Testing zooplankton secondary production models against

Daphnia magna growth

May Gomez1,*, Ico Martınez1, Ismael Mayo1‡, Jose Miguel Morales1, Angelo Santana2,

and Ted T. Packard1

1Biological Oceanography Group, Institute of Oceanography and Global Change, Universidad de Las Palmas de Gran Canaria, Canary Islands,

Spain2Department of Mathematics, Universidad de Las Palmas de Gran Canaria, Canary Islands, Spain

*Corresponding Author: tel: +34 928 452905; fax: +34 928 452922; e-mail: mgomez@dbio.ulpgc.es‡Present address: Institute of Marine Science, CMIMA-CSIC, Barcelona, Spain.

Gomez, M., Martınez, I., Mayo, I., Morales, J. M., Santana, A., and Packard, T. T. Testing zooplankton secondary production models against

Daphnia magna growth. – ICES Journal of Marine Science, doi:10.1093/icesjms/fsr193.

Received 31 March 2011; accepted 15 November 2011.

Modelling secondary production rates in the zooplankton is essential for population ecology studies, but assessing these rates is dif-

ficult and rarely done. Here, five secondary production models are tested by measuring Daphnia magna growth. To provide a range of

growth rates, Daphnia were cultured under three different nutrition regimes (yeast, cornflour, and phytoplankton). Length and

biomass were monitored daily in three simple time-course experiments to provide the growth rates, which ranged from 0.11 to

0.30 d–1 with secondary production rates of 350–643 mg dry mass d21. Secondary production was predicted best by the freshwater

crustacean-based model of Stockwell and Johannsson (1997). Marine copepod-based marine models were totally unsuitable.

Keywords: culture, Daphnia, growth, models, secondary production, zooplankton.

IntroductionSecondary production in zooplankton is growth, the rate of

biomass increase per time. It reflects the net balance between meta-

bolic gains in biomass and the integral of all metabolic losses. One

of the main factors controlling this rate is the quality of food

(Vijverberg, 1989; Martınez et al., 2010), when zooplankton are

fed different types. This leads to changes in growth, longevity,

clutch size, and other life-history characteristics (Schwartz and

Ballinger, 1980; Sterner et al., 1993). Therefore, altering food

quality could serve as an experimental strategy to produce

changes in secondary production in zooplankton cultures. Here,

with the intent of understanding secondary production in

natural zooplankton populations, we use this research strategy to

measure growth and secondary production in cultures of the cla-

doceran Daphnia magna grown on different types of food.

Cladocerans are passive filter-feeders, so cannot select their

food as can calanoid copepods. Daphnia spp. can discriminate

between particles of different length, but not between latex

beads and algal cells (De Mott, 1986). Many investigators

have searched for optimal diets to grow Daphnia fast. Giani

(1991) showed that both quantity and quality of food influence

growth in Daphnia. He found that an optimal diet contained

saturating concentrations of the algae Chlamydomonas sp. and

Scenedesmus sp. Price et al. (1990) found that yeast together

with the alga Selenastrum capricornutum yielded optimal

growth (dry-mass increase) in Daphnia pulex, suggesting that

yeast stimulates the bacterial growth on which D. pulex will

feed. Later, Martınez-Jeronimo et al. (1994) observed that, in

D. magna, the most suitable diet for faster growth was one

containing the alga Scenedesmus incrassatus. In addition,

Munoz-Mejıa and Martınez-Jeronimo (2007) found high

growth rates (m ¼ 0.49–0.58 d21) in three cladocerans (D.

pulex, Ceriodaphnia dubia, and Simocephalus mixtus) growing

on three microalgal species (Ankistrodesmus sp., S. capricornu-

tum, and Scenedesmus incrassatulus). Here, we used three differ-

ent foods, a phytoplankton mixture (Ankistrodesmus sp. and

Scenedesmus sp.), brewers’ yeast, and cornflour. The first two

were high-quality foods (Price et al., 1990; Munoz-Mejıa and

Martınez-Jeronimo, 2007), and the third was a low-quality

food.

