Synchronising and Non-synchronising Dynamics for a Two-species Aggregation Model

TL;DR

This paper analyzes a two-species aggregation model with nonlocal interactions, establishing measure-valued solutions, studying blow-up phenomena, and demonstrating complex aggregate dynamics through numerical simulations.

Contribution

It introduces a measure-valued solution framework for the model, proves existence and uniqueness, and develops a convergent numerical scheme to simulate post-blow-up dynamics.

Findings

Classical solutions blow up in finite time.

Aggregates can synchronize or not upon collision.

Numerical simulations reveal complex post-blow-up behaviors.

Abstract

This paper deals with analysis and numerical simulations of a one-dimensional two-species hyperbolic aggregation model. This model is formed by a system of transport equations with nonlocal velocities, which describes the aggregate dynamics of a two-species population in interaction appearing for instance in bacterial chemotaxis. Blow-up of classical solutions occurs in finite time. This raises the question to define measure-valued solutions for this system. To this aim, we use the duality method developed for transport equations with discontinuous velocity to prove the existence and uniqueness of measure-valued solutions. The proof relies on a stability result. In addition, this approach allows to study the hyperbolic limit of a kinetic chemotaxis model. Moreover, we propose a finite volume numerical scheme whose convergence towards measure-valued solutions is proved. It allows for numerical simulations capturing the behaviour after blow up. Finally, numerical simulations illustrate the complex dynamics of aggregates until the formation of a single aggregate: after blow-up of classical solutions, aggregates of different species are synchronising or nonsynchronising when collide, that is move together or separately, depending on the parameters of the model and masses of species involved.