Ecological Modelling 173 (2004) 313–333
Examination of structurally dynamic eutrophication model
Jingjie Zhang∗ , Sven Erik Jørgensen, Henrik Mahler
a
DFH, Institute A, Environmental Chemistry, University Park 2, DK 2100 Copenhagen 0, Denmark
Received 19 March 2003; received in revised form 16 September 2003; accepted 16 September 2003
Abstract
An ecological modelling software, named Pamolare, for planning and management of lakes and reservoirs was developed by
Jørgensen et al., 2003 [PAMOLARE Training Package, Planning and Management of Lakes and Reservoirs: Models for Eutrophication Management, UNEP DTIE IETC and ILEC, 1091 Oroshimo-cho, Kusatsu, Shiga, 525-0001, Japan, 2003]. The software
has four eutrophication models including application of structurally dynamic models. We tested two models in the Pamolare
program using data from Lake Glumsø, Denmark: one year for calibration, one year for validation and three years for a prognosis
validation. Compared with observations, the simulations by the structurally dynamic approach yielded satisfactory results over all
five years, whilst the two-layer model with trial and error approach in Pamolare did not give acceptable results. We also compared
the results with that of two previously developed models, specifically designed for Lake Glumsø. We concluded that the structurally dynamic model from Pamolare in the Lake Glumsø case gave better results than the two-layer model included in Pamolare,
and yielded equally good results as the two-eutrophication models specifically developed for Lake Glumsø. Furthermore, the
model was far less time consuming to calibrate than the three other models, due to an automatic calibration procedure.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Structurally dynamic model; Pamolare; Two-layer model; Calibration; Validation; Prognosis
1. Introduction
The International Environmental Technology Center of UNEP (IETC-UNEP) and International Lake
Environment Committee (ILEC) has recently developed a modeling software package named Pamolare
(Planning and Management of Lakes and Reservoirs),
offering four eutrophication models with different
complexity levels:
1. The Vollenweider plot (one state variable, the phosphorus or nitrogen loading in g m−2 per year);
∗
Corresponding author. Tel.: +45-35306456;
fax: +45-35306013.
E-mail address: [email protected] (J. Zhang).
2. A model with four state variables; P and N in the
water and in the sediments and several correlations
to calculate other state variables;
3. A two-layer model with 21 state variables; and
4. A structurally dynamic model (SDM) developed
from the two-layer model but applying the structurally dynamic approach (Jørgensen, 1999) for
calibration, validation and prognosis simulations in
addition to an automatic calibration for calibration
of four other important physical–chemical parameters, which are the settling velocity and the decomposed rate of the detritus, and the release rates
of nitrogen and phosphorus in the sediments.
SDM is a recent development in ecological modeling. The parameters are constantly varied to account for the adaptations and the shifts in the species
0304-3800/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2003.09.021
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
composition. The changes of the parameters are either
based on expert knowledge—under such and such
conditions it is known that such and such species will
be best fitted to the conditions or by an optimization of
a so-called goal function that can describe the fitness
to the changed conditions. Exergy has been the most
applied goal function in the development of SDM
(Jørgensen et al., 2002; Zhang et al., 2003a,b). Exergy
expresses the distance from the thermodynamic equilibrium and can therefore be considered a measure
of survival. The more biomass and information the
system possesses, the bigger distance from thermodynamic equilibrium, and the more exergy and the better
survival it has. Survival is biomass and information
(including how to survive by suitable regulation mechanisms and feed backs) (Jørgensen,1999). Exergy can
be approximately calculated as
βi Ci (Jørgensen
and Marques, 2001a; Xu et al., 2001), where βi is a
weighting factor accounting for the information that
the various species are carrying in their genes (βi , for
example, is 1 for detritus and 0 for inorganic components at their highest oxidation states) and Ci is the
concentration of the various organism and component of the ecosystem. Exergy is thereby expressed in
detritus exergy equivalents (Jørgensen, 1999). Fig. 1
shows how a SDM is developed with exergy as a goal
function. At a certain frequency, a test is conducted
to determine if a change in the selected biological
parameters, would give the system a higher exergy. If
1. Use literature range for all
sensitive parameters
Go back
4. Find parameters giving max.
exergy but with less than 45%
deviation of model/observations
it is the case, the parameters are changed accordingly.
The allowed change of the parameters and the frequency of the change are determined by the possible
rate of biological changes—the biological dynamics.
The structurally dynamic approach has been used by
the calibration of biological parameters several times
(Jørgensen et al., 2002; Zhang et al., 2003a) with
some clear improvements of the modeling results. The
SDM-approach uses the thermodynamic function exergy to express the survival as a goal function. The
details about this approach can be found in Jørgensen
et al. (2000, 2002), Jørgensen (1999, 2002) and Zhang
et al. (2003a). For development of prognosis scenarios the SDM approach has been applied in 15 case
studies, most of which, are referred to in Jørgensen
et al. (2000, 2002). It would therefore be interesting
to test the Pamolare SDM against a good database
and compare the results with other models that have
been applied on the same database to assess the differences in applicability and validate the results of the
different modeling approaches. The Glumsø-data applied for development and testing of a eutrophication
model in 1973–1984 would be a candidate for such
examinations.
Lake Glumsø is situated about 70 km south of
Copenhagen, Denmark. The main characteristics are
shown in Table 1. The provided data set has the following advantages (Jørgensen et al., 1978; Jørgensen,
1976, 1986):
2. Calibrate parameters of
physical and chemical importance
by trial and error
3. Introduce size functions to
phytoplankton and zooplankton
parameters.
5. Use parameter combinations as
calibration results if no difference
between 2 and 4.
Fig. 1. The procedure of development of SDM model with exergy as goal function.
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Table 1
Lake Glumsø characteristics
Characteristics
Drainage area (km2 )
Surface area (m2 )a
Maximum depth (m)a
Average depth (m)a
Minimum transparency (cm)b
Retention time (months)a
Temperature (◦ C)b
Dissolved oxygen saturation (%)b
Range
Annual
average
10.9
266000
2.4
1.8
20
5–6
4–25
55–175
Source: a Examination of a lake model (Jørgensen et al., 1978).
b Management of a shallow lake (Jørgensen, 1986).
