A new time-dependent continuous model of biomass size spectra is developed. In this model, predation is the single process governing the energy flow in the ecosystem, as it causes both growth and mortality. Predation is size-dependent. The ratio of predator to prey is assumed to be distributed: predators may feed on a range of prey sizes. Under these assumptions, it is proven that linear size spectra are stationary solutions of the model. The slope of this size spectrum is insensitive to the magnitude of processes in the food web, such as the width of prey size distribution, the volume of water searched while foraging or the assimilation efficiency. Exploited fish communities are simulated by adding a size-dependent fishing mortality to the model: it is found that realistic fishing pressures should not affect the slope of size spectra, but their shape and stationarity.