The Diffusion Limit of Transport Equations II: Chemotaxis Equations

In this paper, we use the diffusion-limit expansion of transport equations developed earlier T. Hillen and H. G. Othmer, SIAM J. Appl. Math., 61 (2000), pp. 751--775 to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modification of the turning rate, the movement speed, or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diffusion, to a drift term in the flux, or to both, in the parabolic limit. We show that the classical chemotaxis equation---which we call the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small. Using this general framework, we derive phenomenological models for chemotaxis of flagellated bacteria, of slime molds, and of myxobacteria. We also show that certain results derived earlier for one-dimensional motion can easily be generalized to two- or three-dimensional motion as well.