Ecological Modelling 139 (2001) 265– 291 www.elsevier.com/locate/ecolmodel

A stoichiometric model relating growth substrate quality (C:N:P ratios) to N:P ratios in the products of heterotrophic release and excretion Franck Touratier a,*, John G. Field b, Coleen L. Moloney c a

Laboratoire d’oce´anographie biologique, Station Marine d’Arcachon, Uni6ersite´ Bordeaux 1, 2 rue du Professeur Jolyet, 33120 Arcachon, France b Zoology Department, Uni6ersity of Cape Town, 7701 Rondebosch, South Africa c Marine and Coastal Management, Pri6ate Bag X2, 8012 Rogge Bay, South Africa Received 27 June 2000; received in revised form 11 January 2001; accepted 26 January 2001

Abstract A stoichiometric model is developed to analyze the influence of growth substrate element composition on the N:P ratios of heterotrophic excretion (true excretion, i.e. metabolic by-products) and release (excretion + feces production) products. The model uses units of C, N, and P, and depicts three types of heterotrophs: the copepods, cladocerans, and bacteria. Most parameters of the model are estimated from experimental data sets representative of these heterotrophs. Net growth efficiencies for N and P vary according to the element composition of growth substrates. The simulated N:P ratios for release and excretion products of copepods and cladocerans are compared to experimental data for heterotrophic regeneration. Results indicate that the simulated N:P ratio for excretion is more representative of the experimental measurements than that for release, especially when the growth substrate N:P ratio is high. Thus the model assumption of adjustable excretion could explain stoichiometric regulation of growth substrates by heterotrophs. Our model gives better fits to observations than two other similar models, mainly because of its ability to simulate the excretion N:P ratio. The C content of growth substrates did not influence stoichiometric excretion and release of N and P for mesozooplankton, but it was important for bacteria. Stoichiometric regeneration of nutrients by heterotrophs affects phytoplankton growth, with large organism-dominated ecosystems accentuating N limitation, whereas small organism-dominated ecosystems favor P limitation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Stoichiometry; Zooplankton; Bacteria; Carbon; Nitrogen; Phosphorus

1. Introduction * Corresponding author. Tel.: + 33-556-223900. E-mail address: [email protected] Touratier).

(F.

The classical C:N:P Redfield ratios (Redfield et al., 1963) of 106:16:1 (all element ratios reported in this paper are in atoms:atoms) are widely used

0304-3800/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 1 ) 0 0 2 3 7 - X

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to convert units of several ecological processes (e.g. nutrient assimilation, primary production), biomasses (e.g. phytoplankton) and concentrations (e.g. nutrients) in both experimental and modeling studies. According to Hoppema and Goeyens (1999), the validity of the Redfield model would depend on the duration of the sampling period over which the ratios are measured and then averaged, i.e. on the time-scale of ecological processes. The Redfield concept is believed to be valid at macroscale (from ca. 1 month to 1 yr), as shown by Le Jehan and Tre´ guer (1983) and Hoppema and Goeyens (1999), but the usefulness of Redfield ratios at mesoscale (from ca. few hours to several weeks) seems weak for marine ecosystems (Banse, 1974; Laws, 1991). In freshwater more than in marine ecosystems, the use of Redfield ratios is questioned; the study of Hecky et al. (1993) on 51 lakes indicates that Redfield ratios are the exception rather than the rule in these systems. As defined by Elser et al. (1996), ‘the elemental stoichiometry considers [the] relative proportions (ratios) of key elements in analyzing how characteristics and activities of organisms influence, and are in turn influenced by, the ecosystem in which they are found’. The stoichiometric approach can be used to take the variability of C:N:P ratios into account, especially at mesoscale in both marine and freshwater ecosystems. The main advantage of stoichiometric theory is that it provides a multi-currency approach (e.g. C, N, and P) for tracking nutrient flows (Reiners, 1986; Sterner and Hessen, 1994; Elser et al., 1996). It follows that not only the quantity, but also the quality of materials (the element ratios) is important in understanding the complexity of ecological interactions. The pioneering stoichiometric model of Sterner (1990) was validated with the experimental data of Le Borgne (1982) for copepods and those of Urabe (1993) for cladocerans. This model showed that the N:P ratios of release products depended on the N:P ratios of growth substrates and zooplankton, increasing as the N:P ratios of growth substrates increased, the shape of the relationship being partly curvilinear. As noted by Urabe (1993), this means that zooplankton do not act as simple nutrient ‘transmitters’. If the N:P ratio of growth substrates is smaller than that of zooplankton, relatively less

N will be released than P, and this could favor N limitation of plankton. The converse would also be true, resulting in P limitation of plankton. Thus zooplankton with a low body N:P ratio could exacerbate P limitation, whereas those with a high body N:P ratio could exacerbate N limitation. Implications of these results are numerous, not fully understood, and need to be explored further. Species composition of phytoplankton communities (Smith, 1983), sedimentation processes (Elser and Foster, 1998) and the overall structure of planktonic food webs (Sterner and Hessen, 1994) could all be influenced by the ratio of N:P in zooplankton release and excretion products. In addition, there are large differences in N:P ratios of nutrients, prey, and predators in freshwater and marine ecosystems (Elser and Hassett, 1994; Hassett et al., 1997), and these could be explained using stoichiometric models. The stoichiometric approach has been used to investigate the influence of the quality of growth substrate on heterotrophic growth, respiration or excretion; several models have been proposed for bacteria (Parnas, 1975; Anderson, 1992; Touratier et al., 1999a) and zooplankton (Sterner, 1990; Anderson, 1992; Anderson and Hessen, 1995; Sterner, 1997; Touratier et al., 1999b). These models all consider two currencies (combinations of C, N and P), whereas Thingstad (1987) and Andersen (1997), respectively, developed three-currency models (C, N and P) for bacteria and zooplankton. These three elements are frequently chosen as model currencies because they might be expected to limit heterotrophic growth. Stoichiometric models that describe the freshwater ecosystems with the three currencies C, N and P, like those of Andersen (1997) and Roelke (2000), progressively appear in the literature. Similar approaches should be used to investigate the functioning of food webs in marine systems. In this study, we develop a stoichiometric model to investigate how the element composition (quality) of growth substrates influences the N:P ratios of heterotrophic excretion and release products. Here, the excretion of an element refers to true excretion (metabolic by-products), whereas release refers to the sum of excretion and feces

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

production, whatever the form of the fecal material (dissolved, particulate, organic, or inorganic). Many factors influence N and P excretion and release (e.g. temperature, size of the heterotroph, concentration of growth substrates, etc.); focusing on the quality of the growth substrates, as was done by Sterner (1990), greatly simplifies the model. Three currencies are used (C, N and P), partly because we want to know whether the C content of growth substrates influences the N:P ratios of excretion and release, and also because some recent experimental studies on bacteria aim to determine whether their growth is C, N, or P limited (Cotner et al., 1997; Rivkin and Anderson, 1997). Three groups of organisms are modeled: heterotrophic bacteria, copepods and cladocerans (the latter two are generally in the size range of 200– 3000 mm, i.e. mesozooplankton). These groups were selected because several experimental data sets are available for them, allowing estimation of most parameters of the model and comparison with simulated results. The three groups are common in freshwater systems, whereas bacteria and copepods play an important role in oceans. Microzooplankton (2–200 mm, e.g. microflagellates), are important contributors to excretion processes (Wheeler and Kirchman, 1986), but are not modeled here because available experimental studies (Goldman et al., 1987a; Nakano, 1994) did not allow us to estimate values of model parameters. Our first objective is to develop a stoichiometric model and to determine, as far as possible, the values of model parameters for the three groups of heterotrophs. The second objective is to compare the release and excretion ratios from the model results for copepods and cladocerans with observed values. From these comparisons, a third objective is to develop and investigate hypotheses about the ways in which heterotrophs manage the variable element composition of their growth substrates. The fourth objective is to compare our simulated results with those of Sterner (1990). Finally, the fifth objective is to discuss the influence of stoichiometric excretion of nutrients on the type of limitation influencing phytoplankton growth.

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2. Description of the model

2.1. Conceptual 6iew of a heterotroph Descriptions, values and units of parameters and variables used in the model are listed in Table 1. A conceptual diagram that describes fluxes of C, N, and P through a heterotroph is shown in Fig. 1. For the C flows, the growth substrates that contain organic C (CS) are ingested (IC) by the heterotroph; one part of IC is assimilated (AC) and the remaining part is egested as fecal material (FC), allowing a gross accumulation of fecal C (CF) in the medium. The assimilated C (AC) is used for production (PC) and respiration (EC), resulting in a gross increase in the C biomass of the heterotroph (CH) and the dissolved inorganic C (CI) in the medium. Equivalent fluxes and concentrations or biomasses can be drawn for N and P (Fig. 1). The sole difference is that N and P net excretion (EN and EP) can be either positive (excreting NI and/or PI into the medium) or negative (taking up NI and/or PI from the medium). The equations used to describe assimilation, feces production, production, respiration, excretion, and release in C, N, and P units are listed in Table 2 (Eqs. (2a), (2b), (2c), (2d) and (2e)). From these, C:N:P ratios for flows (Eqs. (2f), (2g), (2h), (2i) and (2j), Table 2) can be calculated, given C:N:P ratios for ingested materials (IC:IN:IP), and using assumed values for assimilation coefficients (iC, iN, iP) and calculated values for net growth efficiencies (K2C, K2N, K2P; Table 1). The heterotroph represented schematically in Fig. 1 has several flows typical only for bacteria and others found only in zooplankton. However, this single model can be used to simulate both bacteria and zooplankton by adapting the values of parameters. In copepods and cladocerans, ingestion and assimilation are distinct processes, whereas bacteria do not ingest, they directly assimilate growth substrates from the medium. For zooplankton, net excretion is either positive or zero, whereas for bacteria excretion can be positive, zero, or negative; bacteria could outcompete phytoplankton for uptake of inorganic N and P (Currie, 1990; Kirchman, 1994; Rivkin and Anderson, 1997).

