour, 54, 4 Observations show that only a small fraction of the workers in a colony collect building materials (P. J. P. Croucher & N. R. Franks, unpublished data). We henceforth refer to the workers that leave the nest site to collect new building material as external workers. The workers that remain within the new nest site we call internal workers. The building process has a number of distinct stages. (1) The external workers typically depart from the tight cluster of their nestmates inside the nest, collect a single grain of building material, which can be as large as their entire head, and return with it to the nest. The first few operations of this kind often have a rather characteristic pattern. (2) The ant walks back into the nest with its stone and drops it within a distance of one or two of its own body length from the cluster of its nestmates. (3) The ants carrying stones tend to release their stone after they make direct contact with a cluster of their internal worker nestmates or other stones that have been previously deposited. The latter stimulus grows in importance as more and more stones are deposited. Indeed, at this later stage, the ants seem actively to use the stone they are carrying as a battering ram and release their stone only after they have felt the resistance of other stationary stones. uch ants actively bulldoze their stone into others (Franks et al. 1992). (4) The ants that retrieve building material from the outside world rarely if ever pick up a stone that they have dropped inside the nest. (5) The ants that remain in the nest, particularly those on the outside of the tight cluster of workers around the brood are, however, frequently seen to pick up stones that are close to them and bulldoze them outwards again. We quantified these simple behaviour patterns in some detail. We consider first the behaviour of the builders, the external workers, which initially deposit stones, and then of the builders, the internal workers, which relocate such material. Estimating the probabilities of building material being picked up and deposited For brevity, and because separate analysis per colony revealed extremely similar results, we pooled data from the separate experiments with colonies 3, 4 and 9 wherever possible. (1) We recorded the location at which each of 3 stones was initially dropped in terms of its proximity to other items in the nest. We observed such stones for each of the three colonies 3, 4 and 9. Most stones were dropped by (i.e. within one antenna s length), or pushed against, other stones (21 events, i.e. 7% of the total), but stationary nestmates were also used in the first stages of building as cues for dropping stones (27 events, 9%) and spontaneous dropping of stones was also recorded (i.e. dropping with no known stimulus; 63 events, 21%). This study was conducted during the first stages of building when less than 2% of the final number of stones had been carried into the new nest cavity. (2) We recorded how long each of the 3 stones was carried (see (1) above), from the moment the laden ant entered the nest cavity to the moment it dropped the stone. There was an exponential decay in carrying times, that is, the probability of an ant dropping a stone was almost perfectly constant per unit time (Fig. 2). The mean time that a stone was carried was (1/.48), that is 21 s. (3) We made 5 independent observations of haphazardly chosen ants in colony 3 that were carrying stones into the nest. Videotapes of such behaviour were played at slow speed and the number of stones each laden ant passed by, within one antenna s length, before dropping its stone were recorded. Of these 5 ants, seven dropped their stones spontaneously, that is, before encountering a stone. The following analysis is based on data from the remaining 43 ants. Figure 3 shows the natural log of the number of stones still being carried as a function of the number of stones encountered. The relationship is best described by the natural log of the number of stones still being carried= (stones encountered) (r 2 =.92). This suggests that there is a constant probability of between.3 and.5 that a laden ant will drop its load when it encounters a stone. Findings 1, 2 and 3 are entirely consistent with one another. External workers seem to make a fairly random walk within the nest until they encounter their nestmates or stationary stones; the latter are the major cue for the deposition of another stone. In the first stages of nest construction, stones may be encountered fairly haphazardly and there is a constant probability of deposition when a stone is encountered. These two factors explain why the probability of

5 Franks & Deneubourg: elf-organizing nest construction Natural log of the number of stones still being carried Natural log of the number of stones still being carried Duration of stone carrying (s) Figure 2. The natural log of the number of stones still being carried by 3 ants as a function of time since they entered the nest cavity. The relationship is best described by the natural log of stones being carried= t (r 2 =.994; where t is time in s). Note that a progressive negative relationship between these variables is inevitable, hence a high r 2 value is to be expected. However, the very high value observed and the extreme linearity of the plotted relationship suggest that the probability of an ant dropping a stone is almost perfectly constant per unit time, implying that carrying time decays exponentially. dropping a stone is fairly constant per unit time and in turn this accounts for the exponential distribution of carrying times. The behaviour of builders relocating material (4) In the vast majority of cases internal workers that moved stones pushed them outwards from the centre of the nest towards its periphery. In 