Influence of consumer-driven nutrient recycling on primary production and the distribution of N and P in the ocean

In this study we investigated the impact of consumer-driven nutrient recycling (CNR) on oceanic primary production and the distribution of nitrogen (N) and phosphorus (P) in the deep ocean. For this purpose, we used and extended two existing models: a 2-box model of N and P cycling in the global ocean (Tyrrell, 1999), and the model of Sterner (1990) which formalised the principles of CNR theory. The resulting model showed that marine herbivores may affect the supply and the stoichiometry of N and P in the ocean, thereby exerting a control on global primary production. The predicted global primary production was higher when herbivores were included in the model, particularly when these herbivores had higher N:P ratios than phytoplankton. This higher primary production was triggered by a low N:P resupply ratio, which, in turn, favoured the Plimited N2-fixation and eventually the N-limited non-fixers. Conversely, phytoplankton with higher N:P ratios increased herbivore yield until phosphorus became the limiting nutrient, thereby favouring herbivores with a low P-requirement. Finally, producer-consumer interactions fed back on the N and P inventories in the deep ocean through differential nutrient recycling. In this model, N deficit or N excess in the deep ocean resulted not only from the balance between N 2-fixation and denitrification, but also from CNR, especially when the elemental composition of producers and consumers differed substantially. Although the model is fairly simple, these results emphasize our need for a better understanding of how consumers influence nutrient recycling in the ocean. Correspondence to: A. Nugraha (nugraha@univ-brest.fr)

These key processes include the delivery of N and P from the continents and atmosphere, their redistribution via ocean circulation, and their cycling within the ocean and sediments.Riverine inputs and atmospheric deposition contribute to 46-57% of the total N inputs to the ocean (Gruber and Galloway, 2008).N 2 -fixation is an additional significant source of new N to the ocean, especially in oligotrophic gyres where this process contributes approximately 40-52% of new-N inputs to the well-lit surface layer (Codispoti et al., 2001;Gruber, 2004).Phosphorus is mainly delivered to the ocean through rivers (94% of the inputs) as a product of continental weathering, while the eolian flux accounts for the remaining 6% of the total phosphorus flux at present day (Compton et al., 2000).
Once in the ocean, it has long been recognised that biological processes exert a control on N and P cycles (Redfield et al., 1963).Dissolved inorganic N and P are taken up and converted into organic matter by photosynthetic autotrophs in the euphotic zone.Organic N and P are then transferred, either to higher trophic levels in the food web or into the deep ocean.Eventually N and P are remineralised or buried in sediments.
A. Nugraha et al.: Influence of consumer-driven nutrient recycling on primary production In their seminal work, Redfield et al. (1963) found that the stoichiometry of N and P in the bulk plankton biomass converges on a mean ratio of 16N:1P, the so-called Redfield ratio, which closely matches that of the dissolved inorganic pool in the ocean.Based on these observations, Redfield et al. (1963) suggested that marine organisms imprint the biogeochemistry and the distribution of these major elements in the ocean.This finding profoundly influenced the field of marine biogeochemistry over the decades that followed.
The Redfield ratio is still used as a scaling factor within models to derive biological-driven fluxes of elements in the ocean (e.g.Arrigo, 2005;Sarmiento and Gruber, 2006).However, debate remains about the mechanisms which drive the stoichiometry of these elements in the ocean, and its resulting influence on primary productivity (e.g.Tyrrell, 1999;Lenton and Klausmeier, 2007).Significant deviations from Redfield stoichiometry have been reported at different scales, ranging from the organism (e.g.Geider and La Roche, 2002;Quigg et al., 2003;Klausmeier et al., 2004) to the ecosystem (e.g.Karl et al., 2001).Karl et al. (2001) found for example that, in the north Pacific gyre, the N:P ratio in the particulate and dissolved organic pools shows variations in the euphotic zone from < 5 : 1 to > 50 : 1, averaging 19.4:1 (POM) and 19.6:1 (DOM).
Despite these deviations, the stoichiometry of dissolved inorganic N and P in the ocean is nearly constant over large spatial scales (Redfield et al., 1963).This suggests that various processes operate in the surface or the deep ocean to both drive and buffer deviations from Redfield stoichiometry.Several mechanisms capable of this have been identified, for example, the balance between nitrogen fixation and denitrification (Redfield et al., 1963), the existence of a wide range of optimal N:P ratios in phytoplankton (Quigg et al., 2003;Klausmeier et al., 2004), and variations in marine redox conditions that affect the sedimentary sequestration of P (Van Cappellen and Ingall, 1996).
Here we investigate how the stoichiometry of consumerdriven nutrient recycling (Sterner, 1990) may influence primary production and the distribution of N and P in the ocean.We have done this because producer-consumer interactions have been shown to exert a control on nutrient cycling and other various ecosystem functions in numerous freshwater systems (Elser and Urabe, 1999;Sterner and Elser, 2002).The great extent of herbivory in marine systems (Calbet, 2001;Cebrián, 2004;Landry and Calbet, 2004) suggests that these interactions may also be important in the ocean, as has been previously suggested by Corner and Davies (1971).Additionally, the elemental composition of herbivores is not only different from that of their algal food, but it is also often homeostatic, irrespective of the elemental composition of their prey (Sterner, 1990;Sterner et al., 1992;Elser and Urabe, 1999).Homeostatic regulation of whole body elemental composition is maintained by releasing elements which are in excess to requirements, while more efficiently retaining limiting elements.These principles form the ba-sis of the "Consumer-driven nutrient recycling theory", hereafter called CNR (Sterner, 1990;Elser and Urabe, 1999;Sterner and Elser, 2002).
Owing to the low gross growth efficiency (GGE) of aquatic herbivores (∼20-40%; Straile, 1997), most of their ingested food is either egested as particulate organic matter (POM) or excreted as dissolved products (Miller and Landry, 1984;Straile, 1997;Besiktepe and Dam, 2002).The fate of the products released by herbivores, and their relative N:P ratio (known as "resupply ratio", Sterner, 1990), may significantly affect nutrient cycling in the ocean, which, in turn, may feed back on primary producers.To address the question of the possible contribution of CNR in driving the oceanic primary production and the distribution of N and P throughout the ocean, we extended and used two existing models, a 2-box model of N and P cycling in the global ocean (Tyrrell, 1999) and the model of Sterner (1990) that formalised the principles of CNR theory.

