Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics

TL;DR

This paper investigates the dynamics and energy properties of a two-species nonlocal interaction model on graphs, revealing complex behaviors like mixing and phase separation through analytical and numerical analysis.

Contribution

It extends the understanding of nonlocal cross-interaction systems to graph structures, demonstrating their rich pattern formation and energetic characteristics.

Findings

Rich pattern formation on graphs including mixing and phase separation

Analytical and numerical evidence of complex behaviors

Extension of continuous models to discrete graph settings

Abstract

We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of nonquadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof.