Leader formation with mean-field birth and death models
Provides a theoretical framework for modeling leader-follower systems with nonlinear transition rates, but the contribution is primarily mathematical and incremental.
The paper develops a mean-field model for leader-follower dynamics with bidirectional mass transfer, proving existence and uniqueness of solutions and establishing equivalence to a PDE-ODE system. Numerical simulations demonstrate the model's applicability to social interaction dynamics.
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.