Model framework
The model described here combines and extends concepts from previous modelling studies (Sterner, 1990;Tyrrell, 1999).Tyrrell's model describes the cycling of reactive forms of N and P in a global ocean represented by two boxes, a surface box (0-500 m) and a deep box (500-3230 m) (Fig. 1).The vertical distributions of dissolved inorganic N and P are constrained by a constant amount of mixing (K), remineralisation of organic N and P in the surface and deep layers for phytoplankton (SR 0 and DR 0 , respectively), denitrification (DN), and burial in sediments (SF).The constant amount of mixing parameterises ocean overturning, upwelling, and diffusion with a time scale of ∼1000 yr (Broecker and Peng, 1982).The ecosystem focuses on autotrophic organisms with explicit competition between N 2fixers (NF) and other algae (hereafter "non-fixers") (O) in the surface ocean.The former require only phosphate as they can convert N 2 gas into organic N. The latter require both nitrate and phosphate.A single loss term represents all possible sources of natural mortality.This mortality term fuels an implicit detrital pool which is partly remineralised in the surface and deep boxes.The fraction which escapes remineralisation is lost from the system as burial in sediments.In the model presented here, the basic set-up of Tyrell's model is conserved.
In addition, we implemented an upper trophic level (i.e.herbivores) with homeostatic regulation of their N:P body content, following Sterner's (1990) approach (Fig. 2 and Fig. 3).Homeostatic regulation of N and P body content is taken into account by differentiating a GGE for N and P. GGE is classically defined as:

Fig. 1.
A modified version of Tyrell's 2-box model of N and P cycling in the ocean which includes a parameterisation of herbivores.The model parameter abbreviations are described in Table 1.
Fig. 2. Flow diagram for a trophic level i (H i ); ε i represents the proportion of the annual natural mortality rate (M i ) which is associated to predation by the trophic level H (i+1) ; β i represents the assimilation efficiency and (1 − β i ) represents the proportion of ingested food which is not incorporated into biomass; a i is the accumulation efficiency, and (1 − a i ) represents the fraction of nutrient excreted after assimilation.
where I i is the ingestion of the element i, and F i and E i are the release of this element as either faecal pellets or dissolved products.The release of dissolved N or P depends on the differences between the N:P stoichiometry of assimilated food and that of the organism.In other words, the element in excess to requirement will be released (excreted) in higher proportion than the other, while the accumulated fraction will be used for growth.This concept was formalised by Sterner (1990) by defining a different accumulation efficiency for N and P.
The model equations and parameterisations which combine these two model frameworks are detailed below.

Model formulation
There are seven state variables in the model.They correspond to N 2 -fixers (NF), non-fixers (O), herbivores (H 1 ), surface phosphate (P S ), surface nitrate (N S ), deep phosphate (P D ), and deep nitrate (N D ) and are explained below.We will concentrate below on the main modifications made to Tyrrell's and Sterner's models.

Phytoplankton (NF and O)
Phytoplankton are assumed to occur only in the surface ocean.As in Tyrell's model, the two phytoplankton competitors, NF and O, are included.
Both phytoplankton equations are composed of two terms, a growth term and a loss term.The growth rates of NF and O are controlled using a Liebig's Law formulation consisting of a maximum growth rate (either µ NF or µ O ) times a Michaelis-Menten term for nutrient uptake.The last term in the equations above is the loss term represented by a constant specific mortality rate (M 0 ) which covers all possible loss terms pathways from phytoplankton.Mortality for both types of phytoplankton is similar.). ).

Fig. 3.
Fluxes of N and P in herbivores.I i is the ingestion (mmol m −3 yr −1 ), β i is the assimilation efficiency, and a i is the accumulation efficiency (i=N or P).

Herbivores
We used Sterner's model (1990) which describes the fate of N and P from primary producers to consumers, assuming strict homeostasis of the N:P ratio in consumers.In this model, the flux of N and P entering the herbivore pool occurs at a rate equals to the per capita mortality rate of phytoplankton as a result of grazing (ε 0 M 0 , Fig. 2).The fraction of N and P removed from phytoplankton which passed the gut wall is called the assimilation efficiency (β N 1 or β P 1 , Figs. 2  and 3).The unassimilated fraction is then egested as fecal pellets (1-β N 1 or 1-β P 1 ).It should be noted that Sterner (1990) did not explicitly consider the assimilation efficiency and the production of fecal pellets.A fraction of N and P which has passed the gut wall is used to build herbivore biomass.In Sterner's model, this fraction is called the accumulation efficiency (a N for N and a P for P, Figs. 2 and 3).Using Eq. ( 1), it follows that GGE N = a N β N 1 and GGE P = a P β P 1 .The fraction which is not accumulated in herbivore biomass is released (excreted) as dissolved products (Figs. 2 and 3).
In the model, β N 1 and β P 1 are kept constant (see Sect. 2.3).On the other hand, a N and a P are variable and calculated as a function of the difference between the N:P ratio of phytoplankton (R i org , with i = NF or O) and that of the herbivore (R Z org ).Before calculating a N and a P , considering that differential assimilation efficiency for N and P can modify the N:P stoichiometry in the algal food after the gut passage is essential.To take into account this effect, a N and a P are calculated using the N:P ratio of food which has passed the gut wall: Two cases have then to be considered (Fig. 3), following Sterner (1990) there is an excess of P and a deficit of N in the assimilated food compared to herbivore requirement.In this case, the accumulation efficiency of the limiting element, N, is maximal, i.e. a N = L m (the value of the constant L m is discussed in Sect.2.3).Conversely, the accumulation efficiency of the nutrient in excess, P, is lower and proportional to the difference between the N:P ratio in the algal pool and that of the herbivore, i.e.
Using these parameterisations, strict homeostasis of the N:P ratio in herbivore biomass is maintained.Indeed, considering a flux of N and P entering the herbivore pool (I N and I P respectively, mmol m −3 yr −1 ), the N:P ratio in the fraction of nutrients accumulated in herbivore biomass can be written: with org , this yields The second case is when R i org • β N 1 /β P 1 > R Z org , nitrogen is in excess compared to herbivore requirement and its accumulation efficiency is lower than its maximal value, i.e.
Conversely, phosphorus becomes the limiting element and its accumulation efficiency is maximal, i.e. a P = L m .Here again, strict homeostasis of the N:P ratio in herbivore biomass is maintained.
Then the equation for zooplankton in terms of N units is, where β N 1 is the assimilation efficiency of N, ε 0 is the proportion of phytoplankton natural mortality (M 0 ) due to grazing, a NF N and a O N are the accumulation efficiencies of N when herbivores feed on N 2 -fixers (NF) or non-fixers (O), and M 1 is the natural mortality of herbivores.Because R Z org is constant, an explicit equation for zooplankton in term of P is not necessary.Finally, unlike phytoplankton, herbivores are assumed to vertically migrate, something that is parameterised here by assuming that their biomass is distributed between the surface (SD) and the deep layer (DD).This vertical distribution mainly affects the mass balance of dissolved nutrient through the proportions of detritus or dissolved products which are released by zooplankton in either the surface or the deep layer (see below).

Dissolved inorganic nitrogen and phosphorous
The rates of change of total dissolved N and P in the surface (N S , P S ) are described by the equations below: and The first negative terms describe nutrient uptake by phytoplankton.The addition of nutrient to the surface layer is fuelled by the fraction of phytoplankton mortality which is not related to herbivore grazing, the release of fecal pellets by herbivores, herbivore excretion, herbivore natural mortality, mixing, and external nutrient inputs.For both nutrients, a fraction of the detritus is remineralised in the surface layer (fraction SR 0 and SR 1 ).Recycling is also accounted for by herbivores' excretion of dissolved products (SE 1 ).In addition, N is removed from the system by denitrification (DN).Finally, N 2 -fixers and non-fixers have distinct N:P stoichiometry (R NF org and R O org , respectively).The rates of change of total dissolved N and P in the deep ocean (N D , P D ) are given by: Nutrients enter into the deep ocean via the fraction of particulate organic matter (POM) remineralised in the deep layer (DR 0 and DR 1 ) and the excretion of nutrients that are in excess to requirement (DE 1 ).Denitrification (DN) removes dissolved organic N from this reservoir.Mixing processes (K) are also responsible for the transfer of both nutrients from the deep box into the surface box.The fraction of organic material which escapes remineralisation, SF, leaves the system and is buried in the sediments.

