Synchronising and non-synchronising dynamics for a two-species aggregation model

Casimir Emako (1, 2) , Jie Liao (3) , Nicolas Vauchelet (2)

1 MAMBA - Modelling and Analysis for Medical and Biological Applications
2 LJLL - Laboratoire Jacques-Louis Lions
3 East China University of Science and Technology

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.

Mots clés: aggregate dynamics; duality solution; two-species chemotaxis; hydrodynamic limit

Domaines: Equations aux dérivées partielles [math.AP]

Casimir Emako, Jie Liao, Nicolas Vauchelet. Synchronising and non-synchronising dynamics for a two-species aggregation model. Discrete and Continuous Dynamical Systems - Series B, 2017, 22 (6), pp.2121 - 2146. ⟨10.3934/dcdsb.2017088⟩. ⟨hal-01157578⟩