Ecological Modelling 165 (2003) 285–301
A general dynamic model to predict biomass and
production of phytoplankton in lakes
Lars Håkanson a,∗ , Viktor V. Boulion b
a
Department of Earth Sciences, Uppsala University, Villav. 16, 752 36 Uppsala, Sweden
b Zoological Institute of RAS, Universitskaja emb., 1, 199034 St. Petersburg, Russia
Received 13 March 2002; received in revised form 14 January 2003; accepted 12 March 2003
Abstract
This work presents a dynamic model to predict phytoplankton biomass and production. The model has been developed as an
integral part within the framework of a more comprehensive lake ecosystem model, LakeWeb, which also accounts for production
and biomass of bacterioplankton, two types of zooplankton (herbivorous and predatory), two types of fish (prey and predatory),
as well as zoobenthos, macrophytes and benthic algae. The LakeWeb-model is based on ordinary differential equations (the
ecosystem perspective) and gives seasonal variations (the calculation time, dt, is 1 week and Euler’s integration method has
been applied). The sub-model for phytoplankton presented in this work is meant to account for all fundamental abiotic/biotic
interactions and feedbacks (including predation by herbivorous zooplankton) for lakes in general. The model has not been tested
in the traditional way using data from a few well investigated lakes. Instead, it has been tested using empirical regressions based
on data from many lakes. The basic aim of this dynamic model is that it should capture typical functional and structural patterns
in many lakes. It accounts for how variations in (1) lake phosphorus concentrations, (2) water clarity, (3) lake morphometry, (4)
water temperature, (5) lake pH and (6) predation by herbivorous zooplankton influence production and biomass of phytoplankton.
An important demand for this model is that it should be driven by variables easily accessed from standard monitoring programs
and maps (the driving variables are: total phosphorus, colour, pH, lake mean depth, lake area, and epilimnetic temperatures).
We have demonstrated that the new model gives predictions that agree well with the values given by the empirical regressions,
and also expected and requested divergences from these regression lines when they do not provide sufficient resolution. The
model has been tested in a very wide limnological domain: TP values from 3 to 300 !g/l, which covers ultraoligotrophic to
hypertrophic conditions, colour values from 3 to 300 mg Pt/l, which covers ultraoligohumic to highly dystrophic conditions, pH
from 3 to 11, which covers the entire natural range, and lake areas from 0.1 to 100 km2 .
© 2003 Elsevier Science B.V. All rights reserved.
Keywords: Lakes; Models; Phytoplankton; Biomass; Production; Photic zone; Environmental factors
1. Background and introduction
Phytoplankton evidently plays a fundamental role
in lake ecosystems and it has long been known that
∗ Corresponding author. Tel.: +46-1818-3897;
fax: +46-1818-2737.
E-mail address: [email protected] (L. Håkanson).
phosphorus is the nutrient most likely to limit primary
productivity in most (but not all) lakes (Schindler,
1977, 1978; Bierman, 1980; Boynton et al., 1982;
Wetzel, 1983; Persson and Jansson, 1988; Boers et al.,
1993). Several compilations of models, theories and
approaches to the role of phosphorus in lake eutrophication exist (Chapra, 1980; Chapra and Reckhow,
1979, 1983; Vollenweider, 1968, 1976, 1990;
0304-3800/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0304-3800(03)00096-6
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L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Håkanson, 1995a; Nürnberg and Shaw, 1998). Much
research has also been directed to different chemical
forms of phosphorus (organic, Ca-, Al-, Fe-bound
phosphorus, particulate-P) and fractions (Peters,
1981; Bradford and Peters, 1987), exchange processes
between sediments and pore water, and between sediments and lake water (Twinch and Peters, 1984;
Boström et al., 1982; Nürnberg, 1984; Pettersson
and Istvanovics, 1988), and between land and water (Prairie and Kalff, 1986, 1988a,b; Prairie, 1988).
The role of lake phosphorus in wider lake ecosystem
contexts has been discussed in several papers and
books (e.g. Wetzel, 1983; Håkanson and Jansson,
1983; Wetzel and Likens, 1990; Boers et al., 1993;
Håkanson and Boulion, 2002).
Limnologists responded to the eutrophication threat
by developing different types of management models
(see, e.g. Vollenweider, 1968, 1976, 1990; Dillon and
Rigler, 1974, 1975; Schindler, 1974, 1978; OECD,
1982; Schindler et al., 1993; Chapra and Reckhow,
1979, 1983; Straskraba and Gnauck, 1985; Jørgensen
et al., 1986; Jørgensen and Johnsen, 1989; Håkanson
and Peters, 1995; Håkanson, 1995a, 1999). Both experimental and comparative studies of whole lake
ecosystems have been carried out to derive loading
models for lake management. A key factor in this
development was Vollenweider’s (1968) identification of the simple relationship between sedimentation
of phosphorus and water turnover in lakes. Water
turnover is therefore an important factor regulating
the effect of a given nutrient loading.
