Flocking with Multiple Types: Competition, Fluid Limits and Traveling Waves
Abstract
We study a class of interacting particle systems on ℝ with two types. Particles evolve by independent jumps sampled from a fixed distribution, with type-dependent jump rates v+, v- and stochastic type switching driven by non-local order-based interactions. The switching rates depend on the empirical distribution through the proportion of opposite-type particles located ahead, leading to a nonlinear and discontinuous dependence on the empirical measure outside the standard Lipschitz McKean-Vlasov framework. Our first main result is a law of large numbers for the empirical measure process: we prove convergence, along subsequences, to a deterministic measure-valued process characterized by a McKean-Vlasov equation. The proof combines tightness in Wasserstein space with a martingale characterization of limit points. A uniqueness argument based on a Kolmogorov-Smirnov-type distance adapted to the ordering structure yields convergence of the full empirical measure sequence and, in turn, propagation of chaos on finite time intervals. We then study the long-time behavior of the limiting dynamics. Because the system has persistent drift, invariant distributions do not arise; instead, we analyze traveling waves, corresponding to stationary profiles in a moving frame. For exponential jump distributions, the associated non-local integro-differential system admits a local description. In the regime v+>v-=0, this further reduces to a coupled system of non-linear ODEs, allowing a phase-plane analysis that yields a traveling wave as a heteroclinic orbit connecting two equilibria. We also identify the wave speed and mass partition, and derive tail asymptotics by spectral analysis of the linearized system.
Related
- Existence and Asymptotic Stability of Traveling Waves of Two-species Lotka-Volterra Competition-Diffusion Systems via Geometric Singular Perturbations
- Diffusive-dispersive traveling waves and kinetic relations. Generic flux with multiple inflection points
- Large Network limit for Latent Variable Process Models with Endogeneously Evolving Interactions among Multiple Types
- Optimizing Location and Capacity for Multiple Types of Locomotive Maintenance Shops
- BRP-GABLE: An Interpretable Rule-Based Prediction Framework for Multiple Outcome Types
BibTeX
@misc{banerjee2026flocking,
title = {{Flocking with Multiple Types: Competition, Fluid Limits and Traveling Waves}},
author = {Sayan Banerjee and Andrew Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.11279},
eprint = {2605.11279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11279},
}
RIS
TY - UNPB
TI - Flocking with Multiple Types: Competition, Fluid Limits and Traveling Waves
AU - Sayan Banerjee
AU - Andrew Nguyen
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2605.11279
UR - https://arxiv.org/abs/2605.11279
AB - We study a class of interacting particle systems on ℝ with two types. Particles evolve by independent jumps sampled from a fixed distribution, with type-dependent jump rates v_+, v_- and stochastic type switching driven by non-local order-based interactions. The switching rates depend on the empirical distribution through the proportion of opposite-type particles located ahead, leading to a nonlinear and discontinuous dependence on the empirical measure outside the standard Lipschitz McKean-Vlasov framework. Our first main result is a law of large numbers for the empirical measure process: we prove convergence, along subsequences, to a deterministic measure-valued process characterized by a McKean-Vlasov equation. The proof combines tightness in Wasserstein space with a martingale characterization of limit points. A uniqueness argument based on a Kolmogorov-Smirnov-type distance adapted to the ordering structure yields convergence of the full empirical measure sequence and, in turn, propagation of chaos on finite time intervals. We then study the long-time behavior of the limiting dynamics. Because the system has persistent drift, invariant distributions do not arise; instead, we analyze traveling waves, corresponding to stationary profiles in a moving frame. For exponential jump distributions, the associated non-local integro-differential system admits a local description. In the regime v_+>v_-=0, this further reduces to a coupled system of non-linear ODEs, allowing a phase-plane analysis that yields a traveling wave as a heteroclinic orbit connecting two equilibria. We also identify the wave speed and mass partition, and derive tail asymptotics by spectral analysis of the linearized system.
ER -