Modelling is an alternative methodology for estimating growth

and secondary production in zooplankton. Huntley and Lopez

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(1992) found that they could model secondary production from

temperature alone, stirring great interest, and as a result, it has

been used, tested, and compared with other secondary production

and growth models for many parts of the world. Roman et al.

(2000), using data from Arabian Sea mesozooplankton, tested

not only the Huntley and Lopez (1992) model, but also the

Hirst and Sheader (1997) and the Huntley and Boyd (1984)

models, in which growth is a function of both temperature and

body mass, and showed that the Hirst and Sheader (1997)

model gave the higher growth rates. Peterson et al. (2002),

trying to estimate secondary production from the Hirst and

Lampitt (1998) model and measurements of coastal Oregon cala-

noid copepods, came to a similar conclusion that the Hirst and

Sheader (1997) model gave the best estimate. Later Satapoomin

et al. (2004) tested the Huntley and Lopez (1992) model on

Andaman Sea copepods and obtained values that were 18 times

higher than the direct measurements of secondary production.

Leandro et al. (2007) compared the Huntley and Lopez (1992)

and Hirst and Bunker (2003) models on data from the Ria de

Aveiro, a shallow temperate estuary on the Portuguese coast, and

found that secondary production calculated with the Huntley

and Lopez (1992) model was 22% higher than calculations based

on the Hirst and Bunker (2003) model. Miyashita et al. (2009),

working with copepods in subtropical coastal seawater, confirmed

that the Huntley and Lopez (1992) model overestimates secondary

production when compared with the Hirst and Lampitt (1998)

model. It is now clear that in food-limited areas, the Huntley

and Lopez (1992) model overestimates production because it

assumes that, at a given temperature, every organism in a

sample has the same growth rate (Miyashita et al., 2009).

Therefore, its use seems appropriate only in eutrophic regions

such as estuaries and upwelling areas where plankton growth is

not food-limited (Peterson et al., 2002).

The present research is based on laboratory experiments, but

with the intent of understanding secondary production in

natural populations. We assume that a freshwater zooplankter

should serve to test secondary production models just as well as

a marine zooplankter. We take a reductionist approach, measuring

growth in cultures of the cladoceran D. magna, and calculating

secondary production at discrete time-intervals directly from the

product of population biomass and population growth rate

(Benke, 2010; Lehman, 1988). We deliberately chose to investigate

secondary production in the laboratory rather than in the field

despite Redfield’s arguments to the contrary (Redfield, 1958),

because only by combining careful laboratory experimental inves-

tigation with field research can ecology progress. This view is in

keeping with the recommendations of Gomez et al. (2001), who

argued that laboratory investigations with model zooplankters

are needed to calibrate and clarify field measurements of second-

ary production. By a straightforward strategy, the impact of food

quality on growth is characterized, secondary production is calcu-

lated, and the results compared with three marine copepod-based

secondary production models and two freshwater secondary pro-

duction models that can be applied to either copepods or cladocer-

ans. Only the freshwater crustacean-based models served to predict

secondary production in Daphnia.

Material and methodsDaphnia magna was chosen as an experimental model because it is

inexpensive, easy to manage in culture, and its size-spectrum facil-

itates experimentation. Moreover, Daphnia have a short life cycle

and high fecundity, develop directly without larval stages, and

are ubiquitous in freshwater ecosystems. Consequently, the

results from experiments with Daphnia can be applied broadly.

The experiments were designed to test the efficiency of five dif-

ferent secondary production models. To do this, three different

cohort time-course experiments were conducted with D. magna

fed three foods at a daily frequency. The three types of food help

challenge the models because they provide variability in growth

rates and secondary production.

