The diffusion limit of transport equations and applications to aggregating populations (chemotaxis)

It has been observed in many species that the release of a chemical signal can trigger individuals to change their movement behavior such that a swarm or an aggregation is formed (chemotaxis). To model this effect I will start with a transport equation for the population, which comes from a velocity jump process. An appropriate scaling analysis allows to relate this model to the well known parabolic Keller-Segel equations for chemotaxis. I will give some very general assumptions and show an approximation result. It turns out that the diffusion limit in general is non isotropic and necessary and sufficient conditions for isotropy will be derived. Finally I will give several examples, which lead to appropriate models for specific populations.

(Joint work with H. Othmer, Minneapolis.)