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Estimate of the drainage water behaviour in shallow lakes
Ecological Modelling 184 (2005) 219–227
Estimate of the drainage water behaviour in shallow lakes L´ea J. El-Jaick∗ , A.A. Gomes Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, CEP 22290-180 Rio de Janeiro, RJ, Brazil. Received 12 February 2004; received in revised form 8 September 2004; accepted 6 October 2004
Abstract A theoretical estimate of the explicit time dependence of the drainage water in shallow lakes is proposed as an important contribution for understanding the lake dynamics. This information can be obtained applying the mathematical techniques largely used in fitting experimental data, as a sum of Lorentzian functions. These functions were chosen because their centre and width yield a good description of the water basin behaviour. The coefficients of these functions are extracted using results of calculated data for the state variables describing the shallow West Lake, Hangzou. This procedure can also be applied to other shallow lakes generating information about their drainage basin, one of the most important parameters to describe their micrometeorological behaviour. One concludes this work emphasizing the relevance of the explicit time dependence of the drainage variables. This is important to simulate the state functions. It is also necessary to have measured data to validate this approach. Published by Elsevier B.V. Keywords: Drainage basin; Dynamical model; Shallow lakes; Geological information; Lorentzian functions
1. Introduction Different approaches for modelling have been proposed to study the dynamics of lakes in specific environments. The main purpose of this paper is to yield a more detailed description of lakes, taking into account the influence of their environment and the corresponding micrometeorological behaviour. For better understanding, it has been introduced a summary of the literature dealing with theoretical modelling, describ∗ Corresponding author. Tel.: +55 21 2141 7178/7154; fax: +55 21 2141 7540. E-mail address: [email protected] (L.J. El-Jaick).
0304-3800/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.ecolmodel.2004.10.005
ing details of aquatic ecosystems, in several regions of the world. Eutrophication of lakes is a serious problem for water quality. The importance of phosphorus on biological systems is known a long time ago. It is considered as one of the main factors for the eutrophication of aquatic systems. There are simple models describing the behaviour of only one variable in a given problem, or very complex models, using computer software developments. Simplifications are necessary in the modelling processes, but they introduce an uncertainty in the precision of the simulated results. In the absence of experiments, the theory predominated, describing common nutrients, as phosphorus,
220
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227
nitrogen, oxygen, etc., but with the advance of experimental techniques, the models became more sophisticated. This implies the introduction of new mathematical formulations, including other variables, as phytoplankton, macrophyte, and zooplankton, to better deal with specific questions. Theory and experimentation have different importance on the construction of models. A simple dynamic model, successfully describes the release of phosphorus from the sediments to the overlying water in shallow lakes (Van der Molen, 1991). The processes involved in the sediment (advection due to infiltration, diffusion, sedimentation, etc.) were formulated in a steady or an unsteady condition by Jorgensen et al. (1975), Ishikawa and Nishimura (1989), Smits and Van der Molen (1993), Chapelle (1995), Ji et al., (2002) etc. Restoration of lakes has been exhaustively studied in the last years with the development of a variety models (see, for example, Moss et al., 1986; Hosper and Meijer, 1986; Hosper, 1989; Jagtman et al., 1992). More recently, successful restoration was achieved on the Lake Eymir, in Turkey (Beklioglu et al., 2003) by reducing the loading of total phosphorus and dissolved inorganic nitrogen. In spite of this effort, poor water clarity and mainly low-submerged plant coverage persisted. They also found that some species of fish perpetuate the poor water condition. However, a decrease in the inorganic suspended solids, and in the chlorophyll-a concentrations, reduced the fish stock and led to a better water quality. The simulation model of waste stabilization reservoirs developed by Friedler et al. (2003) assumes nonsteady-state conditions and has seven quality variables. Stabilization reservoirs, especially important for warm countries, were conceived in Israel for regulating between treated wastewater inflow and withdrawal of effluent water for irrigation. Thus, improves the quality of the stored wastewater. A combined ecological and hydrodynamic model, considering large number of state variables (three algal species, dissolved oxygen, biochemical oxygen demand and nutrients as phosphorus and nitrogen) and the correspondingly parameters, was successfully applied by Schladow and Hamilton (1997) to predict the water quality in Prospect Reservoir, West of Sidney, Australia. Hamilton and De Stasio (1997) introduced zooplankton algorithms in this previous model so that interactions of phytoplankton and zooplankton could
be simulated. The impact of zooplankton grazing on phytoplankton biomass, studied by Griffin et al. (2001) in the Swan River Estuary, Australia, was shown to be important in attenuating a dinoflagellate bloom. In The Netherlands, where also many lakes are dominated by algae bloom, Van Puijenbroek et al. (2004) developed an integrated modelling for nutrient loading in lakes. Besides the common nutrients, several dynamic models introduced other state variables as phytoplankton, bacterioplankton, zooplankton, macrophyte, fish and algae, to study lakes eutrophication (see, for example, Asaeda and Van Bon, 1997; Clarke and Bennett, 2003; Hakanson and Boulion, 2003; Imteaz et al., 2003; Malmaeus and Hakanson, 2004; Parinet et al., 2004; Elliott and Thackeray, 2004; Hakanson et al., 2004; Srinivasu, 2004). In addition (Cottenie et al., 2003), using data obtained during 3 years for a system of highly interconnected ponds, studied the influences of regional interactions on the local zooplankton communities, showing that environmental constraints are strongly related to the community structure. Different experimental techniques have been used to study the role of fish contamination on lakes and marine waters by chemical products or natural toxins. Liquid chromatography with mass spectrometric detection was developed by Dahlmann et al. (2003) to determine various algal and cyanobacterial toxins extracted from phytoplankton, which can lead to shellfish poisoning. Page and Murphy (2003) also used a geographic information system (GIS) approach to create a database to establish the mercury (Hg) levels in many fishes from remote lakes in Canada, where the Hg quantity exceeds the recommended level for human consumption. This summary gives an overview of some relevant shallow lake studies in ecological literature. The present work is based on the paper by Hongping and Jianyi (2002), where it is used an algal dynamic model to describe the West Lake, Hangzhou. Considering 13 state variables, they show that the model can reasonably respond to the changes of forcing function in lake ecosystem. Here, one uses more general equations for the water column, including explicitly the time dependence of the drainage basin function Q(t), where some micrometeorological effects are taken into account. It should be noted that in the usual description of lake behaviour, the geographical aspects are disregarded in the water flow
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227
dynamics. We have explicitly verified that if Q is taken as time-independent quantity it is not possible to reproduce the experimental data presented by Hongping and Jianyi (2002). This paper contains two different steps: the first one is mostly the theoretical description of lake modelling (Section 2), and the second concerns the calculated results for the lake phyto-zooplankton community observed on an annual time scale (Section 3). To the best knowledge, only the present paper and that of Hongping and Jianyi (2002), consider the geographical details.
2. The model Before explaining the model the relevance of including several distinct water fluxes (e.g., rain water, run-off water, drainage water, etc.), coming from different regions around the considered lake should be emphasized. This comes from the eventual transport of substances and even microorganisms to the lake. The dynamical state equations describe a lake in a given geographical environment and involve several parameters. The rate-equations are the most frequently used in the description of dynamics. They provide quantitatively the time dependence of the process of birth, growth and decomposition of the lake constituents, together with forcing functions like, for example, the time-dependent sunlight incidence. This is a classical approach used in several lake studies, which has appeared in the literature in the last years (e.g., Hamilton and Schladow, 1997; Asaeda and Van Bon, 1997; Griffin et al., 2001; Moisan et al., 2002; Kagalou et al., 2003; Hakanson et al., 2004; Srinivasu, 2004). Many of these rate constants can be measured using physico-chemical techniques, and the values which illustrate this study are those assumed for the Chinese West Lake by Hongping and Jianyi (2002), at their Table 2. Some examples suggesting adequate approaches to hydrobiology as applied to lakes, are given by Ahlgren et al. (1988) and Krivtsov et al. (1999). In the literature, the physico-chemical aspects of the model equations are currently assumed independent of the geographic coordinates of a given lake. However, for very specific situations, e.g., when there exists a connection with the sea, the mathematical structure must be modified to be adequate for this specific re-
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quirement. Consequently, in the case of the Chinese West Lake, it is necessary to introduce explicit time dependence in the drainage water associated with the geographic environment. As already mentioned, the experimental results for the West Lake cannot be reproduced, except if the drainage water value, Q, exhibits intrinsic time dependence. The central point in this work concerns the drainage water, considered here as the main micrometeorological parameter. Contrary to Hongping and Jianyi (2002), one proposes that this parameter depends explicitly on time, via a distribution function Q(t), to be below analysed. Those authors assumed the drainage water as an average, e.g., a time-independent quantity. The motivation to fit the Q(t) functions with a sum of Lorentzians was inspired by the usual approach adopted in techniques of experimental physics, like electron paramagnetic resonance (EPR) and M¨ossbauer effect (ME). The experimental results are very sensitive to the neighbourhood of the probe. This sensitivity is expressed as a combination of functions, like Lorentzians, by the position of their centres and widths. In the present case, the centres stand for the micrometeorological time of rain precipitation and their widths represent the time interval of precipitation and the soil diffusion processes. Starting from theoretical results obtained by Hongping and Jianyi (2002) it was extracted the time dependence of the water drainage. From now on, the dynamics of the model defined below will be considered fixed; thus, only the parameters and forcing functions may be changed to describe different lakes/regions. The state variables used by them are: the biomasses of four species of algae, Cyanophyta, Chlorophyta, Criptophyta and Bacillariophyta, BA1 (t), BA2 (t), BA3 (t) and BA4 (t), respectively, with their respective content of phosphorus PA1 (t), PA2 (t), PA3 (t) and PA4 (t); biomass of the zooplankton, BZ(t), and its content of phosphorus, PZ(t); phosphorus in detritus, PD(t); phosphorus in sediment, PE(t); finally orthophosphate, PS(t). The state equations, from which the information about the drainage water is obtained, have the following general form: dBAi (i) (i) = A1 [t; PS, BAi ] − (QBAi (t)/V ) dt (i)
× BAi − BZ × A2 [t; BAi ]
(1)
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where i = 1, 2, 3, 4 for the four algae and V is the volume of the lake. The main difference in respect to Hongping and Jianyi (2002) concerns the time depen(i) dence drainage water as described as QBAi (t). These distribution functions are strictly time dependent and they intend to describe the contribution from the diverse micrometeorological effects. For the zooplankton, one has the dynamics: dBZ = B1 [t; BAi ] × BZ − (QBZ (t)/V ) × BZ dt
(2)
For orthophosphate and phosphorus in detritus one has respectively dPS = LPS + B2 [t; PD, PS, PE, PAi , BAi ] dt − (QPS (t)/V ) × PS
(3)
dPD = LPD + B3 (t; PZ, PD) − (QPD (t)/V ) × PD dt (4)
(i)
(i)
(i)
(i)
The functions A1 (t), A2 (t), B1 (t), B2 (t), and B3 (t) are non-linear in the state variables and time and include the model parameters. In Eq. (3), PAi (t) correspond to the phosphorus in the algae. These nonlinear functions have the general form F(t; S1 (t), S2 (t), . . ., Sq (t); λ1 , λ2 , . . ., λp ) where λp are the parameters of the model and Sq (t) are the state variables obtained from the experiment as a function of time t. The parameters λp are identical to those of Hongping and Jianyi (2002), but a simulation changing them can be made. The detailed definition of the (i) (i) (i) (i) functionsA1 (t), A2 (t), B1 (t), B2 (t), and B3 (t) are presented in Hongping and Jianyi (2002), with the parameters defined in their Table 2, which one expects to be adequate to the present study. Given the simulated values for the time in the interval from t = 0 and t = 360 days, and performing numerical differentiation of the available time dependence of the state variables, one can extract the corresponding drainage water Q(t)’s that will be fitted by eight Lorentzians. The numerical
Fig. 1. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the algae Cyanophyta, QBA1 (t)/V (see Eq. (1)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227
results of such a procedure are shown in the figures. Another kind of dynamical equation concerns the amount of phosphorus in the four species of algae and in the zooplankton. Their equations of motion are written in general terms: dPAi (i) = Ci [t; BAi , PS, PAi ] − QBAi (t)/V × PAi dt (5) where i = 1, 2, 3, 4 dPZ = D[t; BAi , PAi , BZ] − (QBZ (t)/V ) × PZ (6) dt where, again, different Qi (t) are used since different substances are introduced by the geological drainage distribution. Since no data concerning the internal phosphorus are available, the values for PAi (t) and PZ(t) are assumed proportional to the algae and zooplankton biomasses respectively. This assumption is based on so-
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called Redfield molecule (Redfield et al., 1963), which intends to represent the average content of this element in these biomasses. For the case of phosphorus in the sediment, one has the following equation:
dPE = E[t; PE, PS, PD, PAi ] dt
(7)
It was numerically checked that the shape of the PAi (t) curves are strongly changed accordingly to the maximum and minimum values adopted for the internal phosphorus. It is important to note that, contrary the remainder equations for the lake state variables, the drainage water does not appear explicitly in Eq. (7). This fact seems to be reasonable since drainage only indirectly affects the sediment of the lake via the water column dynamics, and this is expected to have a distinct time scale.
Fig. 2. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the algae Chlorophyta, Cryptophyta and Bacillariophyta, QBAi (t)/V (i = 2, 3, 4), respectively (see Eq. (1)). (These three curves are almost indistinctive.) Thus, it is shown only the eight Lorentzians distribution functions for BA2 (t) (dashed lines), and the fit given by their sum (solid line).
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3. Numerical results and conclusions After obtaining numerically the time-dependent results for the drainage water, their curves have been fitted in a model able to provide relevant information on the Q(t)’s quantities. Due to the complex geometrical topology of water drainage basin, including several distinct water distributions (see Fig. 1 by Hongping and Jianyi (2002)), it has been adopted here a mathematical tool usually applied in fitting EPR and M¨ossbauer data. To study complex lattices configurations in solids one adopts a superposition of several Lorentzians functions. Their centres and widths represent adequately the relative relevance of the involved water paths of the water drainage basin as mentioned above. The numerical procedure goes as follows: the data presented in the figures of Hongping and Jianyi (2002) for BAi (t) (i = 1, 2, 3, 4), BZ(t), PS(t) and PD(t), are taken together with the parameters presented in their Table 2. Given these data, one numerically differenti-
ates the curves adopting the values given by these authors, for the quantities LPS and LPD, which stand for orthophosphate and other forms of phosphorus, which are taken from outside of lake, respectively. Thus, it (i) is possible to extract the curves of QBAi (t) and QM (t) (with M = BZ, PS and PD) using Eqs. (1)–(4), and to fit them using the sum of eight Lorentzians of the general form: H(t) = Y i (t) (8) i=1,8
with Y i (t) = Y0 + (i)
2Ai ∆(i) π (X(t) − X(i) )2 + (∆(i) )2
(9)
The centres of the Lorentzians define the time at which water was introduced/absorbed in the basin. The negative sign of the coefficients A(i) is interpreted as absorption. The widths ∆(i) describe time intervals of rainwater precipitation and/or diffusion. The results of the fitting are presented in the figures. Fig. 1
Fig. 3. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the zooplanktons, QBZ (t)/V (see Eq. (2)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227
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Fig. 4. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the ortophosphate, QPS (t)/V (see Eq. (3)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
shows the decrease in the growth coefficient due to the drainage water, QBA1 (t)/V , for BA1 . In Fig. 2, one can see that QBA2 (t)/V , QBA3 (t)/V and QBA4 (t)/V are almost identical. Note that, except in the cases for zooplankton and orthophosphate (Figs. 3 and 4), which present large negative values, only positive one for the coefficients A(i) of the Lorentzians, do exist. Fig. 5 shows the same, for the phosphorus in detritus. The advantage of the Lorentzian fit is to show at what time and for which time intervals the geological water dynamics for the drainage occurs. The negative terms of the expansion are interpreted as an important indication of absorption of these drainage elements by the lake. This method thus shows, by the large negative values seen in Figs. 3 and 4, the relevance of time-dependent absorption process in the drainage, which describes the dynamics of the lake.
As a final remark, it should be noted that the timedependent drainage water contributions to the lake dynamics are different for each one of the usual state variables. This difference is associated with the several substances drained. Furthermore, these results suggest performing experiments (using, for example, radioactive tracers) in the region of lakes in the North of Rio de Janeiro State. These results would measure the significance of the time dependence in this region associated with micrometeorological effects and the soil diffusion of water. Obviously, for eventual application to Brazilian lakes, the physico-chemical rate processes should be adapted to the local conditions and again extracted from experiment (Carmouze, 1984; Suzuki, 1997). It should be stressed that for other available data corresponding to lakes, in Brazil or else, this proposal can be applied to extract again drainage water results.
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Fig. 5. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the phosphorus in detritus, QPD (t)/V (see Eq. (4)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
This assumption may also be applied to lakes of the southeast of Brazil, but not when studying Brazilian lakes in the north of the country. Acknowledgements The authors acknowledge J. Thadeu Cavalcante for his kind help by developing software for extracting the experimental and theoretical data plotted in figures of Hongping and Jianyi (2002) and to C. da Silva for a critical reading of the manuscript. References Ahlgren, I., Frisk, T., Kamp-Nielsen, L., 1988. Empirical and theoretical models of phosphorus loading, retention and concentration vs. trophic state. Hydrobiologia 170, 285–302. Asaeda, T., Van Bon, T., 1997. Modelling the effects of macrophytes on algal blooming in eutrophic shallow lakes. Ecol. Model. 104, 261–287.
Beklioglu, M., Ince, O., Tuzun, I., 2003. Restoration of the eutrophic Lake Eymir, Turkey, by biomanipulation after a major external nutrient control. Hydrobiologia 490 (1–3), 93–105. Carmouze, J.P., 1984. Gen´eralization dˇıune m´ethode de determination du carbone, mineral total par pH-metrie dans les eaux; son application a` lˇıe´ tude du metabolism aerobie et anaerobie d’une lagune tropicale. Hidrobiol. Prop. 17, 175–189. Chapelle, A., 1995. A preliminary model of nutrient cycling in sediments of a Mediterranean lagoon. Ecol. Model. 80, 131–147. Clarke, L.R., Bennett, D.H., 2003. Seasonal zooplankton abundance and size fluctuations across spatial scales in Lake Pend Oreille, Idaho. J. Fresh. Ecol. 18 (2), 277–290. Cottenie, K., Michels, E., Nuytten, N., De Meester, L., 2003. Zooplankton metacommunity interconnected ponds. Ecology 84 (4), 991–1000. Dahlmann, J., Budakowski, W.R., Luckas, B., 2003. Liquid chromatography–electrospray ionisation–mass spectrometry based method for the simultaneous determination of algal and cyanobacterial toxins in phytoplankton from marine waters and lakes followed by tentative structural elucidation of microcystins. J. Chromatogr. A 994 (1–2), 45–57. Elliott, J.A., Thackeray, S.J., 2004. The simulation of phytoplankton in shallow and deep lakes using PROTECH. Ecol. Model. 178, 357–369.
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227 Friedler, E., Juanico, M., Shelef, G., 2003. Simulation model of wastewater stabilization reservoirs. Ecol. Eng. 20, 121–145. Griffin, S.L., Herzfeld, M., Hamilton, D.P., 2001. Modelling the impact of zooplankton grazing on phytoplankton biomass during a dinoflagellate bloom in the Swan River Estuary, Western Australia. Ecol. Eng. 16, 373–394. Hakanson, L., Boulion, V.V., 2003. A general dynamic model to predict biomass and production of phytoplankton in lakes. Ecol. Model. 165, 285–301. Hakanson, L., Blenckner, T., Malmaeus, J.M., 2004. New, general methods to define de depth separating surface water from deep water, outflow and internal loading for mass-balance models for lakes. Ecol. Mod. 175, 339–352. Hamilton, D.P., Schladow, S.G., 1997. Prediction of water quality in lakes and reservoirs. Part I. Model description. Ecol. Model. 96, 91–110. Hamilton, D.P., De Stasio, B.T., 1997. Modelling phytoplanktonzooplankton interactions in Sparkling Lake, USA. Verh. Int. Verein. Limnol. 26, 487–490. Hongping, P., Jianyi, M., 2002. Study on the algal dynamic model for West Lake, Hangzhou. Ecol. Model. 148, 67–77. Hosper, S.H., 1989. Biomanipulation, new perspective for restoring shallow, eutrophic lakes in The Netherlands. Hydrobiol. Bull. 23, 5–10. Hosper, S.H., Meijer, M.-L., 1986. Control of phosphorus loading and flushing as restoration methods for Lake Veluwe, The Netherlands. Hydrobiol. Bull. 20, 183–194. Imteaz, M.A., Asaeda, T., Lockington, D.A., 2003. Modelling the effects of inflow parameters on lake water quality. Environ. Model. Assesm. 8, 63–70. Ishikawa, M., Nishimura, H., 1989. Mathematical model of phosphate release rate from sediments considering the effect of dissolved oxygen in overlying water. Water Res. 23 (3), 351–359. Jagtman, E., Van der Molen, D.T., Vermij, S., 1992. The influence of flushing on nutrient dynamics, composition and densities of algae and transparency in Veluwemeer, The Netherlands. Hydrobiologia 233, 187–196. Ji, R., Chen, C., Budd, J.W., Schwab, D.J., Beletsky, D., Fahnenstiel, G.L., Johengen, T.H., Vanderploeg, H., Eadie, B., Cotner, J., Gardner, W., Bundy, M., 2002. Influences of suspended sediments on the ecosystem in Lake Michigan: a 3D coupled biophysical modelling experiment. Ecol. Model. 152, 169– 190.
227
Jorgensen, S.E., Nielsen, L.K., Jacobsen, O.S., 1975. A submodel for anaerobic mud-water exchange of phosphate. Ecol. Model. 1, 133–146. Kagalou, I., Petridis, D., Tsimarakis, G., 2003. Seasonal variation of water quality parameters and plankton in a shallow Greek lake. J. Fresh. Ecol. 18 (2), 199–206. Krivtsov, V., Sigee, D., Corliss, J., Bellinger, E., 1999. Examination of the phytoplankton of Rostherne Mere using a simulation mathematical model. Hydrobiologia 414, 71–76. Malmaeus, J.M., Hakanson, L., 2004. Development of a Lake Eutrophication model. Ecol. Model. 171, 35–63. Moisan, J.R., Moisan, A.T., Abbott, M.R., 2002. Modelling the effect of temperature on the maximum growth rates of phytoplankton populations. Ecol. Model. 153, 197–215. Moss, B., Ball, K., Irvine, K., Stanfield, J., 1986. Restoration of two lowland lakes by isolation from nutrient-rich water sources with and without removal of sediment. J. Appl. Ecol. 23, 391–414. Page, K.D., Murphy, J.B., 2003. The geological sources of Hg contamination in Kejimkujik National Park, Nova Scotia, Canada: a GIS approach. Environ. Geol. 43 (8), 882–891. Parinet, B., Lhote, A., Legube, B., 2004. Principal component analysis: an appropriate tool for water quality evaluation and management—application to a tropical lake system. Ecol. Model. 178, 295–311. Redfield, A.C., Ketchum, B.H., Richards, F.A., 1963. The influence of organisms on the composition of the seawater. In: Hill, M.H. (Ed.), The Sea. Interscience, New York, pp. 26–77. Schladow, S.G., Hamilton, D.P., 1997. Prediction of water quality in lakes and reservoirs. Part II. Model calibration, sensitivity analysis and application. Ecol. Model. 96, 111–123. Smits, J.G.C., Van der Molen, D.T., 1993. Application of SWITCH, a model for sediment-water exchange of nutrients, to Lake Veluwe in The Netherlands. Hydrobiology 253, 281–300. Srinivasu, P.D.N., 2004. Regime shifts in eutrophied lakes: a mathematical study. Ecol. Model. 179, 115–130. Suzuki M.S., 1997. PhD Thesis. North Fluminense State University, Brazil, unpublished. Van der Molen, D.T., 1991. A simple, dynamic model for the simulation of the release of phosphorus from sediments in shallow, eutrophic systems. Water Res. 25, 737–744. Van Puijenbroek, P.J.T.M., Janse, J.H., Knoop, J.M., 2004. Integrated modelling for nutrient loading and ecology of lakes in The Netherlands. Ecol. Model. 174, 127–141.
Estimate of the drainage water behaviour in shallow lakes L´ea J. El-Jaick∗ , A.A. Gomes Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, CEP 22290-180 Rio de Janeiro, RJ, Brazil. Received 12 February 2004; received in revised form 8 September 2004; accepted 6 October 2004
Abstract A theoretical estimate of the explicit time dependence of the drainage water in shallow lakes is proposed as an important contribution for understanding the lake dynamics. This information can be obtained applying the mathematical techniques largely used in fitting experimental data, as a sum of Lorentzian functions. These functions were chosen because their centre and width yield a good description of the water basin behaviour. The coefficients of these functions are extracted using results of calculated data for the state variables describing the shallow West Lake, Hangzou. This procedure can also be applied to other shallow lakes generating information about their drainage basin, one of the most important parameters to describe their micrometeorological behaviour. One concludes this work emphasizing the relevance of the explicit time dependence of the drainage variables. This is important to simulate the state functions. It is also necessary to have measured data to validate this approach. Published by Elsevier B.V. Keywords: Drainage basin; Dynamical model; Shallow lakes; Geological information; Lorentzian functions
1. Introduction Different approaches for modelling have been proposed to study the dynamics of lakes in specific environments. The main purpose of this paper is to yield a more detailed description of lakes, taking into account the influence of their environment and the corresponding micrometeorological behaviour. For better understanding, it has been introduced a summary of the literature dealing with theoretical modelling, describ∗ Corresponding author. Tel.: +55 21 2141 7178/7154; fax: +55 21 2141 7540. E-mail address: [email protected] (L.J. El-Jaick).
0304-3800/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.ecolmodel.2004.10.005
ing details of aquatic ecosystems, in several regions of the world. Eutrophication of lakes is a serious problem for water quality. The importance of phosphorus on biological systems is known a long time ago. It is considered as one of the main factors for the eutrophication of aquatic systems. There are simple models describing the behaviour of only one variable in a given problem, or very complex models, using computer software developments. Simplifications are necessary in the modelling processes, but they introduce an uncertainty in the precision of the simulated results. In the absence of experiments, the theory predominated, describing common nutrients, as phosphorus,
220
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nitrogen, oxygen, etc., but with the advance of experimental techniques, the models became more sophisticated. This implies the introduction of new mathematical formulations, including other variables, as phytoplankton, macrophyte, and zooplankton, to better deal with specific questions. Theory and experimentation have different importance on the construction of models. A simple dynamic model, successfully describes the release of phosphorus from the sediments to the overlying water in shallow lakes (Van der Molen, 1991). The processes involved in the sediment (advection due to infiltration, diffusion, sedimentation, etc.) were formulated in a steady or an unsteady condition by Jorgensen et al. (1975), Ishikawa and Nishimura (1989), Smits and Van der Molen (1993), Chapelle (1995), Ji et al., (2002) etc. Restoration of lakes has been exhaustively studied in the last years with the development of a variety models (see, for example, Moss et al., 1986; Hosper and Meijer, 1986; Hosper, 1989; Jagtman et al., 1992). More recently, successful restoration was achieved on the Lake Eymir, in Turkey (Beklioglu et al., 2003) by reducing the loading of total phosphorus and dissolved inorganic nitrogen. In spite of this effort, poor water clarity and mainly low-submerged plant coverage persisted. They also found that some species of fish perpetuate the poor water condition. However, a decrease in the inorganic suspended solids, and in the chlorophyll-a concentrations, reduced the fish stock and led to a better water quality. The simulation model of waste stabilization reservoirs developed by Friedler et al. (2003) assumes nonsteady-state conditions and has seven quality variables. Stabilization reservoirs, especially important for warm countries, were conceived in Israel for regulating between treated wastewater inflow and withdrawal of effluent water for irrigation. Thus, improves the quality of the stored wastewater. A combined ecological and hydrodynamic model, considering large number of state variables (three algal species, dissolved oxygen, biochemical oxygen demand and nutrients as phosphorus and nitrogen) and the correspondingly parameters, was successfully applied by Schladow and Hamilton (1997) to predict the water quality in Prospect Reservoir, West of Sidney, Australia. Hamilton and De Stasio (1997) introduced zooplankton algorithms in this previous model so that interactions of phytoplankton and zooplankton could
be simulated. The impact of zooplankton grazing on phytoplankton biomass, studied by Griffin et al. (2001) in the Swan River Estuary, Australia, was shown to be important in attenuating a dinoflagellate bloom. In The Netherlands, where also many lakes are dominated by algae bloom, Van Puijenbroek et al. (2004) developed an integrated modelling for nutrient loading in lakes. Besides the common nutrients, several dynamic models introduced other state variables as phytoplankton, bacterioplankton, zooplankton, macrophyte, fish and algae, to study lakes eutrophication (see, for example, Asaeda and Van Bon, 1997; Clarke and Bennett, 2003; Hakanson and Boulion, 2003; Imteaz et al., 2003; Malmaeus and Hakanson, 2004; Parinet et al., 2004; Elliott and Thackeray, 2004; Hakanson et al., 2004; Srinivasu, 2004). In addition (Cottenie et al., 2003), using data obtained during 3 years for a system of highly interconnected ponds, studied the influences of regional interactions on the local zooplankton communities, showing that environmental constraints are strongly related to the community structure. Different experimental techniques have been used to study the role of fish contamination on lakes and marine waters by chemical products or natural toxins. Liquid chromatography with mass spectrometric detection was developed by Dahlmann et al. (2003) to determine various algal and cyanobacterial toxins extracted from phytoplankton, which can lead to shellfish poisoning. Page and Murphy (2003) also used a geographic information system (GIS) approach to create a database to establish the mercury (Hg) levels in many fishes from remote lakes in Canada, where the Hg quantity exceeds the recommended level for human consumption. This summary gives an overview of some relevant shallow lake studies in ecological literature. The present work is based on the paper by Hongping and Jianyi (2002), where it is used an algal dynamic model to describe the West Lake, Hangzhou. Considering 13 state variables, they show that the model can reasonably respond to the changes of forcing function in lake ecosystem. Here, one uses more general equations for the water column, including explicitly the time dependence of the drainage basin function Q(t), where some micrometeorological effects are taken into account. It should be noted that in the usual description of lake behaviour, the geographical aspects are disregarded in the water flow
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dynamics. We have explicitly verified that if Q is taken as time-independent quantity it is not possible to reproduce the experimental data presented by Hongping and Jianyi (2002). This paper contains two different steps: the first one is mostly the theoretical description of lake modelling (Section 2), and the second concerns the calculated results for the lake phyto-zooplankton community observed on an annual time scale (Section 3). To the best knowledge, only the present paper and that of Hongping and Jianyi (2002), consider the geographical details.
2. The model Before explaining the model the relevance of including several distinct water fluxes (e.g., rain water, run-off water, drainage water, etc.), coming from different regions around the considered lake should be emphasized. This comes from the eventual transport of substances and even microorganisms to the lake. The dynamical state equations describe a lake in a given geographical environment and involve several parameters. The rate-equations are the most frequently used in the description of dynamics. They provide quantitatively the time dependence of the process of birth, growth and decomposition of the lake constituents, together with forcing functions like, for example, the time-dependent sunlight incidence. This is a classical approach used in several lake studies, which has appeared in the literature in the last years (e.g., Hamilton and Schladow, 1997; Asaeda and Van Bon, 1997; Griffin et al., 2001; Moisan et al., 2002; Kagalou et al., 2003; Hakanson et al., 2004; Srinivasu, 2004). Many of these rate constants can be measured using physico-chemical techniques, and the values which illustrate this study are those assumed for the Chinese West Lake by Hongping and Jianyi (2002), at their Table 2. Some examples suggesting adequate approaches to hydrobiology as applied to lakes, are given by Ahlgren et al. (1988) and Krivtsov et al. (1999). In the literature, the physico-chemical aspects of the model equations are currently assumed independent of the geographic coordinates of a given lake. However, for very specific situations, e.g., when there exists a connection with the sea, the mathematical structure must be modified to be adequate for this specific re-
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quirement. Consequently, in the case of the Chinese West Lake, it is necessary to introduce explicit time dependence in the drainage water associated with the geographic environment. As already mentioned, the experimental results for the West Lake cannot be reproduced, except if the drainage water value, Q, exhibits intrinsic time dependence. The central point in this work concerns the drainage water, considered here as the main micrometeorological parameter. Contrary to Hongping and Jianyi (2002), one proposes that this parameter depends explicitly on time, via a distribution function Q(t), to be below analysed. Those authors assumed the drainage water as an average, e.g., a time-independent quantity. The motivation to fit the Q(t) functions with a sum of Lorentzians was inspired by the usual approach adopted in techniques of experimental physics, like electron paramagnetic resonance (EPR) and M¨ossbauer effect (ME). The experimental results are very sensitive to the neighbourhood of the probe. This sensitivity is expressed as a combination of functions, like Lorentzians, by the position of their centres and widths. In the present case, the centres stand for the micrometeorological time of rain precipitation and their widths represent the time interval of precipitation and the soil diffusion processes. Starting from theoretical results obtained by Hongping and Jianyi (2002) it was extracted the time dependence of the water drainage. From now on, the dynamics of the model defined below will be considered fixed; thus, only the parameters and forcing functions may be changed to describe different lakes/regions. The state variables used by them are: the biomasses of four species of algae, Cyanophyta, Chlorophyta, Criptophyta and Bacillariophyta, BA1 (t), BA2 (t), BA3 (t) and BA4 (t), respectively, with their respective content of phosphorus PA1 (t), PA2 (t), PA3 (t) and PA4 (t); biomass of the zooplankton, BZ(t), and its content of phosphorus, PZ(t); phosphorus in detritus, PD(t); phosphorus in sediment, PE(t); finally orthophosphate, PS(t). The state equations, from which the information about the drainage water is obtained, have the following general form: dBAi (i) (i) = A1 [t; PS, BAi ] − (QBAi (t)/V ) dt (i)
× BAi − BZ × A2 [t; BAi ]
(1)
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where i = 1, 2, 3, 4 for the four algae and V is the volume of the lake. The main difference in respect to Hongping and Jianyi (2002) concerns the time depen(i) dence drainage water as described as QBAi (t). These distribution functions are strictly time dependent and they intend to describe the contribution from the diverse micrometeorological effects. For the zooplankton, one has the dynamics: dBZ = B1 [t; BAi ] × BZ − (QBZ (t)/V ) × BZ dt
(2)
For orthophosphate and phosphorus in detritus one has respectively dPS = LPS + B2 [t; PD, PS, PE, PAi , BAi ] dt − (QPS (t)/V ) × PS
(3)
dPD = LPD + B3 (t; PZ, PD) − (QPD (t)/V ) × PD dt (4)
(i)
(i)
(i)
(i)
The functions A1 (t), A2 (t), B1 (t), B2 (t), and B3 (t) are non-linear in the state variables and time and include the model parameters. In Eq. (3), PAi (t) correspond to the phosphorus in the algae. These nonlinear functions have the general form F(t; S1 (t), S2 (t), . . ., Sq (t); λ1 , λ2 , . . ., λp ) where λp are the parameters of the model and Sq (t) are the state variables obtained from the experiment as a function of time t. The parameters λp are identical to those of Hongping and Jianyi (2002), but a simulation changing them can be made. The detailed definition of the (i) (i) (i) (i) functionsA1 (t), A2 (t), B1 (t), B2 (t), and B3 (t) are presented in Hongping and Jianyi (2002), with the parameters defined in their Table 2, which one expects to be adequate to the present study. Given the simulated values for the time in the interval from t = 0 and t = 360 days, and performing numerical differentiation of the available time dependence of the state variables, one can extract the corresponding drainage water Q(t)’s that will be fitted by eight Lorentzians. The numerical
Fig. 1. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the algae Cyanophyta, QBA1 (t)/V (see Eq. (1)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
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results of such a procedure are shown in the figures. Another kind of dynamical equation concerns the amount of phosphorus in the four species of algae and in the zooplankton. Their equations of motion are written in general terms: dPAi (i) = Ci [t; BAi , PS, PAi ] − QBAi (t)/V × PAi dt (5) where i = 1, 2, 3, 4 dPZ = D[t; BAi , PAi , BZ] − (QBZ (t)/V ) × PZ (6) dt where, again, different Qi (t) are used since different substances are introduced by the geological drainage distribution. Since no data concerning the internal phosphorus are available, the values for PAi (t) and PZ(t) are assumed proportional to the algae and zooplankton biomasses respectively. This assumption is based on so-
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called Redfield molecule (Redfield et al., 1963), which intends to represent the average content of this element in these biomasses. For the case of phosphorus in the sediment, one has the following equation:
dPE = E[t; PE, PS, PD, PAi ] dt
(7)
It was numerically checked that the shape of the PAi (t) curves are strongly changed accordingly to the maximum and minimum values adopted for the internal phosphorus. It is important to note that, contrary the remainder equations for the lake state variables, the drainage water does not appear explicitly in Eq. (7). This fact seems to be reasonable since drainage only indirectly affects the sediment of the lake via the water column dynamics, and this is expected to have a distinct time scale.
