A Hydrodynamic Limit for Chemotaxis in a Given Heterogeneous Environment

Abstract

In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by Koukkous (Stoch. Process. Appl. 84, 297–312, cite.Kou99) and Kipnis and Landim (Scaling limits of interacting particle systems, cite.KL99). Numerical simulations extend and illustrate the theoretical findings.