The diffusion limit of transport equations. II: Chemotaxis equations. (English) Zbl 1103.35098

Othmer, Hans G.; Hillen, Thomas

SIAM J. Appl. Math. 62, No. 4, 1222-1250 (2002).

Summary: 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, 751–775 (2000; Zbl 1002.35120)] 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.

MSC:

35Q80	Applications of PDE in areas other than physics (MSC2000)
92C17	Cell movement (chemotaxis, etc.)

Keywords:

aggregation; chemotaxis equations; diffusion approximation; velocity-jump processes; transport equations

Citations:

Zbl 1002.35120