Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
An
algal growth model for the Vaal River
S.W. Schoombie," A. Cloot," A.J.H. Pieterse,^ J.C. Roos&
" Department of Applied Mathematics^ Department of
Botany and Genetics, University of the Orange Free State,
Bloemfontein, South Africa
ABSTRACT
A dynamic mathematical model was developed to simulate algal growth
and algal blooms in the Vaal river in central South Africa. The model takes
variations in the available light and water temperature into account, and,
in the case of diatoms, also the dissolved silicon content. The model has
been calibrated with available experimental data, and simulations of algal
growth agrees well with observations.
INTRODUCTION
The Vaal river is situated in the central part of South Africa. It is not
only an important source of water for domestic and industrial use, but it is
also used as a pathway for the disposal of industrial and domestic wastes,
leading to pollution and eutrophication of the river. Thus algal blooms occur
frequently, causing acute problems for those authorities responsible for the
treatment and distribution of water to mines, cities and towns (Pieterse
[1]), Walmsey and Butty [2], Roos and Pieterse [3].) Hence the need of a
mathematical model which could be used to predict algal blooms well in
advance, so that appropriate measures can be taken.
The model discussed here is dynamic and deterministic. It follows some
of the ideas which lead to existing models for rivers in other parts of the
world, such as those described in Whitehead and Hornberger [4], Di Toro
et al. [5], Wofsy [6] and Lung and Pearl [7].
It has been observed in the Vaal river that not all algal species are present
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
668 Water Pollution
at all times. A particular algal species would appear whenever conditions
for its growth is favourable, and disappear when adversely affected by its
environment. When several species are present simultaneously, they have
to compete for the available light energy and nutrients. These observations
have been incorporated in our model by distinguishing between various categories of algae. Each category consists of algae with similar sensitivities
to variations in environmental factors which influence their growth. The
interaction of the algal categories with their environment consists of various chemical, physical and biological mechanisms which can be described
by means of deterministic relations. These lead to a number of simultaneous nonlinear ordinary differential equations, which form the mathematical
model.
As pointed out above, we may assume that the Vaal river is eutrophicated and that nutrient availability is not a major variable. Thus, as a
first approximation, we neglected the effects of nutrients in the water in our
model, and took only the under water light climate and the average water
temperature into account. In the case of diatoms, which need silicon in
order to survive and grow, we also considered the interaction between the
diatoms present in the water and the dissolved silicon content.
LIGHT AND TEMPERATURE DEPENDENCE
When taking only variations in light intensity and water temperature into
account, the model consists of N pairs of coupled nonlinear differential equations
(1)
X2- — kf).Xi- — ^5,^2,
(2)
for i = 1,... TV, where Xi,,#2, are the concentration of living and dead
algae of the i-th algal category in suspension respectively. Here x means ^,
and gi is the ratio
7(0
9< = -j—
*optt
between the (time dependent) average light intensity 7 available under the
water surface, and the optimal light intensity 1^. for growth of the z'-th
algal category. The parameter 7 takes the form
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution 669
7=
kext being the vertical light extinction coefficient assumed to take the form
of
N
Wf) = &w + C^f) + I] tc/Zi, + ZgJ •
j=l
S(t) is the total concentration of inorganic material in suspension in the
water. The remaining parameters in the model equations can be classified
as either environmental parameters , or algal parameters which depend on
the characteristics of a particular algal category.
The environmental parameters Imax, //(*), T, S(t), ZQ, k^ and c$ are,
respectively, the maximum light intensity at the water surface, the cosine
of the solar zenith angle, the water temperature, the concentration of suspended inorganic material, the depth of the mixed layer, the light extinction
coefficient for pure water and the light extinction coefficient for suspended
inorganic material.
The algal parameters /opt,, A^,, &D,, &s, and k^ are, respectively, the
optimal light intensity for growth, the maximal algal growth rate, the algal
dying rate, the algal settling rate and the self-shading coefficient for algae.
