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Modelling the multiple nutrient limitation of algal growth
Ecological Modelling, 18 (1983) 99-119 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
MODELLING GROWTH
THE MULTIPLE
NUTRIENT LIMITATION
99
OF ALGAL
W.T. DE GROOT
Centre for Environmental Studies, Rapenburg 127, 2311 GM Leiden (The Netherlands) (Received 3 March 1982; revised 11 November 1982)
ABSTRACT De Groot, W.T., 1983. Modelling the multiple nutrient limitation of algal growth. Ecol. Modelling, 18: 99-119. The way in which simultaneously limiting nutrients are supposed to act upon the algal growth rate is an important aspect of aquatic ecosystem modelling and research. Three different relations between the multiple nutrient limitation and two single nutrient limitations are developed from different biochemical models: a "multiplicative" relation, used in most dynamic ecosystem models, a new "sequential" relation and a "threshold" relation, sensu Liebig. The characteristics and practical consequences of these relations are investigated. By means of three experiments, derived from the literature, it is shown that the multiplicative relation yields the statistically significant worst growth rate predictions.
INTRODUCTION S o m e m a t h e m a t i c a l e x p r e s s i o n o f the algal g r o w t h rate is at the c e n t r e of all d y n a m i c a q u a t i c e c o s y s t e m models. P a r t of this e x p r e s s i o n is the multiple n u t r i e n t limitation, a f u n c t i o n of the single n u t r i e n t limitations; for example, for the two n u t r i e n t s i and j :
Lij=f(L,,Lj) T h e single n u t r i e n t limitations, L i a n d L j, are f u n c t i o n s of the c o n c e n t r a t i o n o f o n e n u t r i e n t only, m e a s u r a b l e w h e n all n u t r i e n t s e x c e p t that u n d e r test are saturated. T h e w a y in which Li a n d Lj are s u p p o s e d to i n t e r a c t in d e t e r m i n i n g L~,j is a m a t t e r o f great p r a c t i c a l i m p o r t a n c e , especially in m o d e l s d e s i g n e d to p r e d i c t the effects of lake r e s t o r a t i o n measures, in which the r e d u c t i o n o f p h o s p h a t e a n d / o r n i t r a t e loadings plays a role. T h e i n t e r a c t i o n of n u t r i e n t s is also a m a t t e r of long s t a n d i n g d e b a t e a m o n g biologists ( R o d h e , 1978; Tailing, 1979); basically, this discussion 0304-3800/83/$03.00
© 1983 Elsevier Science Publishers B.V.
100
centres on the postulate of Blackman (1905), which is an extension of Liebig's "Law of the minimum" from yields to growth rates as the dependent variable:
Li,j=
MIN
( Li, Lj),
vs. Baule's ( 1917) principle: Li, j = t i X L j
The first expression, from which the concept of the limiting factor is directly derived, is probably the most valued one among biologists, while most models probably make use of the second, multiplicative, expression. This paper has three objectives: (1) to develop the above expressions from a common basis, assuming that the algal growth can be modelled by simple biochemical kinetics; in doing so, a third expression, which has, in a way, a unifying character, is generated; (2) to illustrate the differences in the prediction of growth rates with these expressions; (3) to test which of the expressions complies best with experimental data; three experiments, taken from the literature, are used for this purpose. No attention will be paid to the question of how the single nutrient limitations L, and L/ are expressed as functions of the i- and j-concentrations, respectively. L i and L/ are assumed to be known assemblies of measured points, which may fit any saturating curve (e.g., Lederman and Tett, 1981). G E N E R A L EXPRESSION OF LIMITATIONS
The development of the three Li,pexpressions is based on a biochemical kinetic approach. These models can act as a basis for generating other formulae of this kind if the empirical need arises. Furthermore, some insight is gained into the theoretical plausibility of the expressions. However, the expressions' characteristics and empirical value are independent of these generating models. Growth can be regarded as the result of two enzymatic activities: "transporting processes", by which nutrient units are transferred to the places where they are needed, and "building-in processes", by which units are added to the active algal structures. These processes require time and energy. The energy-related limitations (e.g., the light limitation) are supposed to be saturated or constant; hence the kinematic approach, in which only time periods are discussed, seems leasable. The transporting times are supposed to be nutrient concentration dependent, approaching zero at very high nutrient concentrations. The rationale of
101
this is that the transport distance becomes negligible when the nutrient concentration rises. It will also be assumed that the active algal structures contain a constant ratio of nutrient units. The above simplifications are not essential; they permit, however, a rapid mathematical development of the basic options to formulate the multiple nutrient limitation. For the same reason, only two nutrients will be taken into account. The generalized (multi-nutrient) forms of the limitation formulae are presented in Annex I, where it is also shown which form the nutrient limitation takes in terms of concentrations, if Michaelis-Menten kinetics are assumed. The actual growth rate, gact, is defined as the fraction that an existing algal mass adds to itself per day. This rate can be split into two factors: a m a x i m u m growth rate (measurable under all-saturated circumstances) and the multiple nutrient limitation. Hence, this limitation equals:
gact gmax
L,,j-
(1)
This can also be expressed in times needed for the above processes. If tac t is defined as the actual time needed for transporting and building-in enough units of nutrients to add 1 unit of algal mass to an existing one, and /min is defined as the m i n i n u m time to do so, eq. 1 becomes:
ti'j
tmin =
(2)
/act
As stated before, tmi n is reached when all nutrients are saturated and all transporting times approach zero. At this point, the only determinant that remains is the building-in time. Dividing tac t into its transporting and building-in c o m p o n e n t s and defining the single-nutrient limitations as the two extremes of L,,j, we arive at the following general set of equations: rain bt L~,j - bt + tt
(3)
L i = Lim (Li.j)
(4)
j--* oo
Lj = Lim
( Li, ,)
(s)
in which: i,j: nutrients; L,,j: multiple nutrient limitation; Lg, Lj: single nutrient limitations, being equal to L,,j i f j and i become saturated, respectively; rain bt: m i n i m u m building-in time of 1 unit of alga; bt: the actual building-in time of 1 unit of alga; tt: the actual transporting time of 1 unit of alga. The times bt and tt are composed of bt i, tti, btj and ttj, the building-in and transporting times of one unit of i and j, respectively. Furthermore, r is
