Tyrrell replies

Cullen shows that changing the values of PH and NH (KS[PO4] and KS[NO3] in his notation) in my model2 can give rise to a steady-state [NO3]:[PO4] ratio in surface waters that is greater than or equal to 16. Because of this, and because PH and NH are not well known, he questions some of the implications of my model.

Although his analysis is correct, my model must converge to a [NO3]:[PO4] ratio that is slightly less than the ratio at which NO3 and PO4 are equally limiting to growth, regardless of whether the latter ratio is 16 or not. There must therefore be convergence to proximate nitrate (reactive nitrogen) limitation of surface waters.

Equations (1) and (2) of my model2 can be rewritten as


where NF and O are the populations of nitrogen-fixing and other phytoplankton, t is time, M is mortality, and LN and LP represent the growth limitations (range, 0 to 1) caused by [NO3] and [PO4] shortages, respectively, which were expressed as Michaelis–Menten functions in ref. 2 but have been left unspecified here. For steady state, d(NF)/dt and d(O)/dt must both equal zero (both populations are stable in size), and therefore


Because μ′NF is less than μ′O (ref. 1), this equation cannot be satisfied (equilibrium cannot occur) unless LN is less than LP; that is, the proximate limiting nutrient is reactive nitrogen.

This proof makes no assumptions about the values of NH and PH. As Cullen rightly argues, raising NH allows convergence to a surface N:P>16, but then the surface ocean is still most strongly limited by reactive nitrogen, precisely because NH has been raised. But if the simplifying assumption is made that the Michaelis–Menten half-saturation constant for a nutrient is more or less proportional to the rate at which it needs to be taken up to fuel new growth (in which case, NH/PH is about 16), then LN<LP also implies that [NO3]:[PO4]<16 in the surface ocean steady state. Cullen's analysis illuminates the point that a surface ocean could still have nitrogen as the proximate limiting nutrient even if the surface N:P ratio were 100, for instance. It all depends on the values of PH and NH, which need to be better constrained.

This analysis confirms my original point: observations from nutrient-enrichment experiments that reactive nitrogen is more limiting to growth than phosphorus in the surface ocean can be reconciled with phosphorus limitation without recourse to the effects of trace metals. If shortages of iron, for instance, further depress nitrogen fixation and reactive nitrogen concentrations in the open ocean, then my model predicts that adding extra iron could cause only a reduction, not full removal, of the proximate nitrogen limitation.

See also - John J. Cullen