Recently it has been reported that the growth of vesicles is strongly influenced by the presence of pre-added vesicles. In particular, the size distribution of vesicles is sharply centered about the size of the pre-added vesicles: the size-templating "matrix" effect. A description of the dynamics of the concentrations of cluster sizes has been proposed Bolton and Wattis (J. Phys. Chem. B 2003, 107, 7126), in terms of a nonlinear microscopic model that is valid far from equilibrium while agreeing with the equilibrium cluster size distribution data. In this paper we approximate this model by a coarse-graining reduction to a low-dimensional dynamical system, a process introducing two parameters on which the solution depends. We solve the system in two parameter regimes and demonstrate that there is a borderline case between the two types of solution that contain features of each. In each case matched asymptotics are used to derive the solution over a series of time scales, and we consider a range of initial conditions including experimentally relevant cases where a small concentration of vesicles is pre-added. We interpret these results on a phase-plane diagram to illustrate how the matrix effect manifests itself in this model.