102
defined as: r = internal [ i ] / [ j ] ratio so that: one unit of alga = r units of i and 1 unit o f j . This is the starting-point of the next paragraph, in which three different expressions of Li,y (as functions of L~ and Lj) will be derived by assuming different growth kinetics. THREE EXPRESSIONS PREDICTING THE MULTIPLE LIMITATION
In order to develop a first, testable, hypothesis from the general expression of L~,j, the following growth mechanism is assumed. Compared to the time needed to transport and build-in r units of i, the transport and build-in times for one unit of j are negligible, so that bt = r. bt,, tt = r.tt~ and min bt = r. min bti, and eq. 3 becomes: min bt , Li, j = bt i + tt i
Furthermore, nutrient j's only influence is controlling the concentration of an enzyme E: [El =f([j]) in which f is some continuously increasing function, going through the origin of the [I] and [E] axes. Furthermore, bt~ and tt i are influenced by [E] in such a way that bt i = f ' ( [ E ] ) - ( b t i )
E at maximum
tti=f'([E])'(tti)Eatmaximum
in which f ' is some function, continuously decreasing between infinity at [E] = 0 and 1 at [E]'s maximum. The accolade terms denote i's transport and building-in times under maximum [E] circumstances. It follows that: bti=f"([j])'(bti)j__,oo tt i = f " ( [ j ] ) ' ( t t i ) j _ ~ o o
in which f " has the same character as f'. Through this, L~,y, L~ and Lj becomes:
Li, j =
1 rnin bt~ f , , ( [ j ] ) • {bti)j_~ ~ + ( t t i ) j ~ ~
103 L, = Lim ( Li.g) = - 1 . m i n bt i j-~ 1 ( b t i ) j ~ o + (tti)j~o~ L / = Lim (L,,j) - f,,(1 i_,~ [j])
min bt, minbt,+ 0
f"([j])
Hence, (A)
Li, j = L i • tg
A second expression of L;,j can be generated by assuming a "sequential" process in the following manner. One enzyme, constantly at optimum supply, transports and builds-in one unit of nutrient i, repeats this r times and proceeds by transporting and building-in one unit of nutrient j, completing one new algal unit. In terms of the previous paragraph, this implies that min bt equals bt and that bt and tt are made up of ( r . bt, + bt/) and ( r . tt, + tt/), respectively. Thus, eq. 3 becomes: r. bt i + btg Li j =
"
r • bt,
+
big + r
• tt i
+ tt/
Since tt, and bt, approach zero when i and j respectively become saturated, eqs. 4 and 5 become: L, = Lim ( L,,j) j--* o¢
r. bt i + btj r. bt i + btj + r. tt,
r. bt i +btg L / = i~ Lim ( L,,j) ~ r. bt, + btj + ttj
Some algebraic manipulations show that this can be merged as:
1
L,,/= l/L, + l/L~-
(B)
1
Thirdly, a process can be assumed in which the building-in proceeds sequentially, but the transport of i a n d j run parallel. One enzyme transports r units of nutrient i; meanwhile, another enzyme transports one unit of j. After arrival of the last necessary unit (either an i or a j), the building-in process starts, handling the (r + 1) units one after another, completing one new algal unit. E n z y m e concentrations are constantly at optimum. This means that the transport time in the complete process is the maximum of r. tt, and ttj: tt= MAX(r.
tti,
tt/)
while bt and min bt remain as with eq. B. Equation 3 becomes: r . bt~ + btj Li,j --
r. bt i + btg + M A X ( r .
it,, ttj)
104
which equals: L~ j = MIN "
r . bt i + btj
r . bt i + btj
]
r . bt i + btj + r . tt i
r . bt i + btj + ttj
l
As with eq. B, the terms left and right of the comma equal Lj and L i. Hence, L i , j = MIN(Li, L j )
(C)
Obviously, many more expressions for L~,j could be generated from other sets of assumptions. However, the most basic options are now present: the "multiplicative" hypothesis of the ecosystem models (eq. A), a "sequential" hypothesis (eq. B) and the "threshold" hypothesis, sensu Liebig (eq. C). Independently from its empirical merits, any expression derived from a reasonable enzymatic model will have the advantage that it always yields logically possible growth rate predictions. In the discussion, this point will be returned to. COMPARISON O F THE EXPRESSIONS
I n Table I, Li, j is calculated, according to the expressions A, B and C, for five combinations of L i and Lj. In the combinations 1 and 5, in which the concentrations of i and j are set "saturated" or nil, eqs. A, B and C predict the same nutrient limitation, as logically they should do. In the other cases, however, large differences are apparent. Equation A always predicts the lowest and eq. C the highest algal growth rates. Equation B is always in between, but closer to eq. A in the high "eutrophic" region of Li and Lj, and closer to eq. C in the "oligotrophic" case 4. This pattern is generalised in Fig. 1. Here, L~,/ is shown as a set of
TABLEI Multiple nutrient limitation (Li.g) predicted by eqs. A, B and C, for some arbitrary combinations of the single nutrient limitations Li and Lj, functions of the nutrient concentrations i and j , respectively Case
1 "eutrophic" 2 "eutrophic" 3 "mesotrophic" 4 "oligotrophic" 5 "dystrophic"
Li 1.00 0.80 0.50 0.50 0.50
Lj 0.80 0.60 0.40 0.20 0
Li. j
Li. j
Lid
by eq. A
by eq. B
by eq. C
0.80 0.48 0.20 0. I 0 0
0.80 0.52 0.29 0.17 0
0.80 0.60 0.40 0.20 0
105 0
o
o
..5"
Li,j=0.2
',
o
.'s
Li
,'.o
o
.5
EO. A
Li
EO. B
,.o
o
.s
Li
,.o
EQ. C
Fig. 1. Iso-limitation lines of the multiple nutrient limitation, predicted by eqs. A, B and C. For example, if L i = 0.40 and Lj = 0.70, Lt,j is predicted as 0.28, 0.34 and 0.40, respectively. The dotted line refers to Fig. 2.
iso-limitation lines in the L~-Lj-plane. It can be seen that the iso-limitation lines according to eq. B approach eq. A's hyperbolae in the upper right hand ("eutrophic") corner of the Li-Lj-plane, while they approach eq. C's straight lines in the "oligotrophic" region closer to the axes. Figure 2, a cross section through Figs. la, lb and lc, demonstrates this phenomenon. A conclusion can now be drawn. Ecosystem models are often used to predict the effect of lowering nutrient levels in an eutrophicated lake. When eq. A is used in such a model, any lowering of, say, the phosphate level will, through the decreasing Lp, result in a lower algal growth. As will be clear from Fig. 2, in models containing eq. C the algal growth rates will not respond to the phosphate level as long as any other nutrient (e.g. nitrate) is more limiting than phosphate; even below the threshold concentration the predicted ecosystem's reaction will be less. Equations A and C differentiate
.3"
~o~
.25
~~
.50
~'
.75
LI
~.o
Fig. 2. Cross-section of Fig. 1, at Lj = 0.50. Visible is the continuous influence of L, in eq. A, the threshold effect in eq. C and the intermediate and shifting character of eq. B.