Comparing models for phytoplankton and lake
eutrophication, it must be stressed there are major differences among them related to differences in target
variables (from individual species to total biomass),
modelling scales (daily to annual predictions), modelling structures (from empirical/regression models to
approaches based on ordinary and partial differential
equations) and driving variables (whether accessed
from standard monitoring programs, climatological
measurements or specific lake studies). So, to make
meaningful model comparisons is not a simple matter, and this is not the focus of this paper. As far as
the present authors are aware, there are no models for
phytoplankton of the type presented here accounting
to production, grazing, growth, elimination and food
choices in a general, holistic ecosystem framework
designed to achieve practical utility. All modelling
approaches have drawbacks and limitations. Extensive ecosystem models for which all rates and model
variables are empirically calibrated for a given lake
might provide the best descriptions in that lake, but
such models usually fail to predict well in other lakes.
One reason for this is that the total uncertainty of the
model predictions often grow as more processes and
variables are included in the model (the optimal size
dilemma, see Håkanson, 1995b). “Everything should
be done as simple as possible, but not simpler”, according to Albert Einstein, and that statement is valid
also for this work.
Evidently, lake phosphorus concentrations are influenced by many types of emissions: Point sources
(e.g. domestic sewage, industries and fish farms), atmospheric deposition (to the lake surface and to the
catchment), internal loading (linked to resuspension,
diffusion, etc.) and, often most importantly, tributary
input. The characteristics of the catchment, like its
bedrocks, soils, land-use, etc. regulate the phosphorus
concentration in the tributaries.
Modelling of primary phytoplankton production
and phosphorus are central paradigms in limnology.
The basic mass-balance model for phosphorus and
all very simplistic models of the Vollenweider- and
OECD-types (see Vollenweider, 1968; OECD, 1982)
cannot however be used for many important limnological studies (see, e.g. Håkanson and Boulion, 2002).
The basic objectives of the new dynamic model for
phytoplankton presented in this work are:
1. It should give seasonal (weekly) variations.
2. It should account for all important factors known
to influence phytoplankton production in lakes in
general but the driving variables should be few and
readily accessed.
3. It should be compatible with and an integral part of
the LakeWeb-model, where it should provide, e.g.
biotic/abiotic feedbacks.
4. It should give good predictions when tested
against empirical reference equations, but no more
driving variables than those already used in the
LakeWeb-model for other purposes (see Håkanson
and Boulion, 2002) will be accepted. To limit the
number of necessary driving variables is important
for the practical use of a model.
There are many reasons why simplistic regression
models cannot be used to address many fundamental
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
issues in limnology and water management. Most
regression models for the two target variables in
this work, phytoplankton production and biomass,
are based on total phosphorus (TP) concentrations
and such regression models are generally static, i.e.
they do not account for the dynamic behaviour of
phosphorus in lakes, e.g. the transport of TP from
sediments back to the water by advective and diffusive processes, mineralisation of particulate TP,
water mixing, biouptake of dissolved TP and retention of TP in biota. Such processes are handled by
the sub-model for phosphorus in the LakeWeb-model.
But those parts of the LakeWeb-model have been
287
presented elsewhere (Håkanson and Boulion, 2002)
and will not be discussed in this work.
Some models for lake eutrophication use the maximum phytoplankton volume (algal volume, AV) as
an operational target variable. One such alternative,
which is based on an empirical regression between AV
and TP, is given in Fig. 1. Fig. 1 is based on data from
327 measurements from 100 Swedish lakes covering a
broad range concerning lake TP-concentrations (from
3 to 300 !g/l). The trophic states of the lakes included
in this regression range from very oligotrophic to hypertrophic conditions. An AV of 5 mm3 /l is regarded
by, e.g. Swedish authorities as a practical guideline,
Fig. 1. The relationship between total-P (log(TP); TP in !g/l; mean summer values) and maximum volume of phytoplankton during the
summer period (log(AV); algal volume, AV, in mm3 /l). Based on unpublished data from E. Willén (SLU, Uppsala, Sweden). The regression
line and the 95% confidence intervals for the predicted y show that there exists a very strong (r 2 = 0.76, P < 0.0001) general relationship
between the x-variable and the y-variable for these 327 measurements from 100 Swedish lakes, but there is also a substantial residual
variation around the regression line. Some of the variation around the regression line can be related to variations in lake temperature, light
conditions, lake water clarity and predation from herbivorous zooplankton accounted for in this approach, as well as to analytical errors
in determining TP and AV. The critical AV-limit used by, e.g. Swedish environmental authorities is 5 and the alarm limit is 10 mm3 /l.
