A first mathematical model of brood sorting by ants: Functional self-organization without swarm-intelligence

Brood sorting, observed in Leptothorax unifasciatus ant colonies, is a major example of social insects ability to solve problems at the collective level. Two processes characterize this phenomenon: a process of aggregation of all items in a single cluster, coupled with a process of segregation of items in concentric annuli, each containing items of different type and ordered such a way that the smallest are at the center, the largest at the periphery, and prepupæ dispersed in-between. In spite of its influence on algorithmic and robotic methods, no formal explanation of the brood-sorting phenomenon was ever given. We present a first mathematical model devoted to brood sorting. Our hypothesis about ants behavior are consciously minimal: we assume that random rules their acts, not only when they walk but also when they choose a brood item that they pick up, or beside which they deposit the one they carry. The first part of our work deals with the process of aggregation in a single cluster. The main subject of our study is the time evolution of a mathematical function linked to the notion of cluster. We prove that, whatever the number of ants acting, this function tends to decrease until it reaches a threshold that we compute: this threshold matches with the formation of the single cluster. The second part of our work deals with segregation in concentric annuli. Coupling the concept of virtual size of a brood item to the previous conclusions leads to a realistic explanation of the concentric structure observed in ant colonies. Finally, we prove the existence of a feed-back effect, so that our results suggest that brood sorting is a case of self-organization that does not involve swarm-intelligence.