Daphnia were maintained at room temperature (18–218C), in

20-l, well-aerated, aged tapwater aquaria under the natural photo-

period of a north-facing window. Neonates were obtained at three

different times from resting eggs harvested from this mother

culture, according to Peters (1987), then transferred to 5-l culture

tanks. Therefore, each experiment was carried out independently

of each other, at three different times. We aimed to start with a

minimumof 100 resting eggs, but in some experiments (using corn-

flour), we were able to provide more. For this reason, the time-

course for the cornflour experiment is the longest. In the first ex-

periment, the young were fed phytoplankton, in the second,

yeast, and in the third, cornflour, all at saturation levels. Food

types were chosen to provide a simulated natural food of mixed

green algae (Scenedesmus sp. and Ankistrodesmus sp.), a

nutrient-rich artificial diet of Saccharomyces cerevisiae (brewers’

yeast at 7.5 mg ml21), and a nutrient-poor artificial diet of corn-

flour (a saturated solution). These diverse foods should yield differ-

ent growth characteristics. Six Daphnia were sampled daily,

measured under a stereoscopic microscope, then frozen at 2208C

for subsequent dry-mass determination. All measurements were

based on these six replicates. Length (L) from the rostrum (next

to the eye) to the start of the anal spine was measured optically

under themicroscope. Samples for drymass were placed in alumin-

iumcapsules (previouslyweighed andnumbered), dried at 608C for

24 h, and weighed on a Sartorius balance with an ultramicroscale

(+0.1 mg), according to Lovegrove (1966).

Growth data were analysed by a linear mixed-effects model

(Pinheiro and Bates, 2000; Zuur et al., 2009) in which the response

variable was length (or weight) and the explanatory variables were

time and type of food, with individuals considered as random

effects. Initially, a linear fixed-effects model could be adjusted to

the data:

Lij = b0i + b1i dayj + 1ij, 1ij ≈ N(0,s1), (1)

where j is time and i refers to the three types of food (cornflour 1,

phytoplankton 2, and brewers’ yeast 3). Therefore, a linear rela-

tionship results with the intercept and the slope varying with i.

The effect of individuals can be modelled as a random intercept,

so the complete model for the length of individual k with food

type i on day j is the linear mixed effects model:

Lijk = b0i + b1ik + b1i dayj + 1ij,

1ij ≈ N(0,s1),b0ik ≈ N(0,sb).(2)

In thismodel,b1i represents the daily growth ratewith food i,b0i the

mean initial size for Daphnia fed with food type i, and b0ik the de-

parture from the initial size of Daphnia, k. b0ik is considered as a

normal randomvariablewithmean zero and standard deviations1.

Both linear effects models were run using the library nlme

program in the R statistical package (Pinheiro et al., 2009;

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R Development Core Team, 2010). The standard error of sb was so

low (sb ¼ 1.03 × 1026) that when we compared linear

mixed-effects model (2) with linear fixed effects model (1) using

the likelihood ratio test (Zuur et al., 2009), the difference was

not significant (p ¼ 0.3849). Therefore, following the protocol

specified by Zuur et al. (2009), the most appropriate was the

linear fixed-effects model (1).

The model coefficients were reparametrized to facilitate com-

parison between the different food types. In this manner, if we

call g0 the initial mean length and g1 the growth rate when fed

with cornflour, the terms of model (1) are reparametrized as

follows:

b01 = g0, b02 = g0 + g02, b03 = g0 + g03,b11 = g1, b12 = g1 + g12, b13 = g1 + g13,

so the terms gij represent the differences in mean initial length and

slope, respectively, for the diets based on phytoplankton and

brewers’ yeast compared with the diet based on cornflour.

Based on what was learned from the comparison between the

two length-based models, we used the linear fixed-effects model

to analyse the dry-mass time-courses, using the terms 4ij to rep-

resent the differences in mean initial dry mass and slope, respect-

ively, for the diets based on phytoplankton and brewers’ yeast with

respect to the diet based on cornflour.

Growth rates were calculated from an exponential function

fitted to the data:

Wt = W0egt, (3)

whereWt is the dry mass at time t, W0 the dry mass at time zero, t

the time (d), and g the daily (24 h) growth rate (Escribano and

McLaren, 1992). The global rate of growth (gg) was calculated as

the slope of the regression line in a plot of the log growth vs.

time (Kimmerer and McKinnon, 1987).

To calculate secondary production, five models were used based

on three marine field studies, (i)–(iii), and two freshwater field

studies, (iv) and (v).

(i) Huntley and Lopez (1992):g = 0.0445 e0.111 t , where t is the

temperature (8C). This model assumes that the instantan-

eous growth rate is independent of body length and

species, that it is not food-limited, and that it depends

solely on habitat temperature.