- double determinations have consequently been used
for all measurements, frequent measurements (three
times a week) over a period of several weeks during
algae spring blooms are available—the data set is,
in other words, of good quality;
- the lake can be considered as a completely mixed
tank due to its small size (see Table 1). The hydrodynamics are therefore simple;
- significant changes have been recorded during a period of a few years (retention time is less than 6
months; see Table 1);
- a prognosis validation is also possible.
The objectives of this examination are:
1. Which advantages can the SDM-calibration procedure included in the Pamolare model number 4 (abbreviated SDM-p4) offer, compared with a normal
calibration procedure?
2. How do the results from the SDM-p4 model on the
Lake Glumsø case compare with the ones obtained
by other models:
- the previously applied Glumsø model (Jørgensen
et al., 1978; Jørgensen, 1976, 1986), which has
the advantage that it was developed for this particular case, but without the SDM approach (the
model is abbreviated Glumsø-78);
- the previously developed SDM approach model
for Glumsø (Jørgensen et al., 2002)—a model
(abbreviated SDM-Glumsø) was developed for
this particular case but with a simpler structure
and less state variables than the Pamolare SDM
and the earlier Glumsø-78 model; and
315
- the two-layer Pamolare model without employing the SDM approach and automatic calibration
(abbreviated 2L-p3).
3. In summary, the comparison of the results obtained
by all the four models should be able to assess the
advantages and disadvantages that result
• from the application of the SDM-approach for the
calibration, the validation and the prognosis scenarios (comparison of SDM-p4 versus 2L-p3 and
Glumsø-78, and SDM-Glumsø versus Glumsø-78);
• from the use of a specifically developed models versus a general model (comparison of Glumsø-78 and
SDM-Glumsø versus SDM-p4 and 2L-p3); and
• from inclusion of more processes and state variables (Glumsø-78, SDM-p4 and 2L-p3 versus SDMGlumsø).
2. Materials and methods
2.1. The applied models
The details of the two Pamolare models are given
below, while the details about other models including the validation results for Glumsø-78 can be found
in Jørgensen et al. (1978) and for SDM-Glumsø in
Jørgensen et al. (2002).
2.1.1. The Glumsø-78 model and SDM-Glumsø
model
The most characteristic features of Glumsø-78 are:
a two-step description of phytoplankton growth using
intracellular concentrations of nutrients as state variables and allowing a variable P–N ratio, a detailed
sediment–water nutrient exchange sub-model and a
threshold limit for zooplankton. The model has 17
state variables, but 20 state variables when diatoms are
included. If a thermocline (or halocline) is present the
model has by other case studies been expanded with
13 more state variables. The conceptual diagram for
the Glumsø-78 model is shown in Fig. 2.
The SDM-Glumsø model also has a detailed
sediment–water exchange sub-model, but a fixed P–N
ratio for phytoplankton. The model has nine state
variables. As opposed to Glumsø-78 it has only been
applied in one case study, Lake Glumsø, whereas
Glumsø-78 with modifications of the state variables
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
N
P
Z,
NZ, PZ
PF,
NF
Phy, NP,
PP, CP
ND,
PD
PSED,
NSED
PB
PI
Fig. 2. The conceptual diagram of the Glumsø-78 model.
and the equations has been applied to more than 23
case studies.
2.1.2. The two-layer Pamolare model (2L-p3)
The two-layer model program was developed
by IETC-UNEP and ILEC and Prof. H. Tsuno at
Kyoto University. This model (2L-p3) applies a
trial-and-error calibration. It divides the lake into
two layers. For a shallow lake the 0.5 m water close
to the sediment is considered the hypolimnion. The
model has eight state variables in the epilimnion and
eight state variables in the hypolimnion, including
three state variables regarding the nutrients in the
sediment (see the conceptual diagram in Fig. 3), plus
dissolved oxygen in the epilimnion and hypolimnion.
Phytoplankton is divided into three state variables:
blue-green algae, diatoms and other phytoplankton
classes. Totally, the model may have up to 21 state
variables. The basic equations of this model are presented in Appendix A, Tables A.1–A.4.
In this model, growth of three groups of phytoplankton (diatoms, blue-green algae and the other phytoplankton) is by photosynthesis which is governed
by the uptake of inorganic nitrogen and phosphorus
(paths 1–3), growth of and decays to detritus and
inorganic compounds with oxygen consumption and
mortality (paths 4–6). Zooplankton species, which are
filter-feeders, grow via predation on phytoplankton
(paths 7–9) and the process of mortality and decomposition to detritus and inorganic matter with oxygen
consumption (path 10). A residual part of the phytoplankton in filter-feeding predation is directly transformed to detritus. Detritus settles in the sediment
(sedimentation rate, vSD ) and decomposes to dissolved
organics (path 11), which is then degraded to inorganic
matter thus consuming oxygen (path 12).
All of these paths occur in the upper layer of water column (epilimnion), and all paths, except for the
growth of each group of phytoplankton because of
the lack of light, also occur in the lower layer (hypolimnion). Release of inorganic nitrogen and phosphorus and dissolved organics (paths 13–15) from the
sediments, and resuspension from the sediments (path
16), occur in the lower layer of water column. Exchange of the state variables between the upper and
lower layers is expressed by dispersion, KD . The extent of exchange depends on the stratification, reflected
in the value of KD . In the model, artificial circulation
may also be incorporated by adding the circulation
flow rate between the two layers.
Release rates of inorganic nitrogen and phosphorus and dissolved organics (paths 13–15) can be determined by experiments or by a data-fitting method.
These rates can be calculated by the material balance in the sediments. In this case, releasable sediment nitrogen (Nsed ), releasable sediment phospho-
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Fig. 3. The conceptual diagram of the two-layer model.
317
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
rus (Psed ), and releasable sediment organics (Csed )
are considered as state variables and increase and decrease by sedimentation of detritus, and the release
processes from the sediments. The release rates of inorganic nutrients increase by an order of magnitude
under anoxic conditions. Changes of dissolved oxygen
concentrations occur by re-aeration at the water surface, primary production and consumption in the upper
water column, and consumption by processes related
to water–sediment interface.
3.
4.
2.1.3. The SDM-Pamolare model (SDM-p4)
This model deviates from the two-layer model,
2L-p3, by the following features:
5.
1. Phytoplankton covers all classes of phytoplankton,
blue-green algae are found by the equations presented in Appendix A, Table A.5, because the structurally dynamic approach accounts for adaptations
and shifts in species composition.
2. The sizes of phytoplankton and zooplankton are
varied and determined by optimization of the exergy level in the model. Exergy is found as phytoplankton biomass × 3.8 + zooplankton biomass ×
35+detritus biomass (see the equation in Jørgensen
et al., 2000 and Jørgensen, 2002). The exergy optimization takes place with a selected frequency
from 5 to 30 days, and the possible change of the
sizes within a range between 0.02 and 0.25 in log
6.
scale (␮m3 ) can also be selected. A graph showing
the change in size as function of time produced by
the software is shown in Fig. 4.
The sizes of phytoplankton and zooplankton determine the parameters associated with these two state
variables. The equations can be found in Appendix
A, Table A.5.
As the zooplankton mortality by the formulation KdZ = 0.15 − 0.02 × LVZ only covers the
non-predation mortality, an extra parameter covering the predation mortality is introduced.
A carrying capacity of zooplankton is introduced as
a parameter. It was previously found necessary to
introduce this parameter in the Glumsø-78 model
(Jørgensen and Bendoriccho, 2001b).
Four parameters can be calibrated automatically:
sedimentation rate of detritus, decomposition rate
of detritus, sediment release rates of phosphorus
and nitrogen. The criteria are the minimizing sum
square of the standard deviations for all of the
state variables, based on the differences between
observed and simulated values, giving a weighting
factor of 10 to phytoplankton, of 3 to zooplankton,
of 2 to the two nutrients (nitrogen and phosphorus) and of 1 to the remaining state variables. It
is possible to select the state variables that should
be included in the calculations, as some state variables in some lake studies may not be measured
or may be measured with at a frequency or ac-
Fig. 4. The size change of phytoplankton and zooplankton over 1 year simulation in the validation.
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Table 2
Characteristic features of the four models
319
trates the different state variables applied for the four
models.
Model
No. of state
variables
SDM
General/
specific
No. of
layers
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
17–20
9
21
19
No
Yes
No
Yes
Specific
(Specific)
General
General
1 (−2)
1
2
2
2.2. The simulations
curacy that are deemed insufficient for modeling
purposes. For the strategy of the calibration procedure, it is recommended first to test a relatively
wide range of these four parameters and afterwards use the automatic calibration to fine tune the
calibration.
A comparison of the characteristic features of all
the four models is shown in Table 2. Table 3 illus-
All the four models were calibrated using one year’s
observed data set commencing 15 October 1974. Another year’s records between 1 April 1973 and 31
March 1974 were employed for the validation of the
two Pamolare models (2L-3p and SDM-p4). The prognoses resulted in a three years’ period between 1 April
1981 and 1 April 1984, when the wastewater was discharged into the down-stream of the lake, starting 1
April 1981. The phosphorus loading was thereby correspondingly reduced 88%. In order to be fully synchronous with the calibration, the simulation for the
prognoses was conducted for the period between 15
October 1980 and 14 October 1984.
Table 3
The applied state variables in the Glumsø-78, SDM-Glumsø, 2L-p3 and SDM-p4 and models
State variables
Dissolved nitrogen
Soluble reactive phosphorus
Total phytoplankton
Diatom
Blue-green algae
Other phytoplankton
Nitrogen in phytoplankton
Phosphorus in phytoplankton
Zooplankton
Nitrogen in zooplankton
Phosphorus in zooplankton
Nitrogen in fish
Phorsphorus in fish
Detritus
Nitrogen in detritus
Phosphorus in detritus
Dissolved organics
Dissolved oxygen
Nitrogen in sediment
Phosphorus in sediment
Releasable sediment nitrogen
Releasable sediment phosphorus
Releasable sediment dissolved organics
Nitrogen in pore water
Phosphorus in pore water
Biologically released P from sediment
a
Glumsø-78
X
X
X
X
X
X
X
X
X
X
SDM-Glumsø
2L-p3a
SDM-p4a
E
H
X
X
X
X
X
X
X
X
X
X
X
X
S
E
H
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
S
X
X
X
X
X
X
X
X
X
X
X
The two models cover state variables in epilimnion (E), hypolimnion (H) and sediments (S).
X
X
X
X
X
X
320
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
0.500
Phytoplankton (mg Chl. a/l)
0.450
0.400
Obs.
0.350
SDM
0.300
2L-P3
0.250
0.200
0.150
0.100
0.050
0.000
0
50
100
150
200
250
300
350
Time (days)
Fig. 5. The comparison of simulated results obtained by calibrations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
phytoplankton.
Fig. 5 shows the phytoplankton results obtained by
the two Pamolare models (2L-p3 and SDM-p4) after applying exergy optimization calibration for the
SDM-p4 model and using a trial and error calibration for the 2L-p3 model. The calibration is, however,
much more cumbersome with the two-layer model
than with the SDM, because the exergy optimization
procedure and the automatic calibration allow a relatively rapid determination of 10 important parameters. The results of the nutrients calibration as function
3. Results
3.1. The results obtained by the two Pamolare models
The results obtained by Glumsø-78 and SDMGlumsø have previously been published. The details
are given in Jørgensen et al. (1978) and Jørgensen
(1986, 2002). The results will be summarized in tables, where all the four models are compared in the
next section.
Dissolved N (mg N/l)
10.000
9.000
Obs.
8.000
SDM
7.000
2L-P3
6.000
5.000
4.000
3.000
2.000
1.000
0.000
0
50
100
150
200
250
300
350
Time (days)
Fig. 6. The comparison of simulated results obtained by calibrations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
dissolved nitrogen.
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
321
Obs.