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Table 1 Parameters and variables used in the model (wd is without dimension) Description

Values

Units

(C:N)H

C:N atom ratio for the heterotroph

Copepods 4.72

Cladocerans 5.7

Bacteria 5.97

(C:P)H

C:P atom ratio for the heterotroph

113.84

90.77

47.1

K2 LNI LPI h

Net growth efficiency when (C:N:P)A =(C:N:P)H Limitation by NI Limitation by PI Factor for the slope A of K2N

0.4625 0 0 1

0.475 0 0 1

0.478 0.5 0.5 0.064

mmol C (mmol N)−1 mmol C (mmol P)−1 wd wd wd wd

set A

set B

set C

0.8 0.8 0.8

0.6 0.8 0.9

0.6 0.8 0.7

1 1 1

wd wd wd

iC iN iP

Carbon assimilation coefficient Nitrogen assimilation coefficient Phosphorus assimilation coefficient

Variable

Description

units

A

wd

AC, AN, AP CF CH CI (C:N)A

Slope of K2N when (C:P)A is kept constant or K2P when (C:N)A is kept constant Assimilation of organic carbon, nitrogen, and phosphorus Feces concentration in carbon Heterotroph biomass in carbon Dissolved inorganic carbon C:N atom ratio for assimilated growth substrate ((C:N)A =CSiC:NSiN)

(C:N)S

C:N atom ratio for growth substrate ((C:N)S =CS:NS =IC:IN)

(C:P)A

C:P atom ratio for assimilated growth substrate ((C:P)A =CSiC:PSiP)

(C:P)S

C:P atom ratio for growth substrate ((C:P)S =CS:PS =IC:IP)

CS EC, EN, EP FC, FN, FP IC, IN, IP K1 K1C, K1N, K1P K2C, K2N, K2P K2CII, K2CIII KNI

Organic growth substrate concentration in carbon Respiration and net excretion of NI and PI Feces production, in carbon, nitrogen, and phosphorus Ingestion of organic carbon, nitrogen, and phosphorus Gross growth efficiency when (C:N:P)S =(C:N:P)H Gross growth efficiency for carbon, nitrogen, and phosphorus (K1C =PC:IC; K1N= PN:IN; K1P= PP:IP) Net growth efficiency for carbon, nitrogen, and phosphorus (K2C =PC:AC; K2N=PN:AN; K2P= PP:AP) Net growth efficiency for carbon in regions II and III Half-saturation constant for NI uptake

mmol mmol mmol mmol mmol N)−1 mmol N)−1 mmol P)−1 mmol P)−1 mmol mmol mmol mmol wd wd

m−3 day−1 C m−3 C m−3 C m−3 C (mmol C (mmol C (mmol C (mmol C m−3 m−3 day−1 m−3 day−1 m−3 day−1

wd wd mmol N m−3

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

Parameter

Table 1 (Continued) Description

Units

KPI NF NH NI (N:P)A

Half-saturation constant for PI uptake Feces concentration in nitrogen Heterotroph biomass in% nitrogen Dissolved inorganic nitrogen N:P atom ratio for assimilated growth substrate ((N:P)A =NSiN:PSiP)

(N:P)H

N:P atom ratio for heterotrophs

(N:P)R

N:P atom ratio for regeneration, used for experimental data sets

(N:P)S

N:P atom ratio for growth substrate ((N:P)S =NS:PS =IN:IP)

NS PC, PN, PP PF PH PI PS RC, RN, RP TK2CCN

Organic growth substrate concentration in nitrogen Production in carbon, nitrogen, and phosphorus Feces concentration in phosphorus Heterotroph biomass in phosphorus Dissolved inorganic phosphorus Organic growth substrate concentration in phosphorus Release in carbon, nitrogen, and phosphorus Value of (C:N)S when K2C =1

TK2NCN

Value of (C:N)S when K2N =1

TK2PCP

Value of (C:P)S when K2P =1

TK2PNP

Value of (N:P)S when K2P= 1

TNP

Value of (N:P)S when EN:EP is equal to the conceptual model

mmol mmol mmol mmol mmol P)−1 mmol P)−1 mmol P)−1 mmol P)−1 mmol mmol mmol mmol mmol mmol mmol mmol N)−1 mmol N)−1 mmol P)−1 mmol P)−1 mmol P)−1

P m−3 N m−3 N m−3 N m−3 N (mmol N (mmol N (mmol N (mmol N m−3 m−3 day−1 P m−3 P m−3 P m−3 P m−3 m−3 day−1 C (mmol C (mmol C (mmol

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

Variable

N (mmol N (mmol

269

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Fig. 1. Conceptual view of a heterotroph showing all carbon, nitrogen, and phosphorus fluxes describing the processes involved in the stoichiometric model. AC – assimilation of organic carbon; AN – assimilation of organic nitrogen; AP – assimilation of organic phosphorus; CF – feces concentration in carbon; CH – heterotroph biomass in carbon; CI – dissolved inorganic carbon; CS – organic growth substrate concentration in carbon; EC – respiration; EN – net excretion of inorganic nitrogen; EP – net excretion of inorganic phosphorus; FC – feces production in carbon; FN – feces production in nitrogen; FP – feces production in phosphorus; IC – ingestion of organic carbon; IN – ingestion of organic nitrogen; IP – ingestion of organic phosphorus; NF – feces concentration in nitrogen; NH – heterotroph biomass in nitrogen; NI – dissolved inorganic nitrogen; NS – organic growth substrate concentration in nitrogen; PC – production in carbon; PF – feces concentration in phosphorus; PH – heterotroph biomass in phosphorus; PI – dissolved inorganic phosphorus; PN – production in nitrogen; PP – production in phosphorus; PS – organic growth substrate concentration in phosphorus.

Table 2 Equations of the model for fluxes in carbon, nitrogen, and phosphorus (x stands for C, N, or P unit) and their C:N:P ratios Variable

Equations

Assimilation

Ax = Ixix

(2a) AC:AN:AP =ICiC:INiN:IPiP

(2f)

Feces production

Fx = Ix(1−ix)

(2b) FC:FN:FP =IC(1−iC):IN(1−iN):IP(1−iP)

(2g)

Production

Px = AxK2x

(2c) PC:PN:PP =ACK2C:ANK2N:APK2P

(2h)

Ex = Ax(1−K2x)

(2d)

(Rx = Fx+Ex)

(2e) RC:RN:RP =(FC+EC):(FN+EN):(FP+EP)

Respiration and excretion Release

C:N:P ratios

EC:EN:EP =AC(1−K2C):AN(1−K2N):AP(1−K2P)

(2i) (2j)

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2.2. Assumptions used for the construction of the model A number of assumptions were made in developing the model: (i) Only C, N, or P can limit the growth of heterotrophs; other variables like oxygen, vitamins or fatty acids are considered to be nonlimiting. (ii) Growth substrate concentrations are always saturating for bacterial assimilation or zooplankton ingestion, i.e. maximum assimilation and ingestion rates. (iii) The release of dissolved organic matter (DOM) by heterotrophs is insignificant. Although DOM can be excreted by zooplankton (Vidal, 1980) and bacteria (Itturriaga and Zsolnay, 1981; Novitsky and Keplay, 1981), the rates of release are believed to be slow in pelagic systems. (iv) No distinction is made between production and reproduction. (v) Inorganic forms of N and P excreted by heterotrophs and assimilated by bacteria are mainly in the form of ammonium and soluble reactive phosphorus (SRP), respectively. This is in agreement with observations made in several studies (Jawed, 1969; Ferrante, 1976; Urabe, 1993). Four main hypotheses were considered when deciding how to model the way in which heterotrophs deal with variable element compositions in their growth substrates. These hypotheses are used to discuss assumptions (vi)– (ix) of the model. Hypothesis 1 Heterotrophs select suitable growth substrates during ingestion. Copepods can chemically select their prey (Poulet and Marsot, 1978; Donaghay and Small, 1979; DeMott, 1990), but the extent to which this occurs under natural conditions is unknown. The high energetic costs of effective selection (Sierszen and Frost, 1992) and the scarcity of ideal food might constrain copepods to be less selective in the wild (DeMott, 1989). Cladocerans are known to select food particles mainly according to size, not chemical composition (Urabe, 1993). Hypothesis 1 is not well supported, and we assume:

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(vi) The element composition of ingested growth substrates is the same as that available, so that IC:IN:IP = CS:NS:PS = (C:N:P)S. Hypothesis 2 Heterotrophs use adjustable or differential assimilation of C, N, and P from growth substrates. If adjustable assimilation is true, the ratios of assimilation coefficients (iC:iN:iP) should be inverse functions of element ratios of growth substrates ((C:N:P)S). Sarnelle (1992) found no evidence of adjustable assimilation by Daphnia when comparing N:P ratios between seston and settled particles in Zaca Lake. Furthermore, the residence time of food in the gut of copepods is very short (B 30 min; Reinfelder and Fisher, 1991; Wang et al., 1996) compared with the time-scale required for the adaptation of several gut enzymatic activities (between 0.5 and 7 days; Hassett and Landry, 1983). The regulation of element composition solely via adjustable assimilation is therefore unlikely. However, differential assimilation, defined by constant iC:iN:iP ratios that do not equal unity, could still be important in regulation. For example, several studies show that iN \ iC (Landry et al., 1984; Morales, 1987; Daly, 1997) or iP \ iC (Hessen and Andersen, 1990) in zooplankton. However, in contrast to adjustable assimilation, differential assimilation cannot be the sole process by which heterotrophs regulate the stoichiometry of their food because the ratios of assimilation coefficients remain constant. For proportional assimilation, the ratios of assimilation coefficients are constant, but equal unity. This implies that all assimilation coefficients have the same value and that elements are assimilated according to their ratios in the growth substrates, a possibility used by Le Borgne (1982) to analyze his data on zooplankton and then experimentally justified by Reinfelder and Fisher (1991) and Wang et al. (1996) for several species of copepods. This type of assimilation, however, does not allow regulation of stoichiometry, so that heterotrophs must adopt another possibility. Therefore, we assume: (vii) Only differential or proportional assimilation occurs in zooplankton. For bacteria, it is assumed that all assimilation coefficients equal unity. This also implies that production of fecal material by bacteria is zero.

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Hypothesis 3 Heterotrophs vary element ratios in their own biomass. As noted by Sterner (1994), many heterotrophs are able to maintain their element composition at constant levels, despite the variable quality of their growth substrates. Homeostatic regulation of element composition has been demonstrated for cladocerans and copepods (Hessen, 1990; Urabe and Watanabe, 1992; Sterner et al., 1993) living at low and middle latitudes where accumulation of lipids is small or never occurs, in both marine and freshwater ecosystems. Bacteria can adjust their own element composition to a greater extent than zooplankton but less than phytoplankton. For example, under N or P limitation, increased C:N or C:P ratios in bacterial biomass have been observed (Ju¨ rgens and Gu¨ de, 1990; Nakano, 1994) with the P content generally more variable than the N content. Bacteria may possess internal pools for N (ammonium; Brown and Stanley, 1972) and P (polyphosphate granules; Kulaev and Vagabov, 1983). The variability of N and P bacterial contents is relatively small compared with phytoplankton, because of the small size of bacteria and the large proportion of non-replaceable P-rich nucleic acids (Ju¨ rgens and Gu¨ de, 1990) in their composition. However, the mechanisms underlying the variability of C, N, and P contents in bacteria are poorly known and we assume: (viii) Element stoichiometry for all heterotrophs in this study is constant. Hypothesis 4 Heterotrophs adjust the element ratios in excretion and respiration products. These postulated mechanisms are by far the least known, perhaps because of experimental difficulties in measuring the concentrations of true excretion and respiration products. We assume: (ix) Adjustable excretion and respiration of elements (i.e. adjustable net growth efficiencies) is a major process (but not the sole process) by which heterotrophs cope with variable quality in their growth substrates.

2.3. Modeling net growth efficiencies K2C, K2N, and K2P The difference in quality between assimilated and ingested growth substrates is stated by the follow-

ing equation, iC (C:N)A = (C:N)S; iN iN (N:P)A = (N:P)S. iP

(C:P)A =

iC (C:P)S; iP (1)

The assimilation coefficients (i) are assumed constant, whereas respiration, excretion, release and production of C, N and/or P are assumed variable, and depend on the quality of the assimilated substrates. The C:N:P ratios for excretion/respiration and release products (Table 2, Eqs. (2i) and (2j)) are calculated using net growth efficiencies (K2C, K2N, and K2P), which depend on the element ratios ((C:N)A and (C:P)A) of the assimilated growth substrates. In the model, net growth efficiencies are always \ 0 and, contrary to normal application, they can be \1; values of K2N and K2P\ 1 represent negative excretion (i.e. assimilation of inorganic nutrients NI and PI) by heterotrophs. Because all assimilated substrates contain C, there must be respiration, so that K2C cannot be E 1. Different combinations of growth substrate element ratios will result in different combinations of net growth efficiencies for the nutrients. This is represented schematically in a phase diagram (Fig. 2), where five regions (denoted I–V) can be specified. Each of the regions represents a different combination of net excretion of N and P. Region I is defined by positive fluxes for N and P excretion (EN and EP \ 0, Eq. (2d), Table 2): the organism excretes both ammonium and SRP. In region II, only N excretion can be negative or zero (EN 0 0 and EP \ 0: the organism assimilates ammonium and excretes SRP), whereas in region III, only P excretion can be negative or zero (EN \ 0 and EP 0 0: the organism excretes ammonium and assimilates SRP). Both N and P excretion can be negative and/or zero in region IV (EN 0 0 and EP 0 0: the organism assimilates both ammonium and SRP). Since the value of the carbon net growth efficiency (K2C) must necessarily be B1, region V is impossible and, consequently, is not modeled. The thresholds between these regions in the phase diagram (Fig. 2) are denoted TK2CCN, TK2NCN, and TK2PCP (see below and Table 1 for their definitions, and Table 3 for their equations).

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

Fig. 2. Hypothetical example used to describe the structure of the model, showing the position of the regions I –V, delimited by the thresholds TK2CCN, TK2NCN, and TK2PCP. Region I is defined by 0BK2C B1, 0B K2N B1, and 0 BK2PB 1. Region II is defined by 0BK2CB 1, 10 K2N, and 0B K2PB 1. Region III is defined by 0B K2CB 1, 0B K2N B1, and 1 0 K2P. Region IV is defined by 0 B K2CB 1, 1 0 K2N, and 1 0 K2P. Region V is defined by 10 K2C. (C:N)S – C:N atom ratio for growth substrates; (C:P)S – C:P atom ratio for growth substrates;TK2CCN – value of (C:N)S when K2C= 1; TK2NCN – value of (C:N)S when K2N = 1; TK2PCP – value of (C:P)S when K2P = 1.

The thresholds are deduced from the parameterization used for the net growth efficiencies (Table 4). TK2CCN is the function (C:N)S =f{(C:P)S} when K2C=1, TK2NCN is the function (C:N)S = f{(C:P)S} when K2N =1, and TK2PCP is the function (C:P)S = f{(C:N)S} when K2P =1. We choose to refer all thresholds to the quality of ingested growth substrates, i.e. (C:N)S and (C:P)S, to allow comparisons with corresponding threshold values available from the literature. In region I (Fig. 2), variation in the nitrogen net growth efficiency (K2N), for fixed (C:N)H, is described by a plane defined by Eq. (4c) (Table 4, see Touratier et al. (1999a,b) for details). The values in the plane vary with the C:N ratios of the assimilated substrates ((C:N)A), but are independent of (C:P)A. The plane K2N depends on A (the slope of the plane according to the abscissa (C:N)A), and two parameters, (C:N)H and K2.

273

The slope A is calculated from Eq. (4a) (Table 4), using another parameter (h). Therefore, the parameterization of K2N in region I depends on the three parameters (C:N)H, K2, and h that are now explained. Parameter (C:N)H is the C:N ratio for the biomass of the heterotroph (Table 1). To define parameter K2, we have to consider the situation where the quality of assimilated substrates exactly corresponds to the element composition of the heterotroph (i.e. (C:N:P)A = (C:N:P)H) because, in this case, all net growth efficiencies must be equal (K2C= K2N= K2P) in order to keep (C:N:P)H constant. This particular value of net growth efficiency is K2. Two extreme cases of heterotroph physiological behavior in adapting the carbon and nitrogen net growth efficiencies (K2C and K2N, respectively) to variable substrate quality (C:N)A were envisaged by Touratier et al. (1999a,b) for bacteria and copepods. On the one hand, given the fact that the two conditions (K2N\0, and K2N= K2 when (C:N)A = (C:N)H) must be respected, the minimum value for slope A of the plane relying K2N to (C:N)A is equal to 0. This means that K2N is constant, equal to K2, and thus independent of (C:N)A. The proportion of assimilated nitrogen that is allocated to production and excretion remains constant whatever the C:N values of the growth substrates. On the other hand, the maximum value for slope A is K2:(C:N)H. The proportion of assimilated nitrogen that is allocated to production and excretion is variable and highly sensitive to the C:N values of the growth substrates. In order to simplify the interpretation of the slope value, parameter h is used in Eq. (4a) (Table 4). When h= 1, A= 0, and when h= 0, A= K2:(C:N)H. To keep the (C:N:P)H ratio constant, the theory of stoichiometry predicts that the carbon and phosphorus net growth efficiencies (K2C and K2P) must be related to the nitrogen net growth efficiency (K2N) as shown in Eq. (4b) and Eq. (4d) (Table 4; see Table 1 for the description of the variables involved in the parameterization). Bacteria can take up ammonium from the medium (Kirchman et al., 1990; Sanders and Purdie, 1998) when organic N relative to C in growth