(ii) Hirst and Sheader (1997): g = (0.0732× 100.0246 t)/W0.2962.

Here, the intrinsic growth rate (g) is a function of both tem-

perature (t) and body mass (W).

(iii) Hirst and Lampitt (1998): g = (0.0723× 100.0208 t)/W0.3221.

This model also describes the growth rate as a function of

temperature (t) and body mass (W).

(iv) Stockwell and Johannsson (1997): P = 10(a log10 M+b)×

F ×M × N. This is an empirical freshwater model built on

the biomass of four planktonic crustaceans, knowledge of

the Shuter and Ing (1997) model, and water temperature. F

is the correction factor for back-calculated log10-transformed

data regressions (1.12 for cladocerans with temperature

.108C), a and b are the log10–log10 regression coefficients,

M the mean individual dry mass (mg), and N the abundance.

(v) Shuter and Ing (1997): P = 10(a+bt) × B. This equation

results from several analyses in lakes that demonstrated a

strong and consistent correlation between water temperature

(t) and dry-mass-specific production (P). The values of a

and b (specific regression coefficients) for cladocerans are

21.725 and 0.044, respectively, and B is the biomass. For

copepods, these coefficients were different.

In the first three models, secondary production is calculated as

the product of biomass (B) and the modelled growth rate (g). The

final two models generate secondary production directly. For

growth, these estimates of secondary production need to be

divided by biomass.

ResultsGrowth, as length (L) and dry mass (W), are shown in Figure 1.

Length plateaued after 10 d, whereas dry mass continued to in-

crease linearly for at least 12 d. The most rapid length increase

(Figure 1a), greatest length (Table 1), and the best relationship

between length and dry mass (Table 2) were obtained with

Daphnia fed on brewers’ yeast, but the maximum dry mass

(Table 1) was with the phytoplankton mix. This is partially

because the biomass per individual of the Daphnia cohort used

to start the phytoplankton-based culture was greater on the first

day than the cohort used to start the yeast-based culture

(Figure 1b). Moreover, the length per individual of the Daphnia

cohort used to start the phytoplankton-based culture was less on

the first day than the cohort used to start the yeast-based culture

(Figure 1a). This phenotypic plasticity in neonates is well

known. Its cause in Daphnia can be traced to variability in the

age, population density, and nutritional state of the mothers

(Sterner and Robinson, 1994; Boersma, 1997; Ban et al., 2009;

Gorbi et al., 2011). This in turn helps achieve the objectives by pro-

viding variability in growth and secondary production. Cornflour,

as expected, was only half as good as the other two foods.

Table 3 presents the analysis of both length and dry mass time-

courses. The estimate of the residual standard error for the length-

based time-courses was �s1 = 0.104, with an adjusted r2 of 0.962.

The F-statistic for testing the significance of the model gave a

value of 186 on 4 and 25 degrees of freedom (p, 2 × 10216).

The adjusted r2 measures the variance explained by model 1. As

shown, this linear fixed-effects model explains 96.23% of the vari-

ability in the length increases of Daphnia. If time is considered as

the only explanatory variable (L ¼ b0 + b1day + e), the adjusted

r2 is 0.6399. Consequently, time alone explains just 63.99% of the

variance in length, and the difference between both values

(32.24%) can be considered as the variability explained by the diet.

Initially, the Daphnia fed on cornflour and on phytoplankton

had the same length, on average. However, the growth rate in

length on cornflour was g1¼ 0.0875, whereas that on phytoplank-

ton was �g1 + �g12 = 0.0875+ 0.02158 = 0.1091; the difference

was significant (p ¼ 0.003048). This indicates that, despite the

similar starting conditions, the Daphnia eating phytoplankton

grew faster (0.02158 mm d21). When the growth of the cornflour-

fedDaphniawas compared with that of those eating brewers’ yeast,

theDaphnia fed on yeast were initially already larger than those fed

on cornflour. The estimated difference was g01¼ 0.2613 mm

larger on average, and the difference was significant (p ¼

0.00657). However, the larger final biomass for the Daphnia

grown on yeast was not only attributable to the larger initial start-

ing size, but more importantly to the faster daily growth rate. The

difference in the growth rate between the Daphnia grown on corn-

flour and those grown on yeast was 0.05723, significant at a level of

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p ¼ 5.847 × 1025. In other words, for every passing day, Daphnia

fed on yeast grew 0.05723 mm more than those fed on cornflour.

