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

Stimulus input and ant activity update rules for the two-dimensional model.

<p>Stimulus input and ant activity update rules for the two-dimensional model.</p

The skewed activity distribution.

<p>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 1^st indicates the ant was the most active, and a rank of 20^th 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 <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018416#pone-0018416-g002" target="_blank">Figure 2</a> legend. All realisations were run for 50000 time-steps after reaching the steady-state.</p

The distance separating neighbouring ants depends upon the difference in their response-thresholds.

<p>a) Map of ant locations. Symbol sizes are proportional to the threshold of the ant, so sensitive ants have small symbols. b) Mean distance between an active ant and its nearest neighbour (NND), as a function of the <i>difference</i> in sensitivity between the two (<i>θ_i</i> - <i>θ</i>_NN). Error bars are one standard deviation. Drive, <i>δ</i>S = 10, N simulations = 100, N time-steps per simulation = 5×10⁴. The horizontal line shows the expected NND under conditions of complete spatial randomness (Expected NND = 2.56, <i>σ</i> = 1.25, N simulations = 2000).</p

Definition of parameters and response statistics for the two-dimensional model.

<p>Definition of parameters and response statistics for the two-dimensional model.</p

The stimulus landscape percolates at a critical response-threshold.

<p>a) A stimulus landscape as it appears to the outside observer (<i>δS</i> = 1, , N ants/<i>l</i>² = 0.04). The more stimulus a site contains, the darker the grey. b) Threshold-dependent site-occupancy for the same landscape as seen by a sensitive ant (<i>θ_i</i> = 1, <i>p</i> = 0.759). Sites with S<<i>θ_i</i> are white. The largest cluster on the lattice is coloured in red. The cluster ‘percolates’ across the lattice. c) Threshold-dependent site-occupancy for an ant with <i>θ_i</i> = 1.55, here <i>p</i> = 0.594. The occupancy is just above the critical occupancy (<i>p_c = </i>0.5927…), where the mean cluster area displays a phase-transition. d) Threshold-dependent site-occupancy for a less sensitive ant, where and <i>p</i> = 0.32. To this ant most sites do not contain stimulus, clusters of occupied sites do not span the lattice, and hence the landscape does not percolate.</p

Emergence of one-dimensional spatial division of labour.

<p>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.</p

The stages involved in an 'ant bout'.

<p>The position of the ant is indicated by the red square. Each time-step every ant checks its local neighbourhood (the four blue squares) for any stimulus that exceeds its individual response threshold (S><i>θ_i</i>). Here, at <i>t</i>₁ some stimulus arrives in the ant's West square, such that S_W><i>θ_i</i>, so the ant moves onto it, instantaneously reducing the stimulus at that site to its threshold level, <i>θ_i</i>. If more than one neighbouring site has S><i>θ_i</i>, the ant chooses randomly between them (at <i>t_n</i>). At <i>t_n+1</i> the ant has exhausted the stimulus in its four adjacent squares, so it is trapped.</p

The structure of the stimulus landscape- and hence also the bout magnitude- are nonlinear functions of the individual response-threshold.

<p><b>a</b>) Mean cluster size, , normalised by dividing by the maximum cluster possible, <i>l</i>² and b) Mean standardised bout size (size/drive) for individual ant-bouts. Ants were assigned to threshold bins of logarithmically increasing width. The different curves represent different fixed drives (N simulations per drive = 500, ○; δS = 1×10⁻², ⧫; δS = 1×10⁻¹, •; δS = 1×10⁰, ▪; δS = 1×10¹).</p

The scale-free structure of the stimulus landscape.

<p>Both panels depict the survivorship (the complement of the cumulative distribution) function for: a) mean site occupancy, <<i>p</i>> and b) The mean cluster size, , normalised by the maximum cluster possible, <i>l</i>×<i>l</i>. Both <<i>p</i>> and are ensemble-averages, calculated by averaging across all individuals irrespective of threshold. The different curves represent different fixed drives (○; δS = 1×10⁻², ⧫; δS = 1×10⁻¹, •; δS = 1×10⁰, ▪; δS = 1×10¹).</p