  We consider the Saintillan--Shelley kinetic model of active rodlike particles
in Stokes flow (Saintillan & Shelley 2008a,b), for which the uniform, isotropic
suspension of pusher particles is known to be unstable in certain settings.
Through weakly nonlinear analysis accompanied by numerical simulations, we
determine exactly how the isotropic steady state loses stability in different
parameter regimes. We study each of the various types of bifurcations admitted
by the system, including both subcritical and supercritical Hopf and pitchfork
bifurcations. Elucidating this system's behavior near these bifurcations
provides a theoretical means of comparing this model with other physical
systems which transition to turbulence, and makes predictions about the nature
of bifurcations in active suspensions that can be explored experimentally.