The maximal growth rate &G, in Equation (1) varies with the temperature. The following expression for &&, agrees reasonably well with published
experimental data (see Canale and Vogel [8]):
f
= }{
T>
T
-t <
v.
with
-*- o
where Kmaxi,Topt> and T^m, are the overall maximal growth coefficient, the
temperature at which the optimal growth is observed and the minimum
temperature under which no growth of the i-th algal category is possible.
To calibrate the model, we tried tofixas many parameters as possible by
using known properties of the algal categories under consideration whenever
available. Parameters for which we could not determine a reasonable fixed
value a priori, were at most constrained between fixed limits, and were determined byfittingthe model simulations onto actual measurements of algal
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
670 Water Pollution
biomass. Time dependent parameters, such as light intensity, daily average
temperature and concentration of suspended inorganic material were used
as model inputs.
The model was tested on data obtained from one particular site, namely
Stilfontein, during a three year period starting at January 1985. Typically
during the winter and spring (southern hemisphere) of each year two algal
blooms were observed. The one was mainly due to diatoms, and the other
was mainly caused by green algae.
Neglecting the effect of silicon uptake by the diatoms, we used the calibration procedure outlined above for the data at Stilfontein. The model
equations were solved numeri -cally using a fourth order Runge-Kutta method. The weekly averaged temperature and suspended inorganic material
concentration were available as time series, and were updated on a daily
basis for the purposes of computation, using linear interpolation where necessary. For the cosine of the solar zenith angle, approximate formulas were
used (Schoombie and Cloot [9]). Total algal biomass were measured by
determining the chlorophyll-a concentration, and only two algal categories
were considered.
o
o
o
CO-]
\
O)
3.
CO
J_
.c
O
o
C\J
o
o
60.00
30.00 UO.OO 50.00
weeks
Figure 1: Simulated (solid line) and actual (dashed line) total chlorophyll-a
concentration in the Vaal River at Stilfontein during the winter/early spring
period of 1986. (Week 0 = 1 January)
0.00
10.00
20.00
Figure 1 shows the model simulation (solid line) together with the actual
measurements during 1986 at Stilfontein. The agreement between simulated
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution 671
and measured values is fairly good qualitatively as well as quantitatively
during the winter algal bloom period. Deviations at the end of the simulation are due to the appearance of additional algal categories after the
winter period. The use of more than two algal categories would lead to an
improvement. In the next section we shall show that the results can also be
improved by taking into account the uptake of silicon by the diatoms.
DIATOMS AND DISSOLVED SILICON
Unlike other types of algae, diatoms require the element silicon, which is
utilised to form a hard outer shell which protects the diatom against predators and parasites. The diatoms absorb silicon in the form of orthosilicic
acid (Si(OH)4), which in its turn is formed mainly from amorphous SiO2 in
the water. In the absence of diatoms, the concentration of orthosilicic acid
will reach a saturation value which depends on temperature according to
the equation
Sis(T) = Aexp(-^),
(5)
where A// is the enthalpy of the formation of orthosilicic acid, with a value
of 5500 cal/mol., R is the universal gas constant, T is the temperature
expressed in degrees Kelvin, and A is a constant, which, in the case of the
Vaal river, has a value of about 1.9 M/l.
In the presence of diatoms, the actual orthosilicic acid concentration
is regulated by an interaction between the uptake by the diatoms, and
production by chemical means in the water. This can be modeled by the
equation
Ijj-= ksi(Sis(T) - Si) - E
Vixn
(6)
i—diatoTYi
where Si is the actual orthosilicic acid concentration at time Z, and where
only diatom categories are involved in the summation. k$i is the inverse
of the characteristic time of restitution for the orthosilicic acid to a saturated concentration when no diatoms are present, and turns out to have a
value of about 0.1429 day"* in the case of the Vaal river. % is the rate of
silicon uptake by the i-th diatom category per biomass unit, and its dependence on the orthosilicic acid concentration can be modeled by the following
expression:
if
if
if
5. > sir*
Sir" > Si > Sif"
Si < St|"''".
