Chemotaxis and its applications in biology


In the last twenty years one can observe a rapid and consistent growth of interest for bio-mathematical applications. Among them are modeling of tumor invasion and metastasis (Chaplain et al.), modeling of vascular network assembly (Preziosi et al.), pattern formations due to the Turing-type instability (Murray et al.) or chemotaxis-driven processes (Horstmann et al.), protein-protein interaction on the membrane (Goryachev, Bastiaens) and others. These mathematical models, presented as systems of advectionreaction-di usion equations, can take into consideration various biological processes (e.g. transport, random walk, reaction, chemotaxis, growth and decay, etc.). In our talk we focus on chemotaxis-like models, their applications in medicine and biology, and possible developments (chemotaxis := an oriented movement towards or away from regions of higher concentrations of certain chemicals). First, we consider original models for chemotaxis; then, we extend these models to cases, for which chemotaxis-like processes occur on evolving-in-time surfaces. Finally, we present how chemotaxis models are coupled with the Navier-Stokes equations, discuss the obtained model and its numerical solution.