Parameterization
Lists of parameters with their descriptions, units, and values are given in Table 1.We have split the model parameters into two groups, extended Tyrrell's model's and extended Sterner's model's parameters.All material units are expressed in moles, spatial dimensions are in meters, and time is in years.All fraction and efficiency terms are given as percentages and N:P ratios are dimensionless.The physical characteristics of Tyrrell's model were conserved.These include the depth of surface and deep layers (SD and DD), total volume and surface area of the ocean (T vol and T area ), the ocean mixing coefficient (K), riverine input of P (RP) and N (RN) and atmospheric inputs of N (AN).
The biological parameters of primary producers in our model were mostly derived from Tyrell's model.For example, the specific mortality rate (M 0 ), and maximum specific growth rate of N 2 -fixers (µ NF ) and non-fixers (µ O ) were set at 73, 87.6 and 91.25 yr −1 , respectively, and half saturation constant for P and N uptake were also similar to Tyrell's (K P =0.03 mmol P m −3 and K N =0.05 mmol N m −3 ).However, the N:P ratios of N 2 -fixers (R NF org ) and non-fixers (R O org ) were not set to equivalent values.Observations show that this ratio can be extremely variable both within and among species (Quigg et al., 2003;Arrigo, 2005).The N:P ratio of N 2 -fixers tends to be higher than that of non-fixers (review in White et al., 2006).In our standard simulation, we set R NF org equal to 33 (averaged from Hutchins et al., 2007), while R O org was kept equal to the Redfield ratio of N:P=16.The sensitivity of the model to phytoplankton N:P stoichiometry is addressed later.
The specific mortality rate of zooplankton, M 1 , was set at 20 yr −1 .This estimation was based on the average value of the zooplankton mortality from the global study of zooplankton mortality (Hirst and Kiørboe, 2002).We also fixed the N:P ratio of zooplankton (R Z org ) at 25, which represents an average for different marine zooplankton (e.g.Le Borgne, 1982;Ikeda and Mitchell, 1982).The fraction of phytoplankton natural mortality associated with predation by herbivores (ε 0 ) was constrained from global scale assessments of micro-and mesozooplankton grazing (Calbet, 2001;Calbet and Landry, 2004) that suggest that zooplankton consume about 70-80% of the ocean primary production.A value of 80% was chosen here.This critical parameter will be discussed in more detail later in the paper.
As mentioned above, the assimilation efficiency for N and P by herbivores (β N and β P ) were not explicitly considered in Sterner's model.Here we included these parameters because sinking fecal pellets can be a significant component of export to the deep ocean.Nitrogen assimilation efficiencies for copepods are in the ranges 0.70-0.99(Daly, 1997;Landry et al., 1984;Vincent et al., 2007).For those of P, they range from 0.4 to 0.77 for copepods (Butler et al., 1970 Fraction of total zooplankton production remineralized above 500 m % 95 DR 1 Fraction of total zooplankton production remineralized below 500 m % Fraction of zooplankton excretion above 500 m % 90 DE 1 Fraction of zooplankton excretion below 500 m % Mortality rate of zooplankton yr −1 20 c R NF  2007), g Calbet and Landry (2004), Landry and Calbet (2004), Nejstgaard et al. (2007), h Redfield (1934), i Le Borgne (1982), Ikeda and Mitchell (1982), Walve and Larsson (1999), Gismervik (1997), Uye and Kaname (1994), Iguchi andIkeda (2004). et al., 1972), and from 0.54 to 0.82 for cladocerans (Peters and Rigler, 1973;Hessen and Andersen, 1990).A compilation of all these values suggests that N and P assimilation efficiencies are in the ranges 0.70-0.99 and 0.4-0.82,respectively.In the standard simulation, we assigned a slightly lower assimilation efficiency for N (β N = 69%) than for P (β P = 80%) following Anderson et al. (2005).The maximal accumulation efficiency of either N or P (L m ) could be derived from Eq. ( 5) using published values of GGE and the assimilation efficiency.Straile (1997) compiled GGE estimates for several taxonomic groups (nano-or microflagellates, dinoflagellates, ciliates, rotifers, cladocerans, and copepods), and found a mean and median of ∼20-30%, with maximal values of ∼70%.Using an assimilation efficiency of 70-80% for both N and P, we thus derived L m =GGE/β, 0.88-1.00.In the model, we tested different values for L m (0.75 in the base run, and 0.5 and 0.9 in the sensitivity analyses).
Finally, knowledge about the fraction of N and P remineralised or excreted in the surface layer (SR) or in the deep ocean (DR) was essential.The vertical partitioning of remineralisation depended on two parameters, the specific remineralisation rate of detritus and the sinking velocity of particles.In a two box model, we derived estimates of SR and DR using the following relationships: and In this equation, H is the depth of the surface layer, τ is the specific remineralisation rate of detritus (d −1 ) and V is their sinking velocity (m d −1 ), SF is the fraction of organic matter buried in the sediments, and p(z) is a probability distribution for particles in the water column.For simplicity, we assumed that detritus was homogeneously distributed between z = 0 and z = H.Integration of Eq. ( 11) between z = 0 and z = H yields: An estimate of SR required knowledge of the time scale of remineralisation, and of the sinking velocity of detritus.Two studies in temperate and polar systems have reported specific organic matter degradation rates that range from 0.003 d −1 to 0.44 d −1 and average 0.08 ± 0.01 d −1 , with no clear dependence on temperature (Panagiotopoulos et al., 2002;Lønborg et al., 2009).In our model, we used a value of τ = 0.1 d −1 .Sinking velocities depend on the nature of the particles, ranging from <1 m d −1 for single phytoplankton cells (Smayda, 1970) to >700 m d −1 for aggregates or faecal pellets (Uye and Kaname, 1994;Fischer and Karakas ¸, 2009).In the standard simulation, we set V =2.5 m d −1 , which gave an SR=0.95, as in Tyrrell's model.The proportion of organic N and P which escaped remineralisation (SF) was set to 0.3% (Sarmiento and Gruber, 2006), so that DR=1-(0.95 + 0.003)=0.047.Finally, excretion of dissolved nutrients from zooplankton was apportioned between the surface layer (SE 1 =90% in the base run) and the deep layer (DE 1 =10%).

