This paper is devoted to numerical simulations of a kinetic model
describing chemotaxis. This kinetic framework has been investigated
since the 80's when experimental observations have shown that the
motion of bacteria is due to the alternance of 'runs and tumbles'.
Since parabolic and hyperbolic models do not take into account the
microscopic movement of individual cells, kinetic models have become of a
great interest. Dolak and Schmeiser (2005) have then proposed a kinetic
model describing the motion of bacteria responding to temporal gradients
of chemoattractants along their paths.
An existence result for this system is provided and a
numerical scheme relying on a semi-Lagrangian method is presented
and analyzed. An implementation of this scheme allows to obtain
numerical simulations of the model and observe blow-up patterns that
differ greatly from the case of Keller-Segel type of models.