  In the experiment that first demonstrated gyrotactic behaviour of
bottom-heavy swimming microalgae (e.g. Chlamydomonas), Kessler (Nature, vol.
313, 1985, pp. 218-220) showed that a beam-like structure, often referred to as
a gyrotactic plume, would spontaneously appear from a suspension of gyrotactic
swimmers in a downflowing pipe. Such a plume is prone to an instability to form
blips. This work models the gyrotactic plume as a steady parallel basic state
and its subsequent breakdown into blips as an instability, employing both the
Generalised Taylor Dispersion (GTD) theory and the Fokker-Planck model for
comparison. Upon solving for the basic state, it is discovered that the steady
plume solution undergoes sophisticated bifurcations. When there is no net flow,
there exists a non-trivial solution of the plume structure other than the
stationary uniform suspension, stemming from a transcritical bifurcation with
the average cell concentration. When a net downflow is prescribed, there exists
a cusp bifurcation. Furthermore, there is a critical concentration, at which
the cell concentration at the centre would blow up for the GTD model. The
subsequent stability analysis using the steady plume solution shows that the
Fokker-Planck model is inconsistent with what was experimentally observed, as
it predicts stabilisation of axisymmetric blips at high concentration of the
plume and destabilisation of the first non-axisymmetric mode at low flow rates.
