Ants in a labyrinth: a statistical mechanics approach to the division of labour.

Division of labour (DoL) is a fundamental organisational principle in human societies, within virtual and robotic swarms and at all levels of biological organisation. DoL reaches a pinnacle in the insect societies where the most widely used model is based on variation in response thresholds among individuals, and the assumption that individuals and stimuli are well-mixed. Here, we present a spatially explicit model of DoL. Our model is inspired by Pierre de Gennes' 'Ant in a Labyrinth' which laid the foundations of an entire new field in statistical mechanics. We demonstrate the emergence, even in a simplified one-dimensional model, of a spatial patterning of individuals and a right-skewed activity distribution, both of which are characteristics of division of labour in animal societies. We then show using a two-dimensional model that the work done by an individual within an activity bout is a sigmoidal function of its response threshold. Furthermore, there is an inverse relationship between the overall stimulus level and the skewness of the activity distribution. Therefore, the difference in the amount of work done by two individuals with different thresholds increases as the overall stimulus level decreases. Indeed, spatial fluctuations of task stimuli are minimised at these low stimulus levels. Hence, the more unequally labour is divided amongst individuals, the greater the ability of the colony to maintain homeostasis. Finally, we show that the non-random spatial distribution of individuals within biological and social systems could be caused by indirect (stigmergic) interactions, rather than direct agent-to-agent interactions. Our model links the principle of DoL with principles in the statistical mechanics and provides testable hypotheses for future experiments.

Introduction

Both human and animal societies display a division of labour, in which there may be an unequal distribution of effort between or within particular tasks, according to age or experience [1], [2], sex [3], physiology [4] or morphology [5]. Such specialisation has long been known to improve collective productivity [6] because learning allows individuals that focus on a subset of tasks to perform more efficiently than generalists (note however the exception to the rule provided by Dornhaus, 2008). Division of labour is most advanced in the societies of insects such as ants, bees, wasps and termites [7]. Within an insect society, there is typically considerable individual variation in the sensitivity to stimuli associated with particular tasks. One of the simplest models of Division of Labour (DoL), the fixed-threshold model (FTM), invokes this individual variation in sensitivity to such task-related stimuli [8], [9]. There is good evidence for the existence of such response thresholds in ants [10], [11], [12], bumblebees [13], the honey bee [14], [15], [16], wasps [17] and termites [18]. Experiments also provide strong support for the role of response thresholds for the maintenance of colony homeostasis [13], [19], [20]. Individual variation in thresholds has genetic [21], [22], morphological [11], hormonal [23] and developmental [24] components. Although direct evidence for a positive relationship between colony fitness and wide threshold distributions is lacking, there is evidence in the honey bee that genetic variation (the number of patrilines within the colony) positively influences colony fitness [25].

In the FTM, the decision of an individual whether or not to undertake a particular task, such as foraging or brood care, is determined by two parameters; the sensitivity of the individual to stimuli associated with the task (its response threshold), and the level of demand for that task (the stimulus value). When an individual senses that the stimulus exceeds its threshold value, it becomes activated, and performs some work. Through such activity, sensitive (low threshold) individuals reduce the stimulus level such that it often does not reach the threshold of their less sensitive nestmates. This negative feedback loop homeostatically maintains the stimulus level (the task demand) at a steady state, around which it fluctuates. A further consequence of this mechanism, and one that matches the pattern observed in nature, is that the activity distribution becomes right-skewed; a small minority of sensitive individuals perform the majority of the work [13], [26], [27], [28], [29].

Here, we extend the FTM by explicitly including space. This modification induces a spatial 'percolation' effect [30], [31], [32] in which small differences amongst agents in their response thresholds, are related to large differences in their probabilities of performing work. The extension of the original fixed-threshold models of DoL to include space removes the assumptions that individuals and task-associated stimuli are well-mixed. The movement and activity of the individuals in the spatial fixed-threshold model (SFTM) may then be analysed as a case of diffusion in disordered media - a well studied branch of statistical mechanics [33], [34].

The FTM assumes that individuals and stimuli are well-mixed and that each individual experiences the same global stimulus level equally. This is a simplifying assumption. However, it is realistic only for a minority of cases when the stimulus is spatially uniform. For example, honey bees homeostatically maintain the nest temperature and CO2 levels within certain acceptable ranges [13], [20]. When it gets too hot inside the nest, the bees with the lowest threshold to temperature begin to fan their wings, thereby increasing the airflow and reducing the temperature such that it never reaches the thresholds of their less sensitive nestmates. So because temperature and CO2 levels can be expected to be fairly uniform within the nest, the assumption of perfect mixing of stimulus and bees is justified. Therefore modelling this process as a non-spatial process is reasonable. However, when the stimulus in question is heterogeneous over space, perfect mixing can no longer be assumed. To appreciate the importance of modelling DoL without the assumption of perfect mixing, consider the honey bee comb, organised- or rather compartmentalised- into different zones in which the cells contain either brood, pollen or honey [35]. Therefore, tasks are not uniformly distributed in space [36]. Furthermore, individuals themselves are not well mixed. Despite their high potential mobility, individual ants [37], honey bees [38], bumble bees [39] and wasps [40] tend to be faithful to particular parts of the nest and this spatial fidelity persists even when many tasks are removed [41].