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

274

substrates is low (Lancelot and Billen, 1985). Consequently, bacterial growth could become ammonium (NI) limited. To represent the effect of such a limitation on the net growth efficiencies, Eqs. (4e), (4f), (4g), (4h) and (4i) (Table 4) are used for region II. In this region, a plane still describes the nitrogen net growth efficiency K2N (Eq. (4g), Table 4), but the slope A (Eq. (4e), Table 4) is influenced by the external concentration of ammonium (NI) through a limitation called LNI (Eq. (4i), Table 4). When NI is not limiting (LNI = 1), Eq. (4e) and Eq. (4g) are the same as Eq. (4a) and Eq. (4c). When NI becomes limiting (LNI B 1), the nitrogen net growth efficiency K2N is reduced and because K2C and K2P are functions of K2N (Eq. (4f) and Eq. (4h), Table 4), they are also reduced (see Touratier et al. (1999a) for details). The growth of zooplankton is not influenced by inorganic nitrogen, so LNI is set to 0. This implies that slope A = 0 and K2N=1 (nitrogen excretion EN =0), a parameterization proposed by Touratier et al. (1999b) for

copepods. Bacteria are also known to take up phosphate from the medium (Currie and Kalff, 1984; Brussaard and Riegman, 1998) when organic P relative to C in growth substrates is low (Tezuka, 1990). Thus bacterial growth can become phosphate (PI) limited. To represent the effect of such a limitation on the net growth efficiencies, Eqs. (4j), (4k), (4l), (4m) and (4n) (Table 4) are used for region III. The parameterization proposed for region III is kept as simple as possible and is similar to that used for region II. Inorganic phosphorus is the limiting nutrient here, and the net growth efficiency for phosphorus is computed first. The equation for K2P (Eq. (4m), Table 4) in region III is similar to that of K2N in region II (Eq. (4g), Table 4), but calculation of A (Eq. (4j), Table 4) differs (Eq. (4e), Table 4). To explain the origin of slope A in region III, we first substitute for K2N (Eq. (4c), Table 4) in K2P (Eq. (4d), Table 4) of region I:

Table 3 Equations of the model for thresholds between regions I–V Threshold

Boundary

TK2CCN

Between regions I and V

Between regions III and V

TK2NCN

Between regions I and II

Between regions III and IV

TK2PCP

Between regions I and III

Between regions II and IV

Equations TK2CCN =

iNhK2(C:N)H iC[1+(h−1)K2]

(3a)

TK2CCN =

iNhK2(C:N)HLPI iC[1+(h−1)K2LPI−((C:P)H(1−LPI)/(C:P)A)]

(3b)

TK2NCN =

TK2NCN =

TK2PCP =

TK2PCP =

iN(1−hK2)(C:N)H

(3c)

iC(1−h)K2 iN(1−hK2LPI)(C:N)H iC[(1−h)K2LPI+((C:P)H(1−LPI)/(C:P)A)]

 

(3d)

n

iP(C:P)H (C:N)H iCK2 1+h −1 (C:N)A



 

iC K2LNI

(3e)

n

iP(C:P)H (C:N)H (C:N)H(1−LNI) 1+h −1 + (C:N)A (C:N)A

n

(3f)

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275

Table 4 Equations of the model used to compute the net growth efficiencies K2C, K2N, and K2P for the regions I–IV Variable

Region I Region II (TK2NCN0(C:N)S Region III Region IV (TK2NCN0(C:N)S (TK2CCNB(C:N)SBTK2NCN TK2PCP0(C:P)S) (TK2CCNB(C:N)SBTK2NCN (C:P)SBTK2PCP) (C:P)SBTK2PCP) TK2PCP0(C:P)S)

A

A = (1−h)(K2/(C:N)H)

(4a) A = (1−h)(K2LNI/(C:N)H) (4e)

 (C:N) −1n

A = 1+h

K2LPI (C:P)H K2C

K2N

K2C=

K2N(C:N)H (C:N)A

(4b) K2C =

K2N =A[(C:N)A−(C:N)H]

K2N=

− TK2NCN(iC/iN)] +1

K2P

K2P=

K2N(C:N)H(C:P)A (C:N)A(C:P)H

K2P=

NI not limiting

LPI

PI not limiting

 

K2P= 1+ h

(C:N)H −1 (C:N)A

NI KNI+NI

K2 (C:P)A. (C:P)H

K2C(C:N)A (C:N)H



(4i)

+1 NI not limiting

LPI =

(2)

The phosphorus net growth efficiency (K2P) in region I varies according to the C:N and C:P ratios for assimilated substrates (Eq. (2)), but K2P is described by a straight line when (C:N)A is kept constant. The value of A for K2N in region II (Eq. (4e), Table 4) equals the value of A in region I multiplied by the ammonium limitation (LNI). We apply the same method to compute the phosphorus net growth efficiency (K2P) in region III (Eq. (4j), Table 4): the slope of K2P in region I (i.e. Eq. (2) divided by (C:P)A) is multiplied by the phosphate limitation (LPI; Eq. (4n), Table 4). To keep the (C:N:P)H ratio constant, the theory of stoichiometry predicts that the carbon and nitro-

n

K2C(C:P)A i K2P=A (C:P)A−TK2PCP C K2P= iP (C:P)H

K2N(C:N)H(C:P)A (C:N)A(C:P)H

PI not limiting

n

K2N=

(4p)

(4l)

(4h)

LNI =

K2P(C:N)A(C:P)H (C:N)H(C:P)A

(4g)

(4d)

LNI

(4j)

K2N(C:N)H K2P(C:P)H K2C =min(K2CII,K2CIII) =K2CII K2C = =K2CIII (C:N)A (C:P)A (4o) (4f) (4k)

K2N= A[(C:N)A (4c)

+K2

A not necessary

H

(C:N)A

PI KPI+PI

(4q)

(4m) NI KNI+NI

(4r)

PI KPI+PI

(4s)

LNI =

(4n) LPI =

gen net growth efficiencies (K2C and K2N) must be related to K2P using Eqs. (4k) and (4l) (Table 4). Similar to K2N in region II, K2P for zooplankton in region III will equal 1 because they do not take up phosphate under natural conditions, so that LPI = 0. Both ammonium and phosphates can be taken up by bacteria in region IV, so that either NI or PI (Eqs. (4r) and (4s), Table 4) can limit growth. In this study, region IV is in fact considered as a crossing area between regions II and III where both net growth efficiencies K2N and K2PE 1 (i.e. EN and EP 0 0). Using parameterization of region II, we compute the carbon net growth efficiency (K2CII; Eq. (4f), Table 4) as if inorganic phosphorus was never limiting. In the same way, using equations of region III, we compute the

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

276

carbon net growth efficiency (K2CIII; Eq. (4k), Table 4) as if inorganic nitrogen was never limiting. The minimum of the two carbon net growth efficiencies (K2CII and K2CIII) is then chosen for K2C ( Eq. (4o), Table 4). To keep the (C:N:P)H ratio constant, the theory of stoichiometry predicts that the net growth efficiencies K2N and K2P must be related to carbon net growth efficiency (K2C) using Eqs. (4p) and (4q) (Table 4). The parameterization proposed for region IV is compatible with specific characteristics of zooplankton mentioned for regions II and III (i.e. LNI and LPI =0). The equations for N:P ratios of excretion (Eq. (3)) and release (Eq. (4)) that depend on the net growth efficiencies K2N and K2P can easily be deduced from Eqs. (2a), (2b), (2d) and (2e) (Table 2): N:P excretion ratio:

EN i (1 − K2N) =(N:P)S N , EP iP(1 − K2P) (3)

N:P release ratio:

RN FN +EN = RP FP +EP =(N:P)S

(1 − iNK2N) . (1 −iPK2P)

(4)

2.4. Values of parameters for copepods, cladocerans, and bacteria Experimental data sets of Le Borgne (1982), Urabe (1993), and Goldman et al. (1987b) are used to estimate several model parameters. The element composition of mesozooplankton (200 –500 mm, E 73% of weight at all stations was copepods) and their potential food were determined at 42 stations in the Gulf of Guinea between February 1975 and April 1979 by Le Borgne (1982). The N:P ratios of mesozooplankton regeneration products were also measured. This data set will be used to represent the copepods of our model. Similarly, Urabe (1993) measured the element composition of zooplankton and their food in the Funada-ike Pound (Japan) between April and September 1991. The N:P regeneration ratios of zooplankton were also determined during the 13 sampling dates. Cladocerans (especially Daphnia similis and Daphnia magna) dominated zooplank-

ton biomass at all dates except during September when a calanoid copepod was abundant. Consequently, the data set of Urabe (1993) is used to represent cladocerans in our model. For bacteria, no suitable experimental data exist. However, Goldman et al. (1987b), using a natural assemblage of marine bacteria, measured the net growth efficiencies for C and N when bacteria assimilated growth substrates (often amino acids) of different qualities. Touratier et al. (1999a) used experiment B of Goldman et al. (1987b) to estimate most of the parameters for bacteria. Nine parameters determine the behavior of the model: (C:N)H, (C:P)H, h, iC, iN, iP, K2, LNI, and LPI (Table 1). The C:N and C:P values for the three heterotroph groups ((C:N)H and (C:P)H, Table 1) were obtained from Le Borgne (1982), Urabe (1993), and Goldman et al. (1987b). Parameter h was assumed to be 1 for copepods and cladocerans (Table 1), based on estimates by Touratier et al. (1999b) for the copepods Paracalanus par6us and Acartia tonsa using experimental data of Checkley (1980) and Kiørboe (1989). For bacteria, Touratier et al. (1999a) used experimental data sets of Goldman et al. (1987b; experiments A and B) and Lancelot and Billen (1985) to calculate h values between 0 and 0.399. Experiment B of Goldman et al. (1987b) represents the bacteria in this study, so h= 0.064 (Table 1). The assimilation coefficients for C and N are in the ranges 0.54–0.99 and 0.76–0.99, respectively, for the copepod Calanus hyperboreus (Daly, 1997) and in the ranges 0.68–0.85 and 0.73–0.92, respectively, for Calanus pacificus (Landry et al., 1984). For cladocerans, Hessen et al. (1989) estimated a lower value of 0.5 for the C assimilation coefficient of Daphnia longispina, but no value was found in the literature for their N assimilation coefficient. For P, the range of assimilation coefficients for copepods is 0.4–0.77 (Butler et al., 1970; Corner et al., 1972) and 0.54–0.82 for cladocerans (Peters and Rigler, 1973; Hessen and Andersen, 1990). Compiling all these values for both copepods and cladocerans, gives ranges of 0.5–0.99 (C), 0.76– 0.99 (N), and 0.4–0.82 (P). Three sets of assimilation coefficients (sets A, B, and C; Table 1) will be tested for each type of zooplankton. Set A is an example of proportional assimilation, with all as-

