Mean-field Analysis of Multipopulation Dynamics with Label Switching

Abstract

The mean-field analysis of a multi-population agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markow-type jump process on the probability that each agent has of belonging to a given population. A general functional analytic framework for the well-posedness of the problem is established, and some concrete applications are presented, both in the case of discrete and continuous set of labels.In the particular case of a leader-follower dynamics, the existence and approximation results recently obtained in [2] are recovered and generalized as a byproduct of the abstract approach proposed.

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

“…The main objective of this paper is to present a rigorous proof of the formal derivation described above, by means of a multi-step Lagrangian scheme. The scheme we propose is suitable for approximating all equations in the class considered in [31] (we refer to Sect. 2 for the precise details).…”

“…In the subsequent contribution [31], this approach has been suitably extended as an abstract toolbox which is capable of rigorously describing the mean-field limit of a larger class of models which share the following features:…”

“…The framework proposed in [31] couples a nonlinear transport dynamics for the positions x ∈ R d of the agents with a Markov-type jump process for the labels λ ∈ P(U ) (see Sect. 2).…”

“…Since Lagrangian solutions are also Eulerian solutions, the equivalence of the two notions follows if one is able to prove that Eulerian solutions are also Lagrangian. For the model studied in [31], and also for other relevant ones [14], these two notions of solution are equivalent. This has been achieved by means of the superposition principle (see [36], and also [6,Theorem 8.2.1], [8, Theorem 7.1], and [5,Theorem 5.2]), which provides the uniqueness of Eulerian solutions [5,Theorem 5.3].…”

“…In the formal limit for h → 0, we obtain the weak formulation of the nonlinear continuity equation (1). A related heuristic derivation has been outlined also in [2,Remark 4.1], in the context of a leader-follower dynamics which also fits in the setting of [31]. In this case, the R d -component of b Ψ also depends on Ψ , whereas the λ-component acts linearly on λ, modelling a Markov chain on U .…”