The Diffusion Limit of Transport Equations II: Chemotaxis Equations

In this paper we use the di usion-limit expansion of transport equations developed earlier [23] to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modi cation of the turning rate, the movement speed or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic di usion, to a drift term in the ux or to both, in the parabolic limit. We show that the classical chemotaxis equation which we call the Patlak-Keller-Segel-Alt (PKSA) equation only arises when the bias is suÆciently small. Using this general framework, we derive phenomenological models for chemotaxis of agellated bacteria, of slime molds and of myxobacteria. We also show that certain results derived earlier for one-dimensional motion can easily be generalized to twoor three-dimensional motion as well.