Ecological Modelling 151 (2002) 213– 243
www.elsevier.com/locate/ecolmodel
Empirical and dynamical models to predict the cover,
biomass and production of macrophytes in lakes
Lars Håkanson a,*, Viktor V. Boulion b
a
Department of Earth Sciences, Uppsala Uni!ersity, Villa!. 16, 752 36 Uppsala, Sweden
b
Zoological Institute of RAS, Uni!ersitskaja emb., 1, 199034 St. Petersburg, Russia
Received 18 September 2000; received in revised form 23 March 2001; accepted 7 November 2001
Abstract
Macrophytes play several important roles in lake ecosystems, e.g. proving shelter for small fish, binding nutrients
and influencing secondary production by creating habitats for bacteria, benthic algae and zooplankton. However, the
quantitative role of macrophytes in lakes is poorly known because few general, validated models yielding high
predictive power for macrophyte production, cover and biomass have been presented. There are probably many
reasons for this, e.g. related to the costs and efforts required to obtain relevant data. This work is based on a new
database established by us from published sources. Many of the lakes included in this study are situated in the former
Soviet Union. They were investigated during the Soviet period and those results have been largely unknown in the
West. With this new database, we have presented empirical models for macrophyte cover and production yielding
predictions close to the theoretical maximum values, as determined by the uncertainty in the empirical data. Using
data from 35 lakes covering a wide domain in lake characteristics, we have ranked the factors influencing macrophyte
cover and demonstrated that the ratio Secchi depth to mean depth can statistically explain about 40% of the
variability among these lakes in macrophyte cover. Other important factors are latitude (related to lake temperature),
maximum depth and area of the lake shallower than 1 m. A new regression model based on these four factors can
statistically explain 82% of the variation in macrophyte cover among these lakes. We have also presented a dynamic
model for macrophyte production and biomass and several critical tests of that model. The dynamic model gives
better predictions and a more general structure then the empirical model. We have given algorithms for: (1) the
macrophyte production rate; (2) the elimination rate (related to the macrophyte turnover time); (3) the rate of
macrophyte consumption by animals; and (4) the rate of macrophyte erosion. Our results indicate that macrophyte
production is highly dependent on latitude and temperature, morphometry and sediment character, as well as water
clarity, and less dependent on nutrient concentration. Qualitatively, this has been known or suggested before, but this
work gives new quantitative support to such conclusions and also a practically useful model for predictions of
macrophyte production and biomass. © 2002 Elsevier Science B.V. All rights reserved.
Keywords: Lakes; Macrophytes; Cover; Biomass; Production; Primary production; Environmental factors; Latitude; Morphometry
* Corresponding author. Tel.: +46-18-471-3897; fax: +46-18-471-2737.
E-mail address: lars.hakanson@natgeog.uu.se (L. Håkanson).
0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 1 ) 0 0 4 5 8 - 6
214
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
1. Background and aim
(r 2 =0.73; n =44).
For the determination of one of the most fundamental property of lakes, the trophic status,
the basic attention is generally given to phytoplankton production. However, the macrophytes
can make a significant contribution to the total
primary production especially in shallow lakes.
Sometimes the macrophyte production exceeds
the phytoplankton production (Wetzel, 1983).
The macrophytes keep the nutrients bound for
long periods. Consequently, they can help to improve of water quality (Pokrovskaja et al.,
1983). It is also important to emphasize that the
evolution of any lake is closely connected with
overgrowing by root plants (Beeton and Edmondson, 1972). Macrophytes also provide an
important protective environment for small fish.
Although they may not be so important as a
source of food for the fish, macrophytes can still
influence fish production in lakes.
To determinate the relative role of
macrophytes and phytoplankton in primary productivity, it is necessary to study the development of these plant groups relative to the
morphometric and optical properties of the water. The utilization of the macrophyte biomass
in the aquatic food chain is, as we understand
it, poorly investigated. Vorobev (1977) analyzed
data from 229 lakes of the Vologda district
(Russia). He showed that the areal cover by
macrophytes (Maccov, % of lake area) is related
to the ratio between the Secchi depth (Sec in m)
and the mean depth (Dm, in m):
Maccov = 50 (Sec/Dm).
(1)
So, if Sec/Dm = 0.25, the macrophyte cover
should be close to 12.5%, at Sec/Dm =1 it averages 50%, etc.
Duarte and Kalff (1986) quantified the relationship between littoral slope in % and
macrophyte biomass (BMmac). They selected 44
littoral sections in Lake Memphremagog,
Canada. The macrophyte biomass was given by
the maximum values for the growing season in
August. The equation was:
− 0.81
BMmac = 124.47 + 990 slopelit
,
(2)
Duarte and Kalff (1986) also tested this relationship for other lakes and obtained a high coefficient of determination (r 2 =0.81). In this
work, we will also test nor littoral slope since
such data are not available to us, but the role
of mean lake slope on macrophyte cover and
production.
So, one aim of this paper is to critically test
the validity of Eq. (1) and the role of lake
slope. For this purpose, we collected as many
data as possible on macrophyte cover (Maccov)
and macrophyte production (Macprod) from as
many lakes as possible covering as wide lake
characteristics as possible (Table 1). With these
data, we will also carry out statistical analyses
to:
! Rank the factors influencing the variability
among lakes in Maccov and Macprod. Note that
we do not have data to study within-lakes
variations.
! Develop statistical models to predict characteristic lake values of Maccov and Macprod.
From these statistical results, the next aim is to
develop a mechanistic, dynamic model for
macrophyte production and biomass. The focus
of the dynamic model is on lake-typical conditions. We have set the calculation time to 1 week
to obtain seasonal variations. We will also critically test the dynamic model for its descriptive
and predictive power, and discuss its limitations.
The dynamic model includes the following
rates:
1. The initial macrophyte production rate.
2. The erosion rate related to wind/wave erosion
and other types of physical erosion (boats,
etc.).
3. The macrophyte elimination rate related to the
turnover time or characteristic lifespan of
macrophytes.
4. The consumption rate describing how much of
the macrophyte biomass being lost in the lake
foodweb.
In the following, we will address the
parametrization of these rates and we will also
raise some questions for future research.
Glubokoe
Drivjary
Arakhley
Duaja
Big Eravnoe
Sosnovskoe
Little
Eravnoe
Isinga
Big Kharga
Little Kharga Burjatia
Karakhul
Onega
4
5
6
7
8
9
10
12
13
14
15
11
Krasnoe
3
Karelia
Kazakhstan
Burjatia
Burjatia
Burjatia
Burjatia
Burjatia
Lithunia
Chita district,
Siberia
Karelian
Isthmus
Moscow
district
Byelorussia
Vorkuta,
Russia
Big Kharbey
2
Karelia
Region
Chedenjarvi
Lake
1
No.
61.5
43.5
52.7
52.7
52.7
52.7
52.7
52.7
54.5
52.0
55.7
55.8
60.8
67.5
62.0
Latitude (°N)
10 340
1.4
6.5
29.5
30.0
56.2
23.7
99.5
23.2
58.2
32.6
0.59
9.13
21.3
0.65
Area (km2)
Table 1
Macrophyte production and cover, data from 35 lakes
29.5
1.5
0.8
1.0
1.5
1.5
2.5
2.0
14.6
10.4
5.2
9.3
6.6
4.6
3.4
Mean depth
Dm (m)
120
3.8
?
?
?
?
?
?
