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The Nonlinear Dynamics of Survival and Social Facilitation in Termites
J. theor. Biol. (1996) 181, 373–380
The Nonlinear Dynamics of Survival and Social Facilitation in Termites O M† O DS‡ †Department of Biology, Imperial College at Silwood Park, Ascot, Berks SL5 7PY UK and ‡Departamento de Biologia Animal, Universidade Federal de Vic¸osa, 36.570-000 Vic¸osa, MG, Brazil (Received on 31 March 1995, Accepted in revised form on 8 May 1996)
This paper describes a study on termites, investigating the relationship between increasing group size and individual worker longevity under resource-deprived conditions. It was found that survival was significantly lower for isolated individuals and higher for individuals in bigger group sizes, suggesting that social interactions play an important role in the mechanisms leading to longer survival. A computer model, incorporating individual interactions among mobile cellular automata is presented along with experiments. 7 1996 Academic Press Limited
(DeCastro, 1995), to non-crossbreeding hermaphrodite snails whose reproductive output is larger when grouped than when isolated (Vernon, 1995). An extreme example includes chickens that eat more when confronted with video images of other feeding chicken than when the video image shows an empty cage (Keeling & Hurnik, 1993). Social facilitation exists in many invertebrate species such as crabs (Kurta, 1982), centipedes (Hosey et al., 1985), scorpions (El Bakary & Fuzeau-Braesch, 1987) and occurs ubiquitously among the social insects. It has been studied in wasps (Parrish & Fowler, 1983; Fowler, 1992; Reid et al., 1995), bees (Grasse´ & Chauvin, 1944; Chauvin et al., 1985), ants (e.g. Chen, 1937; Chauvin, 1944; Grasse´ & Chauvin, 1944; Lamon & Topoff, 1985; Klotz, 1986; Salzemann & Plateaux, 1988; Ho¨lldobler & Wilson, 1991) and termites (e.g. Grasse´ & Chauvin, 1944; Grasse´, 1946; Lenz & Williams, 1980; Williams et al., 1980; Afzal, 1983; Okot-Kotber, 1983; Springhetti, 1990). Most of the classical studies of social facilitation have focused on task performance by isolated individuals, contrasted with that of individuals in groups (e.g. Chen, 1937). In termites, however, studies of a different nature have been carried out, investigating survival as a function of group size. Williams et al.
1. Introduction Interactions among the many individuals that compose social groups, generate a range of behaviours not present at the individual scale. Large-scale spatiotemporal patterns of behaviour may develop following the concurrent activity of individuals that do not have pre-programmed information for generating these. Since such patterns are the outcome of social interactions, they are not observed when individuals are isolated. One of the most basic manifestations of such phenomena is social facilitation, commonly defined as the ordinary patterns of behaviour that are initiated or increased in pace or frequency by the presence or actions of other animals (Wilson, 1980). Hence, social facilitation is a nonlinear response behaviour whose prerequisites are the capacity of animals to sense, excite and communicate with others and whose relative strength is related to the number of participants involved. Social facilitation is common among several animal species. Examples range from humans who consume more food when in groups than when alone † Author to whom correspondence should be addressed. Present address: Departamento de Sistemas Complejos, Instituto de Fisica, UNAM, Mexico (01000) D.F., Mexico 0022–5193/96/160373 + 08 $18.00/0
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(1980) considered three species of Cryptotermes and investigated the survivorship for groups of one, two, four, eight, 16 and 32 pseudoworkers kept with food supply. They found that survival increased with group size. A different approach was adopted by Lenz & Williams (1980) who investigated the survival of Nasutitermes nigriceps (Hald) in environments rich in food, but with different container sizes and found that survival and wood consumption decreased with increasing container size. This experiment provided strong evidence that interactions between social individuals are crucial to survival, since increasing the container size is equivalent to a reduction of the group density and hence of the rate of contacts and interactions. In these two experiments, however, Cryptotermes and Nasutitermes were provided with food sources, rendering difficult the estimation of the importance and role of social interactions in the mechanisms that lead to increasing survival. Grasse´ (1986) commented on the correlation between group size and longevity in termites, but it is not clear whether his own experiments were carried out with or without food supply (he stated that the termites were kept in optimal conditions, which suggests that food was available). Despite being a behavioural phenomenon widely present across the animal taxa, social facilitation is poorly understood and has received very little attention both experimentally and theoretically. In social insects, the knowledge regarding specific biochemical or neurophysiological mechanisms involved in social facilitation is severely limited despite being one of the most basic and fundamental principles of social behaviour. In this article, we describe a computersimulated model of social interactions where interactions between mobile automata increase the individual longevity and hence that of the group. In addition, we provide experimental evidence that the survival of real termites is significantly correlated with the group size when in resource-deprived environments. The experimental results are useful for parameterising the model, that in turn is used to compare qualitatively the process of termite survival. Results suggest that social interactions play an important role in increasing survival of groups or, conversely, that its absence contributes to a reduction of the longevity of isolated individuals.
2. Survival Experiment Nasutitermes cf. aquilinus workers (third instar and beyond) were collected from a single wild colony from 2.1
Vic¸osa, Minas Gerais in Brazil and were randomly placed in groups of one, two, four, eight and 16 individuals (four replicates). The groups were confined in test tubes made of transparent glass (9.5 cm × 1.4 cm) with hermetically sealing rubber caps. The tubes were kept horizontally separated by plastic foam to prevent stridulation or other mechanically transmitted signals to propagate between the tubes. The workers were allowed to acclimatise for 12 hr and were incubated in the dark in a constant temperature chamber (25°C 2 0.5). No food or water was provided. The samples were kept in darkness and were exposed to light during the counting of survivors only (no more than 5 min). Observations were made at 12 hr intervals. The experiment was continued until 24 hr after all the solitary individuals were dead in order to check the consistency of the trend observed in the multi-individual treatments. Seven observations were made. The data were subjected to survival analysis, using Weibull frequency distribution, as described by Crawley (1993) (see also Pinder III et al. 1978). Significance was assessed by deleting terms from a model until a change of deviance with a P Q 0.001 was observed. Differences between treatments were evaluated by aggregating factor levels until the consequential change in deviance was significant at P Q 0.001). 2.2 All individuals were alive at the time of the first observation and the first dead workers began to appear 24 hr after being confined. The analysis of the mortality figures for the seven observations give significant results that allow us to conclude that mortality was higher for isolated workers and lower for individuals in groups (x 2 = 25.44; 4 d.f.; P Q 0.001). A tendency that is easily identified in the data plotted in Fig. 1. It was clear from the observations, that workers in groups tended to cluster together and to engage in social interactions that included antennations and grooming and that this behaviour was more conspicuous in larger groups.
