Calculus of Variations and Geometric Measure Theory

G. Albi - M. Bongini - F. Rossi - F. Solombrino

Leader formation with mean-field birth and death models

created by solombrin on 17 Dec 2018
modified on 05 May 2019


Published Paper

Inserted: 17 dec 2018
Last Updated: 5 may 2019

Journal: Mathematical Models and Methods in Applied Sciences
Volume: 29
Number: 4
Pages: 633--679
Year: 2019
Doi: 10.1142/S0218202519400025

ArXiv: 1812.07074 PDF


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.