106
in the eutrophic region of L~,j. Because of this feature, a choice in favour of eq. A or C might be arrived at by means of calibrations. This calibration method is less feasable for a decision between eqs. A and B, since eqs. A and B behave essentially the same in the eutrophic region. Yet, eq. B will predict consistently smaller effects of phosphate a n d / o r nitrate loading reductions. This difference becomes even more evident when the rates of loss of algae (sinking, grazing, washout etc.) are subtracted from the predicted rates of growth. EXPERIMENTAL TESTS OF THE EXPRESSIONS
The expressions derived above can be experimentally tested. The design of such experiments would be: (i) establish L, and Lj of an algal species by measuring growth rates under jand/-saturated conditions; (ii) measure concentrations and growth rates under conditions of simultaneous limitation by i and j; (iii) calculate the L i's and L f s corresponding to the concentrations and then the predicted growth rates according to the three equations A, B and C, to test which one best predicts the measured growth rates. In this paragraph, a visualized inspection of the empirical merits of the three equations will precede a statistical treatment. In the literature, three experiments have been found which can be analysed by this method. Since their authors' aims and tools were, in varying degrees, different from those which are relevant here, a re-interpretation of their data was necessary. This is the subject of Annexes II, Ill and IV. In the discussion, two other experiments are briefly mentioned. Droop (1975) investigated the behaviour of Microcystis lutheri in four batch cultures, with phosphate and Vitamin B12 as simultaneously limiting nutrients. From these data, L e, LvIT and the measured multiple nutrient limitation Le, vl a- can be calculated for 17 combinations of [P] and [VIT]. Lp and L v r r can be combined according to eqs. A, B and C, yielding the predicted Lp,vi-r. The complete procedure is given in Annex II. As it turns out, in 13 combinations Lp or Lv~ x is very close to either 0 or 1. In these cases, the three equations predict the same growth rate. These non-discriminating cases can be left out of the analysis. In Fig. 3, measured and predicted growth rates are compared for the three equations and the four remaining discriminating cases. As can be seen, the multiple nutrient limitations predicted by eq. C come closest to those measured, while eq. A gives the worst predictions. Feuillade and Feuillade (1975) experimented with Oscillatoria rubescens in chemostat cultures limited by phosphate and nitrate simultaneously. The N-
107 o ,.: . . . . . . . . . . . . . . . . . . . . . . .
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o
-2],
rv
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i
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M
0'.
1.0
(3":
M 1.o
";
EQ. B
EQ. A
";
M
1.o
EQ. C
Fig. 3. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the single species experiments of Droop (1975). The discriminating cases are represented by the heavy dots. On the diagonal, predicted values equal those measured•
and P-inflows (fluxes) are treated as the independent variables, and the resulting rate of production is measured. The relation between the N- and P-inflows and L N and Lp is not given; they have to be inferred from the dataset before L N and Lp can be determined. In this procedure, fully explained in Annex III, three of the eight experimental cases are used, leaving five cases for which L N and L p can be independently calculated and, through these, the LN, p'S predicted by the three equations. In Fig. 4 these are compared with the limitations measured in the corresponding cases. None of the equations accounts for the zero-growth measured in one case. Accepting this common discrepancy, the visual result is the same as for Droop's experiment: the limitations predicted by eq. C give the best overall fit, while eq. A predicts worst, with eq. B between the two.
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Q
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6 •
+
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./
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,/
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M EQ. A
/
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o
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I
0.25
o
M EQ. B
I
o.bs3
M
0.42
EQ. C
Fig. 4. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the single species experiments of Feuillade and Feuillade (1975). Small crosses indicate experiments used to infer the growth constants, for which M and PR are automatically equal.
108 / ÷ •
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•
9
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I 0.4
o
M EO. B
0!4
0
M
I 0.4
r:a. C
Fig. 5. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the experiments of Shapiro and Glass (1975), with natural algal assemblies, Small crosses indicate experiments used to infer the growth constants, for which M and PR are automatically equal.
Shapiro and Glass (1975) measured growth rates of a natural algal assembly from Lake Superior in batch cultures, with phosphate and taconite (a manganese containing mining waste product) as limiting nutrients. The dataset consists of nine cases of P-tac-combinations. As explained at length in Annex IV, five cases are needed to allow testing of the equations by means of the then remaining four cases. Fig. 5 is the result of the Annex calculations. As in the preceding experiments, it is found that eq. A gives the worst predictions, while eqs. B and C both predict satisfactorily, with errors one quarter of those of eq. A. It can be seen that eq. B's predictions have a better overall direction. F r o m the above analysis it can be hypothesized that eq. A, the multiplicative expression, consistently yields the worst growth rate predictions, while eqs. B and C are close to each other in quality, with eq. C probably slightly better. This hypothesis can be tested statistically. The first step is to calculate the differences between the predicted and measured values of Li. j with regard to the three equations, for the 4 + 5 + 4 experimental cases. Then in order to treat the 13 cases as one dataset, these absolute errors must be made mutually comparable. This is done by making them relative to the largest Li. j, measured in the experiment they are derived from. This scale factor is 1.01 for the errors from Droop (1975), 0.28, 0.060 and 0.47 for the errors in the analysis of the Feuillade and Feuillade data (1975), concerning eqs. A, B and C respectively, and 0.37 for Shapiro and Glass (1975) (ref. Tables II, IV and VI). Thus, the absolute differences become relative differences on the same scale, with n = 13 for all three equations. The averages and standard devia-
109
tions of these series are as follows: eq.A:
d A=0.10;
sA = 0 . 2 0
(n=13)
eq. B:
d B = 0.06;
s~ = 0.12
(n = 13)
eq.C:
dc=0.04;
sc = 0 . 1 2
(n=13)