In that way, the Daphnia fed on yeast ultimately were larger not

only because they were bigger at the start, but also because they

grew significantly faster each day of the experiment.

The results from the linear fixed-effects model (Table 3) using

the dry-mass data reveal that all terms in the model were

significant. This means that dry-mass-based growth in Daphnia

fed on cornflour is slower than in Daphnia fed on either phyto-

plankton or brewers’ yeast. The mean dry mass of the Daphnia

fed on cornflour was initially significantly higher than that of

those fed on the other two foods. However, after 12 d, the dry

mass was less than in the other two groups. Further, the slope of

the line in Figure 1b for the cornflour-fed Daphniawas significant-

ly lower too. With respect to the yeast and phytoplankton experi-

ments, the dry-mass increments with time were similar, and the

Daphnia fed phytoplankton ultimately weighed more than those

that consumed yeast because they initially weighed more.

The estimate of the residual standard error was �s1 = 0.001139.

The F-statistic for testing the significance of the model yielded a

value of 5309 on 5 and 24 degrees of freedom (p, 2.2 ×

10216). That model accounted for 99.9% of the variability in the

dry-mass increases in Daphnia; if time was considered as

the only explanatory variable (dry mass ¼ b0 + b1 day + 1), the

adjusted r2 was 0.7783, so time explained 77.83% of the variance

in dry mass. The difference between both values can then be con-

sidered as the variability explained by diet.

Specific daily growth rates (d21) decreased continuously in the

first 12 d of the experiment, with the steepest decline in Daphnia

grown on brewers’ yeast (Figure 2a). As expected, however, the

fastest global growth rate (d21) during the first 12 d was in the

same yeast-based culture (0.30 d21; Table 2), and the slowest

global growth rates were in Daphnia grown on cornflour

(0.11 d21; Table 2).

On comparing the measured growth rates with those calculated

using the five models (Table 4), there was good agreement with the

Stockwell and Johannsson (1997) model, moderate agreement

with the Shuter and Ing (1997) model, but no agreement with

the others (Figure 3, Table 4). Models (iv) and (v) were based

on freshwater crustaceans (including cladocerans), whereas

models (i)–(iii) were based on marine copepods. In analysing sec-

ondary production rates (Table 4), we found that for Daphnia

grown on phytoplankton, the mean was 643 mg dry mass d21,

on yeast it was 452 mg dry mass d21, and on cornflour, it was

350 mg dry mass d21. Using the best secondary production

model (Stockwell and Johannsson, 1997), secondary production

was calculated to be 719 mg dry mass d21 for Daphnia fed on

phytoplankton, 502 mg dry mass d21 for Daphnia fed on yeast,

and 386 mg dry mass d21 for Daphnia fed on cornflour

(Table 4). The regression equations relating the secondary produc-

tion measurements to the calculated secondary production by the

five models are listed in Table 5. It is clear that the best fit between

the predicted and measured secondary production was in testing

the Stockwell and Johannsson (1997) model when Daphnia were

fed on the phytoplankton mixture and separately on yeast. The

slopes of the relationships between the measured and the modelled

rates lay close and nearly parallel to the 1:1 line (Figure 3). In the

phytoplankton-fed Daphnia, the slope was 0.88, and in the

Figure 1. (a) Three independent growth experiments showing D.magna length increases as functions of time in cultures grown onthree different types of food. Growth equations by food type arephytoplankton, y ¼ 2.082 x/(7.786 + x), r2¼ 0.971; yeast, y ¼3.032 x/(6.653 + x), r2¼ 0.950; cornflour, y ¼ 1.724 x/(6.650 + x),r2¼ 0.904. (b) The same three experiments as in (a), but with growthmeasured by increases in dry mass. Growth equations by the foodtype are phytoplankton, y ¼ 12.09 x + 2.46, r2¼ 0.962; yeast, y ¼10.62 x2 23.76, r2¼ 0.987; cornflour, y ¼ 6.25 x + 20.26, r2¼ 0.919.The key identifies the food type, and the period of the experimentsdepended on the number of Daphnia available. Secondaryproduction, global growth rate, and daily growth rate calculationswere based on the first 12 d of the dataset (Figure 2, Table 2).