288
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Table 1
These empirical equations have been used as normative values (=norms) in the testing of the dynamic model for phytoplankton
y-value
Equation
Range for TP
r2
n
Unit
Reference
Chlorophyll (summer mean)
Chlorophyll (summer maximum)
Chlorophyll (weekly mean)
=0.28 TP0.96
2.5–100
2.5–100
0.77
0.81
77
50
mg ww/m3
mg ww/m3
mg ww/m3
OECD, 1982
OECD, 1982
H & B, 2002
7–200
0.95
38
mg C/m3 day
Peters, 1986
mg C/m3 day
Peters, 1986
mg C/m3 day
Peters, 1986
mg C/m3 day
Peters, 1986
mg ww/m3
mg ww/m3
Peters, 1986
H & B, 2002
Maximum phytoplankton
production (TP > 10)
Maximum phytoplankton
production (TP < 10)
Mean phytoplankton
production (TP > 10)
Mean phytoplankton
production (TP < 10)
Phytoplankton biomass
Phytoplankton biomass
=0.64 TP1.05
=0.5 TP [(0.64 + 0.28)/2 ≈
0.5; (0.96 + 1.05)/2 ≈ 1)]
=20 TP-71
=0.85 TP1.4
=10 TP-79
7–200
0.94
38
=0.85 TP1.4
=30 TP1.4
=30 TP(1.4−0.1·(TP/80−1))
3–80
0.88
27
From Peters (1986), OECD82: (OECD, 1982) and H & B, 2002: (Håkanson and Boulion, 2002); PrimP, primary production (in g ww/m2 year);
n, number of lakes used in the regression; ww, wet weight.
or a “critical” value concerning algae blooming, and
10 mm3 /l as a limit for “alarm” (Persson and Olsson,
1994). Fig. 1 is based on TP-data measured for the
growing season. The spread around the regression
line in Fig. 1 is however considerable and a TP-value
of 35 !g/l can therefore (with a 95% certainty) correspond to AV-values from 0.5 to 12 mm3 /l, a very
wide range indeed.
The basic aim of the new dynamic model presented
in this work is to use a more causal approach to
quantify the most important factors causing the variability around the regression line illustrated in Fig. 1
related to the interactions between phosphorus, water temperature, the depth of the photic zone, predation by zooplankton and the phytoplankton turnover
time. As stressed, there are many empirical models
for chlorophyll and phytoplankton biomass. However,
there only few quantitative dynamic models capturing the most important factors and processes regulating phytoplankton production and accounting for such
fundamental but complex properties as the depth of the
photic zone, predation and seasonal variations. And
no such models have, as far as we know, yielded good
predictions over a wide limnological domain from just
a few readily accessible driving variables.
The empirical regressions used to test the behaviour
of the dynamic phytoplankton model concern the following parts:
• Phytoplankton biomass, which is basically calculated from characteristic lake TP-concentrations, as
given by the empirical reference equations given in
Table 1. These regressions are, like all regressions,
and in fact all models, only applicable in a certain
defined domain, where it has a certain predictive
power, as indicated in Table 1 by the ranges in the
TP-values and by the r2 -values (r2 , the coefficient
of determination; r, the correlation coefficient).
• Mean and maximum phytoplankton production
values (see Table 1). To calculate typical values of
phytoplankton production in kg ww/week for any
given lake from measurements in mg C/m3 day,
we will use modelled values on the volume of the
photic zone, which in turn are calculated from a
new model for the effective depth of the photic
zone (=Secchi depth; the maximum depth of
the photic zone ≈ 2Secchi depth) presented by
Håkanson and Boulion, 2002). The model for the
depth of the photic zone is an important, integral
part of the LakeWeb-model providing biotic/abiotic
feedbacks (increases in phytoplankton production increase phytoplankton biomass which decreases Secchi depth which reduces phytoplankton
production).
The empirical regressions are static and do not provide any seasonal patterns, only characteristic mean
y-values based on the x-variables used in the regressions. We have, however, in some cases included
simple seasonal patterns also in the regressions to target on comparisons between values for the growing
season, since most of the empirical regressions used
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
for the tests are based on data from this period. This
means that divergences between modelled values and
data from the empirical regressions for other seasons
of the year are of less interest. The dynamic model
is, however, meant to provide the best possible estimates of phytoplankton biomass and production for
all seasons of the year. It is evident that the conditions during the growing season are generally more
important for lake foodweb characteristics than the
conditions during the rest of the year.
The part of the LakeWeb-model presented here
is meant to give much more information about the
factors influencing phytoplankton production and
biomass than the empirical models used for the model
tests. Divergences between the modelled values and
the data given by the empirical regressions should be
logical and supported by solid limnological theory.
2. Empirical data and regressions used for
model tests
The empirical basis for the reference regressions in
Table 1 is a large set of data from many lakes covering a very wide range of lake characteristics (see
Håkanson and Boulion, 2002). Many of the data were
collected over several decades by scientists in the former Soviet Union. This “Soviet” data base has been
described in Russian by Boulion (1994). There is also
a large set of data from west European lakes included
in the empirical reference equations. Those data and
regressions have been presented and used by OECD
(1982), Peters (1986), Håkanson and Peters (1995) and
Håkanson (1999).