2.5
Dissolved P (mg P/l)
SDM
2
2L-P3
1.5
1
0.5
0
0
50
100
150
200
250
300
350
Time (days)
Fig. 7. The comparison of simulated results obtained by calibrations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
dissolved reactive phosphorus.
of time for the two models are presented in Figs. 6
and 7.
Figs. 8–10 show the comparisons of the validated
results from SDM-p4 and 2L-p3 with the observations for phytoplankton and nutrients (N and P),
respectively. Compared with the results in the calibration, SDM-p4 gained a better performance in the
validation in simulating the seasonal dynamics of the
nutrients, especially the soluble reactive phosphorus.
The validated result for phytoplankton also produced
an acceptable result with smaller than 50% S.D. (see
Table 5).
Figs. 11 and 12 show the results of the prognoses for
phytoplankton from the SDM-p4 and the 2L-p3 models, respectively. The predicted 3-year transparency
by the SDM-p4 model is presented in Fig. 13. The
results demonstrate that though there were differences
in the calibration and validation results between the
two models, there were little differences between the
models in the prognoses in the first year. But the big
differences were found in the second year, especially
in the third year the 2L-p3 predicted no significant
change compared with the previous years’ results.
Compared with the two-layer model, it is obvious that
0.8
Phytoplankton (mg Chl. a/l)
0.7
Obs.
0.6
SDM
0.5
2L-P3
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
350
T ime (days)
Fig. 8. The comparison of simulated results obtained by validations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
phytoplankton.
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
20
Dissolved N (mg N/l)
18
16
Obs.
14
SDM
12
2L-P3
10
8
6
4
2
0
0
50
100
150
200
250
300
350
Time (days)
Fig. 9. The comparison of simulated results obtained by validations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
dissolved nitrogen.
the validation and prognosis validation using SDM-p4
also gave better results. It can be seen that for transparency, the SDM-p4 model made good predictions,
too. Over all the 3-year simulation, the prognosis validation yielded a very good result for phytoplankton
in the first spring, but it under-estimated the biomass
of phytoplankton in the second and third spring.
However, the SDM-p4 gave a right description of
the trend of the phytoplankton change over 3-year
prediction.
3.2. Comparison of the calibration, the validation
and the prognosis validation results of the four models
Tables 4 and 5 respectively give the standard deviation resulting from the calibrations of the four models and the validation of the 2L-p3 model and the
SDM-p4 model. The standard deviation was calculated
as the average values of the annual phytoplankton values of (the observed phytoplankton concentration −
the simulated phytoplankton concentration) × 100/the
2.5
Obs.
Dissolved P (mg P/l)
2
SDM
2L-P3
1.5
1
0.5
0
0
50
100
150
200
250
300
350
Time (days)
Fig. 10. The comparison of simulated results obtained by validations of SDM-p4 and 2L-p3 models in Pamolare with observed results of
dissolved reactive phosphorus.
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
323
Fig. 11. The prognosis result from 15 October 1980 to 14 October 1984 for phytoplankton obtained by SDM-p4 in Pamolare.
Fig. 12. The prognosis result from 15 October 1980 to 14 October 1984 for phytoplankton (represented as diatom) obtained by 2L-p3 in
Pamolare.
Fig. 13. The prognosis result from 15 October 1980 to 14 October 1984 for transparency obtained by SDM-p4 in Pamolare.
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Table 4
Comparison of standard deviations and correlation coefficients
from the calibrated results among the four models
Table 5
Comparison of standard deviations and correlation coefficients
from the calibrated results between SDM-p4 and 2L-p3 models
Species
Model
Correlation
coefficients
Species
Model
Phytoplankton
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
34
27
47
25
0.86
0.93
0.59
0.80
Phytoplankton
2L-p3
SDM-p4
51
47
Dissolved nitrogen
2L-p3
SDM-p4
>100
54
Negative
0.67
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
50
–
>100
31
Soluble reactive
phosphorus
2L-p3
93
Negative
SDM-p4
42
0.59
Dissolved nitrogen
Soluble reactive
phosphorus
S.D. (%)
0.72
–
Negative
0.86
Glumsø-78
37
0.69
SDM-Glumsø
2L-p3
SDM-p4
–
59
77
–
0.33
0.14
average observed concentration. The calculated correlation coefficients for a plot observed versus simulated phytoplankton in the calibration and validation
years are also presented in these tables. Phytoplankton is the most important state variable for a eutrophication model and it is therefore naturally to focus on
the discrepancy between observed and simulated phytoplankton in the validation.
The calibrated results using the four models can
be found in Table 6, where it compares the obtained
S.D. (%)
Correlation
coefficient
0.39
0.40
values of phytoplankton, nitrogen and phosphorus in
the period between 15 October 1974 and 14 October
1975. The results of the prognosis validation in the
period 1981–1984 for the four models are summarized in Tables 7 and 8.
4. Discussion
4.1. Comparison of the results obtained by the
calibration of the four models
The acceptable standard deviations and high correlation coefficients demonstrate that the calibrated
Table 6
Important calibration results
Species
Methods
Results
Spring
Summer
Autumn
Max.
value
Date
Max.
value
Date
Max.
value
Date
42
58
32
53
52
Early May
Early May
Early May
Early May
Middle April
60
70
37
59
60
Middle July
Late July
Middle July
Middle June
Middle July
47
36
35
26
49
Early September
Early September
Early September
Late August
Early September
Phytoplankton (mg/l)
Observations
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
Soluble nitrogen (mg N/l)
Observations
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
7.0
6.4
8.1
8.4
7.0
Early February
Early March
Middle February
Late February
Early February
–
–
–
–
–
–
–
–
–
–
2.5
4.0
4.7
9.0
0.47
Middle August
Early September
Early October
Middle October
Early October
Soluble reactive
phosphorus (mg P/l)
Observations
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
1.3
0.7
2.6
1.8
1.5
Middle March
Middle March
Middle March
Early February
Early February
2.0
0.8
–
–
–
Early August
Early August
–
–
–
2.0
1.5
1.5
1.7
0.1
Late August
Middle October
Middle October
Middle October
Middle October
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
325
Table 7
The results of 3 year prognosis validation after the treatment of wastewater
Species
Methods
Results
First spring
Second spring
Third spring
Max.
value
Date
Max.
value
Date
Max.
value
Date
Phytoplankton (mg/l)
Observations
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
55
57
42
61
51
Middle April
Early May
Middle April
Early June
Middle April
50
38
36
56
24
Early May
Early May
Early April
Middle May
Early February
38
30
28
55
18
Late March
Early May
Early April
Middle May
Early February
Min. transparency (cm)
Observations
Glumsø-78
SDM-p4
20
20
20
Middle April
Early May
Middle April
25
30
34
Early May
Early May
Early February
50
45
41
Late March
Early May
Early February
results of the phytoplankton are satisfactory for all
the models, except for the two-layer Pamolare model,
where it was difficult to simulate the phytoplankton
concentration as function of time very well within the
time framework, which has been devoted to the trial
and error calibration. It was also very time consuming
to formulate, calibrate and validate the Glumsø-78
model. We might therefore not exclude that the
two-layer model may be able to yield better results,
if more time was devoted to the calibration. This
does, however, not change the facts that the two-layer
model, due to the high number of parameters, is very
difficult and time consuming to calibrate. The two
SDM approaches of SDM-p4 and SDM-Glumsø both
gave a slightly better calibration of phytoplankton, although the calibration results for the soluble reactive
phosphorus concentration and other state variables
such as zooplankton were not better for these two
models than for the Glumsø-78 model. This may
probably be due to the high weighting factor (10)
given to phytoplankton in the calibration, while other
state variables had weighting factors less than 3.
Therefore, it is clear that the SDM-calibration procedure (applied in both SDM-Glumsø and SDM-p4
models) offers better and faster calibration than the
two other approaches, because the results reflect seasonal dynamic changes in the ecosystem. This can be
found in Fig. 14, where the two SDM model gave
phytoplankton higher growth rates in the summer and
lower ones in the winter, as it should also be expected
from an ecological view-point: higher specific surface in the summer when there is more competition
Table 8
Comparison of the four tested models
Criteria
Items
Glumsø-78
SDM-Glumsø
2L-p3
SDM-p4
Calibration
Time consuming
Phytoplankton
Date of max. phytoplankton
Nutrient N
Nutrient P
Phytoplankton
Nutrient N
Nutrient P
Phytoplankton
Date of max. phytoplankton
Min. transparency
Long
Very good
Good
Good
Good
Gooda
Gooda
Gooda
Very good
Good
Very good
Medium
Good
Very good
Good
Very Good
–
–
–
Very good
Very good
–
Long
Good
Acceptable
Acceptable
Good
Acceptable
Not acceptable
Not acceptable
Very good
Not good
–
Short
Very good
Good
Good
(Acceptable)
Good
Good
Good
Good
Good
Very good
Validation
Prognosis
a
Another year applied for the validation than in this study.
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Range of the measured rates
7
The average measured rate
Glumsø-78
Max. Growth rate of Phy (1/d)
6
5
4
3
SDM -P4
SDM -Glumsø
2
2L-P3
1
0
50
100
150
200
250
300
350
Time (days)
Fig. 14. The maximum growth rates of phytoplankton obtained by the four models over 1 year simulation in the calibration.
for the nutrients (Peters, 1983). It was determined in
the Lake Glumsø case (Jørgensen 76 and Jørgensen
et al., 1978) that the measured maximum growth rate
was 4.1 ± 1.8 per day (see Fig. 14). It is noticeable
that the changes of the growth rates for the two SDM
models are in the range of the measured growth rate
(the change of the rate for SDM-Glumsø is almost
covered), whereas the calibrated growth rates for the
Glumsø-78 model is 4.1 and for two-layer model is
in discordance with the measured value.
4.2. Comparison of the results obtained by the
validations and prognoses of the four models
Compared with the observed results, the prognoses
were satisfactory for all the three applied models,
whereas the two-layer model yielded an acceptable
result but with the maximum growth of phytoplankton
appearing too late. It was most probably due to the unsatisfactory calibration. The two structurally dynamic
models generally gave for the prognosis validation a
better accordance for the date of the phytoplankton
peak and the minimum transparency, which was in
accordance with the idea behind the introduction of
the structurally dynamic modeling approach to account for a shift in species composition, included the
seasonal adaptation and structural change of species
in the ecosystem (Zhang et al., 2003a), but it cannot
be excluded that these results were at least partly
due to the application of a weighting factor of 10
for phytoplankton in the automatic calibration procedure for SDM-p4. The SDM-p4 model was tested
with an acceptable result. The over-all results were,
however, not better than the previously developed
Glumsø-78 model, which was specifically developed for the Glumsø conditions. In some respects
the SDM-p4 model performed better: the validation
of the phytoplankton and the prognosis validation
of the dates at which the various peaks of the state
variables occur (in the first and the third years); in
some other respects the Glumsø-78 model gave better
results: the validation of the nutrients observations
and the size of the peaks at the prognosis validation.
The Glumsø-78 model is tailored to Lake Glumsø
while SDM-p4 is a general model, and the same
accuracy should normally not be expected for the
validation and prognosis validation. SDM-Glumsø
has only been tested on the Lake Glumsø data but
in principle it was not developed particularly for
Glumsø, but rather for testing an alternative calibration method by the application of the structurally dynamic approach. Glumsø-78 was developed through
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
testing a number of expressions for the key processes, and some of the sub-models were developed
on basis of in situ or laboratory measurements made
on basis of Glumsø-conditions. It could therefore be
expected (as a hypothesis), that Glumsø-78 would
yield a better validation and prognosis validation
than SDM-p4 that has been developed for a general
use.
For SDM-Glumsø it has previously been concluded that the structurally dynamic approach offers
a better calibration and validation for phytoplankton (Jørgensen et al., 2002), but it can not be concluded that it is the case for SDM-p4, although it
is possible to conclude that the results of the validation and the prognosis validation of phytoplankton were satisfactory for both structurally dynamic
models.
