Qualitative Behavior of Solutions of Chemotactic Diffusion Systems: Effects of Motility and Chemotaxis and Dynamics

Chemotaxis is the oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider the situation of a single bacterial population which responds chemotactically to a nutrient diffusing from an adjacent phase not accessible to the bacteria. The concentration and density of the substrate and cells (resp.)\ satisfy a quasi-linear parabolic system with nonlinear boundary condition. Our first set of results addresses the effects of two important biological parameters λ >0 and χ >0 on the steady states, where λ measures the (random) motility of bacteria and χ the magnitude of chemotactic response (or sensitivity) to the chemical. Our second set of results concerns the global boundedness of time-dependent solutions and stability issues of trivial and nontrivial steady states with small amplitudes.