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.