51 independent observations of haphazardly chosen ants in colony 3, 72% of the ants moved the stone outwards, 12% pushed the stone inwards and 16% pushed laterally. uch internal workers also tended to bulldoze the stones they displaced into one another. (5) To determine the probability that an internal worker will pick up a stone, we played the videotape of the building process forward until we saw a haphazardly chosen stone moved by an internal builder. From that exact moment, we played the videotape backwards and recorded the number of ants passing the stone, within one Number of stones encountered Figure 3. The natural log of the number of stones still being carried, by external workers entering the nest cavity, as a function of the number of stones encountered (for 43 stones that were dropped near other stones). There are only six points in the graph because, of the original 43 stones, some were dropped when the worker encountered one stone, two stones, three stones, etc. Hence, there are few categories of such events. antenna s length, before it was moved. This period of observation terminated when the stone was seen, on the reversing tape, to be moved previously. In this way, the number of ants moving past a stone in the time between it being moved and left and moved again could be recorded. These observations were made in areas of active building. We made 2 independent observations of more-or-less isolated stones lying within the nest of colony 3, in what was to become the corridor, that is, the gap between the edge of the cluster of adult and brood ants and the nest wall. Figure 4 shows the natural log of the number of stones not yet moved against time. The graph can be thought of as a survivorship curve for stationary stones. The relationship is such that the natural log of stones not yet moved= t (r 2 =.95; where T is the time in min; three clear outliers (corresponding to times of 36, 372 and 748 min) were excluded from the analysis). This relationship suggests that the mean lifetime of a stationary stone in this part of the nest is (1/.171 min), that is, almost exactly 1 h. Figure 5 shows the natural log of the number of stones not yet moved against the number of ants passing by each such stone within one antenna s 8

6 784 Animal Behaviour, 54, 4 Natural log of the number of stones not yet moved Time (min) 18 Figure 4. The natural log of the number of not yet moved (more-or-less isolated) stones lying within a nest plotted against time (min). Natural log of the number of stones not yet moved Number of ants passing by each stone Figure 5. The natural log of the number of stones not yet moved against the number of ants passing by each such stone within one antenna s length (from the same set of observations as in Fig. 4). This relationship is best described as the natural log of the number of stones not yet moved= (number of ants passing by) (r 2 =.88). length. The relationship between these variables suggests that the probability of an unladen ant picking up such a stone in the nest is relatively constant and about.1. Figures 4 and 5 show that stones in the same area within the nest are essentially the same in terms of their chance of being picked up. However, this does not imply that the location of a particular stone is unimportant in terms of the likelihood that it will be moved. The more 2 ants passing a stone the more likely it is to be moved in any one moment. In simple terms, it appears that the stones that most frequently get in the way of the ants inside the nest are those that are most often shifted outwards. tones that are deeply buried in the wall can only be passed near their exposed face, whereas stones in the open can be passed by on all sides. Hence stones are likely to remain stationary for long only when they have been incorporated appropriately into a wall. Findings 4 and 5 show that the workers relocating material generally move stones outwards and that stones that are most exposed to ant traffic are those most likely to be moved. In general, these results show that the building workers, both those inside the nest and those coming into it with new building materials, seem to have rather simple behaviour patterns and almost no direct communication with one another. (6) There is no evidence that these Leptothorax builders use either pheromone trails or cement pheromones, as used, for example, by termites (Howse 197; tuart 1972; Bruinsma & Leuthold 1977; Jones 1979, 198) to coordinate their building activity. Pheromone trails are used only a little by Leptothorax workers (but see Aron et al. 1986, 1988; Maschwitz et al. 1986) and their building stones are not porous and are not manipulated in the mandibles of these ants as are the soil particles used by termites which are mixed with cement pheromones (Howse 197; tuart 1972; Bruinsma & Leuthold 1977; Jones 1979, 198). In the case of Leptothorax ants, other forms of positive feedback probably substitute for the absence of positive feedback associated with pheromone building signals becoming stronger and stronger as building work proceeds. In this regard, the observation that both internal and external builders tend to bulldoze stones into other stones is very important. As more stones accumulate in the nest, more active sites become available into which other stones can be pushed. In a sense, through the ants behaviour, stones attract other stones and this is an important source of positive feedback in the building process. Hence building work can speed up. In addition such bulldozing explains how the wall comes to be so tightly packed and densely consolidated (see Fig. 1). A colony is capable of building a new set of walls in about 24 h. The global rate of stone deposition ranges from.3 to.25 granules/s (see Figure 3 in Franks et al. 1992).