Fig. 2. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the algae Chlorophyta, Cryptophyta and Bacillariophyta, QBAi (t)/V (i = 2, 3, 4), respectively (see Eq. (1)). (These three curves are almost indistinctive.) Thus, it is shown only the eight Lorentzians distribution functions for BA2 (t) (dashed lines), and the fit given by their sum (solid line).
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3. Numerical results and conclusions After obtaining numerically the time-dependent results for the drainage water, their curves have been fitted in a model able to provide relevant information on the Q(t)’s quantities. Due to the complex geometrical topology of water drainage basin, including several distinct water distributions (see Fig. 1 by Hongping and Jianyi (2002)), it has been adopted here a mathematical tool usually applied in fitting EPR and M¨ossbauer data. To study complex lattices configurations in solids one adopts a superposition of several Lorentzians functions. Their centres and widths represent adequately the relative relevance of the involved water paths of the water drainage basin as mentioned above. The numerical procedure goes as follows: the data presented in the figures of Hongping and Jianyi (2002) for BAi (t) (i = 1, 2, 3, 4), BZ(t), PS(t) and PD(t), are taken together with the parameters presented in their Table 2. Given these data, one numerically differenti-
ates the curves adopting the values given by these authors, for the quantities LPS and LPD, which stand for orthophosphate and other forms of phosphorus, which are taken from outside of lake, respectively. Thus, it (i) is possible to extract the curves of QBAi (t) and QM (t) (with M = BZ, PS and PD) using Eqs. (1)–(4), and to fit them using the sum of eight Lorentzians of the general form: H(t) = Y i (t) (8) i=1,8
with Y i (t) = Y0 + (i)
2Ai ∆(i) π (X(t) − X(i) )2 + (∆(i) )2
(9)
The centres of the Lorentzians define the time at which water was introduced/absorbed in the basin. The negative sign of the coefficients A(i) is interpreted as absorption. The widths ∆(i) describe time intervals of rainwater precipitation and/or diffusion. The results of the fitting are presented in the figures. Fig. 1
Fig. 3. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the zooplanktons, QBZ (t)/V (see Eq. (2)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
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Fig. 4. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the ortophosphate, QPS (t)/V (see Eq. (3)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
shows the decrease in the growth coefficient due to the drainage water, QBA1 (t)/V , for BA1 . In Fig. 2, one can see that QBA2 (t)/V , QBA3 (t)/V and QBA4 (t)/V are almost identical. Note that, except in the cases for zooplankton and orthophosphate (Figs. 3 and 4), which present large negative values, only positive one for the coefficients A(i) of the Lorentzians, do exist. Fig. 5 shows the same, for the phosphorus in detritus. The advantage of the Lorentzian fit is to show at what time and for which time intervals the geological water dynamics for the drainage occurs. The negative terms of the expansion are interpreted as an important indication of absorption of these drainage elements by the lake. This method thus shows, by the large negative values seen in Figs. 3 and 4, the relevance of time-dependent absorption process in the drainage, which describes the dynamics of the lake.
As a final remark, it should be noted that the timedependent drainage water contributions to the lake dynamics are different for each one of the usual state variables. This difference is associated with the several substances drained. Furthermore, these results suggest performing experiments (using, for example, radioactive tracers) in the region of lakes in the North of Rio de Janeiro State. These results would measure the significance of the time dependence in this region associated with micrometeorological effects and the soil diffusion of water. Obviously, for eventual application to Brazilian lakes, the physico-chemical rate processes should be adapted to the local conditions and again extracted from experiment (Carmouze, 1984; Suzuki, 1997). It should be stressed that for other available data corresponding to lakes, in Brazil or else, this proposal can be applied to extract again drainage water results.
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Fig. 5. The irregular curve corresponds to the numerical solution for the decrease in the growth coefficient due to the drainage water of the phosphorus in detritus, QPD (t)/V (see Eq. (4)). The dashed lines are the eight Lorentzians distribution functions, which the sum gives the fit showed in the solid line.
This assumption may also be applied to lakes of the southeast of Brazil, but not when studying Brazilian lakes in the north of the country. Acknowledgements The authors acknowledge J. Thadeu Cavalcante for his kind help by developing software for extracting the experimental and theoretical data plotted in figures of Hongping and Jianyi (2002) and to C. da Silva for a critical reading of the manuscript. References Ahlgren, I., Frisk, T., Kamp-Nielsen, L., 1988. Empirical and theoretical models of phosphorus loading, retention and concentration vs. trophic state. Hydrobiologia 170, 285–302. Asaeda, T., Van Bon, T., 1997. Modelling the effects of macrophytes on algal blooming in eutrophic shallow lakes. Ecol. Model. 104, 261–287.
Beklioglu, M., Ince, O., Tuzun, I., 2003. Restoration of the eutrophic Lake Eymir, Turkey, by biomanipulation after a major external nutrient control. Hydrobiologia 490 (1–3), 93–105. Carmouze, J.P., 1984. Gen´eralization dˇıune m´ethode de determination du carbone, mineral total par pH-metrie dans les eaux; son application a` lˇıe´ tude du metabolism aerobie et anaerobie d’une lagune tropicale. Hidrobiol. Prop. 17, 175–189. Chapelle, A., 1995. A preliminary model of nutrient cycling in sediments of a Mediterranean lagoon. Ecol. Model. 80, 131–147. Clarke, L.R., Bennett, D.H., 2003. Seasonal zooplankton abundance and size fluctuations across spatial scales in Lake Pend Oreille, Idaho. J. Fresh. Ecol. 18 (2), 277–290. Cottenie, K., Michels, E., Nuytten, N., De Meester, L., 2003. Zooplankton metacommunity interconnected ponds. Ecology 84 (4), 991–1000. Dahlmann, J., Budakowski, W.R., Luckas, B., 2003. Liquid chromatography–electrospray ionisation–mass spectrometry based method for the simultaneous determination of algal and cyanobacterial toxins in phytoplankton from marine waters and lakes followed by tentative structural elucidation of microcystins. J. Chromatogr. A 994 (1–2), 45–57. Elliott, J.A., Thackeray, S.J., 2004. The simulation of phytoplankton in shallow and deep lakes using PROTECH. Ecol. Model. 178, 357–369.
