A cell-growth model with applications to modelling the size distribution of diatoms is examined. The analytic solution to the model without dispersion is found and is shown to display periodic exponential growth rather than asynchronous (or balanced) exponential growth. It is shown that a bounding envelope (hull) of the solution to the model without dispersion takes the same shape as the limiting steady size-distribution (SSD) to the dispersive case as dispersion tends to zero. The effect of variable growth rate on the shape of the hull is also discussed.