Abstract

In recent years the concept of superdiffusion has been found to be a useful tool to describe many phenomenon appearing in nature. In particular, it has been recently discovered that the bacteria E. coli may exhibit a superdiffusive behaviour under certain circumstances. E. coli is a microorganism that swims towards regions of higher chemoatractants, and away from unfavourable environments. This type of behaviour is known as chemotaxis. The movement of E. coli is characterized by a series of run-and-tumble events, which can be modelled via a velocity-jump process. Hence, a possible way to describe it is through kinetic transport equations.

Due to the fact that E. coli was found to have a superdiffusive behaviour, the Keller-Segel model, which is commonly used to describe the movement of E. coli, fails in describing the behaviour of it under this setting. In this talk we shall introduce some kinetic models with a given chemoatractant concentration, and perform the rigorous passage to the macroscopic limit, obtaining fractional-diffusion-advection equations. The coefficients of the latter equations depend on the microscopic quantities governing the movement of the agents.