Here, d is the average relative prediction error, while s is the standard deviation, indicating the degree to which prediction errors vary around their mean. The closer both values approach zero, the better is the predicting equation. As expected, the results of eqs. B and C closely resemble each other. The slight advantage of eq. C can be shown not to be statistically significant in Student's t-test. Equation A seems inferior in terms of both j and s. The significance of the difference in d can only be tested under the assumption that no difference in the standard deviation exists between the populations from which the cases are drawn. Hence, sh must be inspected first against s B and s c. In order to test this standard deviation difference, Cochran's test (Dixon and Massey, 1969) is most suitable. The null-hypothesis (the equations generate relative error populations with the same standard deviation) is tested against the alternative hypothesis that eq. A generates relative errors with a larger standard deviation than both eqs. B and C. As it turns out, Cochran's statistic C ---
s~ 2 + 2+s ~ SA S B
is significant at the 5% level. Since also d A is inferior to both d B and dc, it is inferred that eq. A is the statistically significant worst growth predictor. DISCUSSION
As Tailing (1979) has pointed out, the use of fixed limitation parameters and interaction formulae still has a weak empirical basis. A recent experiment may serve to illustrate this. Young and King (1980) measured growth rates of A n a c y s t i s nidulans under various light, carbon dioxide and phosphate conditions. Under constant light conditions, they found that the impact of a low but constant external phosphate concentration increases at decreasing external carbon dioxide levels. Obviously, none of the three alternative expressions can account for such a phenomenon; at most, a constant influence of phosphate is predicted. The internal nutrient concentration ratio, r, assumed to be constant in the derivation of the alternatives, might be the key factor in explaining the observed facts. Young and King show that the internal C/P-ratio may increase by a factor of 10 when the external
110
[P] decreases, at a constant CO2-1evel. It seems reasonable to suppose that this also takes place at constant [P] and decreasing [CO2]; in other words, the algae need relatively more P when [CO2] decreases. Hence, the P-influence may be stronger, the lower [CO2] becomes. If r is allowed to fluctuate with the external CO2-P-circumstances, all three expressions could be mathematically reformulated and tested. Young and King do not provide enough data for such an analysis. Also the pH, which, through its influence on the growth rate (Golterman, 1975), might be an alternative explanation, is not given. The problems which can be encountered in analysing experimental data can be illustrated in the case of G. Ahlgren (1980). In a chemostat experiment with Oscillatoria agardii it was tried to find a best L N, Lp and LN, p-expression simultaneously. The analysis does not allow for the concluded "limited multiplicative effect". For example, N is supposed to be the (most) limiting factor in cases where clearly L N > Lp; half-value constants found by assuming one Li-formula are applied in another; and very low external N-concentrations which, according to the conclusions, would yield an impossibly low LN, are ignored. One conclusion can be drawn from Ahlgren's data: the measured growth rates are explained by the internal P-concentrations, whatever the other circumstances are. This falls in line with Droop's (1975) findings, but unfortunately is not mentioned by the author. Besides the equations A, B and C, an infinite number of alternative expressions for the multiple nutrient limitation can be formulated. Apart from their empirical value, the L,,/-formulae should meet two "logical" criteria: (i) if any nutrient's concentration is zero (e.g., Lj = O) then the growth should be zero (Li. j = 0), independent of the other nutrients, (ii) if any nutrient concentration is saturated, it should exert no influence; e.g., if Lj = 1, all expressions should yield Li, j = L i. These criteria define the outcomes on the extremes in Figs. 1 and 2, where the lines of eqs. A, B and C meet. All three meet the criteria--as will, in fact, any expression derived from a reasonable biochemical growth model. Jorgensen et al. (1981) apply five formulae in a calibration research, among which are eqs. A, C and two alternatives: 1
L,,j = 1/L, + 1/Lj
(D)
L~.j= ( L~ + L/)/2
(E)
while Scavia and Park (1976), in their model CLEANER, use: 2 L i ' j = 1 / L i q- I / L j
(F)
111
The generalised forms are given in Annex I. None of these expressions meets the second criterion. If nutrient j is in unlimited supply (Lj--- 1), eqs. E and F always predict Li,j's higher than L i, while eq. D always predicts them lower than L~. Moreover, eq. E does not meet the first criterion. If nutrient j is not present at all (Lj = 0), the growth rate is still predicted as positive. Given the many degrees of freedom in most calibration processes, it cannot be excluded that "illogical" equations may yield acceptable results in the calibrated ranges of L~ and Lj. However, for predicting growth rates outside these ranges, e.g. predicting the effect of proposed phosphate control measures, they are inherently weaker tools. Finally, it must be considered along which lines further research might be undertaken. Both laboratory experiments and ecosYstem model calibration procedures may be fruitful in solving the existing discrepancies concerning the multiple nutrient limitation expressions, shown in this paper. Possibly one condition for success might be that one should not try to find all growth influencing factors and growth formulae simultaneously. First, the single nutrient limitation functions must be firmly established, e.g. by measuring them under truly saturated conditions of all nutrients except one. Then, using "logical" formulae only, the multiple nutrient expression can come under test. The direct practical importance in water management justifies a search for an improved empirical grounding of the modelling of nutrient limitation. ACKNOWLEDGEMENTS
Thanks are due to M. Donze and E. Meelis for their general and statistical support. ANNEX
Limitations in terms of concentrations; generalized expressions. Equation 2, for one nutrient only, reads: Li
bti bt, + tt i
The building-in time can be assumed constant, while tt might depend on a constant transport velocity, v, and the average transport distance, d; assuming this distance to be inversely proportional to the concentration (d = a / [ i ] ) , the. nutrient limitation becomes: L,-
bt
a bt + - -
v.[i]
112
Dividing numerator and denominator by bt/[i] it is found that: Li =
[i] a [ i ] +v- •- bt
This is the Michaelis-Menten expression, in which the denominator's second term is measured as the half-value constant, ki. If this type of growth curve is substituted for L i and Lj, the two equations A and B read as follows:
[;].[j] eq.A:
Li,j=Li,LJ-[i].[j]+[i].kj+[j].ki+ki.k
eq.B"
Li,j=l/Li+l/Lj_l=[i].[j]+[i].kj+[j].ki
1
j
[i]. [j]
Note that in terms of concentrations, eq. B shows as a simplification of eq. A. The generalised multi-nutrient forms of the equations, mentioned in this paper, are: eq. A:
L,,j ...... = L~ x Lj x ... X L ,
eq.B"
L,,j ..... " = 1 / L i+ 1 / L j + ... + 1 / L , - ( n -
eq.C:
Li, j ..... , = M I N ( L i , L
eq. D:
Li, j ......
eq.E:
L,,j ...... = ( L , + L j + . . . L , ) / , .