7 Franks & Deneubourg: elf-organizing nest construction 785 Total number of ants per grid square in 15 freeze frames mm Evidence for the use of a template in Leptothorax building There is good evidence that Leptothorax workers use a template partly to organize their building work. (7) It appears that the cluster of adult workers around the carefully sorted brood cluster (see Franks & endova-franks 1992) serves as a mechanical template to determine where the nest wall should be built. Indeed, the behaviour of the first building workers who make contact with the cluster of their nestmates and then pace out a relatively short distance before depositing their building material strongly suggests that the cluster acts as a template (Franks et al. 1992). We recorded the density and location of adult worker ants in nests just before or during the first stages of building. The workers distributed themselves at a fairly high and even density over the brood but away from the brood cluster their density declined in all directions. Altogether, the workers in the nest formed a cluster with a fairly high level of radial symmetry (Fig. 6). The cluster of adult workers and brood in the new nest may produce pheromone signals or cues that help to determine where building material should be deposited Figure 6. The roughly Gaussian density distribution of ants in the nest. The data were obtained from the videotape of colony 4 showing the nest during the building process. The data shown are derived from the pooling of data from 15 separate freeze frames evenly distributed over the period in which most building work occurred. The location of the ants within the nest was recorded on a gridded tracing of 4 4 mm squares. Colonies 3 and 9 (not shown) had similar Gaussian density distributions. mm (8) It is well established that numerous ant species including Leptothorax mark their nest ground and environment and this marking affects the movements of the individuals (see Hölldobler & Wilson 1978, 1986, 199, pp ; Aron et al. 1986; Maschwitz et al. 1986). imply as a by-product of their centralized organizations, a circular gradient of marks can be created with the density of marks decreasing with the distance from the nest centre. It is also clear that clusters of adult ants and their brood produce chemical signals or cues that help other adult ants to orient themselves within nests. For example, Wilson (1962) and Hangartner (1969) have shown that ants can orient to carbon dioxide gradients (see also Hölldobler & Wilson 199, pp. 289, 291). We use the term template, as a shorthand, for the way in which the cluster of adult ants and their brood influences the behaviour of building workers and hence the location of the walls of the nest. This does not imply that the template provides individual workers with global knowledge concerning the structure they help to build. For the individual worker picking up and depositing building blocks, the template (probably through both physical collisions and chemical gradients) provides only local information. The template can be thought of as providing a locally restricted boundary zone where building is most likely to occur. Nevertheless, the use of templates in building does not explain all aspects of the building process. For example, in these leptothoracine ants how could a template explain the production of a single nest entrance? Removal and reunification experiments Figure 7 shows the results of the removal and reunification experiments. In all five replicates in which half the adults were removed, there was no increase in the nest size following reunification (Fig. 7a). This experiment was performed at summer temperatures in the laboratory with ants that would have been overwintering in the field. The lack of rebuilding of the nests on reunification may have been due in part to seasonal influences (see General Discussion). When three-quarters of the adults were removed and later reunited, two nests increased in size in proportion to their new populations, one was unusually large and was slightly reduced

8 786 Animal Behaviour, 54, 4 Internal nest area (cm 2 ) (a) Experiments 1 5 (b) 3 there was an increase in nest size following reunification of the worker population. However, in one the result is completely counterintuitive and in another changes in nest size were not proportional to population size (Fig. 7b). Hysteresis may, in part, explain the multiple outcomes in these results (Fig. 7a, b). Hysteresis occurs if changes in an outcome lag behind changes in (one of) its cause(s). As a consequence, in hysteresis, there may be more than one outcome for a given magnitude of causal variable depending on the history of the system (see below under Properties of the Model). These experiments in which either half or threequarters of the colony were removed and replaced seem to produce enigmatic results. How do the ants first build a nest of an appropriate area for their population size, if they will also tolerate temporarily substantial overcrowding as their population grows? Why does the