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

similation coefficients equal to 0.8 (Table 1). Sets B and C are examples of differential assimilation, with all assimilation coefficients different. Set B has iC B iN B iP, whereas set C has iC BiP B iN (Table 1). We assume that iN and iP are always EiC and that iP can be either \iN (set B) or B iN (set C). All assumed values for iC, iN, and iP fall in the ranges cited above, except the value of iP in set B (0.9, Table 1). For bacteria, all assimilation coefficients equal unity as discussed previously. The model of Sterner (1990) had a parameter L, defined as the maximum accumulation efficiency. Sterner (1990) estimated values for this parameter from experimental data sets, giving estimates of 0.38 for copepods (Le Borgne, 1982) and 0.37 for cladocerans (Urabe, 1993). Parameter L is similar to our variable K1 (Table 1) which is the value of all gross growth efficiencies (K1 =K1C =K1N = K1P) when (C:N:P)S =(C:N:P)H. Values of 0.37 and 0.38 for K1 were estimated by Touratier et al. (1999b) for copepods P. par6us and A. tonsa using experimental data of Checkley (1980) and Kiørboe (1989), respectively. In this study, we therefore assume that K1 =L. The relationship between gross and net growth efficiencies for N (i.e. K1N= iNK2N) can be used to estimate values of K2 for copepods and cladocerans. As assumed above, h =1 for copepods and cladocerans, which implies that the K2N plane is constant and equal to K2, as stated by Eq. (4a) and Eq. (4c) (Table 4) for region I. Since iN is also constant (Table 1), K1N is represented by a constant plane that must be equal to K1, so that we can write K1=iNK2. Replacing iN by its value (0.8 in all sets, Table 1), and K1 by either 0.38 for copepods or 0.37 for cladocerans, parameter K2 is computed: we obtain 0.4625 for copepods and 0.475 for cladocerans (Table 1). Parameter K2 was directly estimated for bacteria by Touratier et al. (1999a) using experiment B of Goldman et al. (1987b), giving a value of 0.478 (Table 1). The limitations LNI and LPI must be zero for copepods and cladocerans because these heterotrophs cannot assimilate inorganic nutrients. For bacteria, the inorganic nutrient concentrations NI

277

and PI are assumed to equal the half-saturation constant for NI and PI uptakes (KNI and KPI), so that both LNI and LPI = 0.5 (Table 1). 3. Results

3.1. Estimation of the net growth efficiencies for copepods, cladocerans, and bacteria The variability of the net growth efficiencies with different element contents of growth substrates for copepods (Fig. 3) and cladocerans (Fig. 4) are very similar, so they are discussed together. In these examples, only set A (Table 1) is used to generate the results. Because h=1 for zooplankton (Table 1), region II does not exist because the threshold TK2NCN is infinite (Eq. (3c), Table 3), and there is no crossover between regions II and III, so region IV does not exist either (region V is not modeled). The value of K2C increases with decreasing C:N and C:P ratios of the substrate whereas K2N=K2 throughout region I and K2NB K2 in region III. The value of K2P increases with decreasing C:N and increasing C:P ratios of the substrate in region I, and equals one for region III. For bacteria (Fig. 5), all thresholds exist and consequently all regions (I–V) appear on Fig. 5(a). This is explained by the small value of h (Table 1). For bacteria, K2NE 1 in regions II and IV, and K2PE1 in regions III and IV because assimilation of inorganic nutrients occurs (LNI and LPI \ 0). As for zooplankton, K2C for bacteria increases with decreasing C:N and C:P ratios for the substrate. The value of K2N increases mainly with (C:N)S, whereas K2P increases with (C:P)S. Traditionally, the assumed range of K2C for bacteria has been 0.25–0.45, but del Giorgio and Cole (1998) showed that the range is in fact much larger for natural bacteria (B0.05–0.6). The range of K2C simulated with our model (Fig. 5(b)) is even larger ( 0–1), but it remains realistic if we consider that K2C values greater than 0.6 only appear for (C:N)S lower than ca. 4, i.e. C:N ratios for labile DOM that are probably never found in natural systems.

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F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

3.2. Estimation of the N:P ratios for release and excretion by copepods and cladocerans There are different definitions of release and excretion in the literature. In some instances, release is defined as the sum of true excretion, nutrient return from feces and food damaged at feeding (Urabe, 1993; Urabe et al., 1995). This definition is especially applied to experimental work, where it is difficult to identify the mechanisms by which ammonium or SRP have been produced by heterotrophs. Here, we use the term ‘regeneration’ to refer to experimental data sets that are used for comparison with model predictions. The N:P regeneration ratio is symbolized by (N:P)R. The model of Sterner (1990) calculates N:P ratios of release products that are directly comparable to those calculated in this study (Table 5). Fig. 4. As in Fig. 3, but for cladocerans.

Fig. 3. Simulated results of the net growth efficiencies K2C, K2N, and K2P for copepods, using set A for the assimilation coefficients. (a) This panel indicates the position of the regions I, III, and V, delimited by the thresholds TK2CCN and TK2PCP (as in Fig. 2), the N:P atom ratio for the growth substrates (i.e. ratio (N:P)S; dotted lines), and the elemental composition of the copepods (black point). (b) Results for K2C. (c) Results for K2N. (d) Results for K2P.

Fig. 5. Simulated results of the net growth efficiencies K2C, K2N, and K2P for bacteria. (a) This panel indicates the position of the regions I – V, delimited by the thresholds TK2CCN, TK2NCN and TK2PCP (as in Fig. 2), the N:P atom ratio for the growth substrates (i.e. ratio (N:P)S; dotted lines), and the elemental composition of the bacteria (black point). (b) Results for K2C. (c) Results for K2N. (d) Results for K2P.

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

Our equations to calculate the N:P ratios of excretion (Eq. (3)) and release (Eq. (4)) products were simplified by setting parameters h = 1, LNI = 0, and LPI = 0 (Table 1). This effectively excluded regions II and IV from the model for zooplankton, and resulted in simplified equations for the N:P release ratio (Eqs. (5d) and (5e), Table 5) and N:P excretion ratio (Eq. (5f), Table 5). Further, to allow future comparisons with the model of Sterner (1990), parameter K2 was replaced by K1 using the formula K1= iNK2. The threshold TK2PNP (Eq. (5c), Table 5) is similar to threshold TK2PCP, but TK2PNP represents the value of (N:P)S when K2P= 1, whereas TK2PCP corresponds to the value of (C:P)S when K2P =1. From Fig. 3(a) and Fig. 4(a), it is clear that threshold TK2PNP is constant, i.e. independent of (N:P)S (Eq. (5c), Table 5). For region I, the N:P ratios for release (Eq. (5d), Table 5) and excretion (Eq. (5f), Table 5) products can be calculated, but for region III only the N:P ratio for release products can be estimated (Eq. (5e), Table 5). It is not possible to compute the N:P ratio for excretion products in region III because P excretion is zero (K2P= 1, Fig. 3(d) and Fig. 4(d)). Our model for zooplankton requires four parameters: K1 (deduced from K2), (N:P)H, iN, and iP, and we select the same values when using Sterner’s (1990) model for common parameters.

