Mathematical model of self-organizing hierarchies in animal societies

Abstract

We propose a mathematical approach to the modelling of self-organizing hierarchies in animal societies. This approach relies on a basic positive feedback mechanism that reinforces the ability of a given individual,to win or to lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. Motivated by experiments carried out on primitively eusocial wasps Polistes, the model, is based on coupled differential equations supplemented with a small stochastic term. Numerical integrations allow many different hierarchical profiles to be obtained depending on the model parameters: (1) the particular form of the probability for an individual to win or lose a fight given its history, (2) the probability of interaction between two individuals, (3) the forgetting strength, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual and (4) two individual recognition parameters, which set the contribution of individual recognition in the process of hierarchical genesis. We compare the results, expressed in terms of a hierarchical index or of the Landau number that describes the degree of linearity of the hierarchy, with various experimental results.