(7)
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
672 Water Pollution
where
Vmaxi represents the maximum rate of silicon uptake at and above the critical
concentration level Si*?** for the particular algal category, while Si™*™ is
the minimum concentration required for the algal category to be able to
absorb silicon. Xi is a tuning parameter which is determined by fitting the
expression for V{ on actual experimental data.
Finally the model equations (1) and (2) are modified by incorporating
silicon uptake in the case of diatoms into the maximal algal growth rate as
follows:
if
if
if
0
Si?"
Si >
Si <
Si >
(9)
where Ar^(T) is defined by Equation (3). Note that the barred parameters
in Equation (9) can differ slightly in value from those in Equation (8). This
is because diatoms are able to take up more silicon than needed, and use
these reserves later during periods of nutrient depletion.
O
O) (JD .~
CO. OO
__
-C ^
O
O
O
O
OJ '
O
O
0.00
~
l
I
I
10.00 20.00 30.00
weeks
I
UO.OO
I
50.00
I
60.00
Figure 2: The same as Figure 1, except that silicon uptake by the diatoms
is taken into account. (Week 0 = 1 January)
Using these modifications whenever diatoms are involved, we found that
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
Water Pollution
673
a better simulation of algal growth can be obtained. Figure 2 shows the
simulated values (solid) line against the observed values of chlorophyll-a
concentration during 1986 at Stilfontein. It is obvious that the first peak,
in which a diatom category was mainly involved, is simulated better than
in Figure 1, where silicon uptake was not taken into account.
CONCLUSIONS
By taking into account only water temperature, under water light climate
and, in the case of diatoms, dissolved silicon , it is possible to simulate algal
growth and algal blooms in the Vaal river fairly well. Calibration can take
place at the beginning of the period of interest, and predictions can be made
about the subsequent behaviour of the algal species present in the water.
The next steps in the development of the model would be to add the
effect of salinity, pH, phosphate and nitrate concentration, and dissolved
oxygen and carbon dioxide. This should give a better quantitative agreement between simulated and observed values, and would also give a better
picture of the total interaction between the algae and their environment.
References
[1] Pieterse, A.H.J. ' Preliminary observations on the removal of phytoplankton from the Vaal River water at the Balkfontein purification
plant,' Chapter IV A, Environmental Quality and Ecosystem Stability
ed. Spanier, E., Steinberger, Y., and Luria, M., pp. 437-448, 1989.
[2] Walmsley, R.D. and Butty, M. 'Guidelines for the control of eutrophication in South Africa' Water Research Commission Report 0798817356,
1980.
[3] Roos, J.C. and Pieterse, A.J.H. 'Diurnal variations in the Vaal, a
turbid South African River: primary productivity and community
metabolism' /IrcA. #%/dro6W., Vol. 124, pp. 459-473, 1992.
[4] Whitehead, P.G. and Hornberger, G.M. 'Modeling algal behaviour in
the Thames River' MW. #es., Vol. 8, pp. 945- 953, 1984.
[5] Di Toro, D.M., O'Connor, D.J. and Thomann, R.V. ' A dynamic model
of the phytoplankton population in the Sacramento-San Joaquim delta'
,%r., /Imer. C%em. &>c., Vol. 106, pp. 131-180, 1971.
Transactions on Ecology and the Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3541
674 Water Pollution
[6] Wofsy, S.C. ' A simple model to predict extinction coefficients and
phytoplankton biomass in eutrophic waters' Limnol. and Oceanogr.,
Vol. 28, pp. 1144-1155, 1983.
[7] Lung, W.S. and Paerl, H.W. 'Modeling blue-green algal blooms in the
lower Neuse River' Wat. Res., Vol. 7, pp. 895-905, 1988.
[8] Can ale, R.P. and Vogel, A.H. 'Effect of temperature on phytoplankton
growth' J. Envir. Engng. Div. Am. Soc. civ. Engnrs. , Vol. 100, pp.
231-241, 1974.
[9] Cloot, A. and Schoombie, S.W. 'Some aspects of the dynamics of a
light-dependent uni-algal growth model' Technical report 1/91, Dept.
Applied Maths., UOFS, 1991.