L.J. El-Jaick, A.A. Gomes / Ecological Modelling 184 (2005) 219–227 Friedler, E., Juanico, M., Shelef, G., 2003. Simulation model of wastewater stabilization reservoirs. Ecol. Eng. 20, 121–145. Griffin, S.L., Herzfeld, M., Hamilton, D.P., 2001. Modelling the impact of zooplankton grazing on phytoplankton biomass during a dinoflagellate bloom in the Swan River Estuary, Western Australia. Ecol. Eng. 16, 373–394. Hakanson, L., Boulion, V.V., 2003. A general dynamic model to predict biomass and production of phytoplankton in lakes. Ecol. Model. 165, 285–301. Hakanson, L., Blenckner, T., Malmaeus, J.M., 2004. New, general methods to define de depth separating surface water from deep water, outflow and internal loading for mass-balance models for lakes. Ecol. Mod. 175, 339–352. Hamilton, D.P., Schladow, S.G., 1997. Prediction of water quality in lakes and reservoirs. Part I. Model description. Ecol. Model. 96, 91–110. Hamilton, D.P., De Stasio, B.T., 1997. Modelling phytoplanktonzooplankton interactions in Sparkling Lake, USA. Verh. Int. Verein. Limnol. 26, 487–490. Hongping, P., Jianyi, M., 2002. Study on the algal dynamic model for West Lake, Hangzhou. Ecol. Model. 148, 67–77. Hosper, S.H., 1989. Biomanipulation, new perspective for restoring shallow, eutrophic lakes in The Netherlands. Hydrobiol. Bull. 23, 5–10. Hosper, S.H., Meijer, M.-L., 1986. Control of phosphorus loading and flushing as restoration methods for Lake Veluwe, The Netherlands. Hydrobiol. Bull. 20, 183–194. Imteaz, M.A., Asaeda, T., Lockington, D.A., 2003. Modelling the effects of inflow parameters on lake water quality. Environ. Model. Assesm. 8, 63–70. Ishikawa, M., Nishimura, H., 1989. Mathematical model of phosphate release rate from sediments considering the effect of dissolved oxygen in overlying water. Water Res. 23 (3), 351–359. Jagtman, E., Van der Molen, D.T., Vermij, S., 1992. The influence of flushing on nutrient dynamics, composition and densities of algae and transparency in Veluwemeer, The Netherlands. Hydrobiologia 233, 187–196. Ji, R., Chen, C., Budd, J.W., Schwab, D.J., Beletsky, D., Fahnenstiel, G.L., Johengen, T.H., Vanderploeg, H., Eadie, B., Cotner, J., Gardner, W., Bundy, M., 2002. Influences of suspended sediments on the ecosystem in Lake Michigan: a 3D coupled biophysical modelling experiment. Ecol. Model. 152, 169– 190.
227
Jorgensen, S.E., Nielsen, L.K., Jacobsen, O.S., 1975. A submodel for anaerobic mud-water exchange of phosphate. Ecol. Model. 1, 133–146. Kagalou, I., Petridis, D., Tsimarakis, G., 2003. Seasonal variation of water quality parameters and plankton in a shallow Greek lake. J. Fresh. Ecol. 18 (2), 199–206. Krivtsov, V., Sigee, D., Corliss, J., Bellinger, E., 1999. Examination of the phytoplankton of Rostherne Mere using a simulation mathematical model. Hydrobiologia 414, 71–76. Malmaeus, J.M., Hakanson, L., 2004. Development of a Lake Eutrophication model. Ecol. Model. 171, 35–63. Moisan, J.R., Moisan, A.T., Abbott, M.R., 2002. Modelling the effect of temperature on the maximum growth rates of phytoplankton populations. Ecol. Model. 153, 197–215. Moss, B., Ball, K., Irvine, K., Stanfield, J., 1986. Restoration of two lowland lakes by isolation from nutrient-rich water sources with and without removal of sediment. J. Appl. Ecol. 23, 391–414. Page, K.D., Murphy, J.B., 2003. The geological sources of Hg contamination in Kejimkujik National Park, Nova Scotia, Canada: a GIS approach. Environ. Geol. 43 (8), 882–891. Parinet, B., Lhote, A., Legube, B., 2004. Principal component analysis: an appropriate tool for water quality evaluation and management—application to a tropical lake system. Ecol. Model. 178, 295–311. Redfield, A.C., Ketchum, B.H., Richards, F.A., 1963. The influence of organisms on the composition of the seawater. In: Hill, M.H. (Ed.), The Sea. Interscience, New York, pp. 26–77. Schladow, S.G., Hamilton, D.P., 1997. Prediction of water quality in lakes and reservoirs. Part II. Model calibration, sensitivity analysis and application. Ecol. Model. 96, 111–123. Smits, J.G.C., Van der Molen, D.T., 1993. Application of SWITCH, a model for sediment-water exchange of nutrients, to Lake Veluwe in The Netherlands. Hydrobiology 253, 281–300. Srinivasu, P.D.N., 2004. Regime shifts in eutrophied lakes: a mathematical study. Ecol. Model. 179, 115–130. Suzuki M.S., 1997. PhD Thesis. North Fluminense State University, Brazil, unpublished. Van der Molen, D.T., 1991. A simple, dynamic model for the simulation of the release of phosphorus from sediments in shallow, eutrophic systems. Water Res. 25, 737–744. Van Puijenbroek, P.J.T.M., Janse, J.H., Knoop, J.M., 2004. Integrated modelling for nutrient loading and ecology of lakes in The Netherlands. Ecol. Model. 174, 127–141.