CAPITULO I: MARCO TEÓRICO
1.12. Los Límites a la Independencia en el Ejercicio de la Función Jurisdiccional
As described previously, in the majority of solutions calculated for the implicit nitrate models, Nnwas set to the mean daily value of Nnobtained from the comparable full model solution.
In order to establish the sensitivity of the implicit models to Nn, a series of three solutions using model 6c3 (an implicit form of the full model) were performed for both OWS \India" and Bermuda Station \S" conditions. In the rst, Nnwas assigned the minimum N
nobserved during a comparable full model solution, in the second, the mean daily value (as previously), and in the third, the maximumNnobserved.
0 100 200 300 0 0.5 1 1.5 Time (days) Concentration Nn = Minimum India 0 100 200 300 0 0.5 1 1.5 Time (days) Concentration Nn = Mean 0 100 200 300 0 0.5 1 1.5 Time (days) Concentration Nn = Maximum
Figure 3.54: Simulated annual cycles of phytoplankton for the full model (solid line) and model 6c3 (dashed line) at OWS \India". Nnin model 6c3 set to full model minimum (left), mean (centre) and maximum (right). Concentrations are in mmol N m;3.
0 100 200 300 0 0.2 0.4 0.6 0.8 Time (days) Concentration Nn = Minimum Bermuda 0 100 200 300 0 0.2 0.4 0.6 0.8 Time (days) Concentration Nn = Mean 0 100 200 300 0 0.2 0.4 0.6 0.8 Time (days) Concentration Nn = Maximum
Figure 3.55: Simulated annual cycles of phytoplankton for the full model (solid line) and model 6c3 (dashed line) at Bermuda Station \S". Nnin model 6c3 set to full model minimum (left), mean (centre) and maximum (right). Concentrations are in mmol N m;3.
Figure 3.54 shows the results of the solutions performed at OWS \India". As can clearly be seen, in all three cases model 6c3 comes very close to the full model solution. In the Nn= minimum case, the departure from the full model solution is at its greatest. However, this only results in a slightly lower spring bloom maximum and somewhat more damped summer predator{prey oscillations. In the two other cases, the t is almost perfect with only slightly more extreme predator{prey oscillations in the
summer. These departures coincide with the lowest concentrations of Nnin the full model.
Figure 3.55 shows the results of the solutions performed at Bermuda Station \S". In these cases, the departures from the full model are very extreme. In the minimumcase, phytoplankton concentrations are below the minimumrecorded by the program performing the solutions (i.e. less than 10;6mmolN m;3). In the mean case, winter levels of phytoplankton fall below those of the full model (Nn concentrations in the full model are close to their maximum then), whilst those in the summer exceed those of the full model considerably. Finally, in the maximum case, winter levels of phytoplankton are closer to those found in the full model, but again summer levels are considerably higher.
3.3.7 Ammonium inhibition of nitrate uptake
As discussed earlier, the formulation of ammonium{inhibition of nitrate uptake used in the full model and the reduced forms is by no means the only formulation for nutrient limitation of phytoplankton growth. To this end, the alternative formulation given in Fasham (1995) (and described previously) was put into the full model so that the signicance of this particular facet to the model can be explored. This model favourably uptakes ammonium, but with no direct inhibition of nitrate uptake.
0 2 4 0 2 4 0 0.5 1 Nn concentration Nr concentration Growth limitation Uptake model 1 0 2 4 0 2 4 0 0.5 1 Nn concentration Nr concentration Growth limitation Uptake model 2
Figure 3.56: Surfaces of phytoplankton growth limitation (model term Q i.e. 0 = totally limited by nutrient availability, 1 = totally unlimited by nutrient availability) produced using dierent models of nitrate and ammonium uptake. Model 1 is that used in Fasham (1993) and in the majority of the work presented here. Model 2 is that used in Fasham (1995). Concentrations are in mmol N m;3.
Figure 3.56 shows the surfaces of growth limitation produced by the two models of nitrate and ammo- nium uptake kinetics. Both models cause the greatest limitation at the origin, and in both this falls away as nitrate, ammonium or both increase. Model 1 has relatively shallow increases in uptake as it moves away from the origin since both half{saturation constants have relatively high values. Model 2, on the other hand, has a sharp increase in uptake as soon as ammonium is present since ammonium's half{saturation constant is very much lower than that in model 1. Consequently, the greatest disparity in uptake between the two models occurs at low concentrations of nitrate and ammonium.