The sampling dates for the “Soviet” data base are
given by Håkanson and Boulion (2002). Depending on
the purposes of the investigations, different scientists
used different schemes of sampling and estimation
of phytoplankton production. Mean concentrations
of chlorophyll-a and phytoplankton production were
determined for the growing season at given sampling
sites. There are also more sporadic observations (e.g.
on route surveys on the Mongolian lakes), then the
samplings took place in the pelagic zone at sites
close to the mean depth of the lakes. It is evident that
although the “Soviet” data base is extensive, there
are only few and scattered data for many variables
for many lakes. Since, however, the lakes cover a
289
wide geographical and limnological domain, these
regressions provide general relationships.
3. The dynamic model for phytoplankton
This section presents and motivates the new dynamic model to calculate production and biomass of
phytoplankton in lakes.
3.1. Basic model
The following ordinary differential equation gives
the changes in the biomass of phytoplankton. The
model is graphically illustrated in Fig. 2:
BMPH (t) = BMPH (t − dt)
+ (IPRPH − CONPHZH − ELPH )dt
(1)
where BMPH , phytoplankton biomass (kg ww); IPRPH ,
initial phytoplankton production (kg ww/week);
CONPHZH , phytoplankton consumption by herbivorous zooplankton (kg ww/week); ELPH , phytoplankton elimination (or turnover) (kg ww/week).
Elimination products and dead phytoplankton are
consumed by bacterioplankton but this is not included
in this differential equation (but it is included in the
overall LakeWeb-model). The initial phytoplankton
biomass is set equal to the norm-value (NBMPH in
kg ww), as calculated by the empirical TP-model given
in Table 1 for mean summer conditions. This regression, however, is only valid for TP-concentration in
the range from 3 to 80 !g/l. At higher TP-values, there
is a likely increase in algal shading, and we have modified the basic equation (see Table 1) to account for
this in the following manner:
If CTP < 80 !g/l then
NBMPH = 10−6 Vol 30 CTP 1.4
NBMPH = 10
−6
Vol 30 CTP
else
(1.4−0.1((CTP/80)−1))
(2)
Fig. 3 illustrates how this algorithm describes the phytoplankton biomass when CTP varies from 3 to 300
!g/l. Similar relationships have been described by, e.g.
Straskraba (1980) and Chow-Fraser and Trew (1994).
Within the overall LakeWeb-model, there is also a
dynamic mass-balance model for phosphorus, which
gives values of CTP from inflow, outflow and internal lake processes regulating fluxes of phosphorus
(like advection, diffusion, biouptake and retention
290
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Fig. 2. Illustration of the phytoplankton sub-model in the LakeWeb-model. The panel of driving variables gives the obligatory driving
variables, which should apply to a specific lake. To run the model one also needs epilimnetic temperatures, which may be accessed from
measurements, climatological tables or models. The panel also lists two critical rates (which are meant to be general model constants
applicable for all lakes) and the three connections to the overall LakeWeb-model (total phosphorus from the mass-balance model, Secchi
depth and the functional group grazing of phytoplankton, herbivorous zooplankton).
of phosphorus in the nine functional groups in the
LakeWeb-model).
The initial phytoplankton production, IPRPH , is
given by:
IPRPH = PrimP · YpH5.5
(3)
where PrimP is the primary phytoplankton production
(kg ww/week) given by the following regression from
Håkanson and Boulion (2001a):
PrimP = (2.13 Chl0.25 + 0.25)4 ,
(r 2 = 0.90; n = 102)
(4)
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
291
Fig. 3. The relationship between lake TP-concentrations and phytoplankton biomass, as given by Eq. (3).
The chlorophyll values are calculated from CTP according to Håkanson and Boulion (2002). So, the
PrimP-value is first given in mg C/m3 day (Eq. (5))
and then transferred to kg ww/week (Eq. (6)). Calculation from g C to g dw is given by 1/0.45; calculation
from g dw to g ww by 1/0.2 (using a water content of
80% for phytoplankton; see Table 2); 7 in Eq. (6) is
the number of days per week.
If Secchi depth > 1 m then:
PrimP = Ytemp (10−6 )((2.13 Chl0.25 + 0.25)4 )
"!
"
!
1
1
7 Area Sec else PrimP
×
0.45
0.2
= Ytemp (10−6 )((2.13 Chl0.25 + 0.25)4 )
!
"!
"
1
1
×
7 Area Sec2
0.45
0.2
(5)
Smith (1979) has described a similar relationship
where algal self-shading governs the upper limit for
Table 2
The following calculation constants (from Håkanson and Boulion,
2002) have been used to transform values for different species
given in kcal, g ww and g dw (1 kJ = 4.19 kcal ∼ 0.42 g C)
For phytoplankton, bacterioplankton, benthic algae,
zoobenthos and fish:
1 kcal ∼ 0.2 g dw ∼ 1 g ww
For zooplankton:
1 kcal ∼ 0.2 g dw ∼ 2 g ww
For macrophytes:
1 kcal ∼ 0.2 g dw ∼ 1.32 g ww
PrimP. The Secchi depth is calculated by a sub-model
within the LakeWeb-model.
