Generalized gradient structures for measure-valued population dynamics and their large-population limit

TL;DR

This paper develops a generalized gradient structure for measure-valued population dynamics models, proves their convergence to a mean-field limit, and establishes a propagation of chaos result with a gradient-flow formulation.

Contribution

It introduces a rigorous generalized gradient structure for measure-valued processes and demonstrates their convergence to a mean-field limit with associated gradient-flow formulation.

Findings

Convergence of the forward Kolmogorov equation to a Liouville equation in the large population limit.

Establishment of a propagation of chaos result for the particle system.

Development of a generalized gradient-flow formulation for the mean-field limit.

Abstract

We consider the forward Kolmogorov equation corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics, including variations of the Bolker-Pacala-Dieckmann-Law model. Under the assumption of detailed balance, we provide a rigorous generalized gradient structure, incorporating the fluxes arising from the birth and death of the particles. Moreover, in the large population limit, we show convergence of the forward Kolmogorov equation to a Liouville equation, which is a transport equation associated with the mean-field limit of the underlying process. In addition, we show convergence of the corresponding gradient structures in the sense of Energy-Dissipation Principles, from which we establish a propagation of chaos result for the particle system and derive a generalized gradient-flow formulation for the mean-field limit.