Figure 3.57 shows the results of runs performed at OWS \India" and Bermuda Station \S" using full models running both uptake models. The results are presented as a phase portrait in nitrate{ammonium space so that comparison with Figure 3.56 is easier.
The major dierence between the two uptake models lies with the concentrations of ammonium they predict. At both OWS \India" and Bermuda Station \S", uptake model 1 predicts ammonium concen- trations considerably greater than those of model 2. The seasonal patterns of nitrate concentration, by
0 2 4 6 8 10 12 0 0.5 1 1.5 2 Nn concentration Nr concentration OWS "India" 0 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 0.2 Nn concentration Nr concentration Bermuda Station "S"
Figure 3.57: Nitrate{Ammoniumphase space portraits showing the trajectories of full model running uptake model 1 (circles) and uptake model 2 (crosses) at OWS \India" (left) and Bermuda Station \S" (right). The circles and crosses mark the locations of the trajectories once every 5 days, and are added to convey the rate of movement along the trajectories. Note that the scale of the Bermuda Station \S" plot is one tenth that of the OWS \India" plot. Concentrations are in mmol N m;3.
contrast, are very similar between the two models. This considerable dierence conceivably represents a means by which the two models could be distinguished and tested.
In both OWS \India" solutions the lowest nitrate and ammonium concentrations are relatively far away from the region of nitrate{ammonium space in which the disparity between the uptake models is great- est. However, in the case of the Bermuda Station \S" solutions, both models spend the entire annual cycle close to the nitrate{ammonium origin.
0 100 200 300 0 0.5 1 1.5 Time (days) Concentration OWS "India" 0 100 200 300 0 0.1 0.2 0.3 0.4 0.5 Time (days) Concentration Bermuda Station "S"
Figure 3.58: Simulated annual cycles of phytoplankton for the full model using uptake model 1 (solid line) and uptake model 2 (dashed line). Concentrations are in mmol N m;3.
Despite the foray of the nitrate{ammonium trajectories into the region close to the origin, gure 3.58 shows relatively little dierence in the phytoplankton time series between the dierent uptake models. The OWS \India" spring bloom is unaected, although the subsequent summer oscillations are some- what damped with uptake model 2. At Bermuda Station \S" the dierences, although smaller, occur
throughout more of the year. The spring bloom is somewhat higher (phytoplankton growth being higher due to the greater uptake of model 2 close to the origin), and the summer dip somewhat lower (due to the concomitantly lower nutrient levels). This lack of dierence in the time series of model compart- ments is repeated for all of the model compartments except ammonium and DON. In the case of DON, uptake model 2 results in autumn levels of DON around twice those produced in uptake model 1 solutions. The similarity between the results is not as good as some of the reduced models have shown, but it is considerably better than most of them. The principal dierences appear to be slightly higher phyto- plankton growth, greater use of ammonium as a growth substrate, and resultant lower concentrations of ammonium (at OWS \India" the maximum ammonium concentration under uptake model 2 is only around a half that when uptake model 1 is used, and around an eighth at Bermuda Station \S").
Z
0.492 0.123 0.630 0.052 0.919 0.289 0.316 0.059 0.562 0.053B
Nd
Nr
Nn
P
D
0.007 0.092 0.492 0.039 0.043 0.000 0.011 0.840 0.079 0.166 0.129 0.643 0.184 0.300 0.210 0.237 1.002Figure 3.59: Annual nitrogen ows (mol N m;2 y;1) of the NH
4 model at OWS \India". Figure 3.59 shows the network of ows for the NH4 model. Since the nutrient uptake model used here does not actively limit uptake in the presence of ammonium, uptake of nitrate and ammonium is greater than in the full model solutions using the standard uptake model. Unsurprisingly then, this leads to greater primary production and slightly higher uxes throughout the network. Since ammonium is now preferred to nitrate there is greater uptake of ammonium and a slightly reduced uptake of nitrate. This leads to lower ammoniumdetrainment and lower nitrate entrainment. This greater uptake of ammonium is also reected in the f{ratios calculated across the year at both OWS \India" and Bermuda Station \S" (see gures 3.22 and 3.29, and tables 3.2, 3.3 and 3.4). At both stations both the annual patterns
3.4 Summary
The primary aim in this chapter was to determine if a \minimum model", which accurately described the behaviour of the full model, could be rebuilt from a na !ve deconstruction of the full model. Fourteen reduced models were constructed, ranging in size from a minimal PZ model to a six compartment model with an implicit representation of nitrate. Each model was reconstructed along rational lines, with at- tention paid to major ecological pathways and to the plausibility and utility of the reduced form. While several models (e.g. models 4c, 4c2 and 5c3) were particularly successful in this regard, the failure of the other models also provided insight into the importance of particular ecological pathways included in the full model.