Table 1. Length, dry mass, and age of D. magna in different culture systems.

Food Maximum length Lmax (mm) Age at Lmax (d) Maximum dry mass Wmax (mg) Age at Wmax (d) Temperature (88888C)

Phytoplankton mixture 1.42+ 0.19 18 153 13 20.9+ 1.1

Yeast 2.01+ 0.14 17 100 12 20.1+ 0.5

Cornflour 1.38+ 0.07 19 150 19 19.2+ 1.2

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yeast-fed Daphnia, it was 1.09. Testing the same model with

cornflour-fed Daphnia yielded a prediction that was too high

because the growth rates were half or less the growth rates in the

other two cultures (Tables 2 and 5, Figure 2), and because

the biomass remained about the same (Table 1). Consequently,

the calculated secondary production was low, and when these

values were plotted against the modelled secondary production,

the resulting slope was steep (2.31; Table 5).

DiscussionThe importance of the role of zooplankton growth and production

in aquatic ecosystems is clear (Pace et al., 2004; Ware and

Thomson, 2005; Johnson et al., 2010), as is the role of Daphnia

in freshwater zooplankton production (McCauley et al., 2008;

Harmon et al., 2009). In the experiments here, Daphnia growth

rates on both yeast and mixed phytoplankton were fast; on corn-

flour, they were low, as expected (Tables 1 and 2). In light of

this, we support mixing Scenedesmus sp. at saturating concentra-

tions with yeast at 7.5 mg ml21 to attain optimum Daphnia

growth, as practiced by Price et al. (1990), Martınez-Jeronimo

et al. (1994), and Main et al. (1997). Here, growing Daphnia sep-

arately on saturating levels of phytoplankton and yeast, the global

growth rates obtained in our experiments (0.2–0.3 d21; Table 2)

are close to the range found by Giani (1991) for D. hyalina and

D. galeata (0.11–0.22 d21), and within the range of that found

by Ojala et al. (1995) for D. longispina cultures growing in

anoxic muds with no food limitation (0.16–0.36 d21), especially

after correcting for temperature differences.

However, Dawidowicz and Loose (1992) found faster growth

rates (0.57 d21) in D. magna growing in a culture containing the

same algae used here, but at a slightly higher temperature

(228C). The temperature difference is not enough to explain the

difference. Vijverberg (1989) and Giani (1991) showed that

growth rates are influenced by food quality, and this is consistent

with the observations and statistical analysis here (Table 3); the

values listed in Tables 2 and 4 here are in the range observed by

the earlier authors.

The initial premise in testing the five secondary production

models (Tables 4 and 5) was that if the models were universally ap-

plicable to planktonic crustaceans, then either a freshwater or a

marine crustacean could serve as a test organism. Ease and eco-

nomics dictated that Daphnia be used. The results argue that the

marine models are not universal; they are probably specific for

copepods. In the study here, the best models for both growth

and secondary production were those for freshwater zooplankton

constructed by Stockwell and Johannsson (1997) and Shuter and

Ing (1997); of the two, Stockwell and Johannsson’s (1997)

model performed best, though both took into account tempera-

ture and biomass. The Huntley and Lopez (1992)

Table 2. Regression analysis between length and dry mass, whereW is the dry mass (mg) and L the length (mm).

Food Equation r2

Global growthrate (d21) r

2

Phytoplankton mixture log W ¼ 1.39 log L

+ 4.51 (11)

0.92* 0.20+ 0.02 (10) 0.89

Yeast log W ¼ 2.68 log L

+ 2.67 (11)

0.98* 0.30+ 0.04 (10) 0.89

Cornflour log W ¼ 1.02 log L

+ 4.46 (17)

0.93* 0.11+ 0.01 (10) 0.97

Global growth rates of the culture systems are calculated as the slope of theregression line in a plot of log growth against time (Kimmerer andMcKinnon, 1987). The number of measurements is given inparenthesis.*p, 0.001.

