Volume Filling Effect in Modelling Chemotaxis

The oriented movement of biological cells or organisms in response to a chemical gra- dient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized mod- els have been proposed to describe the aggregation phase of this process. Most of efforts where concentrated, so far, on mathematical models in which the formation of aggregate is interpreted as finite time blow-up of cell density. In recently proposed models cells are no more treated as point masses and their finite volume is accounted for. Thus, arbitrary high cell densities are precluded in such description and a threshold value for cells density is a priori assumed. Different modeling approaches based on this assumption lead to a class of quasilinear parabolic systems with strong nonlinearities including degenerate or singular diffusion. We give a survey of analytical results on the existence and uniqueness of global-in-time solutions, their convergence to stationary states and on a possibility of reaching the density threshold by a solution. Unsolved problems are pointed as well.