4.3. Comparison of models with the inclusion of
more processes and state variables (Glumsø-78,
SDM-p4 and 2L-p3) with a simple structural model
(SDM-Glumsø)
The results of the SDM-p4 model are as good as
that of the earlier SDM-Glumsø model. Provided that
a good database is available, it might be expected that
a more complex model may yield a better result. In a
comparison of the two models, we found:
1. The calibrated phytoplankton in Table 4 shows that
SDM-Glumsø is slightly better than SDM-p4.
2. The time of appearing peaks is equally good for
the two models, but the sizes of the peaks are
clearly better for SDM-p4 than for SDM-Glumsø
(see Table 6).
3. The prognosis validation is better for SDM-p4
in the first year, but better for SDM-Glumsø in
the second and third year. This better performance is clearly demonstrated in the Glumsø-78
model, where it has more state variables than the
SDM-Glumsø model. However, the SDM-Glumsø
clearly gave a better performance than the 2L-3p
model.
5. Conclusions
The two Pamolare models (SDM-p4 and 2L-p3)
have been tested against a high quality database from
327
the Lake Glumsø. It was found that:
1. The calibration with the SDM models is considerably less time consuming than the usually applied
calibration.
2. The two SDM models yielded a calibration, a
validation and a prognosis validation approximate
to the same quality as the Glumsø-78 model. It
indicates that the general Pamolare SDM model
(SDM-p4) is with other words as effective as the
Glumsø-78 model, a specially developed model
for Lake Glumsø.
3. The Pamolare 2L-3p model did not yield as good
results as the three other models probably owing to
the difficulty to calibrate the model and to the lack
of structurally dynamic approach.
4. The result from the SDM-Glumsø model with nine
state variables reflects that the more processes may
not necessarily be needed, provided that the most
important processes and state variables are already
included in the model.
For all the four models it should not be expected
that they could give very accurate values when applied for a prognosis, but that they only can give relatively acceptable values for scenarios testing different environmental strategies. This is what eutropication models can offer today—not exact prognoses but
relatively acceptable results. The structurally dynamic
approach has demonstrated the advantages over other
methods. It may be possible with the results presented
here to conclude, that the SDM-p4 model—a general
model may be applied equally successfully in other
case studies. However, in order to have a more accurate result it is suggested to develop a specific SDM
model.
Acknowledgements
We thank Mr. Vicente Santiago-Fandino for kindly
providing the Pamolare program for this study. The
authors are grateful to Dr. Gideon Gal and the two
anonymous referees for their critical reading of this
manuscript and giving constructive and valuable comments and suggestions that have improved the clarity
of the paper.
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Appendix A
See Tables A.1–A.5.
Table A.1
Rate equation of each process
Process
Equation
1. Growth of diatom (mg Chl.a/(l·day))
2. Growth of blue-green algae
(mg Chl.a/(l·day))
R1 = µM1 × fT1 × fI1 × fN1 × M1
R2 = µM2 × fT2 × fI2 × fN2 × M2
3. Growth of other phytoplankton
(mg Chl.a/(l·day))
R3 = µM3 × fT3 × fI3 × fN3 × M3
4. Death of diatom (mg Chl.a/(l·day))
R4 = kdM1 θM1
(T −20)
DO
M1
KDO + DO
(T −20)
R5 = kdM2 θM2
DO
M2
KDO + DO
(T −20)
DO
M3
KDO + DO
5. Death of blue-green algae
(mg Chl.a/(l·day))
6. Death of other phytoplankton
(mg Chl.a/(l·day))
R6 = kdM3 θM3
7. Grazing of diatom by zooplankton
(mg Chl.a/(l·day))
R7 = Fmax Z
KmZ
T
M1Z
20 KmZ + (M1 + M2 + M3)
8. Grazing of blue-green algae by
zooplankton (mg Chl.a/(l·day))
R8 = Fmax Z
T
KmZ
M2Z
20 KmZ + (M1 + M2 + M3)
9. Grazing of other phytoplankton by
zooplankton (mg Chl.a/(l·day))
R9 = Fmax Z
KmZ
T
M3Z
20 KmZ + (M1 + M2 + M3)
10. Death of zooplankton (mg DW/(l·day))
R10 = kdZ θZ
11. Decomposition of detritus
(mg DW/(l·day))
R11 = kdD θD
12. Decomposition of dissolved organics
(mg COD/(l·day))
R12 = kdC θC
13. Release of nitrogen from sediment
(mg N/(l·day))
Model (a) R13a = ksrAN
A
1000VL
Model (b) R13b = ksrN Nsed
Hsed
H
14. Release of phosphorus from sediment
(mg P/(l·day))
Model (a) R14a = ksrAP
A
1000VL
Model (b) R14b = ksrP Psed
Hsed
H
15. Release of dissolved organics from
sediment (mg COD/(l·day))
Model (a) R15a = ksrAC
A
1000VL
Model (b) R15b = ksrC Csed
Hsed
H
16. Release of detritus from sediment
(mg DW/(l·day))
R16 = ksrD
17. Re-aeration (mg O2 /(l·day))
R17 = kL
18. Oxygen consumption by sediment
(mg O2 /(l·day))
R18
(T −20)
DO
Z
KDO + DO
(T −20)
D
(T −20)
DO
C
KDO + DO
A
1000VL
A(DOsat − DO)
VL
A
(T −20)
= kDO θDO
1000VL
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
329
Table A.2
Rate equation of each process
Effect of water temperature: fTk (k = 1, 2, 3) (–)
Effect of solar radiation: fIk (k = 1, 2, 3) (–)
Effect of nutrients: fNk (k = 1, 2, 3) (–)
Re-aeration rate constant (m/day)
Saturated dissolved oxygen (mg O2 /l)
fTk = −
(T − ToptMk )2
+1
2
ToptM
k
e
I
I
fIk =
exp −
exp(−εh) − exp −
εh
IoptMk
IoptMk
N
P
KnMk + N KpMk + P
√
KL = max(0.04, 0.782 W − 0.317W + 0.0372W 2 )
8.0
DOsat = 16.5 −
T
22.0
fNk =
Table A.3
Materials balance equations
Upper layer
dCjU
QU CjIN − QUout CjU − δh QUL CjU + (1 − δh )QLU CjL
CjU dHU
Kd A
=
+ FjU −
(CjU − CjL ) −
dt
VU
%HVU
HU dt
QU DIN − QUout DU − δh QUL DU + (1 − δh )QLU DL
vsD DU
Kd A
dDU
DU dHU
=
+ FjU −
−
(DU − DL ) −
dt
VU
HU
%HVU
HU dt
j = N, P, M1, M2, M3, Z, C, DO, Ci = concentrations of j
D: concentrations of detritus
Fi : rate of change of j
U: upper layer, L: lower layer
Lower layer
dCjL
QL CjIN − QLout CjL + δh QUL CjU − (1 − δh )QLU CjL
CjL dHL
Kd A
=
+ FjL +
(CjU − CjL ) −
dt
VU
%HVU
HL dt
QL DIN − QLout DL + δh QUL DU − (1 − δh )QLU DL
dDL
vsD DU
vsD DL
Kd A
DL dHL
=
+ FjL +
−
+
(DU − DL ) −
dt
VU
HL
HL
%HVU
HL dt
j = N, P, M1, M2, M3, Z, C, DO, Ci : concentrations of j
D = concentrations of detritus
Fi : rate of change of j
U: upper layer, L: lower layer
Sediment part
dCjS
vsD DL
= FjS +
γDj fsed j
dt
Hsed
j = Nsed , Psed , Csed , Ci : concentrations of j
Fi : rate of change of j
U: upper layer, L: lower layer
δh = 1: when the thermocline goes up
δh = 0: when the thermocline goes down
In the upper layer, the material balance equation for each state variable consists of input (inflow rate and concentration) from the watershed,
output (flow out) from the layer, the rate of change Fi and the exchange rate between the upper and lower layer. For detritus, the
sedimentation rate is also incorporated.
