Asymptotic Behavior of a Two-Dimensional Keller–Segel Model with and without Density Control

Abstract

We study the Keller–Segel model for chemotaxis, consisting of a drift-diffusion equation describing the evolution of the cell density coupled to an equation for the chemoattractant. It is known that in the classical Keller–Segel model solutions can become unbounded in finite time. We present recent analytical results for this model, and compare its behavior in two space dimensions numerically to the behavior of a model accounting for the finite volume of cells. This modified Keller–Segel model relies on the assumption that cells stop aggregating when their density is too high, and thus allows for the global existence of solutions. We characterize the slow movement of a certain class of plateau-shaped solutions and perform numerical experiments for both models, showing that solutions of the classical (before blow-up) and of the density control model share common features: regions of high cell density are attracted by each other and, under suitable boundary conditions, by the domain boundaries.