Table 3. Analysis by the linear fixed effects model.

Parameter Estimate s.e. t p-value

G0 0.1926 0.04582 4.204 0.0002926

g02 20.03061 0.09333 20.3279 0.7458

g03 0.2613 0.08812 2.965 0.00657

g1 0.0875 0.006795 12.88 1.560 × 10212

g12 0.02158 0.006578 3.281 0.003048

g13 0.05723 0.01186 4.826 5.847 × 10205

40 0.02054 0.0007174 28.63 ,2 × 10216

402 0.00623 0.00009544 65.22 ,2 × 10216

403 20.01779 0.001024 217.36 4.33 × 10215

401 20.04163 0.001106 237.65 ,2 × 10216

412 0.005836 0.0001469 39.73 ,2 × 10216

413 0.004103 0.0001446 28.37 ,2 × 10216

g0 is the initial mean length and g1 the growth rate (in length), and 40 theinitial mean dry mass and 41 the growth rate (in dry mass), both when fedcornflour. The terms gij (length) and 4ij (dry mass) represent thedifferences in mean initial length and dry mass and the slope, respectively,for the diets based on phytoplankton and yeast with respect to the dietbased on cornflour.

Figure 2. (a) Decrease in the daily growth rate, g, as calculated fromthe three growth equations of dry mass ¼ f (time), as given in thecaption of Figure 1b and from Equation (3), for D. magna on threetypes of food. (b) The secondary production trend in D. magnagrown on three different foods, calculated from the daily growthrates shown in (a) and the biomass functions of Figure 1b.

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marine-copepod-based model overestimated secondary produc-

tion, but does not take into account individual biomass, just tem-

perature. Moreover, it assumes that food is not limiting. The Hirst

and Sheader (1997) and Hirst and Lampitt (1998)

marine-copepod-based models underestimated secondary

production.

The success of the Stockwell and Johannsson (1997) model in

predicting secondary production in the Daphnia cultures here is

most likely attributable to the fact that the model was developed

for freshwater zooplankton populations in which cladocerans

play a large role. Using the coefficients for cladocerans, that

model worked well with the Daphnia data here to predict second-

ary production. When Stockwell and Johannsson (1997) tested

their model on cladocerans and copepods, it worked best for

cladocerans, by a large margin. Although the coefficients in the

model were changed for copepods, the secondary production esti-

mates for copepods were higher, by orders of magnitude. It would

be revealing, though, to test the models on other freshwater zoo-

plankton. The failure of the Hirst and Sheader (1997) and Hirst

and Lampitt (1998) models here is most likely explained by the

fact that they were based on marine copepod growth and could

not accommodate the cladoceran data. It would be valuable, there-

fore, to see how well they would predict secondary production in

euphausiids or mysids. Note that none of the models tested have

any mathematical means of incorporating salinity; there is no sal-

inity term, so the models cannot differentiate between marine or

freshwater applications. They are simple constructs based solely

on biomass and/or temperature. As a consequence, their

Table 4. The daily growth rate (d21) and secondary production (mg dry mass d21, in parenthesis) obtained with the five secondaryproduction models, with the number of predictions based on ten samples.

Food Measured rateHuntley and Lopez

(1992)Hirst and Sheader

(1997)Hirst and Lampitt

(1998)Stockwell and

Johannsson (1997)Shuter and Ing

(1997)

Phytoplankton 0.22+ 0.16 (643) 0.48+ 0.07 (1 979) 0.08+ 0.02 (283) 0.06+ 0.02 (208) 0.25+ 0.19 (719) 0.17+ 0.02 (683)

Yeast 0.33+ 0.26 (452) 0.42+ 0.03 (1 000) 0.09+ 0.03 (169) 0.07+ 0.03 (127) 0.37+ 0.32 (502) 0.15+ 0.01 (349)

Cornflour 0.11+ 0.05 (350) 0.36+ 0.05 (1 286) 0.07+ 0.01 (224) 0.05+ 0.01 (169) 0.12+ 0.04 (386) 0.13+ 0.02 (454)

Figure 3. Modelled vs. measured growth rate in D. magna fed phytoplankton, brewers’ yeast, and cornflour. The line in all three panelsrepresents a 1:1 correspondence. The models are identified in the key. Note that in all three experiments, only the secondary productionmodel of Stockwell and Johannsson (1997) approximates the 1:1 relationship. The ellipses signify the observed trend of the different datagroups.

