Leader formation with mean-field birth and death models

TL;DR

This paper develops a mean-field model for leader-follower dynamics with nonlinear state-dependent transition rates, proving well-posedness, establishing PDE-ODE equivalence, and demonstrating convergence of stochastic particle approximations.

Contribution

It introduces a novel mean-field framework for leader-follower interactions with mass transfer, including existence, uniqueness, and convergence results.

Findings

Proved well-posedness of the leader-follower PDE-ODE system.

Established convergence of stochastic particle approximations.

Provided numerical simulations illustrating social interaction dynamics.

Abstract

We provide a mean-field description for a leader-follower dynamics with mass transfer among the two populations. This model allows the transition from followers to leaders and vice versa, with scalar-valued transition rates depending nonlinearly on the global state of the system at each time. We first prove the existence and uniqueness of solutions for the leader-follower dynamics, under suitable assumptions. We then establish, for an appropriate choice of the initial datum, the equivalence of the system with a PDE-ODE system, that consists of a continuity equation over the state space and an ODE for the transition from leader to follower or vice versa. We further introduce a stochastic process approximating the PDE, together with a jump process that models the switch between the two populations. Using a propagation of chaos argument, we show that the particle system generated by these two processes converges in probability to a solution of the PDE-ODE system. Finally, several numerical simulations of social interactions dynamics modeled by our system are discussed.