  Chemotaxis, the directional locomotion of cells towards a source of a
chemical gradient, is an integral part of many biological processes - for
example, bacteria motion, single-cell or multicellular organisms development,
immune response, etc. Chemotaxis directs bacteria's movement to find food
(e.g., glucose) by swimming toward the highest concentration of food molecules.
In multicellular organisms, chemotaxis is critical to early development (e.g.,
movement of sperm towards the egg during fertilization). Chemotaxis also helps
mobilize phagocytic and immune cells at sites of infection, tissue injury, and
thus facilitates immune reactions. In this paper, we study a PDE system that
describes such biological processes in one dimension, which may correspond to a
thin channel, the setting relevant in many applications: for example,
spermatozoa progression to the ovum inside a Fallopian tube or immune response
in a blood vessel.