The pH of the water will influence the phytoplankton production, not just in the sense that pH
influences the depth of the photic zone, as given by
YpH 5.5 (see Section 3.2), but also since pH is likely
to influence phytoplankton production more directly
(Schindler et al., 1985). This is demonstrated by data
given in Table 3 and interpretations of those data
given in Fig. 4. Unfortunately, we do not have data for
the relationship between phytoplankton biomass or
chlorophyll and pH for very basic lakes (pH > 9.5).
But we hypothesise that phytoplankton production
should also decrease at very high pH-levels. If pH
is higher than about 5.5, the data in Fig. 4 indicate
a positive (and logical) correlation between pH and
phytoplankton biomass and chlorophyll. Lakes with
pH higher than 9.5 are assumed to have a lower
phytoplankton production. There are indications that
pH-values up to 9.4 or 9.5 influence species composition but not phytoplankton biomass (see Boulion
et al., 1983; Boulion, 1985).
The pH-influences on phytoplankton production
may be handled by the dimensionless moderators
YpH 5.5 and YpH 9.5 . YpH 5.5 is set to be 1 when pH
5.5. When pH is 2.5, YpH 5.5 is calibrated to give zero
phytoplankton production.
So, if lake pH < 5.5 then:
!
!
""
pH
YpH 5.5 = 1 + 1.85
−1
else
5.5
YpH 5.5 = YpH 9.5
292
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Table 3
Data on pH, lake colour and phytoplankton biomass
No.
Region
Lake
pH
Colour (mg Pt/l)
Phytoplankton
biomass (mg ww/l)
Chl (!g/l)
Reference
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
Vologda region
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Darwin reserve
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Khotavets
Khotavets
Krivoe
Krivoe
Zmeinoe
Zmeinoe
Motykino
Motykino
Dubrovskoe
Dubrovskoe
Dorozhiv
Dorozhiv
Temnoe
Temnoe
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Kostomojarvi
Chernoe
Chernoe
Mollusochnoe
Mollusochnoe
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 1
Goluboe 2
Goluboe 2
Goluboe 2
Goluboe 2
8.12
7.75
6.4
6.6
4.55
4.60
4.80
4.82
4.55
4.58
4.50
4.29
4.50
4.21
6.8
6.6
6.53
6.5
6.48
6.52
6.4
6.54
6.38
6.5
6.5
6.35
6.7
6.8
6.6
6.5
6.83
6.75
6.8
6.05
6.65
6.75
6.5
6.9
6.85
5.43
5.38
5.3
5.6
5.45
5.37
5.4
5.3
5.6
5.2
5.24
5.4
5.3
143
138
412
381
113
97
21
26
177
183
21
14
42
39
Humic
Colour > 100
6.34
6.37
1.83
0.44
0.81
1.99
1.58
0.34
0.41
0.03
0.34
0.17
0.12
0.06
1.34
1.25
1.09
0.23
0.68
2.46
3
3.7
3.87
6.49
2.56
1.61
1.02
0.95
2.6
2.89
3.84
1
5.95
1.3
1.84
1.51
2.07
1.23
0.1
0.46
0.59
1.79
0.84
1.07
0.5
0.49
0.29
1.92
1.19
0.23
0.14
0.06
39.9
22.7
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Korneva, 1994
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Nikulina, 1997
Humic
Humic
Clear
Colour < 7
Clear
Colour < 7
19.4
8.06
5.64
1.09
1.89
5.38
6.04
1.14
2.94
1.66
2.98
14
8.2
1.41
1.43
2.49
11
12.56
12.34
15.6
19.6
10.96
9
4.75
4.48
5.17
4.08
3.12
4.08
6.1
2.6
0.68
1.35
0.62
1.8
0.49
0.55
1.76
0.79
0.48
0.35
0.39
0.33
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
293
Table 3 (Continued )
No.
Region
Lake
pH
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Goluboe 2
Goluboe 2
Goluboe 2
Goluboe 2
Goluboe 2
Goluboe 2
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
Shkolnoe
5.38
5.2
5.37
5.2
5.18
5.3
5.65
5.5
5.9
5.63
5.5
5.5
5.85
5.2
5.8
5.5
Colour (mg Pt/l)
Clear
Colour < 7
If lake pH > 9.5, we hypothesise that YpH 9.5 can be
given by:
!
!