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J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Table A.4
Material balance equations—variable depth in calculation
Appendix B. Notation
State valuables
N (mg N/l)
P (mg P/l)
Phy (mg Chl.a/l)
M1 (mg Chl.a/l)
M2 (mg Chl.a/l)
M3 (mg Chl.a/l)
NP (%)
PP (%)
CP (mg C/l)
Z (mg DW/l)
NZ (%)
PZ (%)
D (mg DW/l)
ND (mg/l)
PD (mg/l)
NF (%)
PF (%)
C (mg COD/l)
DO (mg O2 /l)
Nsed (mg N/l-sed)
Psed (mg P/l-sed)
PI (mg P/l)
PB (mg P/l)
Table A.5
The equations of parameters of phytoplankton and zooplankton
associated with the their sizes
Max. growth rate of phytoplankton: µm = 3.0 − 0.3 × LVP
Max. growth rate of zooplankton: µmZ = 0.8 − 0.13 × LVZ
Michaelis Menten’s half saturation constant for grazing by
zooplankton: KmZ = −0.15 + 0.66 × LVZ
Michaelis Menten’s half saturation constant for nitrogen uptake
by phytoplankton: KnM = 0.047 + 0.067 × LVP
Michaelis Menten’s half saturation constant for phosphorus
uptake by phytoplankton: KpM = 0.0047 + 0.0067 × LVP
Mortality rate of phytoplankton: KdM = 0.8 − 0.13 × LVP
Mortality rate of zooplankton: KdZ = 0.15 − 0.02 × LVZ
Ratio of blue-green algae:
((5 − (concentration of N/concentration of P))/5)2
0; if concentration of N > (5 × concentration of P)
Csed (mg C/l-sed)
Cj (mg/l)
Dissolved nitrogen
Soluble reactive
phosphorus
Total phytoplankton
Diatom
Blue-green algae
Other phytoplankton
Nitrogen in phytoplankton
Phosphorus in
phytoplankton
Carbon in phytoplankton
Zooplankton
Nitrogen in zooplankton
Phosphorus in
zooplankton
Detritus
Nitrogen in detritus
Phosphorus in detritus
Nitrogen in fish
Phosphorus in fish
Dissolved organics
Dissolved oxygen
Releasable sediment
nitrogen
Releasable sediment
phosphorus
Phorsphorus in the pore
water
Biologically released
phosphorus in the
sediment
Releasable sediment
dissolved organics
Concentrations of j
Lake conditions (morphology and mixing condition)
V (m3 )
Volume of each part
H (m)
Water depth of each part
A (m2 )
Surface area of each part
%H (m)
Water depth between each
part
Hsed (m)
Sediment depth
Qc (m3 /(d unit))
Circulation flow
Kd (m2 /d)
Mixing rate
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Appendix B (Continued )
Values of constants and coefficients
Containing ratio
γM1P (mg P/mg Chl.a)
P: Chl.a, diatom
γM2P (mg P/mg Chl.a)
P: Chl.a, blue-green
algae
γM3P (mg P/mg Chl.a)
P: Chl.a, other
phytoplankton
γ ZP (mg P/mg DW)
P: dry weigh,
zooplankton
γ CP (mg P/mg COD)
P: COD, dissolved
organics
γ DP (mg P/mg DW)
P: dry weigh,
sediment
γM1N (mg N/mg Chl.a)
N: Chl.a, diatom
γM2N (mg N/mg Chl.a)
N: Chl.a, blue-green
algae
γM3N (mg N/mg Chl.a)
N: Chl.a, other
phytoplankton
γZN (mg N/mg DW)
N: dry weight,
zooplankton
γCN (mg N/mg COD)
N: COD, dissolved
organics
γDN (mg N/mg DW)
N: dry weigh,
sediment
γDC (mg COD/mg DW)
COD: dry weigh,
sediment
Conversion coefficient
γ M1DO (mg O2 /mg Chl.a) DO: diatom
γ M2DO (mg O2 /mg Chl.a) DO: blue-green
algae
γ M3DO (mg O2 /mg Chl.a) DO: other
phytoplankton
γ CDO (mg O2 /mg COD) DO: dissolved
organics
γ M1Z (mg DW/mg Chl.a) Zooplankton: diatom
γ M2Z (mg DW/mg Chl.a) Zooplankton:
blue-green algae
γ M3Z (mg DW/mg Chl.a) Zooplankton: other
phytoplankton
DO: zooplankton
γ ZDO (mg O2 /mg DW)
γ M1D (mg DW/mg Chl.a) Detritus: diatom
γ M2D (mg DW/mg Chl.a) Detritus: blue-green
algae
γ M3D (mg DW/mg Chl.a) Detritus: other
phytoplankton
331
Appendix B (Continued )
Yield coefficient
YM1D (–)
YM2D (–)
YM3D (–)
YZD (–)
YM1Z (–)
YM2Z (–)
YM3Z (–)
Respiration of diatom
Respiration of blue-green
algae
Respiration of other
phytoplankton
Respiration of
zooplankton
Prediction of diatom
Prediction of blue-green
algae
Prediction of other
phytoplankton
Extinction of sunlight
α (1/m) (m3 /g Chl.a) Extinction coefficient,
algal
)0 (1/m)
Extinction coefficient,
water
Diatom
IoptM1 MJ/(m2 d))
Optimum solar radiation
