A mathematical model for the dynamics of clustering

The formation of several clusters, arising from attracting forces between nonidentical entities or agents, is a phenomenon observed in diverse fields. Think of people gathered through a mutual interest, swarm behaviour of animals or clustering of oscillators in brain cells.We introduce a dynamic model of mutually attracting agents for which we prove that the long-term behaviour consists of agents organized into several groups or clusters. We have completely characterized the cluster structure (i.e. the number of clusters and their composition) by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures.The versatility of the model will be illustrated by discussing its relation to the Kuramoto model and by describing how it applies to a system of interconnected water basins.