32.4
16.7
11.8
32
14.6
18.5
6.6
Maximum
depth Dmax
(m)
4.0
1.5
To bottom
To bottom
To bottom
To bottom
To bottom
To bottom
3.2
6.7
2.0
1.8
2.1
2.5
0.4
Secchi depth
Sec (m)
0.29
50
100
100
91.4
100
100
85.6
13.6
43.7
20
8
7
5
10
1
10 560
1848
1848
1294
1848
1220
924
1188
264
132
178
211
132
37
Maccov (%) Macprod
(g ww/m2
year)
Gorbunov,
1953
Vlasova et al.,
1973;
Kochanova,
1976
Andronikova
et al., 1973
Scherbakov,
1967
Zakharenkova,
1970
Zolotareva,
1981;
Nazarova and
Shishkin,
1981
Manukas,
1973
Neronova and
Karasev, 1977
Neronova and
Karasev, 1977
Neronova and
Karasev, 1977
Neronova and
Karasev, 1977
Neronova and
Karasev, 1977
Neronova and
Karasev, 1977
Khusainova
et al., 1973
Raspopov,
1973;
Dotsenko and
Raspopov,
1982
References
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
215
Tchad
Naroch
Mjastro
Batorino
Kiev reservoir Ukraine
Lacha
Vozhe
Rybinskoe
reservoir
Gorkovskce
reservoir
Nogon-Nur
Vechlen
Sjargozero
18
19
20
21
22
23
24
25
26
27
28
29
51.0
54.8
54.8
54.8
13.0
49.3
53.7
Latitude (°N)
Mongolia
The
Netherlands
Karelia
Volga River
63.6
47.9
52.0
57.0
Vologda
61.0
district, Russia
Vologda
61.0
district, Russia
Volga River
58.5
Byelorussia
Byelorussia
Byelorussia
Africa
Canada
Poland
Marion
Mikolayskoe
16
17
Region
Lake
No.
Table 1 (Continued)
10.54
20
0.047
1610
4550
418
345
925
6.3
13.1
79.6
24 000
0.13
4.6
Area (km2)
7.8
1.0
6.0
5.5
5.6
1.4
1.6
3.5
3.0
5.4
9.0
3.5
2.4
11
Mean depth
Dm (m)
22
1.5
11.9
?
28.0
5.0
5.0
?
5.5
11.3
24.8
5.5
6.0
27
Maximum
depth Dmax
(m)
3.0
1.0
4.0
1.2
1.5
1.1
1.1
2.2
0.8
1.6
5.3
?
?
3.5
Secchi depth
Sec (m)
4.5
50
9
1.4
16.7
48
18.3
32
23
17
30
100
20
19
20
2640
98
33
401
792
462
232
87
178
784
12 672
238
436
Maccov (%) Macprod (g
ww/m2 year)
Klukina and
Freindling,
1983
Ekzertsev,
1958; Ekzertsev
and Dovbnja,
1973
Ekzertsev,
1958; Ekzertsev
and Dovbnja,
1973
Boulion, 1985
Best, 1982
Raspopov, 1978
Efford, 1972
Kajak et al.,
1972
Leveque et al.,
1972
Winberg et al.,
1972
Winberg et al.,
1972
Winberg et al.,
1972
Gak et al.,
1972;
Priymachenko,
1983
Raspopov, 1978
References
216
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Sonozero
Sukkozero
Gormozero
Tuhkozero
Vjagozero
Torosjarvi
30
31
32
33
34
35
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Region
63.6
63.6
63.6
63.3
63.3
63.6
Latitude (°N)
0.61
1.29
1.58
2.36
8.9
9.53
Area (km2)
Note that 1 kcal !0.52 g dw !1.56 g ww for macrophytes.
Lake
No.
Table 1 (Continued)
3.3
1.4
6.2
3.0
1.9
3.7
Mean depth
Dm (m)
?
?
?
?
?
?
Maximum
depth Dmax
(m)
3.0
To bottom
5.0
1.8
To bottom
1.8
Secchi depth
Sec (m)
17.6
21.7
11.3
8.1
3.8
4.9
71
59
53
18
8
11
Maccov (%) Macprod (g
ww/m2 year)
Klukina and
Freindling,
1983
Klukina and
Freindling,
1983
Klukina and
Freindling,
1983
Klukina and
Freindling,
1983
Klukina and
Freindling,
1983
Klukina and
Freindling,
1983
References
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
217
218
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
2. Lakes, data and methods
Many of the data used in this work come from
the former Soviet Union and have not been presented before in the West. We hope that it can be
seen as a good idea to make these data more
generally accessible. The data are compiled in
Table 1 together with literature references. It
should be noted that we have reliable data on
total-P and lake pH from so few lakes (with data
on macrophyte cover and production) that these
limnological state variables are not included in
this study. This evidently will set limits to our
models. However, we have seen it as interesting to
learn how far can one reach in terms of predicting
macrophyte cover and production without such
data. If total-P is a key limiting factor for Maccov
and Macprod then we would obtain poor results.
Our database includes data on Maccov and
Macprod from 35 lakes covering a very wide limnological domain indeed (see Fig. 2 and Table 1),
large and small lakes (from 0.047 to 24 000 km2),
from latitudes 13° N (Lake Tchad in Africa) to
67.5° N, Lake Big Kharbey, Vorkuta, Russia, and
deep and shallow lakes (maximum depth from 1.5
to 120 m). The macrophyte cover varies from 0.29
to 100%, and the characteristic macrophyte production from 1 to about 13 000 g ww/m2 year.
The light conditions in lakes are likely to be
important for the macrophytes, and our data
includes information on Secchi depth, which
varies from 0.4 to 6.7 m. In many lakes the Secchi
depth reaches its maximum value, the maximum
depth.
In the following paragraphs, we will give a brief
methodological description of how our target
variables, macrophyte cover and production, have
been empirically determined.
The removal and weighing of water plants can
give relatively reliable values on the conditions at
the study site, but for our purpose such sitetypical information must be transformed into
lake-typical values, and extrapolations entail
methodological problems. The results depend on
temporal, vertical and areal variations in temperature, substrate (= sediment) type and wind exposure (Vollenweider, 1969). Therefore, to reduce
the uncertainty in the data, one should try to
select large quadrates along transects parallel to
depth gradients. The samples may be collected by
boat or diving. The removal of the plants may be
by hand or sampling equipment. The choice depends on the type of macrophytes collected and
the substrate. The areas for direct sampling are
generally marked with stakes, cords or quadrate
frames (if the area is small). If the plants are
rooted in soft silt, the whole plants can often be
removed in a simple manner. More often, however, the biomass is determined by a method of
harvesting, where the plants are cut by shears or
scythes. Then the roots should be selected separately. The zone of macrophyte cover is generally
contoured on a bathymetric map showing the
boundary depths of the macrophyte distribution.
The percentage cover can also be determined by
aerial photography. This allows relatively exact
determinations of both distribution and structures
of the macrophytes also in large lakes (Raspopov,
1986). After sampling the plants are washed to
remove soil, epiphytes and animals, sorted and
analyzed. Usually, the wet weight is determined
and sub-samples taken for determination of dry
weight, which is generally determined after 24 h at
105 °C (Vollenweider, 1969).
On average, the water content of macrophytes
is 84% (see Table 2) and the ash content 13%.
From Table 2, one can also note that the spread
around the mean ash content is rather high (the
standard deviation is 8%). Bastardo (1979) has
reported an even higher mean ash content of
macrophytes, from 17 up to 28%. Therefore, we
would argue that one could set a typical ash
weight to 15% (dw). For most macrophytes, it is
also possible to set a typical organic carbon content to 46% of the organic matter, and typical
energy content to 4.6 kcal/g of organic matter
(Vollenweider, 1969; Raspopov, 1973). Thus, for
macrophytes 1 kcal! 0.217 g organic matter!
0.250 g dw!1.56 g ww.
