Mathematical models of cell self-organization

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.