Mean-field dynamics of attractive resource interaction: From uniform to aggregated states

TL;DR

This paper analyzes a nonlinear mean-field model of resource distribution among agents, providing a complete description of its long-term behavior, including conditions for uniform or aggregated states, with convergence to a unique equilibrium.

Contribution

It offers a full analytical characterization of the system's dynamics, including fixed points, stability, and asymptotic scenarios, generalizing classical opinion-dynamics frameworks.

Findings

Unique fixed point exists for all parameters

System always converges to equilibrium without oscillations or chaos

Parameter regimes determine uniform or aggregated states

Abstract

We introduce and study a nonlinear discrete dynamical system describing the evolution of a resource distribution among interacting agents. The model generalizes several classical mean-field and opinion-dynamics frameworks and is defined on the standard simplex, where each coordinate evolves according to an interaction rule depending on preference-based mean-field interactions. We provide a complete analytical description of the long-term behavior of the system. First, we establish monotonicity properties and show that the dynamics always remains in a positively invariant region determined by initial conditions. We prove the existence of a unique fixed point for any admissible parameter set and derive an explicit closed-form formula for the equilibrium in arbitrary dimension. We then analyze the local stability of the fixed point and identify parameter regimes leading to aggregation or uniform distributions. Finally, we characterize all possible asymptotic scenarios and show that, despite the nonlinear structure, the system does not exhibit oscillatory or chaotic behavior: every trajectory converges to the unique equilibrium. The results provide a full qualitative theory for this class of monotone resource-interaction models and offer a mathematical explanation for the transition from uniform to aggregated states.