3. A Model of Social Interactions The use of models incorporating discrete interacting entities with behavioural traits that closely match those found in real individual organisms has increased in recent years (DeAngelis & Gross, 1992; Judson, 1994; Kawata & Toquenaga, 1994). These models, so-called Individual Based Models, are very useful for modelling patterns of self-organized collective behaviour that emerge without being explicitly specified
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F. 1. Raw-data from the survival experiment. Per-capita survival was large as the group size increased. The highest survival occurred in groups of 16 individuals. Survival for groups of four and eight individuals was similar but larger than survival for paired and isolated individuals. Each dot represents the total proportion of survivors for each group-size category (four replica per group).
in the individual’s designs or in their rules for interaction. One of the first examples of models with interacting individuals, in the context of social insects, was developed by Goss & Deneubourg (1988) to study the autocatalytic generation of cycles of activity in ants. Recent examples include automata models for self-organized pattern formation in honey bee colonies (Camazine, 1991) and models of selforganized trail formation in ants (Edelstein-Keshet et al., 1995). Automata models of termite behaviour are scarce and, as far as we know, the only model previously published is due to Courtois & Heymans (1991) who studied termite nest building behaviour using cellular automata and based on the idea of ‘‘order through fluctuations’’ developed by Deneubourg (1977). Here we present a model based on the mobile cellular automata concept (Miramontes et al. 1993a, b; Sole´ et al. 1993a, b; Miramontes, 1995) to explore the possible role of social interactions in the pattern of increased termite survival as a function of group size. In our model, a set of interacting individual agents (termites in this case) are defined over a rectangular lattice and follow some of the simple behavioural rules found in real termites. Each cell of the lattice is occupied by only one individual. Termites are able to move randomly to empty adjacent cells, where they interact with those in the eight most immediate cells (reflecting the fact that real termites interact mainly by direct body contact: antennations, grooming and trophallaxis). The lattice has fixed boundary conditions (zero-flux) meaning that no termite leaves the
lattice or appear at the opposite edge when reaching one of the edges. We define a real-valued state variable ak $ [0,1] for each individual k termite indicating, whether that individual is dead or not. A termite is considered to be alive while this variable is greater than zero and dead if zero. This variable increases each time a termite interacts with others, reflecting the ability of termites to excite and respond to the presence of colony mates. This quantity ak inevitably declines to zero after a finite time because termites in isolation do not have access to external inputs. The process of social interactions is defined through a coupling function that reflects the fact that, firstly, individual termites cannot be infinitely excited after repeated interactions, thus the function has to be bounded and, secondly, it is desirable that the activity of an individual increase or decrease according to a single parameter representing the degree of excitability. A continuous function, widely used in models of excitable biological media, that conveniently accommodates these requirements is the hyperbolic tangent function (Amit, 1988). Hence, the coupling between interacting termites is defined as follows:
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where ait is the activity of the i-th termite (maximum of eight) in the first neighbourhood of the ak termite at time t (measured as computer time-steps), gk is the individual excitability parameter and l is a parameter measuring the group-level ability to communicate (see Fig. 2). The simplicity of the model relies on only two parameters that are biologically justifiable.
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F. 2. Temporal evolution of the state variable ak for a single automaton when isolated (curve A) and when in a group of size 16 (curve B) as given by eqn (1). In the first case, the initial value asymptotically decays to zero, while in the second, the decay to zero is delayed each time an interaction occurs. A termite is considered alive when ak q 0 and dead when ak = 0. The threshold value below which ak was regarded as zero was 10 − 10. Initial value of ak was 1.0, gk = 0.7, l = 0.31, lattice size was 7 × 7.
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The parameter gk (often called the gain of the hyperbolic tangent function) accounts for interindividual differences in the capacity to excite and interact (i.e., some individuals are more likely to respond to stimuli of the same intensity than others). For example, it is well known that young adults, in many social species, respond to antennations from nestmates more readily than older ones, a fact that has been quantified in the case of the spontaneous activations of ants in the genus Leptothorax (Cole, 1992). Also, individuals may respond to their mates and behave in different ways depending on their state of health. Parasitised termites, for example, suffer profound morphological modifications (Krishna & Weesner, 1970; Grasse´, 1986) that are likely to provoke changes in their behaviour and modify the way in which they interact with others. An extreme example of this is observed in parasitised bumblebees which may even desert the colony, stopping all forms of interaction with their nestmates (Poulin, 1992; Mu¨ller & Schmid-Hempel, 1992). Hence, young or healthy individuals are characterised by high values of gk , while old or unhealthy individuals would have a lower value. As previously mentioned the second parameter l quantifies the group-level ability to communicate and is equal for each individual in a group. This parameter indicates the strength or intensity with which the members of a given species interact among themselves. A group of non-social individuals, that do not interact between them at all, would have a l of zero. Organisms characterised by a high value of l would be the most socially developed, while those characterised by medium values would be the species of lower degree of socialility, the case of gregarious, presocial species, etc. The experimental results just described were used to find the correct parameter values of the model, as discussed in the next section. 4. Model and Experiment: Results Lattice size was estimated by measuring the average worker area (the projected polygon formed by legs and antenna) and the average walking area inside the tubes. Average worker area was approximately 18.3 mm2 while the effective walking area was 760 mm2, so that about 42 individuals would saturate this space. The length-to-width ratio of the test tubes was 9.5 cm/1.4 cm 1 6.8. A lattice of 14 × 3 = 42 cells was chosen because it contains a maximum of 42 cells and its length-to-width ratio is 4.7, a value close to the one found for the tubes. Both the maximum number of cells and the proportions of the container
are very important for the model’s result, as would be discussed later. In order to estimate gk , we assumed only one individual in the lattice, so that the coupling function (1) reduces to:
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that is clearly independent of l. The average time-to-death of the isolated termites (group size = 1) was found to be of about 42 hours and we found that a value of gk = 0.7 in eqn (2) gave a time-to-death of 42 time steps in the computer simulations. The initial value of the state variable ak0 was set to 1.0. The parameter l was estimated by noting that the average time-to-death in the groups of 16 termites was 123 hours. We found that given gk = 0.7, a l of 0.31 gave a time-to-death of 123 time steps in automata groups of size 16. It is important to remember that since time in the experiment (hr) and time in the model (computer steps) are only roughly equivalent, the parameter values estimated are just good approximations that permit similar qualitative behaviour between the experiment and the simulation. Although when other values of these parameters were used, the qualitative behaviour of the model was quite robust. Survival curves (Weibull distribution) estimated from interacting automata in groups of one, two, four, eight and 16 individuals are shown in Fig. 3(a), as denoted by the numbers beside the curves. Individual mortality differs significantly according to grouping (x 2 = 316.6; 4 d.f.; P Q 0.001), in a way that termites in groups of 16 individuals survived longer than those in groups of eight individuals (x 2 = 238.6; 3 d.f.; P Q 0.001); which in turn survived longer than individuals from smaller groups (1, 2, 4) (x 2 = 47.4; 3 d.f.; P Q 0.001). Individual mortality does not differ for groups of one, two and four (x 2 = 0.34; 1 d.f.; P q 0.50). Survival curves estimated from the data of Nasutitermes cf. aquilinus termites starved to death in groups of one, two, four, eight and 16 individuals (Fig. 1) are represented in Fig. 3(b). Individual mortality differs significantly according to grouping (x 2 = 25.44; 4 d.f.; P Q 0.001), in such a way that termites from larger groups (16 individuals) survived longer than those from groups of eight and four individuals (x2 = 8.89; 1 d.f.; P Q 0.001); which in turn survived longer than individuals from smaller groups (one and two) (x2 = 8.22; 1 d.f.; P Q 0.001). Individual mortality does not differ either for groups of one and two (x 2 = 0.09; 1 d.f.; P q 0.80) or for groups of four and eight (x2 = 0.53; 2 d.f.; P q 0.50).
