Chemotaxis describes the directed movement of cells/organisms in response to chemical signals. If the movement is up to the chemical concentration gradient, it is called attractive chemotaxis, otherwise repulsive chemotaxis. It is observed in many natural systems. For example, myxobacteria produce so-called slime trails on which their cohorts can move more readily.

This talk firstly shows some results about traveling wave solutions to the ODE-Parabolic coupled systems. Secondly, we study the (in-) stability of the homogeneous steady-state solution to the attractive-repulsive chemotaxis model that arises in studying on Alzheimer’s disease. We prove sufficient conditions for destabilization by using the theory of non-negative matrices and graphs. Finally, we consider the attraction-repulsion chemotaxis model arising from Alzheimer’s disease withshow fast diffusive term and nonlinear source subject to Neumann boundary conditions. We show that fast diffusion guarantees global existence of solutions for any given initial value in a bounded domain.