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

Abstract

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 brood 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 items are at the center, the largest at the periphery, and prepupae dispersed in-between. In spite of its influence on algorithmics and robotics 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 behaviour 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 feedback effect, so that our results suggest that brood sorting is a case of self-organization that does not involve swarm intelligence.

Swarm‐based spatial sorting

“…Although we present no formal theoretical analysis of our model, recent related work (Gaubert et al, 2007) supports our observation that a key component in the construction of annular structures is that of the perimeters (referred to by Gaubert et al as the "virtual sizes" of objects). They argue that "...we believe that we have highlighted two of the major causes of the concentric annuli formation in Leptothorax unifasciatus ant colonies, that is: almost minimization of the average of squared-distances between free larvae, coupled with great differences between virtual sizes of brood items."…”

Simplifying and improving swarm-based clustering

“…That is, one ant is able to produce the same clusters that are created by multiple ants. This observation is also confirmed by Gaubert et al [7], who proved that MABC will form clusters almost surely. Note that MABC model was tested on a trivial problem: clustering items with known distinct categories, and no sophisticated distance computation is required.…”

Architecture, space and information in constructions built by humans and social insects: a conceptual review

“…The existence of a spatial segregation of tasks without physical barriers is understood to be the result of simple self-organizing processes of differential aggregation [174][175][176][177]. This suggests that different areas within a nest-with or without physical separation-might specialize in a particular type of task, not because of their intrinsic characteristics, but because of social feedback loops between the workers: the more a task is performed at a location, the more likely it will be performed again at that location.…”