ZoptM1 (◦ C)
Optimum temperature
µM1 (1/d)
Maximum growth rate
Half saturation constant,
KpM1 (mg P/l)
P
KnM1 (mg N/l)
Half saturation constant,
N
kdM1 (1/d)
Mortality rate
θ M1 (–)
Temperature constant
Blue-green algae
IoptM2 (MJ/(m2 d))
ZoptM2 (◦ C)
µM2 (1/d)
KpM2 (mg P/l)
KnM2 (mg N/l)
kdM2 (1/d)
θ M2 (–)
Other phytoplankton
IoptM3 (MJ/(m2 d))
ZoptM3 (◦ C)
µM3 (1/d)
KpM3 (mg P/l)
Optimum solar radiation
Optimum temperature
Maximum growth rate
Half saturation constant,
P
Half saturation constant,
N
Mortality rate
Temperature constant
Optimum solar radiation
Optimum temperature
Maximum growth rate
Half saturation constant,
P
332
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Appendix B (Continued )
KnM3 (mg N/l)
kdM3 (1/d)
θ M3 (–)
Zooplankton
FmZ (l/(d mg DW))
FmZ (mg Chl.a/l)
kdZ (1/d)
θ Z (–)
Detritus
vsD (m/d)
kdD (1/d)
θ D (–)
Dissolved organic
kdC (1/d)
θ C (–)
Floatation of sediment
ksrD (mg DW/(m2 d))
Appendix B (Continued )
Half saturation
constant, N
Mortality rate
Temperature constant
Maximum growth rate
Half saturation constant
Mortality rate
Temperature constant
Sedimentation velocity
Decomposition rate
Temperature constant,
decomposition
ksrC (1/d)
Weather condition
T (◦ C)
I (MJ/(m2 d))
W (m/s)
Q (m3 /d)
Qout (m3 /d)
QUL (m3 /d)
QLU (m3 /d)
Decomposition rate
Temperature constant,
decomposition
δh (–)
Release rate
Oxygen consumption rate by sediment
kDO (mg O2 /(m2 d))
Oxygen consumption
constant
Half saturation
KDO (mg O2 /l)
constant, DO
θ DO (–)
Temperature constant,
decomposition
Release rate from sediment (a)∗
ksrAP (mg P/(m2 d))
Release rate,
phosphorus
ksrAN (mg N/(m2 d))
Release rate, nitrogen
2
ksrAC (mg COD(m d)) Release rate, dissolved
organics
Release rate from sediment (b)∗
fsedP (–)
Fraction ration,
phosphorus
ksrP (1/d)
Release rate,
phosphorus
fsedN (–)
Fraction ration,
nitrogen
ksrN (1/d)
Release rate, nitrogen
fsedC (–)
Fraction ration,
dissolved organics
Rate equations
R1 (mg Chl.a/(l·day))
R2 (mg Chl.a/(l·day))
R3 (mg Chl.a/(l·day))
R4 (mg Chl.a/(l·day))
R5 (mg Chl.a/(l·day))
R6 (mg Chl.a/(l·day))
R7 (mg Chl.a/(l·day))
R8 (mg Chl.a/(l·day))
R9 (mg Chl.a/(l·day))
R10 (mg DW/(l·day))
R11 (mg DW/(l·day))
R12 (mg COD/(l·day))
Release rate, dissolved
organics
∗ Selection of release
rate from sediment—1:
Method (a), 2: Method
(b)
Water temperature
Intensity of sunlight
Wind speed
Inflow rate of each part
Outflow rate of each
part
Flow rate from upper
layer to lower layer
Flow rate from lower
layer to upper layer
1: When the
thermocline goes up, 0:
when the thermocline
goes down
Growth of diatom
Growth of blue-green
algae
Growth of other
phytoplankton
Death of diatom
Death of blue-green
algae
Death of other
phytoplankton
Prediction of diatom by
zooplankton
Prediction of
blue-green algae by
zooplankton
Prediction of other
phytoplankton by
zooplankton
Death of zooplankton
Decomposition of
detritus
Decomposition of
dissolved organics
J. Zhang et al. / Ecological Modelling 173 (2004) 313–333
Appendix B (Continued )
R13 (mg N/(l·day))
R14 (mg P/(l·day))
R15 (mg C OD/(l·day))
R16 (mg DW/(l·day))
R17 (mg O2 /(l·day))
R18 (mg O2 /(l·day))
fTk (–)
fIk (–)
fNk (–)
KL (m/d)
DOsat (mg O2 /l)
Fi (mg Cj /(l d))
t (day)
LVP (log (␮m3 ))
LVP (log (␮m3 ))
Suffixes
U
L
IN
T
333
References
Release of nitrogen
from sediment
Release of phosphorus
from sediment
Release of dissolved
organics from sediment
Release of detritus
from sediment
Re-aeration
Oxygen consumption
by sediment
Temperature affecting
function for Mk
Light intensity affecting
function for Mk
Inorganic nutrient
concentration affecting
function for Mk
Re-aeration rate
constant
Saturated dissolved
oxygen
Changing rate of each
state valuable j
Time
Size of phytoplankton
Size of zooplankton
Upper layer
Lower layer
Inflow
Total
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