Chemotaxis and chemokinesis are fundamental biological processes that can be mathematically modelled by the so called Patlak-Keller-Segel (PKS) systems. In the last decade, the mathematical interest on these problems has grown, giving rise to highly interesting results. One aspect is the observation, that under certain conditions unbounded solutions exist, which have no biologically significance and indicates, that the underlying model is not fully correct; hence regularizations of the model have been proposed in the mathematical literature. Nevertheless, large gradients are still present in the solutions and the numerical treatment of these systems is a non-trivial task.
In addition, in order to describe in a more realistic way biological data, more complicated mathematical models have been proposed, for example taking into account non-constant chemo-sensitivity and chemo-diffusivity coefficients. Also models including multiple bacteria species and multiple chemo-attractants/repellents have been proposed. Not much is known for such systems, neither analytically nor numerically.
In the first part of this project we will provide a robust, stable, highly accurate and positivity preserving finite volume method for the numerical treatment of PKS systems. We will then compare the results obtained by different regularization strategies with existing experimental data.
In the second part we will study numerically the consequences of including non-constant chemo-coefficients into the PKS systems while comparing our numerical results with the biological data.