Critical chemotactic collapse

A Keller–Segel model describes macroscopic dynamics of bacterial colonies and biological cells as well as dynamics of a gas of self-gravitating Brownian particles. Bacteria secret chemical which attracts other bacteria so that they move towards chemical gradient creating nonlocal attraction between bacteria. If bacterial (or Brownian particle) density exceeds a critical value then the density collapses (blows up) in a finite time which corresponds to bacterial aggregation or gravitational collapse. Collapse in the Keller–Segel model has striking qualitative similarities with a nonlinear Schrödinger equation including critical collapse in two dimensions and supercritical collapse in three dimensions. A self-similar solution near blow up point is studied in the critical two-dimensional case and it has a form of a rescaled steady state solution which contains a critical number of bacteria. Time dependence of scaling of that solution has square root scaling law with logarithmic modification.