Analytical solutions
Steady state solutions for the model were calculated to provide insights on the processes and parameters which drive the behaviour of the model.These steady solutions were calculated by setting the solution of differential equations to 0. We used Eqs.( 7) and ( 9) (dP S /dt + dP D /dt=0), and Eqs. ( 8) and ( 10) (dN S /dt + dN D /dt=0) for a system of two equations with two unknowns, NF * and O * (mmol m −3 ).We thus obtained, and + (AN + RN) As in Tyrrell's (1999) model, both phytoplankton types were primarily limited by the rate of supply and loss of the ultimate limiting nutrient, P. Conversely, the supply of N from either the atmosphere or rivers had a positive effect on nonfixers (O * ) and a negative one on N 2 -fixers (NF * ).In other words, external N inputs to the ocean drove the competition between N 2 -fixers and non-fixers.
In our model, the biomass of NF * and O * was also controlled by the assimilation of N and P and by the accumulation efficiencies of N and P in herbivores.High assimilation efficiency and low accumulation efficiency of P (β P 1 and a O P ) favoured both NF * and O * .Conversely, high assimilation efficiency and low accumulation efficiency of N had a negative feedback on NF * and a positive feedback on O * .
Combining Eqs.(14 and 15) gave the total phytoplankton biomass: Summing Eqs. ( 14) and ( 15) cancelled out the role of nitrogen delivery to the ocean and its internal recycling efficiency, so that total phytoplankton biomass (NF only correlated to the delivery of phosphorus (RP), the N:P stoichiometry of non-fixers (R O org ), burial efficiency (SF), the proportion of primary production channelled towards herbivores (ε 0 ), and the assimilation and accumulation efficiencies of P (β P 1 and a O P ).The result of all this was that a high predation rate on a phytoplankton pool that was P-rich compared to the herbivores' requirement resulted in a higher release of P in the dissolved phase which then increased oceanic primary production (see numerical results).
The steady state solution for herbivore biomass (mmol N m −3 ) was given by: Combining Eqs.(14, 15 and 17a) yielded, As expected, the herbivore biomass (in term of units of N) was controlled by the supply of the ultimate limiting nutrient to the ocean (P) and the N:P ratio of phytoplankton.The influence of the supply of P and of the N:P ratio of phytoplankton was however modulated by the assimilation and accumulation efficiencies of N. Unlike for the total phytoplankton biomass, the supply of N, the proximate limiting nutrient of primary production (Tyrrell, 1999), exerted a control on herbivore biomass.Therefore, in the model, both N and P limitation of primary production travelled up the food chain up to herbivores.Indeed, when herbivores fed on a phytoplankton pool with a low N content compared to their requirement, such as non-fixers which had a N:P ratio of 16 (Table 1), N was the limiting element of herbivore growth.In this case, increasing N inputs from either atmosphere or rivers increased non-fixers biomass (Eq.14), which in turn increased herbivore biomass (Eq.17a).This effect was however partly counterbalanced by a decrease in N 2 -fixation when external N inputs increased (Eq.15).All in all, increasing N inputs to the ocean by a factor of two, for example, led to a slight increase of herbivore biomass (∼1%, Eq. 17b).
It should be noted that if a O N = a NF N = a N (corresponding to the case where R O org = R NF org ), the supply of N to the ocean cancelled out from Eq. (17b).The steady state equation then became, Two cases then needed to be considered.
org , then N was the element limiting herbivore growth, a N = L m , and herbivore biomass was positively correlated to cancelled out from Eq. (17c), and herbivore biomass (in term of N unit) reached a plateau given by In this case, all other things being equal herbivore biomass remained constant unless the supply of P to the ocean increased.This general pattern also held when a O N = a NF N (see model sensitivity analysis).
The steady state solutions for nutrients concentrations (mmol m −3 ) were, and with with The surface concentrations of both nutrients depended exclusively on the maximum growth rate (µ NF or µ O ) and the loss term (M 0 ) for phytoplankton (Tyrrell, 1999).The loss term increased the availability of both nutrients in the surface layer and consequently, provided nutrients for the growth of phytoplankton.Deep concentrations of both nutrients, on the other hand, depended on nutrient supply, nutrient remineralisation processes, and the rate of mixing between the two ocean reservoirs.The origin of detritus (phytoplankton, faecal pellets and zooplankton carcasses), together with the .Total primary production (TPP*, Gt C yr −1 ) at steady state was deduced from Eq. ( 16): As with Eq. ( 16), TPP* was primarily controlled by the supply of P to the ocean, as described in Tyrrell (1999), but it was also controlled by recycling through the herbivore parameters.A greater proportion of primary production channelled towards herbivores, together with high assimilation and low accumulation efficiencies for P, increased TPP*.Note that TPP* was converted to Gt C yr −1 for the purpose of comparing model output with observations (see Sect. 3.2).In these cases, the standard Redfield C:N ratio of 106:16 was assumed.
Secondary production (SPP*, Tmol N yr −1 ) could also be derived from Fluxes of various materials at steady state could also be derived (Tmol N yr −1 ), namely Excretion of nitrogen= ( 24) Fecal pellet production = (25) and