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F. 3. Qualitative comparison between survival of simulated and real termites using a Weibull distribution function. The same qualitative patterns are observed for simulated (a) and real termites (b) suggesting that social interactions are important in the mechanism leading to longer survival in grouped termites. Notice that survival for groups of size four and eight do not differ much. It is not clear why this was observed; but is likely to be related to the experiment resolution and further experimental work would be required to solve this. Nevertheless, the general trend of increased survival with group size is confirmed both in the experiment and model. One hundred replicas per group-size category were used in the simulation. Initial value of ak was 1.0, gk = 0.7, l = 0.31, lattice size was 14 × 3. Initial positions were chosen randomly.
It is clear from the shape of the curves, that the instantaneous death rate (i.e., the Hazard function) decreases with increasing group size. Therefore, individuals belonging to smaller groups were subject to a higher risk of death than those belonging to larger groups. Both the size and geometry of the container seems to be important for survival, so we explored the model to assess survival using different lattice configurations. The effects of size were simulated by using square lattices of different sizes, so that different values of density were obtained for each group size. It was observed [Fig. 4(a)] that increased lattice size (or decreasing density value) affects the chances of survival. As the lattice size is increased, the effects of social facilitation are less striking. When the lattice reach the value of 20 × 20 (density = 0.04) the phenomenon of increased survival with increased group size was negligible. This computer experiment
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F. 4. The effects of lattice size for the survival of grouped automata. The intensity of social facilitation decreases as the lattice size increases (four square lattice sizes were used). Notice that for a lattice of size 20 × 20, the effects of social facilitation are negligible because the average rate of contacts between the automata is very small. (b) The effects of the lattice geometry on the automata survival were explored by using different combinations of length-to-width ratio (numbers inside parenthesis) while conserving constant the total number of available cells (42). Social facilitation was maximal when the lattice was square and minimal when it was a one-dimensional array of 42 cells. Initial value of ak was 1.0, gk = 0.7, l = 0.31. Initial positions were random. One hundred replicas per group-size category were used. Error bars are standard deviations.
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using lattices with a constant number of cells (42) but with different values of length-to-width ratio. We found that survival was maximal in square lattices and was minimal when the lattice was reduced to a one-dimensional array of 42 cells [Fig. 4(b)]. In both cases (reduced density and different geometry) the fundamental cause affecting survival, was the decrease of the rate of contacts and hence of the frequency of interactions between individuals. In the first case because spatial dilution of random-mobile individuals means lower number of interactions per unit of time (automata have to travel large distances for an encounter to occur) and in the second case, because a high length-to-width ratio means that automata have less neighbours around for interacting. Thus far, it has been assumed that all automata are identical in their individual characteristics. Nevertheless, in order to bring the simulation closer to the biological reality, inter-individual variability has to be considered. Inter-individual differences are thought to have important consequences for the way insect societies organise, work and behave. Oster & Wilson (1978) argued that individual size (S), age distribution (A) and caste composition (C) are three factors that critically regulate the ergonomic efficiency of the whole society in such a way that the demography may even be of an adaptive nature. It seems reasonable then to assume that the success of insect societies highly depends on the behavioural diversity of its individual components. The introduction of demographic factors (S, A and C) seems then to be a very interesting field to investigate and we used our model to perform an exercise in this direction. Inter-individual variability was tested by comparing the performance of groups of different sizes depending on whether the value of gk was the same or differnt for all individuals. In the case where gk had a constant value (0.7), we observed that social facilitation was quite evident (Fig. 5, curve A). Groups of 16 individuals clearly survived more than groups of smaller size and the individual automata in those groups survived about four times longer on average, than isolated automata. A second simulation was carried out where the individual values were randomly generated on an interval with a flat probability distribution (gk = 0.5 2 0.2), so that there were individuals with high (0.7 maximum), medium and low (0.3 minimum) values of gk intermixed (Fig. 5, curve B). In this last case, while social facilitation is still present, it is of a less intense nature. This could be explained by the fact that individuals with low gk , as said before, are not particularly good at interacting and their participation in the generation of collective behaviour, may be weak. A second result
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F. 5. Social facilitation in automata as a function of group size and inter-individual variability. Social facilitation is quite clear in groups of automata that have the same value of gk (curve A). Automata in larger groups lived longer than those in smaller groups. Automata in groups of 16 lived four times longer, on average, than those in isolation. In this case gk = 0.7 and l was 0.31. (Curve B) Inter-individual variability is introduced with a flat probability distribution of values of gk in the range 0.5 2 0.2. Social facilitation was less intense than in curve A, while the variance of individual time-to-death keeps more or less equal in large groups regardless of individual variability. Initial value of ak was 1.0, gk = 0.7, l = 0.31. Initial positions were random. One hundred replicas per group-size category were used. Error bars are standard deviations. Lattice size was 14 × 3.