The macrophytes have a characteristic seasonal
growth pattern—the biomass attains maximum
values in the summer and minimum values in the
winter. Most shoots of the previous year are not
kept at the time of the maximal biomass. Given
this seasonal pattern, and the fact that non-respiratory losses are relatively small, the maximal
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
219
Table 2
A compilation of data on macrophyte water content, ash weight and content of organic carbon
Species
Water content
(ww, %)
Equisetum
Equisetum
Equisetum
Equisetum
Equisetum
Equisetum+Phragmites
Equisetum+Nuphar
Phragmites
Phragmites+Typha
Nuphar
Nuphar+Polygonum
Polygonum amphibium
Lysimachia
Potamogeton perfoliatus
Potamogeton praelongus
Elodea
Ceratophyllum
Potamogeton pectinatus
Mycriophyllum
Batrachium
Potamogeton perfoliatus
Sagittaria
Acorus
Typha
Eleocharis
Phragmites
Scripus
Equisetum
Carex
Phragmites
Typha
Euguisetum
Scripus
Eleocharis
Sagittaria
Sparganium
Polygonum amphibium
Nuphar
Nymphaea
Potamogeton natans
Potamogeton perfoliatus
Batrachium
Mycriophyllum
Elodea
88
88
91
92
85
81
83
59
77
84
87
85
78
90
86
85
90
89
89
87
90
80
80
80
80
80
80
80
Mean
SD
84
7
Ashes in
(dw, %)
Organic carbon in organic matter
(%)
91
24
19
37
37
18
9
9
9
10
10
10
10
5
5
7
13
7
8
7
15
8
8
10
10
10
13
11
23
48
48
46
45
49
47
48
44
48
44
45
46
43
44
47
43
13
8
46
2
References
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Scherbakov, 1967
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Zakharenkova, 1970
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
Raspopov, 1973
220
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 1. Hypsographic (depth/area) curve for Lake 6, as calculated using the given equation describing changes in lake area with
water depth (D). Several morphometric parameters, like dynamic ratio, form factor, ET- and A-areas, are also defined for Lake 6.
seasonal biomass can be approximated as the
cumulative annual net production. The dead
macrophytes will be consumed by detrivores. Many
studies have dealt with the definition of maximum
macrophyte biomass (Westlake, 1980). Generally,
those results apply to water bodies of the temperate
zone, where the growing season for macrophytes
lasts 4–5 months.
Generalizing the literature data on the seasonal
dynamics of macrophyte biomass, Raspopov
(1973) has shown that the annual production of
water plants usually exceeds the maximum biomass
by 2 – 20%. Raspopov (1973, 1986) suggested that
the annual production of most kinds of emerged
and submerged plants in temperate waters may be
estimated by 1.2 BMmax where BMmax is the maximum biomass during the growing season.
Resuspension is also likely to influence the
macrophytes, and we have calculated several expressions to try to quantify the role of resuspension
and sediment type on macrophyte cover and production (see Fig. 1).
The dynamic ratio (DR=!Area/Dm, where
Area is given in km2 and mean depth, Dm in m) is
used in a traditional way (see Håkanson and
Jansson, 1983) to estimate the areas where fine
sediment erosion (E) and transportation (T) are
likely to dominate the bottom dynamic conditions
and where wind/wave resuspension and slope processes are likely to occur. These are the ET-areas,
i.e., the areas above the wave base (see Fig. 1). We
have calculated DR and ET for all 35 lakes (see
Table 3). ET varies from the theoretical minimum
value of 0.15, to the theoretical maximum value of
Rybinskoe
reservoir
Gorkovskoe
reservoir
Nogon-Nur
Vechten
25
27
28
26
11
12
13
14
15
16
17
18
19
20
21
22
23
Vozhe
Drivjaty
Arakhley
5
6
24
Glubokoe
4
Dusja
Big Eravnoe
Sosnovskoe
Little
Eravnoe
Isinga
Big Kharga
Little Kharga
Karakhul
Onega
Marion
Milkolayskoe
Tchad
Naroch
Mjastro
Batorino
Kiev reservoir
Lacha
Krasnoe
3
7
8
9
10
Chedenjarvi
Big Kharbey
Lake
1
2
No.
Mongolia
The
Netherlands
Volga River
Burjatia
Burjatia
Burjatia
Kazakhstan
Karelia
Canada
Poland
Africa
Byelorussia
Byelorussia
Byelorussia
Ukraina
Vologda
district,
Russia
Vologda
district,
Russia
Volga River
Karelia
Vorkuta,
Russia
Karelian
Isthmus
Moscow
district
Byelorussia
Chita district,
Siberia
Lithunia
Burjatia
Burjatia
Burjatia
Region
Table 3
Calculated data for the lakes
4.47
0.04
7.30
12.05
14.60
3.65
5.43
3.19
0.79
3.45
0.15
0.19
44.26
0.99
0.67
0.84
8.69
11.61
0.33
4.99
1.95
5.00
1.10
0.73
0.08
0.46
0.24
1.00
DR
2.00
1.51
0.60
0.84
0.96
1.18
0.74
1.20
1.22
1.91
1.09
1.43
1.64
1.35
1.32
1.97
0.87
1.36
1.55
0.75
Vd
0.078
9.522
0.048
0.029
0.024
0.095
0.064
0.109
0.440
0.101
2.310
1.780
0.008
0.350
0.518
0.415
0.040
0.030
1.052
0.070
0.178
0.069
0.316
0.473
4.186
0.758
1.464
0.346
20.00
0.01
2520.21
339.07
258.57
1.04
1680.48
0.07
0.96
7222.92
22.03
3.79
2.49
3.37
10.63
6.54
0.20
2.45
0.24
11.60
Slope degrees Area"3 m
(A3, km2)
1.00
0.25
0.55
0.81
0.75
0.75
0.16
0.57
0.21
0.30
0.28
0.29
0.39
0.15
0.33
0.11
0.33
0.27
0.38
0.54
Area"3 m
fraction
6.86
0.005
1312.55
222.19
157.07
0.56
635.03
0.04
0.39
2571.49
9.36
1.59
1.01
1.27
4.63
2.31
0.09
1.02
0.10
6.13
Area"1 m
(A1, km2)
0.34
0.10
0.29
0.53
0.46
0.40
0.06
0.29
0.08
0.11
0.12
0.12
0.16
0.05
0.14
0.04
0.15
0.11
0.16
0.29
Area"1 m
fraction
1.18
4.77
1.88
3.07
3.71
0.97
1.42
0.86
0.26
0.92
0.17
0.16
11.12
0.31
0.23
0.27
2.23
2.96
0.16
1.30
0.55
1.31
0.34
0.25
0.32
0.19
0.15
0.31
ET fraction
2.14
2.73
2.86
3.10
2.41
2.41
2.41
1.94
3.16
2.21
2.48
1.17
2.56
2.56
2.56
2.31
3.10
2.54
2.41
2.41
2.41
2.62
2.37
2.63
3.08
3.21
4.00
90/(90-Lat)
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
221
Sjargozero
Sonozero
Sukkozero
Gormozero
Tuhkozero
Vjagozero
Torosjarvi
29
30
31
32
33
34
35
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Karelia
Region
0.42
0.83
1.57
0.51
0.20
0.81
0.24
DR
1.06
Vd
0.834
0.416
0.221
0.678
1.712
0.428
1.467
3.29
Slope degrees Area"3 m
(A3, km2)
0.31
Area"3 m
fraction
1.44
Area"1 m
(A1, km2)
0.14
Area"1 m
fraction
Dynamic ratio, DR =!Area/Dm; volume development, Vd =3 Dm/Dmax; ET=ET-areas (if ET#1, then E will dominate).
Lake
No.
Table 3 (Continued)
0.18
0.27
0.45
0.20
0.15
0.27
0.15
ET fraction
3.41
3.41
3.37
3.37
3.41
3.41
3.41
90/(90-Lat)
222
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
1. In those cases where ET is higher than 1, the
wave energy is very high and E-areas, rather than
T-areas, are likely to dominate. Such areas generally have very coarse sediments (sand, gravel, etc.)
and little fine sediments. T-areas generally have
mixed deposits, whereas the soft accumulation
areas (A) have continuous sedimentation of fine
materials which settle according to Stokes’ law.
We have also calculated the areas above 1 and
3 m water depth (A1 and A3 in km2) because such
areas are likely to have more macrophytes than
deeper areas. In Table 3, we give data on A1 and
A3 and the fractions of the lake dominated by such
areas. The equation used to calculate A1 and A3 is
given in Fig. 1. It comes from Håkanson (1999).