279

A third model, the ‘conceptual model’, is also used here for comparison. This very simple model, defined by EN:EP or RN:RP = (N:P)S, states that the N:P ratio for excretion or release is equal to that of ingested food. This would imply that the type of limitation (N or P) experienced by phytoplankton could not be influenced by zooplankton regeneration of nutrients. The N:P ratio for excretion of copepods (Fig. 6), when computed with our model, becomes infinite when (N:P)S tends towards TK2PNP. This threshold equals 52.1, 58.6, and 45.6 for copepods, using sets A, B, and C, respectively (Fig. 6(a), (c) and (e)). The values of the TK2PNP thresholds for cladocerans (Fig. 7) are smaller and equal 33.5, 37.7, and 29.3, using sets A, B, and C, respectively (Fig. 7(a), (c) and (e)). Because EP is null when (N:P)S E TK2PNP, EN:EP cannot be computed. The calculated N:P ratios for release behave similarly to EN:EP when (N:P)S B TK2PNP, although lower values are generally obtained, especially when (N:P)S is high. When (N:P)S E TK2PNP, RN:RP is defined and described by a straight line (Eq. (5e), Table 5). The position of the curves EN:EP and RN:RP relative to that of the conceptual model will determine the influence of zooplankton regeneration on phytoplankton growth. By zooming in on low values of (N:P)S (panels in the bottom of Figs. 6 and 7), we can see

Table 5 Equations of models for the N:P ratios of release and excretion by zooplankton Model of Sterner (1990) Condition N:P release ratio

Model of the present study Threshold TK2PNP

(N:P)S0(N:P)H RN (N:P)S(1−K1) = RP 1−(K1(N:P)S/(N:P)H)

TK2PNP =

(N:P)S\(N:P)H RN (N:P)S−K1(N:P)H (5a) = RP 1−K1

iP(N:P)H K1

Condition N:P release ratio

(N:P)SBTK2PNP RN (N:P)S(1−K1) = RP 1−(K1(N:P)S/(N:P)H)

N:P excretion ratio

EN (N:P)S(iN−K1) = EP iP−(K1(N:P)S/(N:P)H)

(5b)

(5c)

(5d)

(N:P)SETK2PNP RN (N:P)S−iP(N:P)H = RP 1−iP

(5f) Not possible

(5e)

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F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

Fig. 6. Comparisons of predicted N:P ratios for release and excretion computed with the three models (our model, the model of Sterner (1990) and the conceptual model), to the regeneration N:P ratios measured by Le Borgne (1982) for copepods; (a and b) – using set A, (c and d) – using set B, and (e and f) – using set C. Lower panels b, d and f, where data of Le Borgne (1982) are added, are zooms of upper panels a, c and e, respectively. Threshold TK2PNP only appears in upper panels, whereas (N:P)H ratio and TNP threshold only appear in lower panels. EN:EP – N:P atom ratio for excretion; (N:P)H – N:P atom ratio for copepods; (N:P)R – N:P atom for regeneration (data of Le Borgne (1982)); (N:P)S – N:P atom ratio for growth substrates; RN:RP – N:P atom ratio for release; TK2PNP – value of (N:P)S when K2P= 1; TNP – value of (N:P)S when EN:EP is equal to the conceptual model.

Fig. 7. As in Fig. 6, but for cladocerans and using experimental data of Urabe (1993).

F. Touratier et al. / Ecological Modelling 139 (2001) 265–291

that RN:RP is equal to the value given by the conceptual model only when (N:P)S =(N:P)H. Ratio EN:EP is equal in the conceptual model only for (N:P)S =

(K1+iP −iN)(N:P)H =TNP. K1

(5)

This threshold is called TNP, and it is equal to (N:P)H only when iN =iP as for the example of proportional assimilation (set A; Fig. 6(b) and Fig. 7(b)). In other cases, TNP\ (N:P)H when iN BiP (set B, Fig. 6(d) and Fig. 7(d)), or TNP B(N:P)H when iN \iP (set C, Fig. 6(f) and Fig. 7(f)). Threshold values for TNP equal 24.1, 30.6, and 17.6 for copepods using sets A, B, and C, respectively. For cladocerans, TNP values equal 15.9, 20.1, and 11.7 using sets A, B, and C, respectively. The influence of iN and iP on model results is obvious for both copepods and cladocerans; compared with proportional assimilation (set A), EN:EP is smaller using set B and larger using set C, but we observe the opposite trends for RN:RP when (N:P)S ETK2PNP. Our model results and those of Sterner (1990) for the ratio RN:RP are exactly the same when (N:P)S 0(N:P)H, as stated by Eq. (5a) and Eq. (5d) (Table 5). The models diverge when (N:P)S \(N:P)H, and our model always gives higher RN:RP values (upper panels in Figs. 6 and 7). The output of the conceptual model compares reasonably well with the measured N:P regeneration ratios, but better matches to the experimental data are generally obtained with our model and that of Sterner (1990). For copepods, best results are obtained using our model and set A (Fig. 6(b)), and predicted EN:EP and RN:RP ratios are of similar quality. For cladocerans, set C provides the best estimate (Fig. 7(f)), but only EN:EP approaches the experimental data of Urabe (1993). A common feature of the experimental data sets of Le Borgne (1982) and Urabe (1993) is that all their data fall in the range (N:P)S B(N:P)H, so that it is not possible to compare their data with the model results for (N:P)S \(N:P)H. However, Urabe et al. (1995) measured the element composition of zooplankton and their food, and the

281

N:P regeneration ratio by zooplankton between June and November 1992 in Lake Biwa (Japan; Table 6). Copepods dominated the zooplankton biomass during the sampling period, except in August when cladocerans were abundant. In all months except in September, (N:P)S \ (N:P)H (Table 6), the range of (N:P)H ratios during the sampling period was very large (18.6–35.4), which is mainly due to the interspecific variability of element composition among the zooplankton species (Urabe et al., 1995). Because of the (N:P)H variability, it is preferable to run the models for each sampling date, using a specific set of parameters representative of that date. The values for K1 (Table 6) were indirectly estimated for each date from Urabe et al.’s (1995) Fig. 3 (not shown here), which provides N production and elimination rates. Urabe et al. (1995) define the elimination of an element as the sum of its regeneration and production (i.e. construction of new biomass for the heterotroph), and we assume that their N elimination rate is close to our N assimilation rate, so that K2 is the ratio of N production rate: N elimination rate, and K1 can be calculated as before. Sets A, B, and C (Table 1) are used for the assimilation coefficients. Values for threshold TK2NNP are given for our model (Table 6) because they specify the values of (N:P)S below which it is possible to compute the EN:EP ratio. The (N:P)R, RN:RP, and EN:EP ratios are plotted against (N:P)H (Fig. 8), a representation adopted by Urabe et al. (1995) to analyze their data. The results from Sterner’s (1990) model are generally smaller than those of Urabe et al. (1995), especially when the difference between ratios (N:P)S and (N:P)H is large in August and November (Table 6 and Fig. 8(a)). The conceptual model produces the worst fit (Fig. 8(a)). Slightly better results are obtained with our model for the RN:RP ratio (Fig. 8(b)), but the difference between predictions and observations remains large for August and November. However, when the EN:EP ratio using set C is used, our model results provide a much better match to the experimental data (Fig. 8(c)). The predicted EN:EP ratios, of course, depend on the values of iN and iP that are used; it is important to obtain good estimates of these.

282

Experimental data of Urabe et al. (1995) for Lake Biwa

Model of Sterner (1990)

Model of the present study Set A (iN = 0.8 and iP = 0.8)

a

Month

(N:P)S

Dominant zooplankton

(N:P)H

(N:P)R

K1

June July August September November

40.7 43.6 39.6 21.9 37.9

Copepods Copepods Cladocerans Copepods Copepods

35.4 29.5 18.6 33.9 26.6

41.4 72.1 273.8 26.1 142.1

0.63 0.23 0.30 0.33 0.43

a

Set B (iN = 0.8 and iP = 0.9)

Set C (iN = 0.8 and iP = 0.7)

RN:RP

TK2PNP

RN:RP

EN:EP

TK2PNP

RN:RP

EN:EP

TK2PNP

RN:RP

EN:EP

49.7 47.8 48.6 18.6 46.4

44.9 102.6 49.6 82.1 49.4

54.6 50.8 76.7 18.6 55.7

91.4 54.0 122.7 17.5 74.8

50.5 115.4 55.8 92.4 55.6

54.6 50.8 76.7 18.6 55.7

39.3 44.3 75.7 14.9 48.8

39.3 89.7 43.4 71.9 43.3

53.7 50.8 76.7 18.6 55.7

np 69.0 323.0 21.1 160.5

K1 values were estimated from Fig. 3 (panel for nitrogen) of Urabe et al. (1995); see text for details.

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Table 6 Comparison of model predictions to experimental data of Urabe et al. (1995) (np: not possible)

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3.3. Estimation of the N:P ratio of excretion by bacteria Parameters h, LNI, and LPI for bacteria are very different from those used for zooplankton (Table 1), and it is not possible to simplify the structure of the model as we did for zooplankton to compute the EN:EP ratio (Eq. (5f), Table 5). This means that the full set of equations (Table 4) must be used to compute the net growth efficiencies

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K2C, K2N, and K2P, in order to predict the EN:EP ratio, using Eq. (3). For zooplankton (Table 5), C does not influence the computation of ratios RN:RP and EN:EP, both by our model and that of Sterner (1990); these ratios depend only on N and P. This is not true for bacteria and the bacterial EN:EP ratio must be plotted against (C:N)S and (C:P)S (Fig. 9(a)), and not (N:P)S as for zooplankton. Since excretion EN and/or EP can be positive or negative for bacteria, the EN:EP ratio can have positive or negative infinite values. In region I, EN:EP \ 0 because both EN and EP are \ 0 (Fig. 9(a)). Regions II and III are characterized by EN:EP B 0 because EN B 0 and EP \ 0, and EN \ 0 and EP B 0, respectively. Finally, EN:EP \ 0 in region IV because both EN and EP B 0. To date, no experimental data are available to test the predictions of our model for bacteria, and the theoretical results obtained here for these heterotrophs must therefore be used with caution.

4. Discussion

Fig. 8. Comparisons of predicted N:P ratios for release and excretion computed with the three models to the regeneration N:P ratios measured by Urabe et al. (1995). (a) The conceptual model and the release N:P ratio computed with the model of Sterner (1990) are compared to the regeneration N:P ratio. (b) The release N:P ratios computed with our model, using sets A, B, and C, are compared to the regeneration N:P ratio. (c) The excretion N:P ratios computed with our model, using sets A, B, and C, are compared to the regeneration N:P ratio. EN:EP – N:P atom ratio for excretion; (N:P)H – N:P atom ratio for zooplankton; (N:P)R – N:P atom for regeneration (data of Urabe et al. (1995)); RN:RP – N:P atom ratio for release.

