We analyse a model of the phosphorus cycle in the ocean given by Slomp and Van Cappellen (Biogeosciences 4:155–171, 2007. 10.5194/bg-4-155-2007). This model contains four distinct oceanic boxes 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.Supplementary InformationThe online version contains supplementary material available at 10.1007/s13137-023-00221-0.