""
pH
YpH 9.5 = 1−3.8
−1
else YpH 9.5 =1 (6)
9.5
This means that YpH 9.5 is 1 when pH 9.5; zero primary
production is obtained if pH > 12. Phytoplankton
consumption by herbivorous zooplankton (CONPHZH
in Eq. (2)) is calculated by the LakeWeb-model by:
CONPHZH = BMPH · CRPHZH
(7)
where BMPH is the actual phytoplankton biomass
(kg ww) and CRPHZH the actual consumption rate
(1/week) quantifying the loss of phytoplankton from
predation by herbivorous zooplankton. CRPHZH is
related to the turnover time of herbivorous zooplankton (TZH ; 6 days or 6/7 weeks; see Håkanson and
Boulion, 2002). That is:
CRPHZH
!
!
""
BMZH
= NCRPHZH + NCRPHZH
−1
(8)
NBMZH
where NCRPHZH is the normal consumption rate
(phytoplankton eaten by herbivorous zooplankton),
as given by 2/TZH (1/weeks); the number of first
order food choices for herbivorous zooplankton is
2, phytoplankton and bacterioplankton; NBMZH ,
Phytoplankton
biomass (mg ww/l)
0.17
0.12
0.45
0.5
1.9
0.5
0.56
0.83
0.35
0.31
0.15
0.18
0.76
0.6
0.67
0.56
Chl (!g/l)
0.76
0.65
0.62
1.2
0.75
0.22
5.38
0.14
0.29
3.36
Reference
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
Nikulina,
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
1997
the normal biomass of herbivorous zooplankton
(kg ww).
Phytoplankton elimination (ELPH ; turnover, etc.) is
given by:
BMPH · 1.386
ELPH =
(9)
TPH
where TPH , the turnover time of phytoplankton, is set
to 3.2 days (see Håkanson and Boulion, 2002).
Phytoplankton production (PRPH in kg ww/week) is
then given by the ratio PRPH = BMPH /TPH .
3.2. Predation pressure
From Eq. (9), we can note that the predation pressure on phytoplankton (or bacterioplankton) from
herbivorous zooplankton is a function of the actual
biomass of herbivorous zooplankton (BMZH ) relative
to the normal biomass of herbivorous zooplankton
(NBMZH ), and the actual zooplankton consumption
rates (CRPHZH and CRBPZH ), and these rates are related to the inverse of the turnover time of herbivorous
zooplankton (TZH ).
Using the LakeWeb-model, Fig. 5 illustrates the difference in predicted phytoplankton biomass if predation by herbivorous zooplankton is accounted for, and
if it is not. Accounting for grazing clearly gives lower
values of phytoplankton biomass during the growing
season.
294
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Fig. 4. Empirical data illustrating the relationship between lake pH and (A) phytoplankton biomass and (B) chlorophyll.
3.3. Changes in pH related to primary
production
It is well known that lake phosphorus concentrations and hence also primary production influences
lake pH via the carbon cycle (see, e.g. Wetzel, 1983):
The higher the primary production, the higher the
lake pH—if all else is constant. This is illustrated
in Fig. 6 using data on lake TP-concentrations and
pH from 936 lakes covering the entire trophic domain. Changes in lake pH, e.g. from acid rain may
also, as already noted, influence phytoplankton production. This will be addressed in this section and
this part of the model is meant to quantitatively describe such complex relationships in a simple manner.
To describe this relationship in a mechanistic way at
the ecosystem scale (i.e. for entire lakes) in a model
designed to predict mean weekly changes in phytoplankton production, is a very complicated task and
beyond the scope of this work. We will, however,
try to capture essential linkages between TP-changes
(as the key nutrient regulating primary production in
most lakes) and pH-changes by creating a dimensionless moderator which expresses the relationship shown
in Fig. 6, i.e. how changes in TP-concentrations are
likely to change lake pH. This means that lake pH is
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
295
Fig. 5. Illustration of predictions of phytoplankton biomass with and without grazing by herbivorous zooplankton using the default conditions
defined for the LakeWeb-model for a 10-year period (521 weeks) (see Håkanson and Boulion, 2002). The tributary TP-concentration has
been changed from 30 to 90 !g/l week 261 to illustrate the dynamic response to a sudden change in nutrient loading.
given by:
pH = pHdef · YpHTP
(10)
where pH, the actual pH-value; pHdef , the default
pH-value for the given lake; YpHTP , the dimensionless moderator expressing how changes in lake
TP-concentrations, and hence also primary produc-
tion, are likely to influence pH-values. From Fig. 6,
one can note (1) that the default TP has been set to
10 !g/l, (2) that the moderator is based not on actual
TP-values but on logarithmic values (to obtain a normal frequency distribution on the x-axis) and (3) that
the amplitude value has been set to 0.1. This means
that if TP changes in the range from 3 to 300 !g/l,
Fig. 6. Regression between lake pH and TP-concentrations [log(TP)] and definition of the dimensionless moderator expressing how changes
in TP are likely to influence pH. Note that there are many complex relationships between TP and pH and that changes in lake production
will also influence lake pH-values. Data from 936 lakes (data from Håkanson and Peters, 1995).
296
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
pH-values are likely to change by a factor of 1.2. So,
YpHTP is given by:
!
!