In the FTM, the distribution of the individual response thresholds within the colony- the Colony Threshold Distribution (CTD)- will bear directly upon the proportion of individuals that are mobilised to respond to a given stimulus level. Indeed, the precise form of the CTD will have significant adaptive consequences [42]. Abrupt discontinuities in the CTD would affect the ability of the colony to produce an appropriate response to small changes in the demand for labour. For example, consider the scenario in which the colony is evenly split between two types of individual; half of the ants have low thresholds and the other half have high thresholds. In that case, the colony will be unable to produce a graded response to fluctuating stimulus levels, because only 0% (both thresholds above stimulus level), 50% (stimulus level above the low threshold but below the high threshold level) or 100% (both thresholds below stimulus level) of the individuals may be active at any one time. If, on the other hand, the CTD has a continuous distribution, the colony will produce a more finely graded response that is proportionate to the stimulus levels.

Experimental data on the form of the CTD is rather limited. To our knowledge, only in the honey bee, Apis melifera is there a quantitative description of the CTD, which is approximately Gaussian [20]. Indeed, several previous simulation studies of the FTM have assumed a Gaussian distributed CTD [19], [43]. For simplicity, we first consider the case of the uniform CTD. Uniform distributions lack any central tendency (they are not humped) and so have a variance, defined by the range of the distribution.

For completeness we also investigate the influence of a Gaussian CTD upon the ability of the colony to minimise both the total task demand and the spatial variation thereof. As well as the aforementioned uniform and Gaussian CTDs, we also explore the consequences of completely removing individual threshold variation.

Methods

Before constructing a biologically relevant individual based model in two dimensions, we will introduce several important concepts and issues using the more abstract but simpler one-dimensional model [44], [45].

i. A model of division of labour in one dimension

Let us imagine a colony of heterogeneous and mobile ants, along with their (stationary) brood, inhabiting a ring-shaped nest in which the ring cross-section is so narrow that ants may not pass one another, although they may pass over the brood. The brood are regularly spaced, so the distance separating brood items is fixed. This scenario is modelled using a one-dimensional cellular automata with 500 grid squares with periodic boundary conditions. Each grid square contains a brood item, and within a single time-step an ant may only move one grid-square, that is, from one brood item to an adjacent brood item. However, there are more brood items than adult ants, so unlike the brood, the ants need not be regularly spaced. Each brood item demands regular labour, for example, grooming and feeding. Let us further assume that the demand of the brood for attention is a stimulus that can be detected by nearby ants. To reflect this, the stimulus grows over time, a single brood item is selected every time-step and its stimulus is increased by a fixed amount - this is the stimulus ‘drive’. Given the ring-shaped nest geometry, and the fact that ants are unable to pass one another, it is reasonable to allow each ant to perceive only local information about its stimulus environment. Thus, each ant may only detect the stimulus level of the brood items within a ‘domain of care’. This domain of care is defined as the brood item that the ant is standing on, plus the two items immediately adjacent to the current item. When an individual ant detects that one of the brood items in its domain of care requires attention (i.e., that the brood item has a stimulus value greater than the threshold of the attending ant), the ant moves to the site and performs work on the brood item. The reduction in the demand for work is reflected by reducing the stimulus value to the threshold level of the attending ant.

There are several possible formulations of response threshold functions in which the probability of response is not binary, that is, not purely deterministic [8], [9], [46], [47]. Although probabilistic response thresholds may be more biologically realistic, they require the choice of an extra parameter that controls the steepness of the sigmoid curve. Hence, we simplify the response thresholds to a step function; when the stimulus is less than the threshold, S<θ, the response probability is zero, and when S>θ, the response probability is equal to one.

It has been shown that collective regulation of task demand is improved when the threshold distribution is broad [48], [49]. Following Page & Mitchell (1998) and Theraulaz (1998) inter-individual threshold variation was generated in the simplest manner possible; the thresholds, θ, were drawn from a uniform distribution so sensitive ants were no more common than insensitive ants.

If the above scenario is initialised with a random spatial distribution of ants, and with all brood having zero stimulus, the total stimulus will grow at first, before reaching a steady state (Figure 1a). At the steady state, the increase of brood-care stimulus is balanced by the work performed by the ants. The collective output (the total work done per time-step) fluctuates intermittently.

