  The Sheldon spectrum describes a remarkable regularity in aquatic ecosystems:
the biomass density as a function of logarithmic body mass is approximately
constant over many orders of magnitude. While size-spectrum models have
explained this phenomenon for assemblages of multicellular organisms, this
paper introduces a species-resolved size-spectrum model to explain the
phenomenon in unicellular plankton. A Sheldon spectrum spanning the cell-size
range of unicellular plankton necessarily consists of a large number of
coexisting species covering a wide range of characteristic sizes. The
coexistence of many phytoplankton species feeding on a small number of
resources is known as the Paradox of the Plankton. Our model resolves the
paradox by showing that coexistence is facilitated by the allometric scaling of
four physiological rates. Two of the allometries have empirical support, the
remaining two emerge from predator-prey interactions exactly when the
abundances follow a Sheldon spectrum. Our plankton model is a scale-invariant
trait-based size-spectrum model: it describes the abundance of phyto- and
zooplankton cells as a function of both size and species trait (the maximal
size before cell division). It incorporates growth due to resource consumption
and predation on smaller cells, death due to predation, and a flexible cell
division process. We give analytic solutions at steady state for both the
within-species size distributions and the relative abundances across species.