To make these calculations, one needs data on
lake area (i.e. maximum lake area, abbreviated
Amax, in Fig. 1), maximum depth, Dmax, and the
form factor (= volume development, Vd=3 Dm/
Dmax). From Table 1, one can note that A1 varies
from 4 (0.04) to 53% in shallow Lake Vozhe,
Russia, which is likely to have a high macrophyte
cover (it is 48%, see Table 1) and macrophyte
production (it is 792 g ww/m2 year, see Table 1).
Table 4
Compilation of characteristic CV-values for different variables
and the corresponding highest reference r 2-values
Variable
CV
r 2r
pH
Secchi
Colour
Chl
TP
PP
DP
PrimP
Fish yield
0.05
0.15
0.20
0.25
0.35
0.46
0.88
0.40–0.60
0.5–0.8
0.998
0.98
0.97
0.96
0.92
0.86
0.49
0.76–0.89
0.58–0.84
Assumed !alues
Maccov
Macprod
0.3–0.45
0.35–0.5
0.87–0.94
0.84–0.92
r 2r ; from Håkanson, 1999; Håkanson and Boulion, 2001. Note
that these CV-values are based on individual samples (not
mean values) and variability within lakes (not among lakes).
Secchi depth in m, colour in mg Pt/l, Chl in !g/l, total-P (TP)
in !g/l, particulate P (PP) in !g/l), dissolved P (DP) in !g/l,
primary phytoplankton production in (PrimP) in !g C/1-day,
fish yield in g ww/m2 year, macrophyte cover in % of lake area
and macrophyte production in g ww/m2 year.
223
As stressed in Eq. (2), the slope conditions in the
littoral have been demonstrated to be important
for macrophyte biomass. We do not have data on
littoral slope, but we have calculated the mean
lake slope and tested if lake slope is important also
for the predictions of Maccov, and Macprod. Lake
slope (in degrees) can be determined in many ways
(see Håkanson, 1981), and we have used an expression, which can be calculated using the data
available to us. This expression comes from
Håkanson (1974) and is given by:
Tan(slope)=Dm/(0.165 !Area).
(3)
We will use statistical methods (regressions,
transformations, etc.) described by Håkanson and
Peters (1995) to quantify relationships among variables and these methods will not be further elaborated in this paper. To evaluate the results of
the regression analyses, it is important to recognize that empirical data are always more or less
uncertain due to problems related to sampling,
transport, storage, analytical procedures, etc. This
will restrict the predictive power of any model.
The theoretically highest reference r 2 (r 2r ; see
Håkanson, 1999, for further information; r is the
correlation coefficient; r 2 is the coefficient of determination) of any model is related to the characteristic CV (CV is the coefficient of variation;
CV=SD/MV; SD, standard deviation; MV, mean
value) of the y-variable. The relationship between
CV for within ecosystem variability and r 2r is given
by:
r 2r =1 −0.66 CV2.
(4)
As a background to the following regressions,
Table 4 gives a compilation of characteristic CVvalues for within-lake variability related to individual samples for some standard lake variables,
and the corresponding r 2r -values. Model validations are not likely to yield r 2-values higher than
the r 2r -values, so these values can be used as
reference values for the following models. One
should also note from Table 4, that we do not
have access to characteristic CV-values for our
two target variables, Maccov and Macprod; and that
the given CV-values for primary production
(PrimP) and fish yield are quite uncertain. Based
224
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 2. Frequency distributions for latitude (A), lake area (B), mean depth (C), maximum depth (D), lake area at water depths above
1 m (E) and Secchi depth (F) using the data given in Tables 1 and 3.
on the existing information on characteristic CVvalues, we assume, however, that a characteristic
CV-value for Maccov should be rather high, about
0.3– 0.45, and even higher for Macprod, 0.35– 0.5.
This would mean that one cannot expect to get
higher r 2-values in predictive model for Maccov
than 0.87–0.94 when modelled values are compared to independent empirical data, and not
higher r 2-values for Macprod than 0.84– 0.92.
Figs. 2 and 3 gives information about two
important issues related to regressions:
1. The frequency distributions and the transfor-
mations yielding as normal distributions as
possible. The normality can be tested in many
ways and here it is simply given by the ratio
between the mean value (MV) and the median
value (M50). This ratio should be close to 1 for
a normal distribution.
2. Empirical models are by definition only valid
in the ranges defined by the model variables,
and the histograms in Figs. 2 and 3 (and the
data in Tables 1 and 3) provide information
about the ranges of all discussed variables from
our database.
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Most lake variables are lognormally distributed
(see Håkanson and Lindström, 1997) and this
implies that the logarithmic transformation would
provide a normal frequency distribution. This is
exemplified for Maccov, Macprod, and A1 in Fig. 3.
This figure also shows that exponential transformations (like the squareroot) may provide MV/
M50-values even closer to 1 than the
log-transformation, e.g. for maximum depth in
Fig. 3H. The best transformation for latitude is
given in Fig. 2G, ((90/(90 −Lat)), yielding a MV/
M50-ratio of 1.07.
225
3. Results and discussion
3.1. Statistical modelling
Using the data from this database, one can see
that the mean optical depth (Sec/Dm; Eq. (1)) only
accounts for about 40% of the variation among
these lakes in macrophyte cover (Fig. 5A). This
means that Eq. (1) tells a significant part of the
‘story’, but not the ‘whole story’.
Table 4 gives a first scanning of interrelationships among the variables, a correlation matrix
Fig. 3. Frequency distributions and transformations yielding more normal frequency distributions for some selected lake variables,
macrophyte cover (A, B and C), macrophyte production (D and E), the ratio Secchi depth to mean depth (F), latitude (G) maximum
depth (H) and lake area at water depths above 1 m (I). The ratio between the mean value (MV) and the median value (M50) is used
as a simple criteria for normality.
1.00
Lat
−0.64
−0.64
1.00
Macprod
0.57
1.00
−0.30
−0.12
0.24
1.00
Area
−0.44
−0.26
0.16
0.74
1.00
Dm
−0.46
−0.21
0.26
0.89
0.95
1.00
Dmax
r-Values higher than 0.5 are bolded and significant (at the 99% level).
Maccov
Macprod
Lat
Area
Dm
Dmax
Vd
DR
A3 (%)
A3 (%)
ET
Slope
See
Maccov
Table 5
Correlation matrix; linear correlations using non-transformed variables
0.38
0.07
−0.55
−0.46
−0.25
−0.44
1.00
Vd
0.30
−0.05
−0.24
−0.27
−0.24
−0.06
−0.44
1.00
DR
0.50
0.41
0.03
−0.12
−0.61
−0.41
−0.30
0.75
1.00
A1 (%)
0.56
0.46
−0.13
−0.15
−0.65
−0.45
−0.06
0.61
0.93
1.00
A3 (%)
0.10
−0.11
0.04
0.15
−0.22
−0.11
−0.23
0.68
0.47
0.36
1.00
ET
−0.30
−0.13
−0.19
−0.17
−0.01
−0.06
0.13
−0.32
−0.29
−0.28
0.44
1.00
Slope
−0.02
−0.17
−0.12
0.14
0.56
0.38
0.09
−0.34
−0.58
−0.63
−0.11
0.19
1.00
Sec
226
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L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
227
Fig. 4. Regressions (regression lines, r 2-values and number of lakes, n) between Secchi depth and macrophyte cover (A) and
macrophyte production (B).