eq. F:
Le, j ...... - 1 / L , + 1 / L j + ... 1 / L ,
1 1)
j ..... L,)
1 1/L~ + 1 / L j + ... 1 / L n
11
in which n is the number of nutrients. ANNEX
I1
An interpretation of the Droop (1975) data. Droop (1975) experimented with Monochrysis lutheri in four batch cultures, with phosphate and Vitamin B12 as simultaneously limiting nutrients. No relation exists between the growth rates and external concentrations, but the internal nutrient concentrations do exert a clear influence. As Droop points out, this influence can be satisfactorily explained by the assumption that only the most limiting nutrient (in terms of internal concentrations) determines the growth rate, that is, by equation C. For the purpose of this
113
article, it makes sense to investigate if the other hypotheses are not better. This is achieved as follows. From the given biomass levels B, the growth rates g (in fraction per day) can be calculated through: ln(1
+ g,) =
In Bt+at- In Bt_at 2 At
in which t and At denote sampling day numbers and their differences, respectively. Also, from the given internal nutrient concentrations, L p and Lvl x can be found, using the expression and parameters mentioned by Droop. Then, setting gmax at 1.5, a dataset is arrived at in which are Lp, Lvi v and the measured Lp.vl r, consisting of 17 cases. The latter can be compared with the Lp,vla-'s, predicted by the eqs. A, B and C, to find a best predicting equation. The data and predictions are given in Table II. In the majority of cases, Lp a n d / o r L v v r turns out to be so close to zero or unity that the three equations do not differentiate (see Table I). In fact, only the cases 11, 12, 15 and 16 offer an opportunity to evaluate the equations. In Fig. 3, these cases are represented by the large dots. Although not relevant for the purpose of
T A B L E II M e a s u r e d a n d predicted growth rates, derived from the four one-species b a t c h cultures of D r o o p (1975), in which p h o s p h a t e a n d vitamin B12 can be simultaneously limiting
Expt. I
Expt. II
Expt. III
Expt. IV
Case
Day
Le
Lvv r
Lp, vl T Lp, vlT, predicted by: measured eq. A eq. B eq. C
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
4 6 8 11 4 6 8 11 15 2 3 5 7 3 5 7 10
0.84 0.39 0.04 - 0.05 0.96 0.96 0.94 0.92 0.88 0.93 0.75 0.40 0.01 0.95 0.82 0.31 0.04
0.98 0.92 0.88 0.85 0.96 0.71 0.56 0.36 -0.03 0.96 0.91 0.76 0.59 0.96 0.83 0.23 -0
1.01 0.39 0.07 - 0 0.92 0.41 0.09 0.07 0.01 0.84 0.66 0.30 0.07 0.99 0.84 0.27 0.03
0.82 0.36 0.04 - 0.04 0.92 0.68 0.53 0.33 -0.03 0.89 0.68 0.30 0.01 0.91 0.68 0.07 -0
0.83 0.38 0.04 - 0.05 0.92 0.69 0.54 0.35 -0.03 0.90 0.70 0.36 0.01 0.91 0.70 0.15 -0
0.84 0.39 0.04 - 0.05 0.96 0.71 0.56 0.36 -0.03 0.93 0.75 0.40 0.01 0.95 0.82 0.23 -0
114
this paper, it may be noted that the growth rates of cases 6, 7 and 8 are grossly overestimated by all three equations. Those cases were characterized by a high total-P concentration, giving rise to high phosphate cell quota. These phosphates evidently act more as "burden-P" than as "luxury-P", and Droop's way of calculating Lp does not account for this. ANNEX III
An interpretation of the Feuillade and Feuillade (1975) data. Feuillade and Feuillade (1975) experimented with Oscillatoria rubescens in chemostat cultures, with N and P as limiting nutrients. Independent variables were the N and P input rates (fluxes); the dependent variable was the rate of production, g. This dataset, comprising eight experimental cases, is represented in Table III. The phosphorus and nitrogen limitations (Lp and LN) are functions of the respective input rates, P-flux and N-flux. In Feuillade and Feuillade, contrary to Droop (1975), these functions are not presented. This implies that the two half-value constants k P and k N (see Annex I) have to be found before the data can be analysed. The same holds for gmax, the maximum growth rate, which determines how a measured growth rate g is translated into a measured Ly, p. These three unknowns have to be inferred from the Feuillade and Feuillade dataset. The following procedure is applied: (i) assume one of eq. A, B or C to hold, (ii) write out this equation in terms of fluxes (Annex I), (iii) substitute the P-flux, N-flux and measured g for three arbitrary experi-
TABLE III The Feuillade and Feuillade (1975) dataset: measured production rates (g), in mg dry matter per liter per hour, and phosphate and nitrate inputs (fluxes) in 8 one-species chemostat experiments Case
P-flux, #g P/1 h
1
7
2 3 4 5 6 7 8
12 22 7 14 24 7 12
N-flux,/xg N / 1 h
g mg/1 h
27 27 27 53 68 64 119 128
0.0 0.80 0.65 i.10 1.30 1.35 1.80 2.40
1 2 3 4 5 6 7 8
Case
0.48 (0.61 0.71 0.48 0.64 0.75 (0.48 (0.61
0.15 0.15 0.15 0.26 0.31 0.30 0.45 0.46 0
0.09) 0.08 0.13 0.15 0.16
0.21) 0.28)
0.07
0.09
0.11 0.125 0.20 0.225
0.21 0.28
LN, P (meas.)
0.08 (0.13 0.21 0.08 0.15 0.23 (0.08 (0.13
0.023 0.023 0.023 0.044 0.056 0.053 0.095 0.101
LN
Lp
LN, P (pred.)
Lp
LN
Testing eq. B
Testing eq. A
0.045 0.060
0.021 0.029 0.042 0.045
0.020
0.018
LN. P (pred.)
0.045) 0.060)
0.016 0.027 0.032 0.034
0.020)
0
LN, P (meas.)
0.35 (0.48 0.62 0.35 0.51 0.65 (0.35 (0.48
Lp
0.15 0.15 0.15 0.26 0.31 0.30 0.45 0.47
L~
Testing eq. C
0.35 0.47
0.15 0.26 0.31 0.30
0.15
0.15
LN. p (pred.)
0.35) 0.47)
0.125 0.21 0.25 0.26
0.15)
0
LN. P (meas.)