derived from contrasting curves A and B in Fig. 5, is that inter-individual variability decrease the variance of the time-to-death when group size is large. It would be highly desirable to have experiments that provide exact demographic parameters so that the values of gk could be arranged accordingly in such a way as to test whether a given probability distribution function may prove to be ergonomically optimal or not. It is clear that the scenarios presented in Fig. 5, while demonstrating the most striking effects of social facilitation, are quite unlike in nature. Nevertheless they provide a first approximation to explore how demographic factors and inter-individual variability possibly affect the performance of social groups, as has been previously suggested by other authors (Oster & Wilson, 1978). 5. Discussion Current hypotheses on social facilitated phenomena rely on the assumption that physiological processes (hormonal, nervous system activity, etc.) are modified because of the mere presence of
conspecifics in the environment. These changes (e.g. hormonal activity) have been measured in animals such as rats and monkeys (Zonjoc, 1975) but no direct observations exist for small and fragile organisms such as termites. Facing the lack of detailed information, regarding biochemical or neurological processes affecting social facilitation in relation to survival, we designed an experiment where termites were predominantly engaged in social interactions. The experiment consisted in groups of termites of different sizes confined in closed containers without supplies of food, water or oxygen (the oxygen available was only the one trapped in the tubes, no air was allowed to circulate). The experiment was designed in this way because the number of variables involved in the process of survival was reduced and because it allowed to assess the importance of resource supplies by contrasting with similar experiments that contained these. In higher animals such as rats, monkeys or chickens, ‘‘interaction’’ means a very complex mixture of inputs simultaneously affecting various senses; however, incomplete or partial clues are often enough for an ‘‘interaction’’ to occur, as is dramatically demonstrated by feeding chickens showing social facilitation (they eat more) when in front of video images of another feeding chicken (Keeling & Hurnik, 1993); so that a physical interaction, as such, was not necessary. In termites, interactions happen mostly by direct body-to-body physical contacts (antennations, food sharing, licking, etc.) so that individuals in large groups are more likely to engage in interactions than those in small groups (and obviously more than those under isolation). In fact, interactions are one of the most common behaviours displayed by social insects, as is evident from the ethograms reported elsewhere. Social insects invest a fairly large amount of the time budget engaging in social interactions. Hence, there is enough motivation for assuming that social interactions are relevant for the mechanisms that regulate important physiological processes at the individual level, including metabolic rates that are closely related to survival chances in resource-deprived environments. In our experiment, no food, water or fresh air supply was available to the confined Nasutitermes workers. Since it is known that cannibalism in termites could provide food supply in stressed colonies (Cook & Scott, 1933), large groups could be thought to be subject to less starvation levels per individual than smaller ones, since more prospective food (in the form of nestmates) was available. However, microscope observations of dead individ-
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uals did not reveal any sign of cannibalism. Another possible source of food, trophallaxis (mouth-tomouth food sharing), is also not important in this context, since it is known to decrease with increasing group size (Afzal, 1983). Therefore, we can assume that all experimental units were reasonably submitted to similar levels of starvation, having theoretically the same average longevity due to food scarcity only. Interestingly, when other authors have found larger groups to profit from social interactions, it seems that the presence of food in the experiments denoted an assumption that survival would result from the well-known increased food intake promoted by social facilitation. Our results, however, suggest that social interactions may enhance survival of grouped individuals, even without resource supply. This supports the view that social facilitation may be involved in triggering physiological changes in such a way, that metabolism may become more efficient so that starving individuals can make better use of body reserves and survive longer. While there is no conclusive evidence that this is the general case in termites (oxygen consumption was not measured in our experiment), other studies provide good evidence that individual metabolic rates, in ants, decrease when individuals are in larger groups (Peakin & Josens, 1978). Nash & Pierce (1994), for example, have shown that Iridomyrmex isolated ants consumed an average volume of 0.0063 ml of oxygen per hour, while grouped ants consumed 0.00168 ml hr − 1. To test the above ideas, we constructed a model whose only biological parameters are the capacity to interact with neighbouring individuals and the inevitable death after some time. The pattern that emerged is essentially the same: survival increases as a function of group size in an environment that lacks resource supply. Moreover, the important result to remark is that the dependence between longevity and group size is nonlinear. In summary, a synergetic phenomenon arises where the basic generating mechanism is facilitation through individual interactions. We would like to thank M. P. Hassell, M. Crawley, E. Vilela, K. Scho¨nrogge, D. Nash, P. Rohani, M. Bonsall, R. Constantino, R. Parentoni and A. Chopps for their help, comments and suggestions. Two anonymous referees provided useful criticisms. O. M. is grateful to the members of the Departamento de Biologia Animal, Universidade Federal de Vic¸osa for their hospitality during a research visit. This work was partially supported by grants from the Universidad Nacional Auto´noma de Me´xico and ORS (O.M.).
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K, A. (1982). Social facilitation of foraging behavior by the Hermit crab, Coenobita compressus, in Costa Rica. Biotropica 14(2), 410. L, B. & T, H. (1985). Social facilitation of eclosion in the fire ant, Selenopsis invicta. Developmental Psychobiology 18(5), 367–374. L, M. & W, E. R. (1980). Influence of container, matrix volume and group size on survival and feeding activity in species of Coptotermes and Nasutitermes (Isoptera: Rhinotermitidae, Termitidae). Material und Organismen 15, 25–46. M, O. (1995). Order-disorder phase transitions in ant societies. Complexity (in press). M, O., S´, R. V. & G, B. C. (1993a). Collective behaviour of random-activated mobile cellular automata. Physica D 63, 145–160. M, O., S´, R. V. & G, B. C. (1993b). Antichaos in ants: the excitability metaphor at two levels. In: Proceedings of the Second European Conference on Artifical Life, 790–807 (Brussels, May 1993). M¨, C. B. & S-H, R. (1992). To die for host or parasitoid. Anim. Behav. 44, 177–179. N, D. R. & P, N. E. (1994). Cost-benefit analysis of a mutualistm: energetic costs to ants of tending lycaenids. Preprint. O-K, B. M. (1983). Influence of group size and composition on soldier differentiation in female final larval instars of a higher termite Macrotermes michaelseni. Physiological Entomology 8, 41–47. O, G. F. & W, E. O. (1978). Caste and Ecology in the Social Insects. Princeton: Princeton University Press. P, M. D. & F, H. G. (1983). Contrasting foraging related behaviors in two sympatric wasps (Vespula maculifrons and Vespula germanica, Ecol. Entomol. 8, 185. P, G. J. & J, G. (1978). Respiration and energy flow. In: Production Ecology of Ants and Termites, (Brian, M. V., ed.), pp. 111–164. Cambridge: Cambridge University Press. P III, J. E., W, J. G. & S, M. H. (1978). The Weibull distribution: a new method of summarizing survivorship data. Ecology 59, 175–179. P, R. (1992). Altered behaviour in parasitised bumblebees: Parasite manipulation or adaptive suicide? Anim. Behav. 44, 174–177. R, B. L., MD, J. F. & R, D. R. (1995). Foraging and spatial-dispersion in protein scavenging workers of Vespula germanica and V. maculifrons (Hymenoptera: Vespidae). J. Insect Behav. 8(3), 315–330. S, A. & P, L. (1988). Sur le mecanisme de l’effet de groupe stimulant la ponte des ouvrieres de la fourmi leptothorax nylanderi. Annales des Sciences Naturelles, Zoologie 9, 37–43. S´, R. V., M, O. & G, B. C. (1993a). Oscillations and chaos in ant societies. J. Theor. Biol. 161, 343–357. S´, R. V., M, O. & G, B. C. (1993b). In: Interdisciplinary Approaches to Non Linear Complex Systems. (Haken, H. & Mikhailov, A., eds), Berlin: Springer-Verlag series on Synergetics. S, A. (1990). Nest digging of Kalotermes flavicollis (Fabr.) (Isoptera Kalotermitidae) by groups of different number of pseudergates. Ethol. Ecol. & Evo. 2, 165–173. V, J. G. (1995). Low reproductive output of isolated, self-fertilizing snails: inbreeding depression or absence of social facilitation. Proc. R. Soc. Lond. B 259, 131–136. W, R. M. C., P-M, J. V. & J, P. (1980). The effect of group size on the survival and feeding economy of pseudoworkers of building damaging Cryptotermes spp. (Isoptera, Kalotermitidae). Social insects in the tropics: Proceedings of the first international symposium organised by the International Union for the Study of Social Insects and the Sociedad Mexicana de Entomologı´ a. Me´xico, pp. 219–234. W, E. O. (1980). Sociobiology. (The abridged edition). Harvard: Belknap Press. Z, R. B. (1965). Social facilitation. Science 149, 271–274.