addition of just a few extra workers unleash an explosion of nest destruction and rebuilding? We suggest one way in which this enigma may be resolved in the next section of this paper when we consider a model of building behaviour for these ants Experiments 1 4 Figure 7. Experiments in which (a) approximately half of the adult ants in each nest were removed and later reunited after the building by the first half of the colony was completed and (b) approximately three-quarters of the adult ant population in each of four colonies was removed and returned after the building by the first quarter of the colony was completed. : The size of nests constructed by each of the reduced colonies; : the final size of the nest after reunification. The percentages of the total adult ant colony populations present in the nests are shown above the bars. In (a) experiments 1 5 were conducted with colonies B, C, D, E and F, respectively; in (b) experiments 1 4 were conducted with colonies A, B, C and D, respectively. when more ants were added and the fourth increased slightly when the colony was reunited (Fig. 7b). Hence, in three of the four colonies 32 A MODEL FOR WALL FORMATION In the previous section we reviewed information on individual behaviour during building in Leptothorax tuberointerruptus. In this section, we develop a model, based on the empirical findings documented above. This so-called bottom-up modelling is a way of seeing if self-organization can account for the pattern of interest, that is, the nest wall with its entrance. The model is based on stone picking up and dropping behaviour, both of which are under the control of amplifying mechanisms and the influence of the template provided by the cluster of brood and workers in the new nest site. The laden or unladen ants show, respectively, a probability of dropping or picking up a stone related to their distance from the brood and worker cluster. These functions for dropping or picking up show, respectively, a maximum or a minimum in relation to a particular density of stimuli (both mechanical and pheromonal) derived from the physical presence of brood and adult ants and also the chemical signals and cues these provide. An unladen ant (U) meets a stone () and can pick it up and become a laden ant (L).

9 Franks & Deneubourg: elf-organizing nest construction 787 The rate at which these events appear in a particular zone (r) isp(r)u. At any moment, the laden ants can drop their stones. D(r)L (1 /K) gives the rate of such dropping. The term (1 /K) is used because an area can accept at most K stones. We consider in this model that the ants are moving randomly in the building zone and we initially assume that it is only the template that affects the behaviour of the builders. If we consider building activity in only a small area, equation (1) describes the dynamics of the population of the stones in such an area. d t =D(r)L(1 /K) P(r)U (1) In this equation, we can neglect loading and unloading times, because these events are relatively brief. The first version of the model does not take account of the changes in the probability of picking up or dropping stones arising from the presence of (increasing numbers of) other stones. To take account of this facilitation effect we replace D(r) by D(r)G(). o the term for dropping becomes: D(r)G()L (1 /K) (2) where G() grows linearly with. This is the simplest possible dependence of G() on and expresses the finding that it is the collision with an obstacle that mostly determines stone dropping. G()=(g 1 g 2 ) (3) where g 1 and g 2 are, respectively, parameters that characterize spontaneous stone dropping and stone dropping near other stones (or stationary ants). imilarly the stone picking-up rate is: P(r)F()U (4) F() decreases with. This means that, as increases, the probability of picking up a stone decreases. To reduce the number of parameters we assume that F() is the inverse of G(): F()=(g 1 +g 2 ) 1 (5) d t =D(r)G()L(1 /K) P(r)F()U (6) At the stationary regime, d 2 =, we obtain the next equation: μ=/(g 1 +g 2 ) 2 (1 /K) with μ=(d(r)l)/(p(r)u) (7) where μ is a measure of the tendency of the ants to build in a particular area. Indeed, if D and L are large or P and U are small, μ is large. Recall that D is the tendency to drop material and L is the number of laden ants, that is, those available to drop material in the area. Conversely P is a measure of the tendency of ants to pick up material and U is the number of unladen ants, that is, those available to pick up material in the area. For simplicity and clarity, equation (7) states the relationship between μ and. Beware, however, that is the dependent variable and μ is the independent variable. We prefer equation 7 to algebraic equivalents, however, because when it is rewritten to state as a function of μ this formulation has many more terms and is less easy to interpret. What is the relationship between the position of the area relative to the value of the four parameters D, L, P and U? In the case of D and P this is straightforward: D and P are related to the