Testing eqs. A, B and C through the Feuillade and Feuillade (1975) data. Method explained in Annex III. The cases used to calculate the parameters k N, kp and gma~, for which the predicted and measured production rates are automatically equal, are put between brackets. Values used for testing are printed in italics
TABLE IV
116
mental cases of Feuillade and Feuillade, thus creating three equations to solve the three unknowns. With the now known kN, kP and gmax, it is possible to calculate LN, Lp and the measured LN, P for the remaining five Feuillade and Feuillade cases, as well as the errors in the predicted LN, P generated by eq. A, B or C. E.g., eq. A, in terms of fluxes, reads: gJgmax =
P-flux P-flux W k P
N-flux N-flux + k N
Substituting the P-flux, N-flux and g of the experimental case 2, one finds: 12
27
0 " 8 / g m a x ----- 12 + k P " 27 + k N
This can be repeated for two other cases, e.g., cases 7 and 8, and kN, k P and gmax are found to be 148, 7.8 and 8.5, respectively. This is used to calculate LN, Lp and the measured LN, P as well as the LN, P predicted by eq. A. The results are shown in Table IV. Repeating this procedure for eqs. B and C, Table IV is completed. F r o m the Table and Fig. 4, it is obvious that none of the equations can account for the zero growth in case 1. Disregarding this c o m m o n discrepancy, it can be seen that in this analysis eq. A comes third in prediction accuracy, while eq. C comes out best. ANNEX IV
A n interpretation of the Shapiro and Glass (1975) data. Shapiro and Glass (1975) noticed a synergistic effect of phosphate and manganese enrichment on algal growth in Lake Superior. This was investi-
TABLE V The Shapiro and Glass (1975) dataset: radiocarbon uptake after 20 days, in thousands Cpm, under various conditions of taconite (mg/1) and phosphate ( # g / l ) additions. Natural algal populations of Lake Superior Additions of phosphate
+0 +5 + 25
Additions of Taconite +0
+0.4
+4
3 7 8
3 8 16
3 10 37
117
gated in batch cultures with natural algal assemblies. Three samples were taken of Lake Superior water, and various amounts of phosphate and taconite, a manganese containing mining waste product, were added. Sufficient specific growth rates are only given for their Sample III. The experiments with this sample will be analysed here. Table V shows the results of the combined effects of the addition of phosphate and taconite, in terms of radiocarbon uptake after 20 days, for nine experiments. The experiments are still in their exponential growth phase after 20 days; hence, the carbon uptakes are indicative for the growth rates (uptake/day), and will be treated as such. The additions will be regarded as indicative for concentrations. As in Annex III, the single nutrient limitations ( L e and LTAC) are not given for the P and taconite additions; neither is g . . . . the maximum growth rate. Again, they have to be inferred from the dataset, using some experimental cases, before eqs. A, B and C can be tested by means of the remaining ones. In Shapiro and Glass, taconite additions over 4 m g / l do not increase growth rates. Hence, over the last column of Table V, the experiment is taconite saturated, implying that, assuming Michaels-Menten kinetics, it can be written, for TAC = 4 mg/1, g -Lpgmax
P-add. P-add. + kP
independent of the equations to be tested. By substituting the P-addition values and the measured g, it is found that gm,x approximates to 100. If the values in Table V are divided by 100, they represent the measured Lp,a-AC (see Table VI). The phosphate limitation Lp equals Lp, TAc over the last column. Hence, for the addition of 0, 5 and 25 mg P / l , Lp equals 0.03, 0.10 and 0.37, respectively. No phosphate limitation exists in any row of Table V. Therefore, as in Annex III, the LTAc that can be found depends on which L P,'rAC equation is assumed to hold true. E.g. for the case (P = 5, TAC = 4), it is known that Lp - 0.10 and the measured Lp, a.AC = 0.08. Now, if eq. A is assumed to hold, LTAC must be 0.80. If eq. C is assumed to hold, LVAc must be 0.08. Using the whole row of P = 5 mg/1, --assuming eq. A to hold: LTAc = 0.7, 0.8 and 1.0 for the three taconite additions, --assuming eq. B to hold: La-AC= 0.19, 0.29 and 1.0, respectively, --assuming eq. C to hold: La-AC= 0.07, 0.08 and 1.0, respectively. For the row and column used, the predicted Lp,TAc will now automatically coincide with the measured Lp,a-AC, leaving four cases to test the equations, by comparing the predicted and measured L e,a-Ac's. This is represented in Table VI. To check the limitations stated above, the cases of the column TAC = + 4 and P = + 5 are given between brackets.
119
From Table VI, it can be concluded that in this analysis eqs. B and C both yield far better predictions than eq. A. The sum of the absolute errors of prediction by eq. A is 0.37, while this amounts to 0.09 for both eqs. B and C. REFERENCES Ahlgren, G., 1980. Effects on algal growth rates by multiple nutrient limitation. Arch. Hydrobiol., 89: 43-53. Baule, B., 1917. Zu Mitscherlichs Gesetz der physiologische Beziehungen. Landwirtsch. Jahrb., 51 : 363-385. Blackman, F.F., 1905. Optima and limiting factors. Ann. Bot., 19: 281-295. Dixon, W.J. and Massey, Jr., F.J., 1969. Introduction to statistical analysis, 3rd edn. McGraw Hill, New York, 638 pp. Droop, M.R., 1975. The nutrient status of algal cells in batch culture. J. Mar. Biol. Ass. U.K., 55: 541-555. Feuillade, J.B. and Feuillade, M.G., 1975. Etude des besoins en azote et en phosphore d'Oscillatoria rubescens D.C. h l'aide de cultures en chemostats. Verh. Int. Ver. Limnol., 19: 2698-2708. Golterman, H.L., 1975. Developments in Water Science, Vol. 2, Elsevier, Amsterdam, 489 pp. J4rgensen, S.E., J4rgensen, L.A., Kamp-Nielsen, L. and Meijer, H.F., 1981. Parameter estimation in eutrophication modelling. Ecol. Modelling, 13:111-129. Lederman, T.C. and Tett, P., 1981. Problems in modelling the photosynthesis-light relationship for phytoplankton. Bot. Mar., 24: 125-134. Rodhe, W., 1978. Algae in culture and nature. Mitt. Int. Ver. Limnol., 21: 7-20. Scavia, D. and Park, R.A., 1976. Documentation of selected constructs and parameter values in the aquatic model CLEANER. Ecol. Modelling, 2: 33-58. Shapiro, J. and Glass, G.E., 1975. Synergistic effects of phosphate and manganese on growth of Lake Superior algae. Verh. Int. Ver. Limnol., 19: 395-404. Tailing, J.F., 1979. Factor interactions and implications for the prediction of lake metabolism. Arch. Hydrobiol. Beih., 13: 96-109. Young, T.C. and King, D.L., 1980. Interacting limits to algal growth: light, phosphorus and carbon dioxide availability. Water Res., 14: 409-412.