The Nonlinear Dynamics of Survival and Social Facilitation in Termites O M† O DS‡ †Department of Biology, Imperial College at Silwood Park, Ascot, Berks SL5 7PY UK and ‡Departamento de Biologia Animal, Universidade Federal de Vic¸osa, 36.570-000 Vic¸osa, MG, Brazil (Received on 31 March 1995, Accepted in revised form on 8 May 1996)
This paper describes a study on termites, investigating the relationship between increasing group size and individual worker longevity under resource-deprived conditions. It was found that survival was significantly lower for isolated individuals and higher for individuals in bigger group sizes, suggesting that social interactions play an important role in the mechanisms leading to longer survival. A computer model, incorporating individual interactions among mobile cellular automata is presented along with experiments. 7 1996 Academic Press Limited
(DeCastro, 1995), to non-crossbreeding hermaphrodite snails whose reproductive output is larger when grouped than when isolated (Vernon, 1995). An extreme example includes chickens that eat more when confronted with video images of other feeding chicken than when the video image shows an empty cage (Keeling & Hurnik, 1993). Social facilitation exists in many invertebrate species such as crabs (Kurta, 1982), centipedes (Hosey et al., 1985), scorpions (El Bakary & Fuzeau-Braesch, 1987) and occurs ubiquitously among the social insects. It has been studied in wasps (Parrish & Fowler, 1983; Fowler, 1992; Reid et al., 1995), bees (Grasse´ & Chauvin, 1944; Chauvin et al., 1985), ants (e.g. Chen, 1937; Chauvin, 1944; Grasse´ & Chauvin, 1944; Lamon & Topoff, 1985; Klotz, 1986; Salzemann & Plateaux, 1988; Ho¨lldobler & Wilson, 1991) and termites (e.g. Grasse´ & Chauvin, 1944; Grasse´, 1946; Lenz & Williams, 1980; Williams et al., 1980; Afzal, 1983; Okot-Kotber, 1983; Springhetti, 1990). Most of the classical studies of social facilitation have focused on task performance by isolated individuals, contrasted with that of individuals in groups (e.g. Chen, 1937). In termites, however, studies of a different nature have been carried out, investigating survival as a function of group size. Williams et al.
1. Introduction Interactions among the many individuals that compose social groups, generate a range of behaviours not present at the individual scale. Large-scale spatiotemporal patterns of behaviour may develop following the concurrent activity of individuals that do not have pre-programmed information for generating these. Since such patterns are the outcome of social interactions, they are not observed when individuals are isolated. One of the most basic manifestations of such phenomena is social facilitation, commonly defined as the ordinary patterns of behaviour that are initiated or increased in pace or frequency by the presence or actions of other animals (Wilson, 1980). Hence, social facilitation is a nonlinear response behaviour whose prerequisites are the capacity of animals to sense, excite and communicate with others and whose relative strength is related to the number of participants involved. Social facilitation is common among several animal species. Examples range from humans who consume more food when in groups than when alone † Author to whom correspondence should be addressed. Present address: Departamento de Sistemas Complejos, Instituto de Fisica, UNAM, Mexico (01000) D.F., Mexico 0022–5193/96/160373 + 08 $18.00/0
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(1980) considered three species of Cryptotermes and investigated the survivorship for groups of one, two, four, eight, 16 and 32 pseudoworkers kept with food supply. They found that survival increased with group size. A different approach was adopted by Lenz & Williams (1980) who investigated the survival of Nasutitermes nigriceps (Hald) in environments rich in food, but with different container sizes and found that survival and wood consumption decreased with increasing container size. This experiment provided strong evidence that interactions between social individuals are crucial to survival, since increasing the container size is equivalent to a reduction of the group density and hence of the rate of contacts and interactions. In these two experiments, however, Cryptotermes and Nasutitermes were provided with food sources, rendering difficult the estimation of the importance and role of social interactions in the mechanisms that lead to increasing survival. Grasse´ (1986) commented on the correlation between group size and longevity in termites, but it is not clear whether his own experiments were carried out with or without food supply (he stated that the termites were kept in optimal conditions, which suggests that food was available). Despite being a behavioural phenomenon widely present across the animal taxa, social facilitation is poorly understood and has received very little attention both experimentally and theoretically. In social insects, the knowledge regarding specific biochemical or neurophysiological mechanisms involved in social facilitation is severely limited despite being one of the most basic and fundamental principles of social behaviour. In this article, we describe a computersimulated model of social interactions where interactions between mobile automata increase the individual longevity and hence that of the group. In addition, we provide experimental evidence that the survival of real termites is significantly correlated with the group size when in resource-deprived environments. The experimental results are useful for parameterising the model, that in turn is used to compare qualitatively the process of termite survival. Results suggest that social interactions play an important role in increasing survival of groups or, conversely, that its absence contributes to a reduction of the longevity of isolated individuals.