Table 6
Stepwise multiple regression analyses using the data for the parameters given in Tables 1 and 3 to calculate:
Step
r2
x-variable
Model
(A) y-variable=!Maccov; n=19 lakes
1
18
0.52
2
7
0.67
3
4
0.74
4
4
0.84
Sec/Dm
90/(90-Lat)
!Dmax
log(A1)
y=1.944+4.825x1
y=6.757+3.83x1−1.57x2
y=8.31+2.57x1−1.50x2−0.286x3
y=10.49+1.502x1−1.993x2−0.422x3+0.490x4
(B) y-variable =log(Macprod); n= 35 lakes
1
90
0.73
2
11
0.80
log(Maccov)
90/(90-Lat)
y=0.678+1.328x1
y=2.472+1.028x1−0.516x2
F
(A) the macrophyte cover (Maccov in % of lake area); and (B) the macrophyte production (Macprod in g ww/m2 year) using
methods (transformations, etc.) given by Håkanson and Peters (1995)
(showing linear correlation coefficients, r, for nontransformed variables). In lake ecosystems there
are, as shown in Table 4, no ‘independent’ variables. Table 4 indicates that high and interesting
correlations exist between Maccov and Macprod,
latitude (Lat) and A3 or A1. It is also interesting
to note that Secchi depth, an operational expression for the depth of the photic zone, and a
potential limiting factor for macrophyte production, does NOT in itself correlate well with either
Maccov or Macprod, (see Fig. 4).
The mechanistic reasons for the high correlations between Maccov and Macprod, latitude and
A3 or A1 are evident: high latitude means low
temperature, and low production; large areas of
A1 and A3 mean a high potential production; and
high macrophyte cover evidently leads to high
macrophyte production.
Lake slope does not seem to be as important as
littoral slope (as given by Eq. (2)), and this is
understandable. About 80% of the variability
among lakes in macrophyte biomass can be related to littoral slope, whereas lake slope only
explains (statistically) less than 10% of the variation in Maccov among our lakes and less than 2%
of the variation in Macprod (Table 5).
Results of stepwise multiple regressions are
given in Table 6. This table gives a ranking of the
factors influencing first Maccov then Macprod. One
can see that the highest r 2-values are obtained for
Maccov rather than for Macprod, which is in agreement with the assumed higher CV-value for
228
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Macprod (see Table 4). From Table 6, one can also
note:
! Sec/Dm is the most important factor to statistically explain the variation among these lakes in
Maccov; r 2 = 0.52 for n =19. Fig. 5A gives the
regression for the 25 lakes for which we have
pair-wise data on Maccov and Sec/Dm. Then the
r 2-value is 0.38. The reason for the difference in
the number of lakes is that we only have a
‘complete’ data set for the stepwise multiple
regressions for 19 lakes.
! The next important factor for Maccov is latitude. If latitude is added, 67% of the variation
in Maccov among the lakes can be statistically
accounted for.
! The third factor is maximum depth; the deeper
the lake the smaller Maccov, which is logical.
! The fourth factor is the area of the lake at
depths smaller than 1 m, A1; r 2 =0.84.
It is interesting that none of the other variables
(and transformations) tested (see Tables 1 and 3),
like lake slope and ET-areas, add significant predictive power to this regression model, that the
included model variables are not highly inter-related and that the obtained r 2-value after four
steps (0.84) is very close to the assumed theoretical
reference r 2-value for Maccov of 0.87– 0.94.
Fig. 6 gives a direct comparison between actual
(non-transformed) empirical values of Maccov and
modelled values. The r 2-value is 0.82 and the data
are evenly spread around the regression line,
which has a slope very close to the ideal of 1 (it is
0.97).
Fig. 7 gives a 3D-diagram relating the two most
important model variables, Sec/Dm and latitude,
to Maccov. This diagram shows how these two
model variables influence Maccov when Dmax and
A1 are held constant. It is interesting to note the
very strong influence of latitude on Maccov. The
importance of Sec/Dm has been stressed before (see
Eq. (1)), but these results indicate that Eq. (1) does
not explain as much as previously anticipated.
Fig. 5. Regressions (regression lines, r 2-values and number of lakes, n) between macrophyte cover (!Maccov) and the most
important factors influencing Maccov-variability among these lakes, the ratio Secchi to mean depth (A), latitude (B), maximum depth
(D) and the lake area above 1 m water depth (D).
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
229
Maccov alone. This is logical.
When the next important factor, latitude, is
added, the r 2-value increases to 0.80, which is
close to the assumed theoretical reference value
of 0.84– 0.92 (Table 4).
Fig. 8 gives regressions between Macprod and
Maccov and Macprod and latitude. These relationships are highly significant (P#0.0001). Fig. 9
gives two comparisons between empirical data
and modelled values for Macprod, the first for
log-transformed values, the other for actual values. The first figure indicates that the model is
very good. The scatter around the regression line
is even; the slope is 1, the intercept close to zero
and the r 2-value close to r 2r . However, Fig. 9B
gives the regression for the actual data, and these
results are not as good. The r 2-value has dropped
to 0.68 and the slope is 1.5. This means that the
model on average gives 50% lower values than the
empirical data, and that the difference between
empirical data and modelled values can be high.
Still, this is the first empirical model for Macprod,
as far as we know, and the results for the logtransformation are good. Naturally, macrophyte
!
Fig. 6. A comparison (regression) between empirical data and
modelled values of macrophyte cover (using the empirical
model). (n= 19 lakes).
The results of the statistical analyses for Macprod
are given in Table 6B. We can note that:
! 73% of the variability in Macprod among these
35 lakes can be statistically explained by
Fig. 7. A 3D-diagram illustrating how the ratio Sec/Dm and latitude influence macrophyte cover, as given by the empirical model
in Table 6A, for a lake with a maximum depth of 20 m and A1 =2 km2 (lake area above 1 m water depth).
230
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 8. Regressions (regression lines, r 2-values and number of lakes, n) between macrophyte production [log(Macprod); values in g
ww/m2 year] and macrophyte cover (A) and latitude (B).
Fig. 9. A comparison (regression) between empirical data and modelled values of macrophyte production (using the empirical model
given in Table 6B) for logarithmic values (A) and actual (non-transformed) values; (n=35 lakes).
cover also influences the relationship between
macrophyte and phytoplankton productions. In
Eravno-Kharga
lakes
(Buryatiya),
the
macrophytes cover 85– 100% of the lake areas
(Neronova and Karasev, 1977), in Lake Karakul,
Kazakhstan, about 100% (Khusainova et al.,
1973), and in Lake Lacha in the Vologda district,
47% (Raspopov, 1978). In such macrophyte lakes,
the annual production of water plants attains high
values (500–2000 kcal/m2; 1 kcal !1.56 g ww)
and exceeds phytoplankton production. In lakes
with covering from 5 up to 20%, the macrophyte
production is usually less than 200 kcal/m2 year
and comparable or smaller than the phytoplankton production.
To illustrate how Maccov and latitude influence
Macprod, Fig. 10 gives a 3D-diagram for the
Macprod-model. One can see that the macrophyte
production is likely to be very low at latitudes
higher than 70° N.
In the next section, we will address macrophyte
production with a dynamic approach, and the
question is if the dynamic model can predict
better or worse than the statistical model?
3.2. Dynamical modelling
Fig. 11 gives a general outline of the dynamic
model, and the panel of driving variables. This is
a box model based on one ordinary differential
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
equation. This model is incorporated in a more
holistic lake ecosystem model (LakeWeb), but
LakeWeb will not be discussed in this paper.
Subsequently, we will treat all the fluxes regulating macrophyte biomass and production, as
given by this model. It should also be stressed
that the model is meant to predict weekly values
of macrophyte biomass and production. It gives
lake-characteristic values, and it is meant to be
driven by parameters easily accessed from standard monitoring programs and maps. The driving variables are: total phosphorus, lake pH,
Secchi depth, latitude, maximum depth, mean
depth and lake area. These factors are assumed
to be important for the prediction of
macrophyte biomass and productions, but not
equally important. The model also uses epilimnetic temperatures, which could either be measured or predicted by a model driven by data
on latitude, altitude and continentality (see Ottosson and Abrahamsson, 1998). For the following simulations, we have used a modified
231
version of the temperature model giving weekly
temperatures. The modified model has been presented by Håkanson and Boulion (2001). It is
shown in Fig. 12, and it will not be further
elaborated in this paper.
