Nonlinear Models for Subdiffusive Transport in Chemotaxis

We are studying subdiffusive transport in inhomogeneous media governed by the fractional Fokker-Planck equation with space dependent anomalous exponent. We are also interested in the study of chemotaxis within a subdiffusive environment, where the diffusion is influenced by a gradient of concentration of an external chemical signal. As well as the diffusion being influence by an external signal, we also study nonlinear reactions which cause the diffusion to be influenced by its own concentration gradient. Chemotaxis is caused by the sensitivity of diffusing particles/cells/beings to an external factor. This external factor causes particles to either be attracted or repulsed from regions of higher concentration of the factor, causing a persistence of movement. In inhomogeneous media, under the CTRW framework, the escape rate of a random walker from a point is not constant throughout the domain. This introduces a dependence of the anomalous exponent on the spatial co-ordinate. Any spatial fluctuation in the anomalous exponent can lead to the phenomenon of anomalous aggregation.