2. Survival Experiment Nasutitermes cf. aquilinus workers (third instar and beyond) were collected from a single wild colony from 2.1
Vic¸osa, Minas Gerais in Brazil and were randomly placed in groups of one, two, four, eight and 16 individuals (four replicates). The groups were confined in test tubes made of transparent glass (9.5 cm × 1.4 cm) with hermetically sealing rubber caps. The tubes were kept horizontally separated by plastic foam to prevent stridulation or other mechanically transmitted signals to propagate between the tubes. The workers were allowed to acclimatise for 12 hr and were incubated in the dark in a constant temperature chamber (25°C 2 0.5). No food or water was provided. The samples were kept in darkness and were exposed to light during the counting of survivors only (no more than 5 min). Observations were made at 12 hr intervals. The experiment was continued until 24 hr after all the solitary individuals were dead in order to check the consistency of the trend observed in the multi-individual treatments. Seven observations were made. The data were subjected to survival analysis, using Weibull frequency distribution, as described by Crawley (1993) (see also Pinder III et al. 1978). Significance was assessed by deleting terms from a model until a change of deviance with a P Q 0.001 was observed. Differences between treatments were evaluated by aggregating factor levels until the consequential change in deviance was significant at P Q 0.001). 2.2 All individuals were alive at the time of the first observation and the first dead workers began to appear 24 hr after being confined. The analysis of the mortality figures for the seven observations give significant results that allow us to conclude that mortality was higher for isolated workers and lower for individuals in groups (x 2 = 25.44; 4 d.f.; P Q 0.001). A tendency that is easily identified in the data plotted in Fig. 1. It was clear from the observations, that workers in groups tended to cluster together and to engage in social interactions that included antennations and grooming and that this behaviour was more conspicuous in larger groups.
3. A Model of Social Interactions The use of models incorporating discrete interacting entities with behavioural traits that closely match those found in real individual organisms has increased in recent years (DeAngelis & Gross, 1992; Judson, 1994; Kawata & Toquenaga, 1994). These models, so-called Individual Based Models, are very useful for modelling patterns of self-organized collective behaviour that emerge without being explicitly specified
Survival
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F. 1. Raw-data from the survival experiment. Per-capita survival was large as the group size increased. The highest survival occurred in groups of 16 individuals. Survival for groups of four and eight individuals was similar but larger than survival for paired and isolated individuals. Each dot represents the total proportion of survivors for each group-size category (four replica per group).
in the individual’s designs or in their rules for interaction. One of the first examples of models with interacting individuals, in the context of social insects, was developed by Goss & Deneubourg (1988) to study the autocatalytic generation of cycles of activity in ants. Recent examples include automata models for self-organized pattern formation in honey bee colonies (Camazine, 1991) and models of selforganized trail formation in ants (Edelstein-Keshet et al., 1995). Automata models of termite behaviour are scarce and, as far as we know, the only model previously published is due to Courtois & Heymans (1991) who studied termite nest building behaviour using cellular automata and based on the idea of ‘‘order through fluctuations’’ developed by Deneubourg (1977). Here we present a model based on the mobile cellular automata concept (Miramontes et al. 1993a, b; Sole´ et al. 1993a, b; Miramontes, 1995) to explore the possible role of social interactions in the pattern of increased termite survival as a function of group size. In our model, a set of interacting individual agents (termites in this case) are defined over a rectangular lattice and follow some of the simple behavioural rules found in real termites. Each cell of the lattice is occupied by only one individual. Termites are able to move randomly to empty adjacent cells, where they interact with those in the eight most immediate cells (reflecting the fact that real termites interact mainly by direct body contact: antennations, grooming and trophallaxis). The lattice has fixed boundary conditions (zero-flux) meaning that no termite leaves the
lattice or appear at the opposite edge when reaching one of the edges. We define a real-valued state variable ak $ [0,1] for each individual k termite indicating, whether that individual is dead or not. A termite is considered to be alive while this variable is greater than zero and dead if zero. This variable increases each time a termite interacts with others, reflecting the ability of termites to excite and respond to the presence of colony mates. This quantity ak inevitably declines to zero after a finite time because termites in isolation do not have access to external inputs. The process of social interactions is defined through a coupling function that reflects the fact that, firstly, individual termites cannot be infinitely excited after repeated interactions, thus the function has to be bounded and, secondly, it is desirable that the activity of an individual increase or decrease according to a single parameter representing the degree of excitability. A continuous function, widely used in models of excitable biological media, that conveniently accommodates these requirements is the hyperbolic tangent function (Amit, 1988). Hence, the coupling between interacting termites is defined as follows:
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akt = tanh gk l a ait − 1 + akt − 1 i=1
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where ait is the activity of the i-th termite (maximum of eight) in the first neighbourhood of the ak termite at time t (measured as computer time-steps), gk is the individual excitability parameter and l is a parameter measuring the group-level ability to communicate (see Fig. 2). The simplicity of the model relies on only two parameters that are biologically justifiable.
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F. 2. Temporal evolution of the state variable ak for a single automaton when isolated (curve A) and when in a group of size 16 (curve B) as given by eqn (1). In the first case, the initial value asymptotically decays to zero, while in the second, the decay to zero is delayed each time an interaction occurs. A termite is considered alive when ak q 0 and dead when ak = 0. The threshold value below which ak was regarded as zero was 10 − 10. Initial value of ak was 1.0, gk = 0.7, l = 0.31, lattice size was 7 × 7.
