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An equation for the dynamics of the vesicle supply center model of tip growth in fungal hyphae is derived. For this we analytically prove the existence and uniqueness of a traveling wave solution which exhibits the experimentally observed behavior. The linearized dynamics around this solution is analyzed and we conclude that all eigenmodes decay in time. Numerical calculation of the first eigenvalue gives a timescale in which small perturbations will die out

Existence and linear stability of solutions of the ballistic VSC model