Figure 1

Emergence of one-dimensional spatial division of labour.

a) Black line: The development of the steady state (world circumference = 500, N ants = 20, stimulus drive = 0.1 stimulus units per time-step). Red line: The total work done per time-step. b) The positions of the ants in the ring nest as a function of time. The ants measure their position clockwise from a fixed but arbitrarily chosen point along the ring. There is a transition from a random initial configuration, to one in which ants are aggregated into a few clusters, with low threshold ants shuttling between the clusters. The clusters are represented by the straight lines.

The most interesting outcome of the one-dimensional model concerns the spatial distribution of the ants. Initially, the ants are distributed at random positions along the ring nest, however, as the model self-organises towards the steady-state the ants become aggregated into a few clusters. In Figure 1b at t = 0 the ants are randomly (uniformly) distributed around the ring, but when the simulation reaches the steady state at t∼50,000, five stable clusters form. Interestingly, the distance separating the clusters is fairly constant, that is, on average the clusters are regularly spaced (overdispersed).

The mechanism behind this pattern is revealed when the average distance from every ant to its two neighbouring ants is plotted as a function of its response threshold (Figure 2). The greater the sensitivity of an ant (i.e. the lower its response threshold), the greater the distance separating the ant from its neighbouring ants. The clusters consist of ants with relatively high thresholds, while ants with relatively low thresholds shuttle between these clusters.

Figure 2

The mean distance to the left and right neighbours is an inverse function of the response threshold.

Periodic boundary conditions apply. World circumference = 500, N ants = 20, stimulus drive = 0.1 stimulus units per time-step, N simulations = 500.

The observation that sensitive ants ‘box in’ their less sensitive nestmates, implies that the activity distribution is skewed. Indeed, when the total work done by each of the ants within a given period is accumulated, and divided by the time elapsed, one arrives at a useful measure of the individual activity; the work done per time-step per ant. As in real social insect colonies, the individual activity distribution is highly right-skewed so that a minority of the ants perform the majority of the work (Figure 3). As there are many alternative functions for plotting distributions [50], [51], for ease of comparison, Figure 3 shows the skewed activity distribution plotted using two common methods; a Zipf-type rank distribution [27], and a survivorship (the complement of the cumulative distribution) function [26].

Figure 3

The skewed activity distribution.

Individual ant activity is measured on a per-ant basis, as the work done per time-step. Main panel; an activity-rank plot. A rank of 1st indicates the ant was the most active, and a rank of 20th indicates the ant was the least active. Panel insert: the same data as the main panel plotted as the survivorship of the individual ant activity. The distribution is exponential-like. Model parameters as in Figure 2 legend. All realisations were run for 50000 time-steps after reaching the steady-state.

In summary, when considering the task of attending brood items situated on each grid square in a one-dimensional nest, the activity of ants with a uniform distribution of threshold values and a finite domain of care is highly skewed. Relatively few ants with low thresholds will attend a majority of brood items while ants with higher thresholds cluster into groups and attend only a small fraction of the brood items. This skewness is induced by the spatial aspect and the indirect interactions among ants.

ii. A model of division of labour in two dimensions

Pierre de Gennes' Ant in a Labyrinth

We will introduce the more biologically realistic two-dimensional model with a brief discussion of the similarity between the concept of a spatially explicit DoL in social insects and the classic ‘Ant in a Labyrinth’ model [32]. The importance of de Gennes’ model cannot be overstated; it laid the foundation for an entire new field in statistical physics, known as ‘diffusion in disordered media’. We will use the statistical tools of this approach [30], [52] to demonstrate the influence the inclusion of space has on the DoL when the 'disordered medium' is a collective environment, such as a social insect nest, filled with stimuli that vary in intensity across space. The aim of this approach is to explore the consequences of removing the assumption of well-mixed stimuli and agents.

In de Gennes' (1976) model, each site on a two-dimensional square lattice is occupied with probability p and hence unoccupied with probability 1-p. The probability p is known as the probability of occupancy. For example, if the occupation probability is p = 0.5, then, on average, half of the lattice sites will be occupied, and half unoccupied. A single ‘blind ant’ is then randomly dropped onto an occupied site on the lattice. The ant follows a simple rule: choose randomly one of the four adjacent (nearest-neighbour) sites (NESW). If the chosen site is occupied, the ant moves onto it, however, if it is empty, the ant does not move. Either way, the time is incremented by one unit. The ant is termed ‘blind’ because the initial four-way choice is made irrespective of those sites' occupancies.

A cluster is defined as a set of occupied sites which are connected by one of their four nearest neighbours such that any two sites in a cluster may be reached by a series of consecutive steps to the North, East, South or West. The mean cluster size, , is the average cluster size to which an occupied site belongs (excluding the infinite percolating cluster, see later). If p is increased from p = 0, the mean-cluster size incresases and a critical point is reached (p) where the largest cluster spans the lattice, and the average cluster size, diverges (goes to ∞). The lattice is then said to ‘percolate’. When p