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The parameter gk (often called the gain of the hyperbolic tangent function) accounts for interindividual differences in the capacity to excite and interact (i.e., some individuals are more likely to respond to stimuli of the same intensity than others). For example, it is well known that young adults, in many social species, respond to antennations from nestmates more readily than older ones, a fact that has been quantified in the case of the spontaneous activations of ants in the genus Leptothorax (Cole, 1992). Also, individuals may respond to their mates and behave in different ways depending on their state of health. Parasitised termites, for example, suffer profound morphological modifications (Krishna & Weesner, 1970; Grasse´, 1986) that are likely to provoke changes in their behaviour and modify the way in which they interact with others. An extreme example of this is observed in parasitised bumblebees which may even desert the colony, stopping all forms of interaction with their nestmates (Poulin, 1992; Mu¨ller & Schmid-Hempel, 1992). Hence, young or healthy individuals are characterised by high values of gk , while old or unhealthy individuals would have a lower value. As previously mentioned the second parameter l quantifies the group-level ability to communicate and is equal for each individual in a group. This parameter indicates the strength or intensity with which the members of a given species interact among themselves. A group of non-social individuals, that do not interact between them at all, would have a l of zero. Organisms characterised by a high value of l would be the most socially developed, while those characterised by medium values would be the species of lower degree of socialility, the case of gregarious, presocial species, etc. The experimental results just described were used to find the correct parameter values of the model, as discussed in the next section. 4. Model and Experiment: Results Lattice size was estimated by measuring the average worker area (the projected polygon formed by legs and antenna) and the average walking area inside the tubes. Average worker area was approximately 18.3 mm2 while the effective walking area was 760 mm2, so that about 42 individuals would saturate this space. The length-to-width ratio of the test tubes was 9.5 cm/1.4 cm 1 6.8. A lattice of 14 × 3 = 42 cells was chosen because it contains a maximum of 42 cells and its length-to-width ratio is 4.7, a value close to the one found for the tubes. Both the maximum number of cells and the proportions of the container
are very important for the model’s result, as would be discussed later. In order to estimate gk , we assumed only one individual in the lattice, so that the coupling function (1) reduces to:
0 1
akt = tanh gkakt − 1
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that is clearly independent of l. The average time-to-death of the isolated termites (group size = 1) was found to be of about 42 hours and we found that a value of gk = 0.7 in eqn (2) gave a time-to-death of 42 time steps in the computer simulations. The initial value of the state variable ak0 was set to 1.0. The parameter l was estimated by noting that the average time-to-death in the groups of 16 termites was 123 hours. We found that given gk = 0.7, a l of 0.31 gave a time-to-death of 123 time steps in automata groups of size 16. It is important to remember that since time in the experiment (hr) and time in the model (computer steps) are only roughly equivalent, the parameter values estimated are just good approximations that permit similar qualitative behaviour between the experiment and the simulation. Although when other values of these parameters were used, the qualitative behaviour of the model was quite robust. Survival curves (Weibull distribution) estimated from interacting automata in groups of one, two, four, eight and 16 individuals are shown in Fig. 3(a), as denoted by the numbers beside the curves. Individual mortality differs significantly according to grouping (x 2 = 316.6; 4 d.f.; P Q 0.001), in a way that termites in groups of 16 individuals survived longer than those in groups of eight individuals (x 2 = 238.6; 3 d.f.; P Q 0.001); which in turn survived longer than individuals from smaller groups (1, 2, 4) (x 2 = 47.4; 3 d.f.; P Q 0.001). Individual mortality does not differ for groups of one, two and four (x 2 = 0.34; 1 d.f.; P q 0.50). Survival curves estimated from the data of Nasutitermes cf. aquilinus termites starved to death in groups of one, two, four, eight and 16 individuals (Fig. 1) are represented in Fig. 3(b). Individual mortality differs significantly according to grouping (x 2 = 25.44; 4 d.f.; P Q 0.001), in such a way that termites from larger groups (16 individuals) survived longer than those from groups of eight and four individuals (x2 = 8.89; 1 d.f.; P Q 0.001); which in turn survived longer than individuals from smaller groups (one and two) (x2 = 8.22; 1 d.f.; P Q 0.001). Individual mortality does not differ either for groups of one and two (x 2 = 0.09; 1 d.f.; P q 0.80) or for groups of four and eight (x2 = 0.53; 2 d.f.; P q 0.50).
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is, indeed, in agreement with the experiment by Lenz & Williams (1980) where the survival of termites was found to decrease as the container size was increased. The geometry of the container also affected the outcome of simulations. We explored this aspect by
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F. 3. Qualitative comparison between survival of simulated and real termites using a Weibull distribution function. The same qualitative patterns are observed for simulated (a) and real termites (b) suggesting that social interactions are important in the mechanism leading to longer survival in grouped termites. Notice that survival for groups of size four and eight do not differ much. It is not clear why this was observed; but is likely to be related to the experiment resolution and further experimental work would be required to solve this. Nevertheless, the general trend of increased survival with group size is confirmed both in the experiment and model. One hundred replicas per group-size category were used in the simulation. Initial value of ak was 1.0, gk = 0.7, l = 0.31, lattice size was 14 × 3. Initial positions were chosen randomly.
It is clear from the shape of the curves, that the instantaneous death rate (i.e., the Hazard function) decreases with increasing group size. Therefore, individuals belonging to smaller groups were subject to a higher risk of death than those belonging to larger groups. Both the size and geometry of the container seems to be important for survival, so we explored the model to assess survival using different lattice configurations. The effects of size were simulated by using square lattices of different sizes, so that different values of density were obtained for each group size. It was observed [Fig. 4(a)] that increased lattice size (or decreasing density value) affects the chances of survival. As the lattice size is increased, the effects of social facilitation are less striking. When the lattice reach the value of 20 × 20 (density = 0.04) the phenomenon of increased survival with increased group size was negligible. This computer experiment
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F. 4. The effects of lattice size for the survival of grouped automata. The intensity of social facilitation decreases as the lattice size increases (four square lattice sizes were used). Notice that for a lattice of size 20 × 20, the effects of social facilitation are negligible because the average rate of contacts between the automata is very small. (b) The effects of the lattice geometry on the automata survival were explored by using different combinations of length-to-width ratio (numbers inside parenthesis) while conserving constant the total number of available cells (42). Social facilitation was maximal when the lattice was square and minimal when it was a one-dimensional array of 42 cells. Initial value of ak was 1.0, gk = 0.7, l = 0.31. Initial positions were random. One hundred replicas per group-size category were used. Error bars are standard deviations.
