  We analyze the stability of biological membrane tubes, with and without a
base flow of lipids. Membrane dynamics are completely specified by two
dimensionless numbers: the well-known F\"oppl--von K\'arm\'an number $\Gamma$
and the recently introduced Scriven--Love number $SL$, respectively quantifying
the base tension and base flow speed. For unstable tubes, the growth rate of a
local perturbation depends only on $\Gamma$, whereas $SL$ governs the absolute
or convective nature of the instability. Furthermore, nonlinear simulations of
unstable tubes reveal an initially localized disturbance results in propagating
fronts, which leave a thin atrophied tube in their wake. Depending on the value
of $\Gamma$, the thin tube is connected to the unperturbed regions via
oscillatory or monotonic shape transitions -- reminiscent of recent
experimental observations on the retraction and atrophy of axons. We elucidate
our findings through a weakly nonlinear analysis, which shows membrane dynamics
may be approximated by a model of the class of extended Fisher--Kolmogorov
equations. Our study sheds light on the pattern selection mechanism in axonal
shapes by recognizing the existence of two Lifshitz points, at which the front
dynamics undergo steady-to-oscillatory bifurcations.
