Deep-Sea Research I 46 (1999) 1793}1808 Improved estimation of f-ratio in natural phytoplankton assemblages Marc Elskens! *, Leo Goeyens!, Frank Dehairs!, Andrew Rees", Ian Joint", Willy Baeyens! !Laboratory of Analytical Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium "NERC Centre for Coastal and Marine Sciences, Plymouth Marine Laboratory, Prospect Place, The Hoe, Plymouth PL1 3DH, UK Received 2 February 1998; received in revised form 19 October 1998; accepted 28 December 1998 Abstract Statistical properties in non-linear regression models of f-ratio versus nitrate relationships were investigated using a case study conducted at the European continental margin (Project OMEX). Although the OMEX data "t within the family of empirical models introduced by Platt and Harrison (1985) Nature 318, 55}58), it is shown that the discrepancy between experimental and "tted values is larger than expected, assuming that the data are accurate. Since the ambient ammonium concentration plays a leading role in regulating new production, the present analysis was extended to include explicitly the e!ect of ammonium on f-ratio versus nitrate plots. Experimental results based on controlled ammonium additions were used to express the f-ratio as a function of both nitrate concentration and ammonium inhibition, i.e. H f(,- ?).I(,& @). The estimation behaviour of the data set/model combination was analysed by testing the appropriateness of various model functions for f H and I. The best "t was obtained with a sigmoid curve, for which mean values of the random #uctuation are almost commensur- ate with the estimated uncertainty of the measurements with natural phytoplankton assem- blages. Precision of the f-ratio estimates was further assessed from contour diagrams of constant likelihood. The signi"cance, validity limits of the estimated parameters, and the relevance of the proposed model as a predictive tool are discussed. Overall, this empirically determined model results in improved precision of f-ratio estimations. ( 1999 Elsevier Science Ltd. All rights reserved. * Corresponding author. Tel.: 0032-2-6292716; fax: 0032-2-6293274. E-mail address: [email protected] (M. Elskens) 0967-0637/99/$- see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 3 7 ( 9 9 ) 0 0 0 2 3 - 0 1794 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1. Introduction Knowledge of the relationship between nutrient supply and phytoplankton produc- tion is essential for understanding the major processes in marine ecosystems. It de"nes the sequestration of atmospheric CO and its transport into deep water via the biological pump (Longhurst and Harrison, 1989). Dugdale and Goering (1967) pro- vided an outstanding model, which distinguishes new and regenerated primary production, according to the origin of the nitrogenous nutrient taken up by phyto- plankton. This model assumes a steady state in the surface layer, which relates the concentration of the limiting nutrient to the corresponding speci"c uptake rate and the loss rate of phytoplankton (Dugdale, 1967). The major nitrogen stock in the ocean is nitrate, which is assimilated by phyto- plankton in the euphotic zone and incorporated into the food chain. The supply rate of nitrate to the euphotic zone regulates primary production (Dugdale and Goering, 1967) and determines the maximal export rate due to settling of particles (Eppley and Peterson, 1979). In order to address the spatiotemporal variability in nitrogen uptake by the autotrophic community and particle export to the deep sea, various models, using f-ratio as a key parameter, have been suggested (Eppley and Peterson, 1979; Platt and Harrison, 1985; Dugdale and Wilkerson, 1989). The f-ratio, usually de"ned as the ratio of nitrate uptake to the uptake of nitrate and ammonium (plus occa- sionally urea), increases asymptotically with increasing nitrate concentration. It is assumed that this relation applies throughout the World Ocean (Harrison et al., 1987; Sathyendranath et al., 1991; Vezina, 1994) but breaks down in oceanic regions where nitrogen does not limit primary production, such as the high-nutrient low-chlorophyll areas (HNLC, Minas et al., 1986). Recently, sea-surface temperatures have been used as a proxy to model surface nitrate distributions, as well as total and new production (Dugdale et al., 1997). The latter approach provides an interesting tool to extend productivity estimates to large ocean basin surfaces, and to integrate productivity over entire growth seasons. It must be emphasised, however, that f-ratio versus surface nitrate concentration always shows considerable scatter, and the relationship is signi"cantly a!ected by additional parameters such as the availability of other nutrients and phytoplankton species composition. The present study aims to reduce the uncertainty associated with the estimation of f-ratio and provides con"dence regions for the f-ratio versus