influence of the template. Hence, if the ants are in an area in which building is authorized by the template, D is large and P is small whereas if the ants are in an area not authorized for building by the template D is small and P is large. μ is also influenced by the relative number of laden (L) and (U) unladen ants. The unladen ants are essentially ants moving outwards from inside the nest as described in the Results. In addition, there are the ants that must leave the nest for foraging and to look for building material. These ants also act like a pressure : if the wall perimeter is small and the number of workers inside is large, the pressure is large and U is large, leading to a small value of μ. If the wall is built further away, for the same population, U will be smaller and so μ will be large for the corresponding zone. For the laden ants the situation is the converse. Any factor favourable to a high value of L leads to a high value of μ and vice versa. Numerical Values From the experiments of Franks et al. (1992) and subsequent ones described in this paper, we can estimate the parameters for our model of the building process. What are the values for the numbers of laden and unladen workers? The

10 788 Animal Behaviour, 54, 4 fraction of laden ants has been estimated (by individually marking all the adult ants with paint: see endova-franks & Franks 1994, 1995) tobe only about 1% of the total population and these individuals are scattered over all of the area (P. J. P. Croucher & N. R. Franks, personal observation). The unladen ants, which are more or less equal to the total population, are clustered in the neighbourhood of the brood. Hence L/U<.1 in the zone of active building. The number of unladen ants (U) in the building zone is low, typically about 2/cm 2. From the data analysis presented in Fig. 3 the probability of a laden ant dropping its stone when it collides with a stationary stone or a stationary ant, in an area of active building, is approximately.4. This can serve as an approximate estimate of the maximum for D. The ratio g 1 /g 2 corresponds to the ratio of dropping a stone spontaneously/dropping a stone near another stone or a stationary ant. Recall that the ratio of such events is 63 to 237 (i.e. 63 spontaneous drops, 21 near stones and 27 near other ants). These events have been recorded when less than 2% of the final number of stones are present in the nest. The value of at this stage is around 8 /cm 2. Hence we can estimate g 1 /g 2 from g 1 /g 2 =63/237 where =. This implies a value of g 1 of approximately 25 times the value of g 2. Taking account of the speed of the laden ants (around 1 mm/s), the width of the search path of the ants (.5 mm) plus the size of the granules (.5 mm) and the area of activity of each ant (/mm 2 ) [hence, ((1 (.5+.5))/)]: we estimate g 2 to be around 1 2 /s. The stones we used were approximately.5 mm in diameter. o per cm 2 the capacity K (the number of granules that can be placed in a given area) is less than about 4 (see Franks et al. 1992). The area on which the ants moved was between 4 and 1 mm 2 (for the case where ants must go a little way outside the nest to pick up granules; for more information see Franks et al. 1992). The total number of granules offered to the ants in the experiments of Franks et al. (1992) was between 4 and. Estimating the thickness of the walls the ants build to be around 4 mm, we have good agreement between the number of granules needed to build such walls and the number of granules provided. Calculations based on Fig. 4 give a mean time until stones are picked up of 1 h. From this we calculate a new term Π which has a value of 1/1 h and corresponds to PU/(g 1 +g 2 ). These observations were made at a time when the estimated number of stones was around 8/cm 2. We further estimate g 1 to be 25g 2, g 2 to be 1 2 and U to be approximately 2/cm 2. This leads to P being approximately Π/2. P, expressed in terms of seconds, is thus around 1 4. This value can be cross checked by estimating the walking speed of the ants, which is less than 1 mm/s (the unladen ants are typically much less active than the laden ants), taking account of the size of the stones (.5 mm diameter) and the probability of picking up a stone when the ant encounters one (.1; see Fig. 5). By multiplying these three values together (and converting to cm 2 ) we obtain a value for P of s: fairly similar to the earlier estimate. Properties of the Model Equations 6 and 7 exhibit classical multistationary regimes. We can analyse the link between and μ for different values of g 1 and g 2. Figure 8a, b summarizes the behaviour of equation 7 which gives the stationary properties of equation 6, for the link between μ and. When g 1 /g 2 >.125 K, the solution of is unique: this means that as μ grows, grows more or less abruptly (Fig. 8a). When g 1 /g 2 <.125 K, (7) exhibits multistationary behaviour (Fig. 8b). The lower the ratio of g 1 /g 2, the larger is the region in which multistationary regimes are exhibited. Our estimates of the parameter values suggest