Pattern formation in a suspension of swimming microorganisms: equations and stability theory

A model for collective movement and pattern formation in layered suspensions of negatively geotactic micro-organisms is presented. The motility of the organism is described by an average upward swimming speed U and a diffusivity tensor D . It is shown that the equilibrium suspension is unstable to infinitesimal perturbations when either the layer depth or the mean concentration of the organisms exceeds a critical value. For deep layers the maximum growth rate determines a preferred pattern size explicitly in terms of U and D . The results are compared with observations of patterns formed by the ciliated protozoan Tetrahymena pyriformis.