Numerical solutions and model assessment
Numerical solutions were derived by substituting the parameter values (Table 1) into the analytical solutions calculated in Sect.3.1.These numerical solutions and comparisons to published values are listed in Table 2.
The predicted total phytoplankton biomass (8.5 Tmol N) fell within the literature range (7.5-10.7 Tmol N: Gasol et al., 1997;Tyrrell, 1999), and most of it (99%) was accounted for by non-fixers.Total zooplankton biomass (12.9 Tmol N) was also roughly consistent with the observed range of 8.1 ± 1.9 Tmol N for oceanic systems (Gasol et al., 1997; using a C:N ratio of 5.0 for the conversion).In the meta-analysis of Gasol et al. (1997), protozooplankton accounted for 3.0 ± 0.7 Tmol N and mesozooplankton for 5.1 ± 1.2 Tmol N.This contribution of mesozooplankton is similar to the estimate of Hernandez-León and Ikeda (2005), who gave an estimate of 4.4 ± 0.4 Tmol N.
Our model's prediction for nutrient concentrations was compared to real ocean values using the NOAA-NODC-WOA05 database (Garcia et al., 2006).Matching the scale of integration to the model resolution, we calculated the observed nutrient concentrations in each reservoir.Surface concentrations for both nutrients are lower than given in the literature.This is mainly due to the rough division between the surface (500 m) and the deep layer (>500 m), as previously noted by Yool and Tyrrell (2003).The predicted N:P ratio in the surface layer (N * S : P * S =13.4) is slightly higher than observed (N S : P S =12.5), but still N-deficient compared to the canonical Redfield ratio of N:P=16.
Unlike surface nutrients concentrations, deep concentrations of both nutrients were close to the literature value.For the default parameter values listed in Table 1, however, the model converged on a numerical solution in which the deep N:P ratio is lower (N * D : P * D =12.2) than the observed average (N D : P D =14.3).From Eqs. (20 and 21), it appears that the deep N:P ratio was controlled primarily by the parameters associated with herbivores, i.e. the assimilation and accumulation efficiencies of N and P (β N 1 , β P 1 , and a j i ), the fraction of zooplankton excretion which occurred in the deep ocean (DE 1 ), and the fraction of detritus produced by herbivores which was remineralised in the deep reservoir (DR 1 ).The sensitivity of the model predictions to these parameters is explored later in Sect.3.3.
The predicted total primary production, 48.97 Gt C yr −1 , falls within the range of literature values.Global estimates of primary production from satellite radiometer data suggest that 36-61 Gt C are fixed each year by oceanic microalgae (Longhurst, 1995;Antoine et al., 1996;Dunne et al., 2007).
Our model estimated a denitrification of 128 Tg N yr −1 falls into the upper range of recent estimates of denitrification process in water column.Current estimates differ quite significantly, ranging from 65 ± 20 Tg N yr −1 (Gruber, 2004) to 116-150 Tg N yr −1 (Codispoti et al., 2001;Galloway et al., 2004).
Our model exported 2.45 Gt C yr −1 (0.43 Gt N yr −1 ), or 5.2% of TPP*, to depths deeper than 500 m.This sits at the lower end estimates from the literature (3-25% TPP*) (Schlesinger, 1977;Tyrrell, 1999;Dunne et al., 2007).In our model, most of the export was related to herbivores (0.34 Gt N yr −1 ) through the production of faecal pellets (0.10 Gt N yr −1 ) and carcasses (0.13 Gt N yr −1 ) at depth.Part of the export was also accounted for by excretion below 500 m (0.11 Gt N yr −1 ).
The predicted total excretion of N (1.2 Gt N yr −1 ) is at the lower range of global estimates of ammonia excretion by mesozooplankton (1.18-2.38Gt N yr −1 , Hérnandez- León et al., 2008).In addition, it should be note that global estimates of excretion in the ocean would be even higher if microzooplankton excretion was accounted for in observations and if the release of urea and dissolved organic nitrogen (DON) www.biogeosciences.net/7/1285/2010/Biogeosciences, 7, 1285-1305, 2010  were also considered.Although urea and DON excretion are generally considered to be lower compared to NH + 4 excretion (Bidigare, 1983), Steinberg and Saba (2008) found that their contribution to total excretion of N varies from 7% to 53% depending on the variety of planktonic crustacean.Excretion of an even greater proportion of N as DON have been reported in Acartia tonsa, for example (62-89%, Miller and Glibert, 1998).
In the model, excretion by herbivores depended on the N and P assimilation and accumulation efficiencies.Using the parameters values in Table 1, when zooplankton fed on nonfixers (R O org = 16), the predicted zooplankton GGE for N and P were (Table 2) and, Conversely, when zooplankton fed on N 2 fixers (R NF org = 33), GGE for N and P were and, GGE P = a N F P β P 1 = L m β P 1 = 0.60.Given that the N:P ratio of herbivores was greater than that of non-fixers (owing to the fact that the biomass of non-fixers was considered to be dominant), zooplankton tended to accumulate N more efficiently than P, which resulted in the release of products with a low N:P ratio (N:P=5.95,calculated from Table 2).DeMott et al. (1998) showed that Daphnia magna have a lower GGE when fed on Scenedemus with a high P content than when fed on Scenedemus having low P content (GGE P =7% and 50% respectively).A study conducted by Vincent et al. (2007) on Acartia discaudata reported a GGE N of 30%, a value lower than estimated by our model, but the N:P ratio of the algal food was not documented in Vincent et al. study.It should be pointed out that our modelling assessment was intended to show that our model produces a reasonable fit to the main features of ocean N and P cycling.We have not intended to make a perfect fit with observations as it would be unrealistic to do so with a 2-box model.Rather, our study was conceived to explore the impact of the higher trophic level on N and P cycling at the ocean level as a whole.) plotted against the ratio of nitrogen to phosphorus (N:P) in non-fixers, using four different N:P ratios in herbivores; The N:P ratio of the products released by herbivores plotted against the N:P ratio in non-fixers; (c) Gross growth efficiency of N (GGE N ) and P (GGE P ) plotted against the N:P ratio in non-fixers (R O org ) using the following set of parameters: assimilation efficiencies, β N 1 = 0.69 and β P 1 = 0.80, maximum accumulation efficiency, L m = 0.9, and N:P ratio in herbivores, R Z org =20.

Sensitivity analyses
The main objective of the sensitivity analysis was to determine how ecosystem functions and nutrients inventories respond when (1) the nitrogen to phosphorus ratio (N:P) in phytoplankton varies and when ( 2) most of primary production is channelled towards herbivores.Attention has also been paid to variations in herbivore parameters.
Herbivore biomass and nutrient GGE -Figure 4a shows how herbivore biomass responds to changes in the N:P ratio of the algal pool.Here we considered only the N:P ratio in phytoplankton that do not fix N 2 because variations of the N:P ratio in N 2 -fixers had a smaller impact on model solutions (results not shown).This partly resulted from the low contribution of N 2 -fixers to total phytoplankton biomass (∼1%, see Tyrrell, 1999).As predicted by Eq. ( 17d), herbivore biomass increased until the N:P ratio in the algal pool reached a threshold value above which herbivore biomass remained constant unless the N:P ratio of herbivores increased (Fig. 4a).This threshold ratio corresponded to the case where .80 and β N 1 =0.69, this threshold ratio is equal to 23.188 (Fig. 4c).This was a condition for which the growth of herbivores shifted from N to P limitation.Below this threshold ratio, the excess of P in the phytoplankton pool compared to the herbivore requirement was excreted at a N:P ratio lower than that of phytoplankton (Fig. 4b).In parallel, GGE P was lower than its maximal value, while, on the other hand, GGE N was maximal (Fig. 4c).Conversely, above this threshold ratio, phytoplankton nitrogen content was in excess compared to herbivore requirement, the N:P ratio of the excreted products increased (Fig. 4b), and GGE P is maximal while GGE N decreased exponentially (Fig. 4c).In Fig. 4c, it should also be noted that GGE N and GGE P reached simultaneously their maximal value when the N:P ratio is the algal pool reached the threshold ratio of 23.188, that is when the N:P ratio of algal food which has passed the gut wall matched the N:P ratio of herbivore.
Phytoplankton biomass and production -Shifts in the accumulation efficiency of N and P maintained the condition of strict homeostasis in the N:P stoichiometry of herbivores, as predicted by Sterner's model (1990).This homeostatic regulation of the elemental composition of herbivores had an impact on the production rates of both N 2 -fixers and non-fixers.When the N:P ratio in the algal pool was lower than that of herbivores, the excess P released to seawater favoured the growth of N 2 -fixers (which were phosphorus-limited).As a consequence, N 2 -fixation increased compared to the case where herbivores were not considered (Fig. 5a, b).This effect was more pronounced when the difference between the N:P ratio in phytoplankton and herbivores was higher, and/or when the maximal accumulation efficiency (L m = a P ) of phosphorus was lower (Fig. 5a, b).
Figure 5 also shows that the N:P ratio in the algal pool that determined whether N 2 fixation would occur R O org fix decreased when herbivores were included in the model.This ratio was derived from the steady state solution for N 2 -fixers (Eq.14), and satisfied the following condition:   increased exponentially, and became undefined when R Z org is lower than 5.264.The greater fixation of N 2 when herbivores fed on a P-rich algal pool resulted in an input of new N into the surface of the ocean.This new N was eventually available for the Nlimited non-fixers (O * ), so that their growth also benefited indirectly from the release of P by herbivores (Fig. 5c, d).

