Motivated by the observation of highly unstable flowing states in suspensions of
microtubules and kinesin, we analyse a model of mutually propelled filaments
suspended in a solvent. The system undergoes a mean-field isotropic–nematic
transition for large enough filament concentrations when the nematic order
parameter is allowed to vary in space and time. We analyse the model
in two contexts: a quasi-one-dimensional channel with no-slip walls and a
two-dimensional box with periodic boundaries. Using stability analysis and
numerical calculations we show that the interplay between non-uniform nematic
order, activity, and flow results in a variety of complex scenarios that include
spontaneous banded laminar flow, relaxation oscillations and chaos