The behaviour of three of the simplest models, 2c, 3c and 3c3, at OWS \India" revealed the rst aw of the reduced models. At OWS \India", nutrient limitation is always less signicant than irradiance limitation during full model solutions, and consequently all three reduced models degenerate to rapid PZ cycles. Although these cycles are conned to the summer months, as are the oscillations of the full model, their severity is greater. These results contrast with those of model 3c2, which has no nutrient limitation, but which has a detrital compartment which the zooplankton can graze on. This latter model produces a series of damped summer oscillations, similar to those of the full model.
Another aw of the simpler reduced models, as well as the implicit nitrate models, was revealed by solutions determined for Bermuda Station \S". At this station, nutrient limitation of the full model is more severe than irradiance limitation for a considerable period of the simulated year. Models 2c and 3c2, which have neither explicit nor implicit nutrient limitation, both produce extreme solutions, with oscillations in the phytoplankton and zooplankton populations which take them to values more than one order of magnitude greater than these populations reach in the full model. In the case of model 3c and the implicit nitrate forms, nutrient limitation is considerably more muted than in the full model (in model 3c, this is because regeneration is so rapid in the implicit nitrate forms, this is because nitrate cannot be depleted). Consequently, the summer slump in phytoplankton and zooplankton populations found in the full model do not occur in these models, leading to high summer production and even oscillatory behaviour (model 3c3).
As suggested during the original formulation of the reduced models, the reinstatement of the bacterial compartment poses a number of problems. The failure of model 5c, and the diculties found with models 5c2 and 6c would appear to conrm this. In the case of model 5c, the bacteria utilise only general nutrient for their metabolism. Since they have no seasonal limitations placed on them (unlike the phytoplankton) they make use of general nutrient throughout the year, raising their own populations at the expense of phytoplankton who spend their year considerably more nutrient limited than in the
full model. While the bacterial populations produced across the year by model 5c are never much more than twice the annual maximum found in the full model solutions, these low populations hide bacterial production (annual nitrogen uptake) more than 130 times that found in the full model (for OWS \In- dia"). In model 5c, the N{B{Z pathway almost entirely surplants the N{P{Z pathway.
Models 5c2 and 6c (and, at OWS \India", model 5c4 { the implicit nitrate version of model 6c) also exhibit slightly more active bacterial populations, but nowhere nearly on the scale of model 5c. In both cases, as an alternative to using DON as a growth substrate, bacteria are given access to detritus as a carbon source (note that detritus is also assigned ow inputs which formerly ran to DON). This pathway makes sense since bacteria are known to occur in association with particulate material as well be free{living (Totterdellet al., 1993). However, while ecologically the tying of bacteria to detritus may be unobjectionable, in the context of the Fasham (1993) model it appears that this step is misjudged. In both reduced models, the bacterial populations now have access to a much larger carbon source. Consequently their populations are more pronounced across the year. This leads to a larger zooplankton population, and also to a larger phytoplankton population (supported by regenerated excretion and pre- dation losses from the bacterial and zooplankton populations respectively). Since most detritus becomes regenerated by bacteria, its sinking ux out of the mixed layer is shifted markedly (in both models the ux is lower than the full model ux by at least a third). Note that both models produce very similar results despite diering in the nitrogen source their bacterial utilise (general nutrient in model 5c2, am- monium in model 6c).
Although their results at Bermuda Station \S" have already been singled out as erroneous, the per- formance the implicit nitrate models 4c3 and 6c3 at OWS \India" is considerably better. This is unsurprising however, since both models are respectively the implicit nitrate forms of reduced model 5c3 and the full model itself. Model 5c3, along with model 4c2, is one of the most successful of the reduced forms (note that 4c2 is the parent model to 5c3). Both of these models accurately caricature the full model's dynamics and ow network, and both predict values of NPP and sinking ux close to those produced by the full model.
In the context of the primary aim of this chapter, the reduced forms known as 4c2 and 5c3 appear to be candidates for the title of \minimum model". The slightly increased complexity of model 5c3 allows it to capture certain facets of the full model that are not addressed by model 4c2 (the f{ratio for instance), and where those facets are required, it is undoubtedly superior.