that μ will be in the range of less than 8 to 4 and g 1 /g 2 will be less than.125 K. Hence, the dynamics of the system will be in the domain in which hysteresis will occur. Our estimation of the experimental parameters suggests that active building occurs under conditions in which hysteresis is likely to occur. uch behaviour is therefore a clear prediction of the modelling. There is experimental evidence that this prediction is valid. Recall the one-half and three-quarters removal and reunification experiments. In the former, new building was not initiated by the return of the other half of the colony. uch experiments show that a full size colony can occur in two different sizes of nest (normal or half size) and suggest that this depends, at least in part, on the history of the building. The three-quarters removal and reunification experiment shows that new building

11 Franks & Deneubourg: elf-organizing nest construction (a) 4 3 (b) 4 3 (c) µ 4 µ 4 Time (min) 3 Figure 8. The behaviour of equation (7). is the number of stones and μ is a measure of the tendency of the ants to build in a particular area. (a) The relationship between and μ when g 1 =.75, g 2 =.1, K=4 is simply sigmoidal. The system does not have multiple stationary states. (b) The relationship between and μ when g 1 =.25, g 2 =.1, K=4 is a classic hysteresis curve. The system has multiple stationary states. (c) The temporal dynamics of stone dropping in the zone for the set of parameters corresponding to (a) (i.e. g 1 /g 2 =75>.125 K). g 1 =.75, g 2 =.1, K=4, D=.4, P=.2, L=.2, U=2. Initial conditions =. is the number of stones in a focal zone, μ is the tendency of the ants to build in the focal zone, g 1 and g 2 are, respectively, parameters that characterize spontaneous stone dropping and stone dropping near other stones (or stationary ants). K is the maximum number of stones that can be placed in an area. D is the tendency to drop stones. P is the tendency to pick up stones. L is the number of laden ants available to drop stones. U is the number of unladen ants available to pick up stones. can be initiated by the return of more ants to a smaller nest. It is noteworthy, however, that there is great variability in the rebuilding response when three-quarters of the worker population are returned to a nest built by one-quarter of their colony (see Fig. 7a, b). uch variability is not unexpected in a system such as this which can exhibit multiple outcomes. A relatively small amount of random variation in such a complex system may lead to large differences between different replicates of the same experiment. Figures 9, 1 and 11 explain this type of result in terms of what happens in the two zones in which building is observed. The two notional zones 1 and 2 are, respectively, nearer and further from the centre of the nest. When 5% of the colony is taken away and then returned, after building had been completed by the first 5%, the structure that is used by the now complete colony is different from that originally built by the whole colony. Hence a colony can be housed in two different sizes of nest corresponding to two different solutions to the building problem. Figure 9a corresponds to the situation when there is no hysteresis. μ 5,1 and μ 5,2 correspond to the μ for the 5% population and for the two zones in which active building may be observed (Fig. 9a). imilarly in Fig. 9b, μ,1 and μ,2 correspond to the μ for the complete (%) population. In the case of the 5% colony it is zone 1, closer to the centre of the nest, that has the greatest value of μ, and in which the walls appear, μ 5,1. Here zone 2 is characterized by a low value of μ and no walls appear, μ 5,2. What happens when we add the other 5% of the colony? The value of μ for zone 1 decreases because the number of unladen ants increases and the template effect modifies the probability of picking up and dropping material. Inversely, the μ corresponding to zone 2 increases and this leads to a shift of the walls from zone 1 to zone 2 (Fig. 9c). Figure 1 corresponds to the range of parameters in which hysteresis is observed. The single difference between Fig. 1a and b is the relative position of the values for μ and μ 5. Figure 1a describes the quantity of stones in the two zones when only 5% of the colony is present. As in the case of Fig. 9a the μ corresponding to zone 1 is greater than the μ corresponding to zone 2. We use Fig. 1b, not to explore all of the possibilities, but to show how easily hysteresis can occur when we reintroduce the missing 5% of the colony. If the shift of the μ corresponding to zone 1 and zone 2 is such that the system remains for both zones within the domain in which both zones remain on their respective branches (i.e. close to the upper and lower asymptotes) we have no modification of the location of the walls. This is confirmed by Fig. 1c showing the temporal dynamics of this case. Even though the preferred zone for building changes (at 6 min) from zone 1 to zone 2, the number of stones in the two zones changes very little. By contrast, Fig. 11a,b shows how, when the modification of the respective μ is