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using lattices with a constant number of cells (42) but with different values of length-to-width ratio. We found that survival was maximal in square lattices and was minimal when the lattice was reduced to a one-dimensional array of 42 cells [Fig. 4(b)]. In both cases (reduced density and different geometry) the fundamental cause affecting survival, was the decrease of the rate of contacts and hence of the frequency of interactions between individuals. In the first case because spatial dilution of random-mobile individuals means lower number of interactions per unit of time (automata have to travel large distances for an encounter to occur) and in the second case, because a high length-to-width ratio means that automata have less neighbours around for interacting. Thus far, it has been assumed that all automata are identical in their individual characteristics. Nevertheless, in order to bring the simulation closer to the biological reality, inter-individual variability has to be considered. Inter-individual differences are thought to have important consequences for the way insect societies organise, work and behave. Oster & Wilson (1978) argued that individual size (S), age distribution (A) and caste composition (C) are three factors that critically regulate the ergonomic efficiency of the whole society in such a way that the demography may even be of an adaptive nature. It seems reasonable then to assume that the success of insect societies highly depends on the behavioural diversity of its individual components. The introduction of demographic factors (S, A and C) seems then to be a very interesting field to investigate and we used our model to perform an exercise in this direction. Inter-individual variability was tested by comparing the performance of groups of different sizes depending on whether the value of gk was the same or differnt for all individuals. In the case where gk had a constant value (0.7), we observed that social facilitation was quite evident (Fig. 5, curve A). Groups of 16 individuals clearly survived more than groups of smaller size and the individual automata in those groups survived about four times longer on average, than isolated automata. A second simulation was carried out where the individual values were randomly generated on an interval with a flat probability distribution (gk = 0.5 2 0.2), so that there were individuals with high (0.7 maximum), medium and low (0.3 minimum) values of gk intermixed (Fig. 5, curve B). In this last case, while social facilitation is still present, it is of a less intense nature. This could be explained by the fact that individuals with low gk , as said before, are not particularly good at interacting and their participation in the generation of collective behaviour, may be weak. A second result
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F. 5. Social facilitation in automata as a function of group size and inter-individual variability. Social facilitation is quite clear in groups of automata that have the same value of gk (curve A). Automata in larger groups lived longer than those in smaller groups. Automata in groups of 16 lived four times longer, on average, than those in isolation. In this case gk = 0.7 and l was 0.31. (Curve B) Inter-individual variability is introduced with a flat probability distribution of values of gk in the range 0.5 2 0.2. Social facilitation was less intense than in curve A, while the variance of individual time-to-death keeps more or less equal in large groups regardless of individual variability. Initial value of ak was 1.0, gk = 0.7, l = 0.31. Initial positions were random. One hundred replicas per group-size category were used. Error bars are standard deviations. Lattice size was 14 × 3.
derived from contrasting curves A and B in Fig. 5, is that inter-individual variability decrease the variance of the time-to-death when group size is large. It would be highly desirable to have experiments that provide exact demographic parameters so that the values of gk could be arranged accordingly in such a way as to test whether a given probability distribution function may prove to be ergonomically optimal or not. It is clear that the scenarios presented in Fig. 5, while demonstrating the most striking effects of social facilitation, are quite unlike in nature. Nevertheless they provide a first approximation to explore how demographic factors and inter-individual variability possibly affect the performance of social groups, as has been previously suggested by other authors (Oster & Wilson, 1978). 5. Discussion Current hypotheses on social facilitated phenomena rely on the assumption that physiological processes (hormonal, nervous system activity, etc.) are modified because of the mere presence of
conspecifics in the environment. These changes (e.g. hormonal activity) have been measured in animals such as rats and monkeys (Zonjoc, 1975) but no direct observations exist for small and fragile organisms such as termites. Facing the lack of detailed information, regarding biochemical or neurological processes affecting social facilitation in relation to survival, we designed an experiment where termites were predominantly engaged in social interactions. The experiment consisted in groups of termites of different sizes confined in closed containers without supplies of food, water or oxygen (the oxygen available was only the one trapped in the tubes, no air was allowed to circulate). The experiment was designed in this way because the number of variables involved in the process of survival was reduced and because it allowed to assess the importance of resource supplies by contrasting with similar experiments that contained these. In higher animals such as rats, monkeys or chickens, ‘‘interaction’’ means a very complex mixture of inputs simultaneously affecting various senses; however, incomplete or partial clues are often enough for an ‘‘interaction’’ to occur, as is dramatically demonstrated by feeding chickens showing social facilitation (they eat more) when in front of video images of another feeding chicken (Keeling & Hurnik, 1993); so that a physical interaction, as such, was not necessary. In termites, interactions happen mostly by direct body-to-body physical contacts (antennations, food sharing, licking, etc.) so that individuals in large groups are more likely to engage in interactions than those in small groups (and obviously more than those under isolation). In fact, interactions are one of the most common behaviours displayed by social insects, as is evident from the ethograms reported elsewhere. Social insects invest a fairly large amount of the time budget engaging in social interactions. Hence, there is enough motivation for assuming that social interactions are relevant for the mechanisms that regulate important physiological processes at the individual level, including metabolic rates that are closely related to survival chances in resource-deprived environments. In our experiment, no food, water or fresh air supply was available to the confined Nasutitermes workers. Since it is known that cannibalism in termites could provide food supply in stressed colonies (Cook & Scott, 1933), large groups could be thought to be subject to less starvation levels per individual than smaller ones, since more prospective food (in the form of nestmates) was available. However, microscope observations of dead individ-
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uals did not reveal any sign of cannibalism. Another possible source of food, trophallaxis (mouth-tomouth food sharing), is also not important in this context, since it is known to decrease with increasing group size (Afzal, 1983). Therefore, we can assume that all experimental units were reasonably submitted to similar levels of starvation, having theoretically the same average longevity due to food scarcity only. Interestingly, when other authors have found larger groups to profit from social interactions, it seems that the presence of food in the experiments denoted an assumption that survival would result from the well-known increased food intake promoted by social facilitation. Our results, however, suggest that social interactions may enhance survival of grouped individuals, even without resource supply. This supports the view that social facilitation may be involved in triggering physiological changes in such a way, that metabolism may become more efficient so that starving individuals can make better use of body reserves and survive longer. While there is no conclusive evidence that this is the general case in termites (oxygen consumption was not measured in our experiment), other studies provide good evidence that individual metabolic rates, in ants, decrease when individuals are in larger groups (Peakin & Josens, 1978). Nash & Pierce (1994), for example, have shown that Iridomyrmex isolated ants consumed an average volume of 0.0063 ml of oxygen per hour, while grouped ants consumed 0.00168 ml hr − 1. To test the above ideas, we constructed a model whose only biological parameters are the capacity to interact with neighbouring individuals and the inevitable death after some time. The pattern that emerged is essentially the same: survival increases as a function of group size in an environment that lacks resource supply. Moreover, the important result to remark is that the dependence between longevity and group size is nonlinear. In summary, a synergetic phenomenon arises where the basic generating mechanism is facilitation through individual interactions. We would like to thank M. P. Hassell, M. Crawley, E. Vilela, K. Scho¨nrogge, D. Nash, P. Rohani, M. Bonsall, R. Constantino, R. Parentoni and A. Chopps for their help, comments and suggestions. Two anonymous referees provided useful criticisms. O. M. is grateful to the members of the Departamento de Biologia Animal, Universidade Federal de Vic¸osa for their hospitality during a research visit. This work was partially supported by grants from the Universidad Nacional Auto´noma de Me´xico and ORS (O.M.).
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