Comparisons of RN:RP ratios to (N:P)R experimental data would be valid only if all fecal materials produced by organisms are mineralized very quickly into dissolved inorganic nutrients (NI and PI). This assumption is unrealistic (Urabe, 1993), because the remineralization of particulate organic matter is a very slow process (Andersen et al., 1986; Tezuka, 1989) in comparison with the excretory activity of heterotrophs. Therefore, RN:RP should be a bad predictor of (N:P)R. The results shown in Fig. 6(b), Fig. 7(f) and especially Fig. 8(c) suggest that EN:EP could provide a better estimate of (N:P)R. However, food damaged at feeding (Lampert, 1978; Urabe, 1993), might counteract this observation because some inorganic nutrients could flow into the medium from broken phytoplankton cells, and thus influence (N:P)R. For low values of (N:P)S (Fig. 6(b) and Fig. 7(f)), the difference between RN:RP and EN:EP ratios is not important, so that the RN:RP ratio could also be considered as a good approximation of (N:P)R, but this does not hold when (N:P)S values are greater than (N:P)H, as shown in

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Fig. 9. Results obtained with our model for bacteria. (a) Simulated excretion N:P ratio. (b) This panel shows the regions where bacteria favor phosphorus (P) or nitrogen (N) limitation of phytoplankton growth. EN:EP – N:P atom ratio for excretion; (N:P)H: N:P atom ratio in bacteria.

Fig. 8(b). In general, the results presented here suggest that EN:EP is probably a better predictor of (N:P)R than RN:RP. For marine copepods, proportional assimilation (set A, Fig. 6(b)) is more realistic than differential assimilation (sets B and C, Fig. 6(d), (f)), when EN:EP is compared to (N:P)R. This agrees well with the experimental work of Reinfelder and Fisher (1991) and Wang et al. (1996) for marine calanoid copepods. For cladocerans, the opposite is true, and the hypothesis of differential assimila-

tion where iN \ iP (set C, Fig. 7(f)) provides the best results, and this is true also when the model is compared to the data of Urabe et al. (1995) (Fig. 8(c)). Variability in food quality, after processing by zooplankton, should be evident in the release products, i.e. excretion and/or fecal materials. When proportional assimilation occurs (Fig. 6(b)), all the variability is found in the excretion products because fecal materials then have an element composition that is similar to that of the food. For differential assimilation (Fig. 7(f), Fig. 8(c)), the variability is partially but predominantly found in the excretion products. Therefore, the present study provides evidence that the hypothesis of adjustable excretion could be an effective tool by which zooplankton regulate the net element assimilation of food. Parameter h is a key parameter in our model. The assumption that h=1 for zooplankton seems justified by the model outputs for copepods and cladocerans (Figs. 6–8). For bacteria, parameter h is always B 1, as shown by Touratier et al. (1999a). The present study does not provide any supplementary information for bacteria because experimental data could not be used. As discussed in Touratier et al. (1999a,b), there is no apparent explanation for variability in h, neither among the bacterial experimental data sets, nor between bacteria and zooplankton. Most parameters of the present model are involved in the computation of thresholds TK2CCN, TK2NCN, and TK2PCP (Table 3). Parameter h is particularly influential because, according to its value, threshold TK2NCN may exist (when hB 1 for bacteria; Fig. 5(a)) or not (when h= 1 for zooplankton; Fig. 3(a) and Fig. 4(a)). Whatever the heterotroph or the value of h, however, there is always a threshold TK2PCP (Fig. 3(a), Fig. 4(a) and Fig. 5(a)). It is, therefore, interesting to review the literature in order to discuss the value and the existence of these thresholds. Thresholds TK2NCN and TK2PCP are close to the so-called ‘threshold elemental ratios’ or TERs found in the literature. Hereafter, our symbolism used for thresholds is maintained, even when the threshold value originates from the literature.

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For cladocerans, an approximate range of 130– 350 for TK2PCP is deduced from Olsen et al. (1986), Urabe and Watanabe (1992), and Sterner (1993). Threshold TK2PCP is known to be dependent on food concentration (Sterner, 1997) and (C:P)H (Urabe and Watanabe, 1992). Our model predicts that TK2PCP also depends on (C:N)S (Fig. 4(a) and Eqs. (3e) and (3f); Table 3). No experimental evidence exists of such a dependence, but it could explain the large range of TK2PCP found in the literature (see above). Threshold TK2PCP computed with our model for cladocerans is always \ 90, and it increases with (C:N)S (Fig. 4(a)). Interestingly, we did not find any estimate of TK2PCP for copepods in the literature. Perhaps, threshold TK2PCP simply does not exist for these organisms, but it is true that all the literature reviewed here for the thresholds is mainly concerned with cladocerans, possibly because cladocerans are often more P limited than freshwater or marine copepods. From the results of our model for copepods (Fig. 3(a)), threshold TK2PCP is always \ 115, and it increases more rapidly with (C:N)S than for cladocerans, showing that cladocerans may be more often P limited than copepods. According to our model, there is no threshold TK2NCN for zooplankton (h =1), and consequently only regions I and III appear on Fig. 3(a) and Fig. 4(a). The experimental results of Checkley (1980) and Kiørboe (1989) for copepods clearly demonstrate that the nitrogen gross growth efficiency (K1N) is constant whatever the food C:N ratio, so that K1N could never reach 1. Assuming proportional or differential assimilation, this also implies that K2N is also a constant B1, so that N excretion is never zero. No experimental evidence was found in the literature showing that N excretion could be zero, or that K2N could reach unity for mesozooplankon. The three models of Anderson (1992), Urabe and Watanabe (1992), and Anderson and Hessen (1995), however, suggest that threshold TK2NCN may exist. Models of Anderson (1992) and Anderson and Hessen (1995) predict that TK2NCN = 10.35 and \10 for copepods, respectively, but the results of their models are not validated by the experimental results of Checkley (1980) and Kiørboe (1989). Urabe and Watanabe (1992) imply the existence of a threshold TK2NCN

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because they hypothesize that zooplankton must utilize N as efficiently as possible to realize the maximum net production for a given concentration of C in the food. This hypothesis implies that K1N must increase with increasing (C:N)S (hB1), so that K1N could reach 1. Based on this hypothesis, the authors developed a model where TK2NCN = (C:N)H:K1C, from which threshold TK2NCN is computed (range= 13.7–29.9), using experimental data of K1C and (C:N)H for two cladocerans species. However, no experimental evidence yet supports this. For bacteria, the existence of thresholds for TK2PCP and TK2NCN has been clearly demonstrated. Tezuka (1989) observed that the bacterial P excretion stopped when (C:P)S \ 60, indicating that TK2PCP was ca. 60. The results of our model show similar values (Fig. 5(a)), although TK2PCP varies with (C:N)S as for zooplankton. Thresholds for TK2NCN measured by Lancelot and Billen (1985) and Goldman et al. (1987b), and predicted with the model of Anderson (1992), are in the range 10–13, i.e. close to the results provided by our model (Fig. 5(a)). This is not surprising since parameters of our model for bacteria were estimated from Goldman et al. (1987b). Among the three models tested in the present study for zooplankton, the conceptual model provides the worst results (Fig. 6(b), Fig. 7(f), and Fig. 8(a)), especially when (N:P)S \ (N:P)H. Since RN:RP predicted by our model and that of Sterner (1990) is probably not representative of (N:P)R, it is not possible to assess the quality of the models for RN:RP ratios (experimental data for ratio RN:RP would be required to compare the model of Sterner (1990) and ours). It is, however, interesting to compare the structure of these models (Eqs. (5a), (5b), (5c), (5d) and (5e), Table 5), because RN:RP predicted by these models is exactly the same if we assume that K1= iP in our model. No experimental evidence suggests such a relationship between K1 and iP. The prediction of our model for EN:EP gives realistic results when compared with the (N:P)R ratios measured in the experimental studies (Le Borgne, 1982; Urabe, 1993; Urabe et al., 1995) for copepods and cladocerans. This is true for both small and large values of (N:P)S, but the iN:iP ratio