""
log(TP)
YpHTP = 1 + 0.1
−1
(11)
log(10)
With this, the dynamic model has been described and
motivated and we will focus on how it behaves.
4. Model tests and model simulations
4.1. Set-up
The aims of this section are:
(1) To test the dynamic model using the general empirical regressions (see Table 1) as references. So,
we will study how the dynamic model predicts
selected target variables compared to the empirical regressions, and we will specifically focus
on the parts where the empirical models cannot,
and should not, provide realistic predictions. This
would be for situations outside the ranges of the
empirical models and for changes in variables not
included in the regressions.
(2) To illustrate how the model behaves in situations
where gradients are created in a systematic manner for all pertinent driving variables. That is:
• TP-concentrations will be changed from 3, 10,
30, 100 to 300 !g/l. This covers the entire range
from ultraoligotrophic to hypertrophic conditions;
• lake colour values, i.e. allochthonous influences, will be changed from ultraoligohumic
to hyperhumic conditions by creating colour
gradients from 3, 10, 30, 100 to 300 mg Pt/l;
• lake pH-values will be altered from extremely
acidic to extremely basic conditions; pH-values
from 3 to 11 will be tested;
• morphometric influences will be tested by
changing the mean depth from 1, 2, 4, 8 to
16 m and the lake area from 0.1, 0.5 1, 10 to
100 km2 .
(3) All these changes will be calculated for the following two target variables and to get full comparability, we will use data for the hypothetical
default lake. The characteristics of this lake are
given in the insets in Figs. 7 and 8.
• Phytoplankton production (PRPH ). We will relate
the calculated values to two empirical reference values: (1) the maximum phytoplankton production, as
given by Table 1, and (2) the mean phytoplankton
production, as given by Table 1. The basic idea is
that the model should provide relevant values for
the growing season for all types of lakes.
Normative empirical maximum values for phytoplankton production (NPRPH max in kg ww/week)
are then given by:
If CTP > 8 !g/l then NPRPH max
!
"
1
= (20CTP − 71)10−6
0.45
!
"
1
×
7 A Sec else NPRPH max
0.2
!
"!
"
1
1
= (3.95 CTP 1.5 )10−6
7 A Sec
0.45
0.2
(12)
Normative empirical mean values (NPRPHMV ) are
given by:
If CTP > 10 !g/l then NPRPHMV
!
"
1
= (10CTP − 79)10−6
0.45
!
"
1
×
7 A Sec else NPRPHMV
0.2
!
"!
"
1
1
= (0.85CTP 1.4 )10−6
7 A Sec
0.45
0.2
(13)
• Phytoplankton biomass (BMPH ). Empirical reference models are given in Table 1 using two different measures of lake volume, either the entire lake
volume (NBMV in kg ww), which should yield too
high values of phytoplankton biomass (maximum
reference values), since phytoplankton biomass is
not produced in the entire lake volume but in the
photic zone, and empirical values calculated by using the volume of the photic zone (NBMVS ). NBMV
is given by NBMPH in Eq. (3), and the other empirical reference equation is calculated from:
If CTP < 80 !g/l then NBMVS
= (10−6 ) A Sec 30 CTP 1.4
= (10
−6
) A Sec 30CTP
else NBMVS
1.4−0.05((CTP /80)−1))
(14)
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
297
Fig. 7. Critical model testing (sensitivity analyses) for phytoplankton production using the default conditions defined for the LakeWeb-model
(see Håkanson and Boulion, 2002 and inset). The driving variables have been varied in 2-year steps (e.g. in Fig. A, TP is changed in five
2-year steps from 3 to 300 !g/l), while all else is constant. (A) Along a trophic state gradient (TP = 3, 10, 30, 100 and 300 !g/l). (B)
Along a humic state gradient (lake colour = 3, 10, 30, 100 and 300 mg Pt/l). (C) Along an pH-gradient (pH 3, 5, 7, 9 and 11). (D) Along
a gradient of mean depths (1, 2, 4, 8, 16 m). (E) Along a lake size gradient (area = 0.1, 0.5, 1, 10 and 100 km2 ).
The results of all these tests will be summarised in
the following multi-diagram figures, where the driving
variables have been varied in 2-year steps (e.g. TP is
changed in five 2-year steps from 3 to 300 !g/l), while
all else is constant.
4.2. Phytoplankton production
• Fig. 7A shows how the model predicts phytoplankton production (kg ww/week) along the TP-gradient.