Using the default parameters values listed in
Overall, total primary production (TPP*) increased when herbivores were explicitly considered (Fig. 6).From the steady state solution for TPP* (Eq.16), the effect of herbivores on TPP* was driven by the term SF 1 − ε 0 β P 1 1 − a O P .All other things being equal, the increase in TPP* was minimal when the accumulation efficiency of P was maximal (a O P = L m ).This is illustrated on Fig. 6a, b (and also on Fig. 5) by the "straight line" just above the model solution obtained without herbivores (which corresponds to Tyrell's model).All the model solutions converged towards this straight line when R O org ≥ R Z org • β P 1 /β N 1 , i.e. when P became the limiting nutrient for herbivore growth (Fig. 4c, d).
Nutrient concentrations and the deep N:P ratio -Figure 7 shows the steady state solutions for nutrient concentrations plotted against the N:P ratio of the algal pool.Surface nutri- ent concentrations were unaffected by variations of the N:P ratio in the algal pool, as expected from the steady state solutions (Eqs.18 and 19).Deep nutrient concentrations, on the other hand, were affected by variations of the N:P ratio in non-fixers, and by the herbivore parameters (Eqs.20 and 21).When the N:P ratio in the algal pool increased, the deep N concentration increased and the deep P concentration decreased nonlinearly until Other algae N:P ratio  6. Global primary production plotted against the ratio of nitrogen to phosphorus (N:P) in non-fixers pool using two different maximum accumulation efficiencies in herbivores (L m =0.9 or L m =0.5), and four different N:P ratios in herbivores (R Z org ).The assimilation efficiency of nitrogen (β N 1 ) and phosphorus (β P 1 ) were set at 0.69 and 0.80, respectively.Solutions for the Tyrell's model without herbivores are also shown (dashed line).Using the default parameter set listed in Table 1, this threshold ratio was equal to 28.99.Above 28.99, deep N concentrations increased linearly, while deep P concentrations remained constant.The relative importance of each parameter driving the deep N:P ratio was evaluated using a sensitivity index (SI), calculated as follows, In this Eq., X R O org std is the model solution for a variable X (deep nutrient concentrations and N:P ratio) using the default parameter set listed in Table 1 and the variable N:P ratio in non- −50% is the model solution for the same variable when a model parameter has been decreased by 50%.We tested values for R O org from R O org =1 to R O org =50.The results (Table 3) show that deep nutrient concentrations were primarily controlled by the mixing rate (K) and herbivores' parameters, notably L m , DR 1 , ε 0 , β P 1 , and SE 1 .Deep nutrient concentrations were less sensitive to the other parameters, DR 0 , R Z org , DN and β N 1 .The predicted deep N:P ratio was mainly controlled by two parameters: SE 1 and DR 1 .
Figure 8 illustrates the sensitivity of the predicted deep ocean N:P ratio to variations of both the N:P ratio of nonfixers and SE 1 .In contrast to Tyrrell's original model, changes in the deep nutrient N:P ratio were nonlinearly related to the phytoplankton N:P ratio.Higher SE 1 (i.e. a lower proportion of excretion occurring in the deep ocean) tended to reduce changes in deep nutrient N:P ratios when phytoplankton stoichiometry varied, notably when In this case, deviations from Redfield stoichiometry in phytoplankton were partly absorbed in the surface of the ocean through the excretion of the excess of N. Conversely, a lower SE 1 amplified the increase in the deep nutrient N:P ratio when R O org was higher than R Z org • β P 1 /β N 1 and reduced it when R O org was lower.Figure 9 shows the estimated deep ocean N* for different N:P ratios in herbivores (R Z org ) and for different values of SE 1 .Gruber and Sarmiento (1997) introduced this N* parameter to derive estimates of N 2 -fixation and denitrification in the ocean, defining it as: As shown on Fig. 9, the actions of herbivores may alter the deep ocean N*.Low Fig. 8. Simulated deep ocean N:P ratio vs. non-fixers N:P ratio (R O org ), using a maximum accumulation efficiency for herbivores of (a) L m =0.9 and (b) L m =0.5.Different values of the fractions of excretion occuring in the surface of the ocean (SE 1 ) were also used.The other parameters were kept similar with base run (Table 1).
Fig. 9.Estimated deep ocean N* plotted against the N:P ratio in herbivores (R Z org ), using different fraction of herbivore excretion occuring in the surface layer (SE 1 = 0.80, 0.90, 0.95 or 1.0).Parameters other than R Z org and SE 1 are kept similar with base run (Table 1).The N* value was corrected from N* in Tyrrell's original model without herbivores.and high SE 1 (SE 1 ≥ 0.95) induced positive N* values.Strong negative N* values were predicted when R Z org increased (>13.8) and SE 1 decreased (SE 1 ≤ 0.95).When 100% of the excretion occurred in the surface, and R Z org was high, N* values became positive.This implies that differential recycling of N and P by herbivores in the surface of the ocean can induce a N deficit (negative N*) or an excess of N (positive N*) in the deeper layers that do not result from either denitrification or N 2 -fixation.