We have made rather crude guesses about
continentality and altitude for the lakes included
in the testing of the dynamic model. These are
the 19 lakes for which we have a ‘complete’
data set and these lakes were used for the
derivation of the empirical model for Maccov.
All these lakes have been ‘placed’ at an altitude
of 100 m above the sea level and lakes 1– 7 (see
Table 1) have been given a continentality of
5, 000 km, the rest a continentality of 500 km. It
will be shown later that the model predictions
are rather independent of these values.
The differential equation for macrophyte
biomass (BM; values in kg ww) is given by:
BM(t)=BM(t − dt) +(Ini− Con− Eli− Ero) dt,
(5)
Fig. 10. A 3D-diagram illustrating how macrophyte cover and latitude influence macrophyte production, as given by the empirical
model in Table 6B.
232
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 11. Illustration of the dynamic macrophyte model and its panel of driving variables. The upper part of the panel shows the
lake-specific variables that should be available for each lake. The lower part gives the four model constants. Latitude, continentality
and altitude are used to calculate epilimnetic temperatures.
where Ini is the initial macrophyte production (kg
ww/week); Con, food chain consumption of
macrophytes (kg ww/week); Eli, elimination of
macrophytes related to macrophyte turnover (kg
ww/week); and Ero, physical erosion of
macrophytes related to, e.g. wave action (kg ww/
week).
We will treat these processes one by one in the
following parts, starting with initial macrophyte
production.
3.2.1. Initial production
The initial macrophyte biomass [BM(0) in kg
ww] is basically given by the empirical model, i.e.
BM(0)
=0.001 Area 1/52 10 $ (2.472
+ 1.028 log10(Maccov)−0.516 90/(90 −Lat)),
(6)
where 0.001 is the dimensional adjustment of g
ww to kg ww; Area, lake area in m2; 1/52, dimen-
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
sional adjustment of year to week; Maccov, Maccov
in %; from the empirical model in Table 6A; Lat,
latitude in °N.
The initial macrophyte production (Ini in kg
ww/week) gives the theoretical maximum value
for the macrophyte production without any loss.
Ini is given by:
Ini = Rprod DelpH DelTP Area (Maccov 0.01) Ytemp,
(7)
where Rprod is the initial macrophyte production
rate (kg ww/m2 week); DelpH, a delay factor for
pH; changes in lake pH from one day to the next
cannot cause corresponding rapid changes in
macrophyte production. The delay factor is based
on the characteristic turnover time for
macrophytes (TMac), which is set to 300 days
233
(from Raspopov, 1973). DelpH is given by:
DelpH =SMTH(YpH, TMac,YpH),
(8)
where SMTH stands for ‘smoothing function’.
This smoothing function has been described by
Håkanson (1999). It is illustrated in Fig. 13 and it
gives an exponential smoothing of the input (here
YpH), by means of an averaging time (here TMac).
The initial value is YpH, a dimensionless moderator for pH-influences on macrophyte production
(see Håkanson and Peters, 1995, for more information on dimensionless moderators).
So, it is assumed that lake pH will influence
macrophyte production. The changes in lake pH
and the influence of pH on macrophyte production is given by the following algorithms:
Fig. 12. The temperature sub-model (from Ottosson and Abrahamsson, 1998, as modified by Håkanson and Boulion, 2001). The
empirical model to estimate the duration of the growing season from latitude comes from Håkanson and Boulion (2001).
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
234
Fig. 13. Illustration and definition of the smoothing function. Modified from Håkanson (1999).
if lake pH " 6.5
then YpH = (1+ 2 (pH/6.5 − 1)) else YpH 8.5
(9)
if lake pH # 8.5
then YpH 8.5 = (1−5.7 (pH/8.5 −1)) else YpH
= 1.
(10)
This means that YpH =1 for all pH-values in
the range from 6.5 to 8.5, and that YpH approaches 0 when pH approaches 3.25 and 10.
Note that these algorithms have not been critically tested because we lack reliable data on lake
pH. They are logical constructs, or working hypotheses. It would be very interesting to test this
approach. It is evident that there are major differences among the 19 tested lakes in characteristic
pH-values. In the following tests, we have set
YpH = 1 for all these lakes. We will show that this
simplification actually gives good predictions.
This indicates that these 19 lakes likely have
pH-values in the interval between 6.5 and 8.5.
The delay factor for total phosphorus (DelTP) is
defined in the same manner as DelpH.
DelTP =SMTH(YTP, TMac, YTP),
(11)
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
where YTP is a dimensionless moderator quantifying the influence of total phosphorus (TP) on
macrophyte production. YTP is given by:
YTP = (1+ 0.05 (TP/15 −1)),
(12)
where TP is the actual value of TP in !g/l; 15, the
norm-value for TP in !g/l; if the actual TP-value
is equal to the norm-value then YTP should be 1;
0.05, the amplitude value expressing how changes
in TP will influence macrophyte production. In
this model, it is assumed that TP is not a limiting
factor for macrophyte production, so the amplitude value is set low. If the actual TP-value is 300,
which is typical for hypertrophic lakes, YTP =
1.95, and macrophyte production about two times
higher than at TP=15, if all else is constant.
Note that this algorithm has not been critically
tested due to a lack of relevant data. It would be
interesting to test also this approach.
The influences of water temperature on
macrophyte production is given by Ytemp, which is
defined by the ratio:
Ytemp = EpiTemp/Reference temperature,
(13)
where EpiTemp is weekly epilimnetic temperature
(°C), as calculated from the temperature submodel given in Fig. 12, i.e. from latitude, altitude
and continentality. The reference temperature is
set to 9 °C (see Håkanson and Boulion, 2001).
This means that Ytemp is a simple dimensionless
moderator.
With this, all equations giving the initial
macrophyte production have been presented. The
model calibrations have focussed on the value for
the Rprod, the initial macrophyte production rate.
The initial tests have used values for Rprod in the
range 1/TMac to 10/TMac. It is likely that the
calculated biomasses and production values (=
biomass/turnover time) would be too low if the
Rprod-value is set to 1/TMac. The aim of the calibrations is to seek a general, default value for
Rprod, which could be used as a model constant
for all lakes. The calibrations and model tests are
described in a following section.
3.2.2. Consumption
The section addresses the question: how much
of the macrophyte biomass is lost per week due to
235
consumption by animals (mainly zoobenthos)?
This loss of biomass is given by:
Con=BM Rcon,
(14)
where Rcon (dimension 1/week) is simply assumed
to be very small compared to the erosion rate
(Rero,) and set to:
Rcon =Rero/100.
(15)
3.2.3. Elimination
The elimination loss is traditionally (see
Håkanson, 1999) given by:
Eli=BM 1.386/TMac,
(16)
where 1.386 is the half-life constant (= −ln(0.5)/
0.5), and TMac is, as mentioned, set to 300 days
(i.e. 300/7 weeks).
3.2.4. Erosion
Physical erosion of macrophytes is a complicated process (see Leclerc et al., 2000) involving
wind speed, duration, fetch, wave characteristics,
slope processes, erosion related to boating, etc.
This means that the erosion rate (Rero; 1/week)
has been in focus in the calibrations of the model.
Basically, Ero is given by:
Ero=BM Rero.
(17)
The questions are: which value should be given
to Rero and what factors govern Rero? This will be
addressed in the next section.
3.3. Model calibrations and !alidations
Validations, i.e. the model testing against independent data, are, evidently, the ultimate test for
predictive power (Håkanson and Peters, 1995).
We have selected eight lakes (2, 6, 13, 17, 22, 28,
30, and 33) of the 19 lakes with ‘complete’ data
for the initial calibrations of the initial
macrophyte production rate. The aim of the calibrations has been to derive general algorithms
and rate constants for the four regulatory rates in
this model. Since the rate of macrophyte consumption is set low (0.01 of the erosion rate), and
since we have reliable information about the
turnover time (TMac) and hence also the elimina-
236
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
tion rate (1/TMac), we have focused our calibrations first on the initial production rate, and then
on the erosion rate. As already stressed, we have
tested values for the initial consumption derived
from the inverse of the macrophyte turnover time,
which should give the proper order of magnitude
of the production rate. So, the main reason why
we have only used eight lakes for the calibrations
is that the initial production rate can be estimated, as far as order of magnitude is concerned.