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influences the quality of the predictions (Figs. 6– 8). Our model can be considered as an attractive alternative to the model proposed by Sterner (1990), but two additional parameters (iN and iP) must be known to predict both RN:RP and EN:EP with our model. Two models are proposed by Thingstad (1987) to describe the utilization of N, P and organic C substrates by bacteria: a Monod type model considers constant bacterial element composition, whereas a Droop type model assumes a variable element composition in bacterial biomass. Not only the absence of experimental data for bacteria, but also the utilization of radically different assumptions (e.g. C, N and P substrates are assumed to be chemically and physically independent) for model construction make the comparison with our model very difficult. For zooplankton, a model with three currencies (C, N and P) is proposed by Andersen (1997). This model is not compared to the present one because of its resemblance to the model of Sterner (1990). Stoichiometric regeneration of nutrients by heterotrophs could influence phytoplankton growth. Elser and Hassett (1994) studied 36 lakes and 17 marine and estuarine sites, and measured element ratios for seston ((N:P)S) and zooplankton ((N:P)H). For lakes, average values of (N:P)S and (N:P)H are approximately of 39 and 19, respectively (thus (N:P)S \(N:P)H). For marine and estuarine sites, values of 20 and 26 were computed, respectively (thus (N:P)S B(N:P)H), showing large differences between freshwater and marine ecosystems in the element composition of potential prey and predators. Using these values and assuming K1=0.75, Elser and Hassett (1994) predicted RN:RP ratios with the model of Sterner (1990): 100 for lakes, and 20 for marine and estuarine sites. Using the same values for (N:P)S, (N:P)H, and K1, but using set A (i.e. iN =iP =0.8) to run our model, our predictions for RN:RP are 119 for lakes and 11.8 for marine sites. If the EN:EP ratio is computed with our model, it is infinite for lakes (this means that (N:P)S \TK2PNP, i.e. P excretion is null) and 4.4 for marine sites. No major difference appears between models for the RN:RP ratio. However, if we accept that EN:EP is a better predictor of (N:P)R, the difference between lakes

and marine systems is predicted to be infinite using our model, and only of a factor of 5 using Sterner’s (1990) model (Elser and Hassett, 1994). The results obtained here are speculative, especially because parameter K1 may be overestimated. However, it is possible that the influence of stoichiometric regeneration on P limitation in lakes or N limitation in marine or estuarine systems could be much more pronounced than previously believed. A way to assess whether the EN:EP ratio for zooplankton favors N or P limitation in phytoplankton is to compare this ratio to our conceptual model used in Figs. 6–8. Our conceptual model predicts that EN:EP = (N:P)S, so that P or N limitation of phytoplankton is not influenced by excretion, assuming that the (N:P)S ratio reflects the NI:PI ratio of the medium. Consequently, for a certain value of (N:P)S, if the EN:EP ratio predicted by our model is smaller than the conceptual model, N limitation of phytoplankton is favored because relatively less N is excreted than P. Conversely, if the EN:EP ratio is higher than the conceptual model, P limitation of phytoplankton is favored because relatively less P is excreted than N. The value of thresholds TNP, previously computed and represented on Figs. 6 and 7 for zooplankton, are therefore important because they specify the boundaries between favorable N or P limitation (if (N:P)S B TNP, N limitation is favored, and if (N:P)S \ TNP, P limitation is favored). For bacteria, let us assume that results obtained in Fig. 9(a) are valid. By comparing Fig. 9(a) to Fig. 5(a), it is possible to determine the (C:N)S and (C:P)S, or the (N:P)S values for which bacteria favor N or P limitation in phytoplankton, as indicated in Fig. 9(b). In region I, when EN:EP \ (N:P)H, P limitation is favored, and conversely when EN:EP B (N:P)H, N limitation is favored (Fig. 9(b)). In region II, NI is assimilated, and PI is excreted, so that N limitation is favored. In region III, NI is excreted and PI is assimilated by bacteria, resulting in an increase of P limitation. In region IV, when EN:EP \ (N:P)H or EN:EP B (N:P)H, N or P limitation is favored, respectively, because both EN and EP are negative. In fact, Fig. 9(b) shows that (N:P)H for bacteria is similar to

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the threshold TNP for zooplankton, because it specifies the boundary between two areas where N or P limitation is favored. All these considerations allow us to evaluate the role played by the N:P ratio of excretion products of heterotrophs in influencing phytoplankton growth. This is done in Table 7, using only the experimental results of Le Borgne (1982) and Urabe (1993) for which the best values of threshold TNP for copepods and cladocerans are 24.1 (Fig. 6(b)) and 11.7 (Fig. 7(f)), respectively. For bacteria, the (N:P)H ratio is 7.8, computed from Table 1. By comparing TNP thresholds for copepods and cladocerans, and (N:P)H for bacteria, we can write: (N:P)BAC BTNPCLA BTNPCOP. Table 7 shows that for very low (N:P)S values, all heterotrophs cause N limitation for phytoplankton, and when (N:P)S is very high all heterotrophs cause P limitation. For intermediate values of (N:P)S, heterotrophs can cause N or P limitation. For the overall tendency, we imagine a hypothetical mixing of bacteria, copepods, and cladocerans that ingest or assimilate growth substrates of the same quality. For intermediate values of (N:P)S, it is only possible to conclude what type of limitation occurs if one knows the N and P growth substrate concentrations, the biomass of heterotrophs and their specific rates of excretion. Table 7 shows also that N deficient growth substrates (i.e. low (N:P)S) reinforce N limitation

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of phytoplankton through the excretory activity of heterotrophs, and the converse is true for P, so that phytoplankton may become more and more N or P limited, especially during periods when primary production is based on regenerated nutrients. The overall interpretation of our results, given in Table 7, must be considered very cautiously. Decreasing concentrations of growth substrates, for example, is known to reduce the influence of their quality on heterotrophic processes (Sterner and Robinson, 1994; Mu¨ ller-Navarra, 1995; Rothhaupt, 1995). At low substrate concentrations, therefore, the qualitative influences of excretion on phytoplankton growth are thought to be limited. Furthermore, bacteria and zooplankton do not have access to the same pool of growth substrates. Bacteria assimilate DOM and inorganic nutrients, whereas zooplankton ingest particulate organic matter. Both the quality and the quantity of these pools may be radically different, but all can occur in the same ecosystem. All these arguments lead to the picture of an ecosystem where the influence of excretion by heterotrophs is very complex. This complexity increases if we consider that bacteria are excluded from the hypothesis of strict homeostasis, as discussed above. In Table 7, only the feedback of excretion to phytoplankton growth is taken into account, but excretion acts directly or indirectly on almost all constituents of an ecosystem.

Table 7 Influence of growth substrate quality and excretion by heterotrophs on the element that limits phytoplankton growtha

a N – the heterotroph favors nitrogen limitation of phytoplankton growth; P – the heterotroph favors phosphorus limitation of phytoplankton growth. The overall tendency refers to a hypothetical mixing of the three types of organisms (copepods, cladocerans, and bacteria).

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No experimental data could be used in the present study to estimate the parameters of our model for microzooplankton (2– 200 mm). It is generally thought that microzooplankton are responsible for most of the NI and PI excreted in marine systems (Wheeler and Kirchman, 1986; Caron and Goldman, 1990; Glibert et al., 1992). Bacteria have potentially faster specific excretion rates than microzooplankton, because of their small size (Moloney and Field, 1991). However, bacteria are believed to be often N or P limited (Williams, 1986), so that net assimilation of NI and/or PI by bacteria should be the rule for many aquatic ecosystems. The specific rate of excretion is generally faster for small microzooplankton than for larger mesozooplankton (Moloney and Field, 1991). Goldman et al. (1987a) showed that N regeneration efficiency of the microflagellate Paraphysomonas imperforata decreases with increasing prey C:N ratio, suggesting that K1N increases with (C:N)S, and that parameter h could be B1. If true, microzooplankton could have excretory behavior intermediate between those of bacteria and mesozooplankton. Nakano (1994) found no significant relationship between the N:P ratio of nutrients released by microflagellates and that of their prey (bacteria). Taking these two studies together, this means that microflagellates could be characterized by h B1, and that no significant relationship exists between EN:EP and (N:P)S. This is exactly what we found for bacteria with our model; the C content of growth substrates influences bacterial N and P excretion. Bacteria tended to cause P limitation for most values of (N:P)S, whereas copepods tended towards N limitation (Table 7). This is supported by Hassett et al. (1997) and Skjøldal (1993), who believe that phytoplankton– zooplankton dominated environments may accentuate N limitation, whereas microbial web dominated environments may cause P limitation. Sterner et al. (1997) propose that the element composition of seston could influence the importance of microbial versus grazing processes. Elser et al. (1996) hypothesize that body size could be positively correlated with the N:P ratio of heterotrophic biomass. From the present study, we know that the N:P ratio for bacteria is lower than that for cladocerans or

copepods, supporting this hypothesis. Bacteria dominate biomass in oligotrophic waters (Fuhrman et al., 1989; Azam and Smith, 1991). Thus, oligotrophic conditions could indirectly favor P limitation above N limitation, by selecting the smallest heterotrophs. Many oligotrophic regions are often P limited, e.g. the Mediterranean Sea (Thingstad and Rassoulzadegan, 1995; Thingstad et al., 1998) and the Sargasso Sea (Cotner et al., 1997; Rivkin and Anderson, 1997). Conversely, eutrophic or mesotrophic conditions should indirectly favor N limitation, as occurs in the Benguela upwelling (Andrews and Hutchings, 1980), the central North Sea (Brockmann et al., 1990), and the Greenland Sea (Kattner and Bude´ us, 1997). When discussing the nature of the nutrient that limits a particular ecosystem, however, it is clear that numerous other factors are involved (e.g. geochemical factors, circulation, etc.) Because of their potential impacts, however, factors like the stoichiometry of the regenerated nutrients produced by heterotrophs, the organism size and the trophic status of an ecosystem (oligotrophic versus eutrophic conditions) should be further explored.

Acknowledgements This work was supported by a grant from the Benguela Ecology Programme (BEP), funded by the South African Foundation for Research Development and the Department of Environmental Affairs and Tourism, and a grant from the University of Cape Town. We are very grateful to J. Urabe who kindly provided his data sets from the Funada-ike Pound and Lake Biwa of Japan. We also thank the two anonymous referees and the Editor-in-Chief S.E. Jørgensen for their most useful comments and suggestions.

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