Discussion
The main question addressed in this study was to what extent does CNR drive oceanic primary production and the distribution of N and P in the global ocean?To answer this question, we used a model built upon foundations provided by Tyrrell (1999) and Sterner (1990).The model was kept admittedly simple in term of the N and P metabolism of phy-toplankton and herbivores.Regarding herbivores, however, Sterner's approach has been proven robust in estimating observed patterns of CNR (Sterner and Elser, 2002).
Our combination of these two models has shown that marine herbivores may exert a control on global primary productivity through the resupply and the stoichiometry of N and P in the ocean.
In the model, P remained the ultimate limiting nutrient of marine productivity, as in Tyrell's (1999) original model.When herbivores were included in the model, however, variations of marine primary productivity resulted not only from variations in the inputs of P from rivers (Tyrrell, 1999), but also from CNR. Levels of global primary production were higher particularly when herbivores had higher N:P ratios than phytoplankton (Fig. 6).This higher primary production was triggered by a low N:P resupply ratio from herbivores (Fig. 4b), which in turn favoured the P-limited N 2fixation (Fig. 5a, b).Eventually, the N-limited non-fixers benefited from the input of this "new N" fuelled by N 2fixation (Fig. 5c, d).
This influence of CNR on N 2 -fixation has been demonstrated in lakes, with reduced N 2 -fixation when herbivores were composed of Daphnia pulex, a species with low wholebody N:P ratios, while N 2 -fixation was higher with herbivores having higher N:P ratios (MacKay and Elser, 1998).In the ocean, evidences for P-limitation of N 2 -fixation and for low N:P resupply ratios from migrating zooplankton have been also documented in the Pacific subtropical gyres (Moutin et al., 2005;Hannides et al., 2009).Additionally, it has been shown that marine species of microzooplankton have usually a low N:P resupply ratio, in the range 2:1 to 8:1 (Dolan, 1997).
Our results, therefore, support the idea that in the oceanic P-limited systems, grazing by herbivores, notably microzooplankton, may alleviate P-limitation of marine productivity.
The model has also revealed that CNR can influence the N:P ratio in non-fixers that trigger N 2 -fixation ( R O org fix , Biogeosciences, 7, 1285Biogeosciences, 7, -1305Biogeosciences, 7, , 2010 www.biogeosciences.net/7/1285/2010/Eq. 27).Using a modified version of Tyrrell's (1999) model, Lenton and Klausmeier (2007) found that competitive dynamics between N 2 -fixers and non-fixers set the N:P threshold for N 2 -fixation.Here, we found that when the N:P ratio in herbivores (R Z org ) increased above Redfield stoichiometry, R O org fix converged towards a constant value of ∼4-5, while R O org fix increased sharply when R Z org decreased below Redfield stoichiometry.
Values for R O org fix of ∼4-5 fall in the lower range of observed N:P ratios in phytoplankton (Quigg et al., 2003).These values are, to our knowledge, rarely encountered in natural marine phytoplankton communities.Similarly, a review of the N:P ratio in marine zooplankton has shown that most of these organisms, excepted fish larvae, have a N:P ratio usually higher or close to 16:1 (Table 4).Therefore, our results would suggest that N 2 -fixation should be rarely triggered by the N:P ratios of non-fixers in most of the cases encountered in the ocean.
The model has also shown that phytoplankton with higher N:P ratios increased the yield of herbivores until R O org ≥ R Z org • β P 1 /β N 1 (Fig. 4), when P became the element limiting herbivore growth.As predicted by Sterner's model, this case favoured herbivores with a low P-requirement (or high N:P ratios; Fig. 4).Such situations have been reported in lakes where zooplankton community assemblages with lower P requirements are correlated with high C:P ratios in the seston (Gulati et al., 1991).In the ocean, P-limitation has been reported in various systems, notably in the oligotrophic gyres (Karl et al., 2001;Moutin et al., 2005).Such systems would be useful for testing whether or not phytoplankton communities with higher N:P ratios, such as Synechoccocus or Prochlorochoccus (Bertilsson et al., 2003), also favour zooplankton assemblages with low P requirement in the ocean.
Finally, the model suggested that CNR both drove and buffered the impact of deviations from Redfield stoichiometry in non-fixers on the deep ocean N:P ratio.This influence of CNR was driven by three mechanisms, the importance of herbivory which accounted for 80% of the annual primary production in the model, the homeostatic regulation of herbivore's N:P stoichiometry, and the resulting excretion of N and P in various proportions.
When most of the excretion occurred in the surface of the ocean, CNR reduced the impact of variations of the N:P ratio in non-fixers on the deep N:P ratio (Fig. 8).Then, The deep N:P ratio tended to mimic the N:P ratio of herbivores rather than that of phytoplankton.Additionally, N deficit (N*<0) or N excess (N*>0) can result not only from shifts in the balance between N 2 -fixation and denitrification (Gruber and Sarmiento, 1997), but also from CNR.When R Z org was higher than Redfield stoichiometry, this induced N excess in the deep ocean if more than 95% of the excretion occurred in the surface of the ocean (Fig. 9).Conversely, N deficit was found when more than ∼5% of the excretion occurred in the deep ocean (Fig. 9).
In other words, if primary production is mainly consumed by non-migrating zooplankton with high N:P ratios, such as microzooplankton, most of the excretion will occur in the surface of the ocean, thereby inducing N excess in the deeper layers (Fig. 9).In this case, estimates of denitrification in the deep ocean from N* calculations could be underestimated.Conversely, migrating herbivores, such as mesozooplankton, will excrete part of the nutrients in the deep layers with a low N:P resupply ratio.This active transport of nutrients could induce N deficit in the deeper layers (Fig. 9), and a possible overestimation of denitrification.
In the North Pacific gyre, Hannides et al. (2009) have shown that P fluxes to the deeper layers mediated by excretion from migrating zooplankton was a significant component of P cycling in this area.This active downward flux of P was shown to be equal in magnitude to the P flux mediated by particles sinking, thereby reinforcing P-limitation of marine productivity in the upper layers.Additionally, this P-rich flux mediated by migrating zooplankton may induce apparent N deficit in the deeper layers that would not result from denitrification.
All together, our model results and recent studies in the subtropical gyres suggest that marine herbivores and CNR may be important in driving the stoichiometry of N and P in the deep ocean, and should be included in efforts to understand biogeochemical cycles.Although simple, the proposed parameterisation of the influence of CNR on oceanic N and P cycles is driven by some critical parameters which need further investigations.
(1) Proportion of primary production channelled towards herbivores -A key parameter in the model was the proportion of primary production channelled towards herbivores (set at ε 0 =0.8) rather than exported.However, this parameter may vary in space and time.As noted previously, grazing estimates based on dilution experiments have shown that heterotrophic protists consume an average of 67% of the daily phytoplankton growth, with higher proportions found in the open ocean (Calbet and Landry, 2004).This is in line with the idea that, in the ocean, the dynamics of picoand nanophytoplankton communities are notably controlled by grazing pressure (Sherr andSherr, 1994, 2002), which eventually results in efficient recycling of nutrients in surface waters (Azam et al., 1983;Legendre and Rassoulzadegan, 1995).
Large microphytoplankton, on the other hand, can more easily escape from grazing.Calbet (2001) found for example that mesozooplankton (200-20 000 µm in size) consume an average of 12% of the annual oceanic primary production.Locally, salps or large krill swarms can consume a significant proportion of the daily primary production on a short time scale.Von Bodungen (1986)   primary production over a period of 3 weeks in the Brandsfield Strait (Southern Ocean).
The proportion of primary production channelled towards herbivores will thus vary both temporally and spatially, according to the dominance of phytoplankton communities by either small or large organisms.Summing the contribution of protists and mesozooplankton gives an estimate of 79% −87% of the oceanic primary production being exported towards herbivores.
A process, not included in the model, which would also result in an efficient remineralisation of organic matter within the upper ocean, is cell lysis associated with virus infections.Cell lysis rates can be high (up to 87.8 d −1 ) in the ocean and may account for a significant part of small phytoplankton natural mortality (Marbá et al., 2007).Higher latitudes, coastal areas, or fronts are, conversely, characterised by seasonal blooms dominated by large phytoplankton such as diatoms.Bloom terminations often result in the export and mass sedimentation of large aggregates which contains various particles, including exopolymers like mucus, transparent exopolymer particles (TEP), phytoplankton cells, detritus, faecal pellets, minerals and mucus (Thornton, 2002).
(2) Release of unassimilated nutrients between soluble (excreted) and particulate forms (egested) and proportion of excretion occuring in the surface of the ocean -The release of unassimilated nutrients between soluble (excreted) and particulate (egested) forms together with the proportion of excreted to egested nutrients remineralised in the surface of the ocean are also critical model parameters.
To our knowledge, few things are known about the proportion of the excretion occurring in the surface of the ocean.A significant proportion of excretion is likely accounted for by small organisms (nano-and microzooplankton) which have high turnover rates.These organisms do not perform large vertical migrations, and should excrete N or P mostly in the surface of the ocean.Also, they produce minipellets which are mostly remineralised in the surface (Gowing and Silver, 1985;Nöthig and Von Bodungen, 1989;González, 1993).A study on microzooplankton grazing in the Southern Ocean suggested, for example, that during winter, the efficiency of the biological carbon pump is reduced when a greater proportion of the photosynthetically fixed carbon is passed to the microzooplankton compared to summer when the larger cells dominate phytoplankton biomass and the bulk of the photosynthetically fixed carbon are consumed by meso-and macrozooplankton (Froneman and Perissinotto, 1996).
Unlike nano-and microzooplankton, large organisms like copepods and other large crustaceans can vertical migrate on short time scales.Various studies show that excretion by these organisms below the euphotic zone can account for a significant part of the biological pump and the vertical flux of nutrients in a variety of open-ocean environments (Longhurst et al., 1990;Dam et al., 1995;Le Borgne and Rodier, 1997;Steinberg et al., 2000;Al-Mutairi and Landry, 2001;Steinberg et al., 2002).Hernandez-León and Ikeda (2005) also reported that ∼10-15% of mesozooplankton respiration (13.0 ± 4.2 Gt C yr −1 ) occurs at depths deeper than 500 m.Part of this deep excretion is accounted for by organisms which feed in surface layers and excrete part of their assimilated nutrients in the mesopelagic zone.
If we combine the contribution of protists and mesozooplankton to grazing and global excretion, we can reasonably suggest that ∼90% of the excretion occurs at depths shallower than 500 m.This vertical partitioning needs further investigation with regards to its importance in driving the deep nutrient inventories, as illustrated by the model presented herein.
Considering the influence of CNR in driving marine ecosystem functioning, an important issue which would need further investigations is the influence of CNR on iron cycle in the Ocean.Iron limits primary production in several oceanic environments (Martin and Fitzwater, 1988;Hutchins and Bruland, 1998), and N 2 -fixers are known to have high Fe-requirement compared to other algae (Finkel et al., 2010).Thus, the influence of CNR on N 2 -fixation could be triggered not only by P but also by Fe resupply from herbivores.Several studies have suggested that grazing by herbivores enhances iron recycling (Barbeau et al., 1996(Barbeau et al., , 2001;;Twining et al., 2004a;Sato et al., 2007;Sarthou et al., 2008).Barbeau et al. (1996Barbeau et al. ( , 2001) ) found for example that digestion of colloidal iron in the acidic food vacuoles of protozoan grazers may be a mechanism for the regeneration of bioavailable iron from the refractory iron phases.Similarly, a study of the impact of grazing on Fe regeneration in a naturally iron-fertilised area also revealed that copepod grazing resulted in a 1.7-2.3-foldincrease in Fe regeneration (Sarthou et al., 2008).It was also found that Fe regeneration accounted for 42-61% of the total Fe demand, and the presence of copepods increased Fe regeneration by 48%.Sato et al. (2007) have suggested that organic Fe-binding ligand formation during microzooplankton and copepod grazing on phytoplankton may be responsible for the observed increase of Fe regeneration.
Applying CNR theory principles to iron implies that enhanced iron recycling from grazing should result from Feexcess in the phytoplankton pool compared to the herbivore requirement.To our knowledge, few things are known about the iron requirement of zooplankton.Twining et al. (2004b) measured the C:P:Fe stoichiometry of individual cells of heterotrophic flagellates (H flag ) during the Southern Ocean Iron experiment.They found that H flag and phytoplankton had similar Fe:P ratios under low Fe-conditions, while H flag had lower Fe:P ratios than phytoplankton under high-Fe conditions.Low Fe:P ratio in the consumer pool compared to the algal pool would be consistent with enhanced iron recycling from herbivores.This influence of CNR on iron cycling would need further investigations, especially regarding the Fe requirement of consumers.