The correspondence between model-predicted values and the values given by the two empirical
models (curve 1, the maximum production and
curve 3, the mean production) is excellent. Note
that the empirical models may not give realistic seasonal patterns. The modelled values should
be compared with the order of magnitude values
given by the empirical models. Since our dynamic
model gives mean weekly values, curve 2 should
fall between curves 1 and 3. The reason why the
298
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Fig. 8. Critical model testing (sensitivity analyses) for phytoplankton biomass using the default conditions defined for the LakeWeb-model
(see Håkanson and Boulion, 2002 and inset). The driving variables have been varied in 2-year steps, while all else is constant. (A) Along
a trophic state gradient (TP = 3, 10, 30, 100 and 300 !g/l). (B) Along a humic state gradient (lake colour = 3, 10, 30, 100 and 300 mg
Pt/l). (C) Along an acid state gradient (pH 3, 5, 7, 9 and 11). (D) Along a gradient of mean depths (1, 2, 4, 8, 16 m). (E) Along a lake
size gradient (area = 0.1, 0.5, 1, 10 and 100 km2 ).
empirical models do not predict appropriate seasonal pattern is that the values are calculated using
the model-predicted seasonally variable data on the
effective depth of the photic zone (=Secchi depth),
but not seasonally variable information on other
factors regulating phytoplankton production (like
temperature).
• Fig. 7B shows the results along the colour gradient.
The correspondence is also excellent. Note that the
TP-value in the default lake is 10 !g/l.
• Fig. 7C illustrates the interesting correspondence
between the model-predicted values and the two reference curves along the pH-gradient. Note that the
model predicts a low phytoplankton production if
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
pH 11, and that the model predicts a slightly higher
phytoplankton production if pH 5 (for TP = 10 !g/l
and colour = 50 mg Pt/l) because the water clarity is then higher than at pH 6 or 7. The empirical
models do not account for any pH dependency, so
they predict unrealistically high phytoplankton production values for pH 3.
• Fig. 7D illustrates how phytoplankton production
varies among lakes with different mean depths. One
can see the logical and excellent correspondence
between the three curves.
• Fig. 7E shows how phytoplankton production is
likely to vary among lakes with different areas—
if all else is kept constant. Also in this case, there
is a fine and logical correspondence between the
curves.
To conclude: it is evident from the results given in
Fig. 7 that the model predicts phytoplankton production in good agreement with the empirical models.
4.3. Phytoplankton biomass
Fig. 8 gives the results for phytoplankton biomass.
• Fig. 8A demonstrates modelled values along the
TP-gradient. The correspondence between modelpredicted values and the values given by the two
empirical models (curve 2, the maximum biomass
based on the entire lake volume and curve 3, the
mean biomass based on the volume of the photic
layer) is excellent for all lakes within the range of
the empirical models (that is for TP < 80 !g/l).
Curve 2, gives unrealistically high values for eutrophic and hypertrophic lakes. Note again that the
empirical models may not give realistic seasonal
patterns.
• Fig. 8B shows the results along the colour gradient.
The correspondence is logical. In ultraoligohumic,
clear-water lakes with a high Secchi depth, the actual phytoplankton biomass is higher than suggested
by the empirical models, which are not based on
data from such lakes.
• Fig. 8C gives the good correspondence between
the model-predicted values of phytoplankton
biomass and the two references (marked 2 and 3)
along the pH-gradient. Note that the model predicts a low but realistic phytoplankton biomass if
pH 11.
299
• Fig. 8D illustrates how phytoplankton biomass
varies among lakes with different mean depths.
There is an excellent logical correspondence between the three curves.
• Fig. 8E shows phytoplankton biomasses in lakes
with different areas. Also in this case, there is a
good correspondence between the curves.
To conclude: these results show that, under the
given presuppositions, the model predicts phytoplankton biomass very well.
5. Concluding remarks
The topics discussed in this paper concern the
“base” of the lake foodweb, and are of fundamental
importance in limnology. The aim has been to present
the new dynamic model for phytoplankton which is
meant to capture the most important, general structural and functional characteristics relevant for this
part of the lake foodweb. A very important demand
for the model is that it must only be driven by readily
accessible variables. This model can be run if data are
at hand on lake TP-concentration, pH, colour, mean
depth, lake area and epilimnetic temperature. An important aspect of this work concerns the critical tests of
the model. We have used regression models based on
extensive data sets in the testing of the new dynamic
model. This type of testing stresses the basic aim of the
dynamic model. It is meant to quantitatively describe
general, typical interrelationships regarding phytoplankton in lakes. If these fundamental interrelationships are properly described, then divergences from
these normal patterns can be quantified and related to
factors not accounted for in the model, e.g. specific
contamination situations. The overall LakeWeb-model
is meant to be a practical tool for science, engineering
and management aspects of limnology.
An important aspect of these tests is that one does
not need to speculate about the explanations to the
various observed phenomena. They are, in fact, mathematically defined. This is evidently not the case for
most observed phenomena in natural ecosystems. So,
if the model can capture the essential interactions in
natural ecosystem at the given scale, it is an excellent
tool to gain understanding about often very complex
and contradicting observations.
300
L. Håkanson, V.V. Boulion / Ecological Modelling 165 (2003) 285–301
Acknowledgements
The work has partly been carried out within the
EU-project “Phytoplankton on-line” (contract number: RTD Project: EVK1-CT1999-00037) and with
support of the Russian Foundation for Basic Research
(Project 00-15-97825).
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