12 79 Animal Behaviour, 54, 4 4 (a) 4 (a) 3 Zone 2 Zone 1 3 Zone 2 Zone 1 µ 5, 2 µ 5, 1 4 µ µ 5, 2 µ 5, 1 4 µ 4 3 (b) Zone 2 Zone (b) Zone 2 Zone 1 µ, 1 µ, 2 µ 3 (c) (c) µ, 1 µ, 2 µ Time (min) Figure 9. Theoretical sigmoidal curve describing the relationship between and μ for (a) a halved (5%) and (b) a restored (%) population of ants. There is no hysteresis and hence there are no multiple stationary states. The two zones 1 and 2 are, respectively, nearer and further from the centre of the nest. Building is authorized in zone 1 but not in zone 2 in (a) and in zone 2 but not zone 1 in (b). Further explanation in text. (c) The temporal dynamics of stone dropping in two zones for a set of parameters corresponding to Fig. 8a (g 1 /g 2 =75>.125 K). During the first 6 min (1 h), when the ant population is halved, zone 1 is preferred with μ 1 =75 and μ 2 =12.5. During the next 1 h, when the population of ants is restored to full, zone 2 is preferred (μ 1 =12.5 and μ 2 =75) and the walls are shifted from zone 1 to zone 2 (g 1 =.75, g 2 =.1, K=4, D1=.15, D2=.25, P=.2, L=.2, U=2). During the 1 h, D1=.25 and D2=.15. To simplify, all the other parameters (P, L and U) were kept constant. Zone 1 is represented by the thin line, zone 2 by the thick line. Terms are defined in the legend to Fig Time (min) Figure 1. Relationship between and μ showing hysteresis. (a) The quantity of stones in the two zones when only 5% of the colony is present. (b) hows how easily hysteresis can occur when we reintroduce the missing 5% of the colony. Further explanation in text. Building is authorized in zone 1 but not in zone 2. (c) The temporal dynamics of stone dropping in the two zones for a set of parameters corresponding to Fig. 8b (g 1 / g 2 =25<.125 K). During the first 6 min (1 h), when only 5% of the ant population is present, zone 1 is preferred with μ 1 =15 and μ 2 =5. During the next 1 h, zone 2 is preferred (with μ 1 =9 and μ 2 =). The thin line represents zone 1 and the thick line zone 2. The wall remains at its initial position, despite the fact that zone 2 is more attractive. Initially, g 1 =.25, g 2 =.1, K=4, D1=.3, D2=.1, P=.2, L=.2, U=2. During the second 1 h, D1=.18 and D2=.2. To simplify, all other parameters (P, L and U) were kept constant. Terms are defined in the legend to Fig. 8.

13 Franks & Deneubourg: elf-organizing nest construction (a) (b) (c) µ 25, 2 µ 25, 1 4 µ µ, 1 µ Zone 2 Zone 1 Zone 2 Zone 1 µ, Time (min) Figure 11. Relationship between and μ showing hysteresis. Further explanation in text. When the modification of the respective μ is much greater, as is likely to occur when the colony population is increased from 25 to % compared with 5 to %, the new values of μ for the two zones can both fall outside the domain in which hysteresis would occur. (a) Building is authorized in zone 1. (b) Building is authorized in zone 2. (c) The temporal dynamics of stone dropping in two zones for a set of parameters corresponding to Fig. 11b (g 1 / g 2 =25<.125 K). During the first 6 min (1 h), with 25% of the ant population, zone 1 is preferred with μ 1 =15 and μ 2 =5. During the next 1 h, with the colony restored to %, zone 2 is preferred (μ 1 =5 and μ 2 =15). The thin line represents zone 1 and the thick line zone 2. The modification of the parameters is sufficiently strong to shift the walls from zone 1 to zone 2 (g 1 =.25, g 2 =.1, K=4, D1=.3, D2=.1, P=.2, L=.2, U=2). During the second 1 h, D1=.1 and D2=.3. To simplify, all the other parameters (P, L and U) were kept constant. Terms are defined in the legend to Fig. 8. much greater, as is likely to occur when the colony population is increased from 25 to %, the new values of μ for the two zones can both fall beyond the limits of this window. In this case a complete rebuilding of the walls further from the centre of the nest can occur. This is confirmed by Fig. 11c showing the temporal dynamics of this case. There are three sets of possible limitations that may apply to this model. First, the model does not take explicit account of the space in which building occurs. econd, the model does not take account of the limitation of building material. Third, the model does not consider that the parameters pertinent to one zone may be influenced by the status of such parameters in other zones. The third possibility is largely a by-product of the first two possible limitations. However, our purpose was to try to capture, and to give an opportunity to discuss and understand, the important properties of this system. The local description that we have employed is sufficient to realize these goals. A spatial model would introduce the mobility of ants and stones between the different zones and should be able to examine the relationship between the movements of these entities. We now consider together and in more detail points 2 and 3 because they are associated with one another. In Figures 9, 1 and 11 we have considered that μ 1 and μ 2 can be modified completely independently of one another. However, if we take account