Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Myeongju Chae, Young-Pil Choi

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.

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

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.
Paper Structure (17 sections, 11 theorems, 219 equations)

This paper contains 17 sections, 11 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of paper
  4. Preliminaries
  5. Finite-particle system and Liouville equation
  6. Nonlinear McKean--Vlasov process
  7. One-dimensional spin-flip dynamics for the internal state
  8. Reflection coupling
  9. Uniform-in-time propagation of chaos
  10. Coupling estimate
  11. Proof of Theorem \ref{['thm_propa']}
  12. Long-time exponential contraction
  13. Proof of Theorem \ref{['thm_lt']}
  14. Bifurcation for large $\eta$
  15. Fourier basis on the even subspace
  16. ...and 2 more sections

Key Result

Theorem A

Let $\mu^N$ be the empirical measure associated to the interacting diffusion--jump system eq0--eq1, that is, where $(X_t^i,Y_t^i)$ is a solution of eq0--eq1. Let $\bar{\rho}$ be a solution of intro_new_pde. Assume that $F$ satisfies hyp_F and the interaction strength $\eta$ is sufficiently small. Then there exist constants $c>0$ and $C>0$, independent of $t$ and $N$, such that for all $t\ge0$, a

Theorems & Definitions (18)

  • Theorem A: Uniform-in-time propagation of chaos
  • Theorem B: Long-time exponential contraction
  • Theorem C: Bifurcation criterion for large interaction strength
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 8 more