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taxonomic origin determines their applicability. The findings here

support this deduction.

Earlier, we suggested that model zooplankters in the laboratory

could be used to test secondary production models. Certainly, if a

model cannot predict secondary production in a well-controlled

laboratory aquarium, then it is unlikely to be able to predict sec-

ondary production at sea or in a lake. Here, a cladoceran, D.

magna, was used to test models of secondary production, and it

became clear that only freshwater-based models at least partially

based on cladocerans can predict secondary production in

Daphnia. However, if we had applied any of the five models to zoo-

plankton other than cladocerans or copepods, they would most

probably have failed regardless of their marine or freshwater

origin. This suggests future avenues of research are needed in

which new models, based on other zooplankton organisms,

should be constructed and in which new and old models are chal-

lenged by zooplankton other than copepods and cladocerans.

To conclude, the best secondary production prediction found

here used the Stockwell and Johannsson (1997)

freshwater-crustacean-based model. The Hirst and Sheader

(1997) and Hirst and Lampitt (1998) marine-copepod-based

models underestimated secondary production; as expected, the

Huntley and Lopez (1992) model overestimated it. Finally, our

opinions are that the differences in model performance were at-

tributable to their taxonomic rather than to their ecological base.

AcknowledgementsWe thank M. Victoria Zelada Garcıa Duran and Maria Guerrero

Claros for technical assistance and Consuelo Enrıquez for gram-

matical correction. The work was supported by Project 18/95 of

the Canary Autonomous Goverment and project EXZOME

(CTM2008-01616/MAR) granted to MG. I. Martınez and

I. Mayo were supported by a grant from Fundacion

Universitaria de Las Palmas, and TTP by contract EXMAR

SE-539 10/17 (Proyecto Estructurante en Ciencias Marinas).

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Table 5. The relationship between model-predicted rates of secondary production and calculations of secondary production based on themeasured growth rates of Daphnia (x).

FoodHuntley andLopez (1992)

Hirst andSheader (1997)

Hirst andLampitt (1998)

Stockwell andJohannsson(1997)

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Phytoplankton 14.99 x 2 766.73,

r2 ¼ 0.48

(slope ¼ 14.99)

1.72 x2 82.51,

r2 ¼ 0.57

(slope ¼ 1.72)

1.24 x 2 59.26,

r2 ¼ 0.58

(slope ¼ 1.24)

0.88 x + 14.90,

r2 ¼ 0.76

(slope ¼ 0.88)

5.23 x 2 268.41,

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(slope ¼ 5.23)

Yeast 4.30 x2 104.65,

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(slope ¼ 4.30)

0.61 x2 12.07,

r2 ¼ 0.71

(slope ¼ 0.61)

0.32 x + 0.14,

r2 ¼ 0.64

(slope ¼ 0.32)

1.09 x + 0.99,

r2 ¼ 0.78

(slope ¼ 1.09)

1.50 x 2 36.69,

r2 ¼ 0.64

(slope ¼ 1.50)

Cornflour 24.6 x2 733.89,

r2 ¼ 0.57

(slope ¼ 24.6)

3.16 x2 88.35,

r2 ¼ 0.55

(slope ¼ 3.16)

2.31 x 2 64.25,

r2 ¼ 0.54

(slope ¼ 2.31)

2.31 x 2 42.56,

r2 ¼ 0.48

(slope ¼ 2.31)

8.76 x 2 261.67,

r2 ¼ 0.57

(slope ¼ 8.76)

The underlined slopes in the tests with the Stockwell and Johannsson (1997) model attest to the agreement between the measured and predicted rates. Inboth cases, the slopes are close to the 1:1 line.

Testing zooplankton secondary production models against Daphnia magna growth Page 7 of 8

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Handling editor: Andy Payne

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