of the constraints in the real system this cannot be the case. First, if there is a modification of the ants behaviour, in terms of the probability of dropping and picking up material, in one zone, changes in these probabilities are likely to occur in the second zone as a consequence and vice versa. We have seen that the probabilities of dropping and picking up material are under the influence of the physical and chemical cues provided by the adult and the brood ants; such cues decrease from the colony centre towards the periphery. A modification of the composition of the colony leads to a modification of the spatial distribution of such cues and in turn to a modification of the probabilities of dropping and picking up material. There is a second link between the μ of the different zones which results from the limitation of the abundance of building material and of workers. μ depends upon the local ratio between laden and unladen workers. For example, in the case of the relocation of the walls, the stones are prisoners of the pre-existing walls. This may

14 792 Animal Behaviour, 54, 4 lead to the local ratio L/U being lower than it would be in other circumstances. Consequently, this could reinforce the hysteresis effect because it could prevent another zone from having a higher value of μ. There is one last point, which we consider now, that we wish to discuss in the context of the different values of μ. Formation of the Entrance Observations show the strong influence of the original distribution of stones on the final pattern. Placing a large quantity of material in one area tends to result in an exceptionally thick wall near that area. The walls are constructed progressively away from this site in a pincer movement (Figure 1b in Franks et al. 1992). The result is that the wall becomes progressively thinner away from the original building site and the nest entrance tends to be diametrically opposed to the original site of the building materials. It is even possible to see the emergence of a small number of entrances with a surprisingly regular spacing between them (see Figure 1g and h in Franks et al. 1992). uch observations lead us to discuss a small modification of the model to take into account the dynamics of entrance formation. In the previous equations we considered an isolated area of wall. However, there can also be competition between different parts of the same wall. When ants are leaving the nest for foraging and returning there is considerable traffic inand out of the nest. On both outward and inward journeys these ants may meet stones and try to avoid these obstacles and search for other paths. This searching behaviour may also be associated with stone picking-up behaviour: ants that meet stones along their path try to push these stones out of their path. Taking account of this traffic, we modify equation (6) to obtain equation (8): d t =D(r)G()L (1 /K) P(r)F()U T(r) (8) We do not modify the term for dropping, but only the term for picking up stones. This idea is related to our previous discussion. We consider that the intensity and organization of the traffic are the key factors and that no specialized behaviour pattern is involved in the formation of the door. T(r) is the fraction of traffic in the focal zone. What is this fraction? It is related to the proportion of obstacles in this zone compared with all the other areas. If the quantity of stones is, respectively, large or small in this focal zone, compared with the quantity in the other areas, the proportion of the traffic will be, respectively, small or large. We adopt a very simple way to describe this: T(r)=A/(A+A ) (9) where A and A are, respectively, a measure of the attractiveness of the focal zone and all the other zones combined. We have chosen A to be inversely proportional to : A=(g 3 +) 1 (1) A is proportional to the ease with which the ants can cross the other zones. A is small or large when the obstacles are, respectively, numerous or rare (g 3 is simply an estimated constant which is proportional to the ease with which ants can cross an area with no stones). We replace, in equation (8), T(r) with its representation in (9) which in turn incorporates (1). Hence: d t =D(r)G()L(1 /K) P(r)F()U (g 3 +) 1 /((g 3 +) 1 +A ) (11) Equation (11) at the stationary regimes is characterized by the next relation between μ and : μ=/((g 1 +g 2 ) 2 (1 /K)(1+A g 3 +A )) (12) This equation (12) exhibits similar properties to those of (7), with the exception of certain differences attributable to the numerical values of the parameters (see Fig. 12). This version of the model is pertinent to three phenomena: (1) regulation of the number of entrances; (2) their relative location; and (3) the entrance as a focal site for nest reconstruction. We consider these in the General Discussion that follows. GENERAL DICUION Our goal has been to explore, through both experiments and a mathematical model, the mechanisms and individual behaviours used to produce the observed patterns of building in these ants. Many studies have looked at building