Conclusion and perspectives
By linking the two important works of Tyrrell (1999) and Sterner (1990), this study has shed light on the possible role of herbivores in driving N and P cycles in the ocean, following the pathways of N and P from food source, incorporation into herbivorous zooplankton, the resulting metabolic products, and, finally, nutrient inventories.Here, the importance of the N:P resupply ratio depended on the fate of nutrients that are bound into consumer biomass.If all of the herbivore biomass was eventually remineralised in the surface layer and became available for uptake by phytoplankton, the differential ratios by which herbivores excrete nutrients would be cancelled out by the N:P ratios of herbivore biomass remineralisation.In our model however, the nutrient resupply ratio from herbivores affected the dissolved inorganic N:P ratios at steady state through two processes: a fraction of detritus was removed from the surface of the ocean through settling, and a fraction of herbivore excretion occurred in the deep ocean.Although simple, the model used here emphasises the need for us to better understand how consumers influence nutrient recycling in the ocean.This study should be extend to higher trophic levels, beyond herbivores.As pointed at by Sterner and Elser (2002), the fate of N and P which are channelled towards herbivores versus carnivores remain poorly constrained.
The model presented here does not discriminate between herbivorous protists and mesozooplankton, the later which are able to feed actively on protists (Gismervik and Andersen, 1997).Also, there is evidence that predation accounts for a significant proportion of natural mortality in small marine invertebrates, up to 70-80% in copepods for example (Hirst and Kiørboe, 2002).
The elemental stoichiometry of marine predators is still poorly documented compared to freshwater organisms and needs further investigations.A study in the Baltic Sea has shown for example that the biomass of pelagic fishes could represent a significant pool of P (Hjerne and Hansson, 2002), thereby influencing nutrient cycling in this system.
In conclusion, the two models used here support the idea that CNR may influence nutrient cycling and inventories at global scale and that this process should be account for in ocean physical-biological models which often assume Redfield stoichiometry throughout the food chain.
Fig. 4. (a)Global herbivore biomass (mmol N m −3 ) plotted against the ratio of nitrogen to phosphorus (N:P) in non-fixers, using four different N:P ratios in herbivores; The N:P ratio of the products released by herbivores plotted against the N:P ratio in non-fixers; (c) Gross growth efficiency of N (GGE N ) and P (GGE P ) plotted against the N:P ratio in non-fixers (R O org ) using the following set of parameters: assimilation efficiencies, β N 1 = 0.69 and β P 1 = 0.80, maximum accumulation efficiency, L m = 0.9, and N:P ratio in herbivores, R Z org =20.

Fig. 5 .
Fig. 5. (a) and (b) N 2 -fixation plotted against the ratio of nitrogen to phosphorus (N:P) in non-fixers using two different maximum accumulation efficiencies for herbivores (L m =0.9 or L m =0.5), and four different N:P ratios in herbivores (R Z org ).The assimilation efficiency of nitrogen (β N 1 ) and phosphorus (β P 1 ) were set at 0.69 and 0.80, respectively.(c) and (d) same as (a) and (b), but for non-fixers.Solutions for the Tyrell's model without herbivores are also shown (dashed line).

Fig. 7 .
Fig. 7.Nutrient concentrations plotted against the N:P ratio in non-fixers using the default parameters set listed in Table1, except for L m .

Table 1 .
Definitions of model parameters, values and literature sources.

Table 2 .
Comparison of model solutions at steady-state with values from the literature.

Table 3 .
Sensitivity of the deep nutrient concentrations to model parameters.Values represent the mean sensitivity index (SI, %) as a percentage of variation of model solutions compared to the standard simulation when a parameter has been decreased by 50%.See text for details on the calculation of SI.

Table 4 .
Ecological C:N:P stoichiometry of various marine pelagic invertebrates and vertebrates (mean and standard deviation in brackets).