  We analyse a model of the phosphorus cycle in the ocean given by Slomp and
Van Cappellen (2007, https://doi.org/10.5194/bg-4-155-2007). This model
contains four distinct oceanic basins and includes relevant parts of the water,
carbon and oxygen cycles. We show that the model can essentially be solved
analytically, and its behaviour completely understood without recourse to
numerical methods. In particular, we show that, in the model, the carbon and
phosphorus concentrations in the different ocean reservoirs are all slaved to
the concentration of soluble reactive phosphorus in the deep ocean, which
relaxes to an equilibrium on a time scale of 180,000 y, and we show that the
deep ocean is either oxic or anoxic, depending on a critical parameter which we
can determine explicitly. Finally, we examine how the value of this critical
parameter depends on the physical parameters contained in the model. The
presented methodology is based on tools from applied mathematics and can be
used to reduce the complexity of other large, biogeochemical models.
