Uniform-in-time propagation of chaos and bifurcation in two-type adhesion systems

TL;DR

This paper analyzes a two-phenotype tumor cell adhesion model, proving uniform propagation of chaos and long-term stability in weak interactions, and exploring bifurcations in strong interaction regimes.

Contribution

It establishes uniform-in-time propagation of chaos for a nonlocal adhesion model and investigates bifurcation phenomena in strong interaction regimes.

Findings

Proves exponential long-time contraction in the nonlinear McKean--Vlasov equation.

Establishes uniform-in-time propagation of chaos in the weak-interaction regime.

Identifies bifurcation leading to loss of stability in the strong interaction regime.

Abstract

We study a nonlocal adhesion model for two interacting tumor cell phenotypes, combining diffusion, pairwise interactions, and random phenotypic switching. The system admits a microscopic diffusion--jump particle description whose mean-field limit is a nonlinear McKean--Vlasov equation on a product space encoding position and internal state. We first establish uniform-in-time propagation of chaos in the weak-interaction regime using a coupling approach that combines reflection coupling for the diffusion with an optimal coupling of the spin-flip dynamics. As a byproduct, we obtain exponential long-time contraction for the nonlinear McKean--Vlasov equation in the first-order Wasserstein distance, implying uniqueness of the stationary distribution. We also investigate the complementary regime of strong interactions, where the homogeneous equilibrium may lose stability through a bifurcation mechanism.