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We analyze the stability of biological membrane tubes, with and without abase flow of lipids. Membrane dynamics are completely specified by twodimensionless numbers: the well-known F\"oppl--von K\'arm\'an number $\Gamma$and the recently introduced Scriven--Love number $SL$, respectively quantifyingthe base tension and base flow speed. For unstable tubes, the growth rate of alocal perturbation depends only on $\Gamma$, whereas $SL$ governs the absoluteor convective nature of the instability. Furthermore, nonlinear simulations ofunstable tubes reveal an initially localized disturbance results in propagatingfronts, which leave a thin atrophied tube in their wake. Depending on the valueof $\Gamma$, the thin tube is connected to the unperturbed regions viaoscillatory or monotonic shape transitions -- reminiscent of recentexperimental observations on the retraction and atrophy of axons. We elucidateour findings through a weakly nonlinear analysis, which shows membrane dynamicsmay be approximated by a model of the class of extended Fisher--Kolmogorovequations. Our study sheds light on the pattern selection mechanism in axonalshapes by recognizing the existence of two Lifshitz points, at which the frontdynamics undergo steady-to-oscillatory bifurcations.

Absolute vs Convective Instabilities and Front Propagation in Lipid Membrane Tubes