When a reasonable generic value for the initial
production rate was established, we developed the
expression for the erosion rate. Then the questions were: which value for the erosion rate provides best fit to the empirical data for macrophyte
production and biomass in each lake? Is it possible to explain the variability in the established
lake-specific erosion rates by means of statistical
analyses? Are the results of the statistical analyses
logical and physically sound?
The calibrations have been carried out in a
standard way by numerous iterations and the aim
has been to obtain as generic values as possible,
and/or algorithms for the initial production and
the erosion rate. These features will then be incorporated in the model and after that the model will
be validated.
The model-predicted values for the macrophyte
production and biomass in each lake have been
compared to empirical data, as exemplified in Fig.
14 for Lake 6, Lake Arkhley, Siberia. Fig. 14A
gives three curves. Curve 1 is the initial
macrophyte production in kg ww/week, which
should be higher than the value given by line 2,
the empirical value and curve 3, the model-predicted values, which contrary to curve 1, also
accounts for erosion, consumption and turnover.
The empirical reference value for the macrophyte
biomass (BMemp in kg ww) is given by:
BMemp =PRyr 0.001 GS/(7 TMac),
Fig. 14. Illustration of the calibration procedure using the
dynamic model for Lake 6. (A) Gives initial macrophyte
production (curve 1), which should be higher than the modelled macrophyte production (cuve 3), which should vary
around the measured data (line 2). (B) Illustration of how
modelled values of macrophyte biomass depend on the rate of
macrophyte production, Rpro, and that Rpro =0.1 gives the
best fit to the empirical data in this lake. (C) Illustration of
how modelled values of macrophyte biomass vary with the
erosion rate, Rero. A Rero-value of 0.05 was used for this lake
in the derivation of the model for Rero.
(18)
where PMyr is the measured values, as given in
Table 1 of the macrophyte production in the lake
in g ww/year (= Area Macprod); 0.001 is the calculation constant g ww to kg ww; TMac, the turnover
time of macrophytes; 300 days; 7 is a calculation
constant (from days to weeks); GS, the duration
of the growing season (days). GS is calculated
from latitude (Lat in °N) by the following empirical model (as presented by Håkanson and
Boulion, 2001):
GS = −0.058 Lat2 +0.549 Lat+365,
(r 2 =0.996; slope= 1.0; n =8).
(19)
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
237
Table 7
Obtained best-fit values for the erosion rate in the 19 tested lakes
No.
Lake
Region
Erosion rate best fit
1
2
3
4
5
6
7
14
15
17
19
20
21
23
24
25
27
28
29
Chedenjarvi
Big Kharbey
Krasnoe
Glubokoe
Drivjaty
Arakhley
Dusja
Karakhul
Onega
Mikolayskoe
Naroch
Mjastro
Batorino
Lacha
Vozhe
Rybinskoe reservoir
Nogon-Nur
Vechten
Sjargozero
Karelia
Vorkuta, Russia
Karelian Isthmus
Moscow district
Byelorussia
Chita district, Siberia
Lithunia
Kazakhstan
Karelia
Poland
Byelorussia
Byelorussia
Byelorussia
Vologda district, Russia
Vologda district, Russia
Volga River
Mongolia
The Netherlands
Karelia
0.1500
0.0050
0.0200
0.0050
0.1000
0.0500
0.0001
0.0001
0.3000
0.0010
0.0001
0.0500
0.1500
0.0500
0.0100
0.0100
0.0001
0.1200
0.1500
Empirical values of the weekly macrophyte production (PRemp in kg ww/week) are calculated
from measured annual values (PRyr; PMemp =
PMyr/52).
Fig. 14B illustrates a typical calibration for
Lake 6 where different values of the production
rate (Rpro) have been tested to get the best possible fit relative to the empirical value (Eq. (18)). In
this example, the best result was given by Rpro =
0.1, which is also the generic value used in the
model. Fig. 14C illustrates the same calibration
procedure for the erosion rate when Rpro is set to
0.1. If we compare the results in Fig. 14B and C
for this lake, we can see that the predictions are
more sensitive to the value used for the production rate than the erosion rate, which is logical,
and that the best fit is obtained for a low erosion
rate in this lake Rero =0.05 (1/week).
This calibration routine has been carried out
for the 19 lakes for which we have a ‘complete’
data set using a generic value of 0.1 for the
production rate. This gave 19 lake-typical values
for the erosion rate, which are given in Table 7.
Using the same statistical methods as described
before to derive the empirical models for
macrophyte cover and production (Table 5), we
have tested to see if there is any general and
logical pattern among the values for the erosion
rates given in Table 7. The results are shown in
Table 8. Sixty percent of the variability in the
values for the erosion rates can be statistically
explained by differences among these lakes in
macrophyte cover and the form factor (Vd; see
Fig. 1 for definition). The larger the macrophyte
cover, the smaller the erosion rate.
This is logical since the exposed parts of the
macrophyte cover could be regarded as wave protection for the sheltered, inner parts. Fifty percent
of the variability in the established erosion rates
Table 8
Stepwise multiple regression analyses using the data for the
parameters given in Tables 1 and 3 to predict the obtained
best-fit lake values for the erosion rate (ER) using methods
(transformations, etc.) given by Håkanson and Peters (1995)
Step F
1
2
r2
x-variable
Model
17 0.49 log(Maccov) y=0.185−1.089x1
4 0.60 Vd
y=0.112−0.134x2+0.077x2
y=ER; n=19 lakes.
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L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 15. Illustration of predictive power of the dynamic model. (A) A comparison between empirical data and modelled data for
logarithmic values for macrophyte production. Note that the r 2-value is 0.89 and the slope 0.94. (B) A comparison between actual
empirical and modelled data for macrophyte production. The r 2-value is very high, 0.997, and the slope 1.02. These results could
be assumed to depend on one outlier (the highest value). However, if this value is omitted the r 2-value only drops to 0.92 and the
slope is 0.98 (see Fig. 16). (C) A frequency distribution showing the relative difference between empirical data (E) and modelled
values (M). Note that the mean value is close to the ideal (zero), but that the standard deviation (SD) is high (1.01). (D) A similar
comparison as in C, but for absolute differences. Note that in one lake the modelled value is three times lower than the empirical.
This may be explained by errors in the model and/or by uncertainties in the empirical data. The median value (M50) for the absolute
differences is 0.63 and the mean absolute error (MV) is 0.7.
can be related to variations in macrophyte cover.
The next most important factor of the tested (see
Tables 1 and 3) is the form factor. Lakes with
large shallow areas (V-shaped lakes) logically
have higher erosion rates than U-shaped lakes.
For the following validations, we have used the
empirical model for the erosion rate given in
Table 8, and the empirical model for macrophyte
cover given in Table 6A in the calculations of
macrophyte production and biomass using the
dynamic model. We have held all model variable
related to the initial production rate, the turnover
rate, the consumption rate and the erosion rate
unchanged and only altered the lake-specific variables for area, mean depth, maximum depth, Sec-
chi depth and latitude. The values for total-P,
lake pH have been constant for all lakes (15 !g
P/1 and 7, respectively). The altitudes have been
constant (100 m.a.s.l.) for all lakes and the continentalities have been set, as mentioned, either to
5, 000 km (for lakes 1– 7) or to 500 km (for the
rest of the lakes). So, the questions are: from these
presuppositions — what predictive power (r 2) will
the dynamic model give? How far from the theoretical maximum, as given by r 2r in Table 4, will be
obtained when modelled values are compared to
empirical?
The results of the validation are given in Fig.
15. For logarithmic values of macrophyte production, the r 2-value is 0.89 and the slope is close to
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
the ideal of 1 (it is 0.94). The spread around the
regression line is even. However, if we use not
logarithmic but actual values, the spread around
the regression line is quite different (Fig. 15B).
