Macroscopic behavior of active systems with a dynamic preferred direction

We present the derivation of the macroscopic equations for active systems with a dynamic preferred direction, which can be axial or polar in nature.

In addition to the usual hydrodynamic variables we introduce the mean angular velocity [1] or the macroscopic velocity [2] associated with the motion of the active units as a new variable and discuss their macroscopic consequences [1,2]. Such an approach is expected to be useful for a number of biological systems including, for example, the formation of dynamic macroscopic patterns shown by certain bacteria such as Proteus mirabilis, shoals of fish, flocks of birds and migrating insects.

As a concrete application we set up a macroscopic model of bacterial growth and transport based on a polar dynamic preferred direction — the collective velocity of the bacteria [3]. This collective velocity is subject to an isotropic-nematic like transition modeling the density-controlled transformation between immotile and motile bacterial states. The approach can be applied also to other systems spontaneously switching between individual (disordered) and collective (ordered) behavior, and/or collectively responding to density variations, e.g., bird flocks, fish schools etc. We observe a characteristic and robust stop-and-go behavior of the type also observed for the growth of bacteria experimentally [4].