MODELLING GROWTH
THE MULTIPLE
NUTRIENT LIMITATION
99
OF ALGAL
W.T. DE GROOT
Centre for Environmental Studies, Rapenburg 127, 2311 GM Leiden (The Netherlands) (Received 3 March 1982; revised 11 November 1982)
ABSTRACT De Groot, W.T., 1983. Modelling the multiple nutrient limitation of algal growth. Ecol. Modelling, 18: 99-119. The way in which simultaneously limiting nutrients are supposed to act upon the algal growth rate is an important aspect of aquatic ecosystem modelling and research. Three different relations between the multiple nutrient limitation and two single nutrient limitations are developed from different biochemical models: a "multiplicative" relation, used in most dynamic ecosystem models, a new "sequential" relation and a "threshold" relation, sensu Liebig. The characteristics and practical consequences of these relations are investigated. By means of three experiments, derived from the literature, it is shown that the multiplicative relation yields the statistically significant worst growth rate predictions.
INTRODUCTION S o m e m a t h e m a t i c a l e x p r e s s i o n o f the algal g r o w t h rate is at the c e n t r e of all d y n a m i c a q u a t i c e c o s y s t e m models. P a r t of this e x p r e s s i o n is the multiple n u t r i e n t limitation, a f u n c t i o n of the single n u t r i e n t limitations; for example, for the two n u t r i e n t s i and j :
Lij=f(L,,Lj) T h e single n u t r i e n t limitations, L i a n d L j, are f u n c t i o n s of the c o n c e n t r a t i o n o f o n e n u t r i e n t only, m e a s u r a b l e w h e n all n u t r i e n t s e x c e p t that u n d e r test are saturated. T h e w a y in which Li a n d Lj are s u p p o s e d to i n t e r a c t in d e t e r m i n i n g L~,j is a m a t t e r o f great p r a c t i c a l i m p o r t a n c e , especially in m o d e l s d e s i g n e d to p r e d i c t the effects of lake r e s t o r a t i o n measures, in which the r e d u c t i o n o f p h o s p h a t e a n d / o r n i t r a t e loadings plays a role. T h e i n t e r a c t i o n of n u t r i e n t s is also a m a t t e r of long s t a n d i n g d e b a t e a m o n g biologists ( R o d h e , 1978; Tailing, 1979); basically, this discussion 0304-3800/83/$03.00
© 1983 Elsevier Science Publishers B.V.
100
centres on the postulate of Blackman (1905), which is an extension of Liebig's "Law of the minimum" from yields to growth rates as the dependent variable:
Li,j=
MIN
( Li, Lj),
vs. Baule's ( 1917) principle: Li, j = t i X L j
The first expression, from which the concept of the limiting factor is directly derived, is probably the most valued one among biologists, while most models probably make use of the second, multiplicative, expression. This paper has three objectives: (1) to develop the above expressions from a common basis, assuming that the algal growth can be modelled by simple biochemical kinetics; in doing so, a third expression, which has, in a way, a unifying character, is generated; (2) to illustrate the differences in the prediction of growth rates with these expressions; (3) to test which of the expressions complies best with experimental data; three experiments, taken from the literature, are used for this purpose. No attention will be paid to the question of how the single nutrient limitations L, and L/ are expressed as functions of the i- and j-concentrations, respectively. L i and L/ are assumed to be known assemblies of measured points, which may fit any saturating curve (e.g., Lederman and Tett, 1981). G E N E R A L EXPRESSION OF LIMITATIONS
The development of the three Li,pexpressions is based on a biochemical kinetic approach. These models can act as a basis for generating other formulae of this kind if the empirical need arises. Furthermore, some insight is gained into the theoretical plausibility of the expressions. However, the expressions' characteristics and empirical value are independent of these generating models. Growth can be regarded as the result of two enzymatic activities: "transporting processes", by which nutrient units are transferred to the places where they are needed, and "building-in processes", by which units are added to the active algal structures. These processes require time and energy. The energy-related limitations (e.g., the light limitation) are supposed to be saturated or constant; hence the kinematic approach, in which only time periods are discussed, seems leasable. The transporting times are supposed to be nutrient concentration dependent, approaching zero at very high nutrient concentrations. The rationale of
101
this is that the transport distance becomes negligible when the nutrient concentration rises. It will also be assumed that the active algal structures contain a constant ratio of nutrient units. The above simplifications are not essential; they permit, however, a rapid mathematical development of the basic options to formulate the multiple nutrient limitation. For the same reason, only two nutrients will be taken into account. The generalized (multi-nutrient) forms of the limitation formulae are presented in Annex I, where it is also shown which form the nutrient limitation takes in terms of concentrations, if Michaelis-Menten kinetics are assumed. The actual growth rate, gact, is defined as the fraction that an existing algal mass adds to itself per day. This rate can be split into two factors: a m a x i m u m growth rate (measurable under all-saturated circumstances) and the multiple nutrient limitation. Hence, this limitation equals:
gact gmax
L,,j-
(1)
This can also be expressed in times needed for the above processes. If tac t is defined as the actual time needed for transporting and building-in enough units of nutrients to add 1 unit of algal mass to an existing one, and /min is defined as the m i n i n u m time to do so, eq. 1 becomes:
ti'j
tmin =
(2)
/act
As stated before, tmi n is reached when all nutrients are saturated and all transporting times approach zero. At this point, the only determinant that remains is the building-in time. Dividing tac t into its transporting and building-in c o m p o n e n t s and defining the single-nutrient limitations as the two extremes of L,,j, we arive at the following general set of equations: rain bt L~,j - bt + tt
(3)
L i = Lim (Li.j)
(4)
j--* oo
