Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems

TL;DR

This paper rigorously derives population cross-diffusion PDEs from stochastic particle systems, establishing the mean-field limit and local diffusion models, and proves global existence of solutions with numerical validation.

Contribution

It provides a rigorous derivation of cross-diffusion systems from particle models and proves global existence of solutions in a new scaling regime.

Findings

Derivation of cross-diffusion PDEs from particle systems

Proof of global existence of solutions for small initial data

Numerical simulations validating the particle-level models

Abstract

Population cross-diffusion systems of Shigesada-Kawasaki-Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.