The r 2-value has increased to 0.997 and the slope
is almost ideal (1.02), but one could assume that
this depends on the outlier (Lake 26 with the
highest macrophyte production). This is, however, not the case. If we omit this lake, the
results are also very good indeed (see Fig. 16);
r 2 = 0.915; slope= 0.98. Fig. 15C gives the histogram illustrating the error, as defined by the
difference between empirical data (E) and modelled values (M), as (E− M)/M. One can note
that the mean error is close to zero, but that the
spread is large (the standard deviation, SD, is
about 1). Fig. 15D gives the same results for the
absolute error. The median absolute error, which
is the most relevant statistical measure for this
skewed frequency distribution, is 0.63, the mean
absolute error is 0.71 and the standard deviation
is 0.7.
Note that these are excellent results, but there
is also room for improvements. It is likely that
the model would have predicted even better had
reliable data been available on total-P, pH, conti-
Fig. 16. A comparison (regression) between empirical data and
modelled values of macrophyte production using the dynamic
model when Lake 26 has been omitted. (n =18 lakes).
239
nentality and altitude. From these validation results, however, one can also hypothesize that the
best way for further improvements would rather
be to access more reliable empirical data from
more lakes, than improving the model structure.
3.4. Model tests
This section will give results of many sensitivity tests carried out to illustrate how the model
works, and to highlight the fact that various
components in the model influence the target predictions of macrophyte biomass and production
differently. There are evident uncertainties with
all the structural components of the dynamic
model, and with all the lake-specific driving variables. But all of these uncertainties do not influence the predictions equally. The basic aim of
this section is to demonstrate the relative role of
all important model components in predicting
macrophyte biomass and production.
Fig. 17 gives the results of sensitivity tests (according to procedures given by Håkanson and
Peters, 1995) when all obligatory diving variables
have been varied, one at the time, while all else is
constant. The first figure (Fig. 17A) gives the
results when lake total-P concentrations have
been altered from 3 to 300 !g/l. This covers the
entire range of lake trophic categories, from oligotrophic to hypertrophic conditions. Note that
curve 5 illustrates how the model predicts if the
TP-concentrations are suddenly changed from
300 to 100 !g/l week 261. This may not be a
realistic event— it is meant to illustrate how the
model work, nothing else. The delay function
then gives a realistic response to the change in
lake TP-concentrations. One should also note
that macrophyte production is not, according to
this model and prevalent knowledge, limited by
lake nutrient levels but by other factors illustrated in Fig. 17.
Fig. 17B gives a similar sensitivity analysis for
lake pH. The model predicts that there is no
macrophyte production if pH is lower than 3.25
or higher than 10, and macrophyte production is
not pH-dependent if pH-values fall in the range
from 6.5 to 8.5. The response to a sudden change
in pH, for example from a lake liming, is given
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L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
Fig. 17. Sensitivity analyses using the dynamic model to predict macrophyte biomass (kg ww) in Lake 6. (A) Here the total
phosphorus concentrations (TP) have been varied from 3 to 300 !g/l, while all else were kept constant. Curve 5 illustrates how the
model predicts a hypothetical change in TP-concentrations from 300 to 100 !g/l week 261. (B) A similar simulation for different
pH-values. (C) Here we have changed the latitude from 20 to 70° N, while all else were kept constant. This influences macrophyte
production very much. (D) In this simulation, we have altered Secchi depth from 0.5 to 16 m. (E) Here we have altered the mean
lake depth from 7 to 11 m, while all else is constant. (F) Results for different hypothetical maximum depth (12 –120 m) in this lake,
while all else are constant.
by the delay function (curve 7), which is similar
to the delay for TP-changes.
So, variations in TP and pH do not influence
macrophyte production very much for all ‘normal’ lakes. Lake temperature, as related to latitude, on the other hand, has a profound impact
on macrophyte production and biomass. In Fig.
17C, we have ‘placed’ Lake 6 at latitude from
20 to 70° N, and the model predicts great
changes in macrophyte biomass.
The Secchi depth is also important for
macrophyte production and biomass (Fig. 17D),
especially if the Secchi depth, and hence also
the depth of the photic zone, attains high
values. In lakes with turbid waters and
Secchi depth lower than 2 m, the model predictions are not so sensitive to small alterations in
Secchi depth. Such lakes are generally shallow
with frequent resuspension events, and this
model does not account for such changes, i.e.
changes from site to site and hour to hour
within the lake.
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
The mean depth of the lake is one of several
morphometrical factors regulating resuspension
(Håkanson and Jansson, 1983), and Fig. 17E
gives results from sensitivity analyses where the
mean depth have been altered from 7 to 11 m
using data for Lake 6, which has a mean depth of
10.4 m. One can note that this would drastically
change macrophyte production and biomass. So,
it is important to use an accurate value for the
lake mean depth when macrophyte predictions are
carried out with this model.
Fig. 17F gives similar results for changes in
maximum depth. In this example, the maximum
depth for Lake 6 has been altered from 12 to 120
m; the actual value is 16.7 m. It is, we think, very
interesting to note that maximum biomasses are
obtained for Dmax-values of 30 m in this lake. If
the lake had been shallower or deeper, the
macrophyte biomass would have been lower. This
is related to the fact that the model accounts for
morphometric influences on macrophyte production and biomass in several ways. Both slope
processes on inclining bottom areas and wind/
241
wave-induced resuspension are incorporated in
the model, although we do not use any wind data
at all.
It is evident that the values or algorithms given
to the four fundamental rates in the dynamic
model regulate the model predictions, and Fig. 18
gives sensitivity analyses to illustrate the relative
role of the given processes. Fig. 18A illustrates
how uncertainties in the initial production rate, as
given by the three curve for 1.5 def (curve 1;
def =default value=0.1 kg ww/m2 week), def and
0.5 def, influence macrophyte biomass. The same
uncertainty factor of 50% relative to the default
value has been used for all four rates. From these
presuppositions, Fig. 18D illustrates that the uncertainty in the consumption rate plays a negligible role. This is because the consumption rate is
very small and influences macrophyte biomass
very little. Also the value given to the turnover
rate (1/300 days) plays a minor role in the overall
predictions of macrophyte biomass (Fig. 18C),
whereas the value used for the erosion rate is
more important (Fig. 18B).
Fig. 18. Sensitivity analyses using the dynamic model to predict macrophyte biomass (kg ww) in Lake 6. (A) Results for different
initial production rates; 1.5 def value, default value ( =0.1) and 0.5 def value, while all else is kept constant. (B) Results for similar
changes in the erosion rate. (C) Results related to different macrophyte turnover times. (D) Results for different consumption rates.
242
L. Håkanson, V.V. Boulion / Ecological Modelling 151 (2002) 213–243
4. Conclusions
This work has presented a new database for
macrophyte cover and macrophyte production.
The database gives information for 35 lakes,
which cover a very wide domain of lake characteristics. It includes lake characteristic data on morphometry, and many expressions for lake form
associated with sediment type and bottom dynamic conditions. We also have data on Secchi
depth and we have used validated models to
predict the duration of the growing season and
lake temperatures from information on lake latitude, and assumed values for altitude and continentality. Empirical models have been presented
for macrophyte cover and production (Table 5)
and the factors influencing macrophyte cover and
production have been ranked. We have also presented a dynamic model for macrophyte production and biomass, which gives weekly values from
information on: (1) the initial production rate; (2)
macrophyte turnover rate; (3) rate of macrophyte
consumption by animals; and (4) physical erosion
rate. The model has been critically tested and
yielded very high predictive power. The r 2-value
obtained when modelled values for macrophyte
production were compared to empirical data is
0.997 (n= 19). This sounds better than it actually
is and depends on a few very high values. The
median relative absolute error is 63%. It is probable that the model cannot be significantly improved until more comprehensive sets of reliable
data are available. We hope that this publication
can help to establish such data.
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