HYPERBOLIC MODELS FOR CHEMOSENSITIVE MOVEMENT
Chemosensitive movement describes the active orientation of individuals on chemical signals. In cases of cellular slime molds or flagellated bacteria, chemosensitive movement leads to aggregation and pattern formation. The classical mathematical model to describe chemosensitive movement is the diffusion based Patlak–Keller–Segel model. It suffers from the drawback of infinite propagation speeds. The relevant model parameters (motility and chemosensitivity) are related to population statistics. Hyperbolic models respect finite propagation speeds and the relevant model parameters (turning rate, distribution of new chosen velocities) are based on the individual movement patterns of the species at hand. In this paper hyperbolic models (in 1-D) and a transport model (in n-D) for chemosensitive movement are discussed and compared to the classical model. For the hyperbolic and transport models the following topics are reviewed: parabolic limit (which in some cases leads to the Patlak–Keller–Segel model), local and global existence, asymptotic behavior and moment closure. The moment closure approach leads to models based on Cattaneo's law of heat conduction (telegraph equation).