Mathematical models of cell self-organization

• Main Author: Benoît Perthame
• Format: Article
• Language: English
• Published: SpringerOpen 2011-04-01
• Series: Journal of the Egyptian Mathematical Society
• Subjects: 35B45; 35K55; 35K57; 82B40; 92C17
• Online Access: http://www.sciencedirect.com/science/article/pii/S1110256X11000083

Various classes of Partial Differential Equations have shown to be successful in describing the self-organization of bacterial colonies, a topic also sometimes called socio-biology. For instance parabolic systems are standard; the classical Patlak–Keller–Segel system and Mimura’s system are able to explain two elementary processes underlying qualitative behaviors of populations and complex patterns: oriented drift by chemoattraction and colony growth with local nutrient depletion. More recently nonlinear hyperbolic and kinetic models also have been used to describe the phenomena at a smaller scale. We explain here some motivations for ‘microscopic’ descriptions, the mathematical difficulties arising in their analysis and how kinetic models can help in understanding the unity of these descriptions.