On a Chemotaxis Model with Saturated Chemotactic Flux

We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function—its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states to model cell aggregation. After obtaining local and global existence results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very complicated spiky structures. 2000 Mathematics Subject Classification. Primary: 92C17, 35K50, 35B36, 76M12, 76M20; Secondary: 35B32.