nitrate relation. Since most of the models discussed are non-linear, con"dence regions and intervals were calculated according to the likelihood method. Although this method is computation- ally demanding, it produces con"dence limits with observed coverages that are closer to the nominal probability than those obtained using linear methods (Donaldson and Schnabel, 1987). This study is an attempt to investigate both the nitrate and ammonium uptake regimes, since recent work has shown that nitrate uptake may be substantially reduced in the presence of ammonium, even at nanomolar concentra- tions (Wheeler and Kokkinakis, 1990; Harrison et al., 1996). The data used to inves- tigate these relationships are nutrient distribution pro"les and nitrogen uptake data obtained during a 3-year study in the northeastern Atlantic Ocean margin (Elskens et al., 1997; Rees et al., 1999). M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 1795 2. Methods 2.1. Data collection and treatment The data were obtained between March 1993 and October 1995 during seven cruises on board R.V. Belgica, R.R.S. Discovery, R.R.S. Charles Darwin and R.V. Valdivia conducted as part of the Ocean Margin Exchange (OMEX) project 1993} 1995. Special care was exercised in computing nitrogen uptake rates with respect to the isotope mass balance (Collos, 1987), excess tracer addition (Harrison et al., 1996), and isotope dilution e!ects (Glibert et al., 1982). Full details of the method used are set out in publications by Elskens et al. (1997) and Rees et al. (1999). In the present study, only measurements made in the upper 20 m layer are considered, yielding a set of about 200 data distributed among "ve sites located across the ocean margin of the northeast Atlantic (Joint et al., 1999): an area representative of the Celtic Sea Shelf (493}503N103}10.53W), the OMEX 1 site on Goban Spur (113}123W), the OMEX 2 site (123}12.73W), the OMEX 3 site (133}16.53W) and La Chapelle Bank (473}483N 63}83W). 2.2. Estimation of the probable error awecting measurements with natural assemblages To gauge whether the value of the residual mean variance is su$ciently small in a given non-linear regression model (see below), it is helpful to determine the e magnitude of the experimental uncertainties (&) that can a!ect the measurements. An estimate of the standard deviation was "rst derived from the variability associated with regression lines for the nutrient standards. This method yields a conservative l ( \ ( l estimate of the standard deviation with values of 0.05 M for 0 [NO ] 10 M l ( > ( l and 0.04 M for 0 [NH ] 1 M. The variability of N abundance measure- ments was estimated from replicate determinations when available. Samples with N abundance ranging from 0.5 up to 3%, as generally observed in this study, show a coe$cient of variation of about 5%. Propagation of these experimental uncertain- ties on N-uptake rates and f-ratio was estimated according to Miller and Miller (1988). e ' l It follows that the mean (&) is 0.082 (CV 15%) for nitrate 1 M, but is relatively higher at lower nutrient concentrations, reaching 0.053 (CV 39%) when both nitrate and ammonium are below 0.1 lM. 2.3. Non-linear regression modelling Computerised non-linear curve "tting was used throughout this study to obtain the parameters of the independent variables that give the best "t between the anticipated equation and the data. The "tter used a Marquardt}Levenberg algorithm, which seeks the observed and predicted values of the dependent variable by an iterative procedure (SigmaPlot software, Jandel Scienti"c). Unless otherwise stated, 1796 M. Elskens et al. / Deep-Sea Research I 46 (1999) 1793}1808 e the stochastic term () in the model functions refers to an additive error assumption. The loss function to minimise is L h " + ! h RSS( ) (>G f (XG; )) , G h where RSS( ) is the residual sum of squares, >G the dependent variable, XG the explanatory variable(s) and h a vector of p parameters to be estimated. The goodness of "t was assessed by comparing the magnitude of the residual variance: s"RSS(hK )/(n!p), amongst various model functions (Healy, 1984; Ratkowsky, 1990), where RSS(hK ) corresponds to the least-square estimate of the p parameters and n is the sample size. e e It was considered to be satisfactory when () approached (&). However, additional information is required to determine which of several compet- ing models is the most appropriate. This was achieved (i) by examining the residuals e after "tting the models: the normality assumption for () being ascertained by the Kolmogorov}Smirnov test (Bradley, 1968) and (ii) by computing asymptotic standard errors on the parameter estimates, as well as parameter dependencies (SigmaPlot software, Jandel Scienti"c): variance of the parameter, other parameters constant D"1! . variance of the parameter, other parameters changing Parameters with dependencies near 1 are strongly interdependent.