Description
In this talk, I will present a series of works on the approximation of the doubly parabolic Keller-Segel model using stochastic interacting particle systems in various spatial dimensions. We will introduce two types of approximations. The first, based on the Duhamel formulation of the system, is a singularly interacting non-Markovian particle system that approximates cell positions, from which we can recover the concentration field. The second is a two-species particle system that simultaneously tracks cell positions and chemo-attractant particles (in the spirit of Stevens, 2000).
After a brief review of the 1D setting, we focus on the 2D case. We obtain the convergence of both systems to the macroscopic limit equation as the number of particles tends to infinity, provided the sensitivity parameter is sufficiently small. We will discuss also results showing that the two-species system converges to the non-Markovian system when accelerating the production rate of the chemo-attractant while keeping the number of cell particles fixed. The proofs rely in part on a novel functional inequality applied to particle trajectories.
This talk is based on joint works with N. Fournier, J.-F. Jabir, and D. Talay.