Lj = Lim
( Li, ,)
(s)
in which: i,j: nutrients; L,,j: multiple nutrient limitation; Lg, Lj: single nutrient limitations, being equal to L,,j i f j and i become saturated, respectively; rain bt: m i n i m u m building-in time of 1 unit of alga; bt: the actual building-in time of 1 unit of alga; tt: the actual transporting time of 1 unit of alga. The times bt and tt are composed of bt i, tti, btj and ttj, the building-in and transporting times of one unit of i and j, respectively. Furthermore, r is
102
defined as: r = internal [ i ] / [ j ] ratio so that: one unit of alga = r units of i and 1 unit o f j . This is the starting-point of the next paragraph, in which three different expressions of Li,y (as functions of L~ and Lj) will be derived by assuming different growth kinetics. THREE EXPRESSIONS PREDICTING THE MULTIPLE LIMITATION
In order to develop a first, testable, hypothesis from the general expression of L~,j, the following growth mechanism is assumed. Compared to the time needed to transport and build-in r units of i, the transport and build-in times for one unit of j are negligible, so that bt = r. bt,, tt = r.tt~ and min bt = r. min bti, and eq. 3 becomes: min bt , Li, j = bt i + tt i
Furthermore, nutrient j's only influence is controlling the concentration of an enzyme E: [El =f([j]) in which f is some continuously increasing function, going through the origin of the [I] and [E] axes. Furthermore, bt~ and tt i are influenced by [E] in such a way that bt i = f ' ( [ E ] ) - ( b t i )
E at maximum
tti=f'([E])'(tti)Eatmaximum
in which f ' is some function, continuously decreasing between infinity at [E] = 0 and 1 at [E]'s maximum. The accolade terms denote i's transport and building-in times under maximum [E] circumstances. It follows that: bti=f"([j])'(bti)j__,oo tt i = f " ( [ j ] ) ' ( t t i ) j _ ~ o o
in which f " has the same character as f'. Through this, L~,y, L~ and Lj becomes:
Li, j =
1 rnin bt~ f , , ( [ j ] ) • {bti)j_~ ~ + ( t t i ) j ~ ~
103 L, = Lim ( Li.g) = - 1 . m i n bt i j-~ 1 ( b t i ) j ~ o + (tti)j~o~ L / = Lim (L,,j) - f,,(1 i_,~ [j])
min bt, minbt,+ 0
f"([j])
Hence, (A)
Li, j = L i • tg
A second expression of L;,j can be generated by assuming a "sequential" process in the following manner. One enzyme, constantly at optimum supply, transports and builds-in one unit of nutrient i, repeats this r times and proceeds by transporting and building-in one unit of nutrient j, completing one new algal unit. In terms of the previous paragraph, this implies that min bt equals bt and that bt and tt are made up of ( r . bt, + bt/) and ( r . tt, + tt/), respectively. Thus, eq. 3 becomes: r. bt i + btg Li j =
"
r • bt,
+
big + r
• tt i
+ tt/
Since tt, and bt, approach zero when i and j respectively become saturated, eqs. 4 and 5 become: L, = Lim ( L,,j) j--* o¢
r. bt i + btj r. bt i + btj + r. tt,
r. bt i +btg L / = i~ Lim ( L,,j) ~ r. bt, + btj + ttj
Some algebraic manipulations show that this can be merged as:
1
L,,/= l/L, + l/L~-
(B)
1
Thirdly, a process can be assumed in which the building-in proceeds sequentially, but the transport of i a n d j run parallel. One enzyme transports r units of nutrient i; meanwhile, another enzyme transports one unit of j. After arrival of the last necessary unit (either an i or a j), the building-in process starts, handling the (r + 1) units one after another, completing one new algal unit. E n z y m e concentrations are constantly at optimum. This means that the transport time in the complete process is the maximum of r. tt, and ttj: tt= MAX(r.
tti,
tt/)
while bt and min bt remain as with eq. B. Equation 3 becomes: r . bt~ + btj Li,j --
r. bt i + btg + M A X ( r .
it,, ttj)
104
which equals: L~ j = MIN "
r . bt i + btj
r . bt i + btj
]
r . bt i + btj + r . tt i
r . bt i + btj + ttj
l
As with eq. B, the terms left and right of the comma equal Lj and L i. Hence, L i , j = MIN(Li, L j )
(C)
Obviously, many more expressions for L~,j could be generated from other sets of assumptions. However, the most basic options are now present: the "multiplicative" hypothesis of the ecosystem models (eq. A), a "sequential" hypothesis (eq. B) and the "threshold" hypothesis, sensu Liebig (eq. C). Independently from its empirical merits, any expression derived from a reasonable enzymatic model will have the advantage that it always yields logically possible growth rate predictions. In the discussion, this point will be returned to. COMPARISON O F THE EXPRESSIONS
I n Table I, Li, j is calculated, according to the expressions A, B and C, for five combinations of L i and Lj. In the combinations 1 and 5, in which the concentrations of i and j are set "saturated" or nil, eqs. A, B and C predict the same nutrient limitation, as logically they should do. In the other cases, however, large differences are apparent. Equation A always predicts the lowest and eq. C the highest algal growth rates. Equation B is always in between, but closer to eq. A in the high "eutrophic" region of Li and Lj, and closer to eq. C in the "oligotrophic" case 4. This pattern is generalised in Fig. 1. Here, L~,/ is shown as a set of
TABLEI Multiple nutrient limitation (Li.g) predicted by eqs. A, B and C, for some arbitrary combinations of the single nutrient limitations Li and Lj, functions of the nutrient concentrations i and j , respectively Case
1 "eutrophic" 2 "eutrophic" 3 "mesotrophic" 4 "oligotrophic" 5 "dystrophic"
Li 1.00 0.80 0.50 0.50 0.50
Lj 0.80 0.60 0.40 0.20 0
Li. j
Li. j
Lid
by eq. A
by eq. B
by eq. C
0.80 0.48 0.20 0. I 0 0
0.80 0.52 0.29 0.17 0
0.80 0.60 0.40 0.20 0
105 0
o
o
..5"
Li,j=0.2
',
o
.'s
Li
,'.o
o
.5
EO. A
Li
EO. B
,.o
o
.s
Li
,.o
EQ. C
Fig. 1. Iso-limitation lines of the multiple nutrient limitation, predicted by eqs. A, B and C. For example, if L i = 0.40 and Lj = 0.70, Lt,j is predicted as 0.28, 0.34 and 0.40, respectively. The dotted line refers to Fig. 2.
iso-limitation lines in the L~-Lj-plane. It can be seen that the iso-limitation lines according to eq. B approach eq. A's hyperbolae in the upper right hand ("eutrophic") corner of the Li-Lj-plane, while they approach eq. C's straight lines in the "oligotrophic" region closer to the axes. Figure 2, a cross section through Figs. la, lb and lc, demonstrates this phenomenon. A conclusion can now be drawn. Ecosystem models are often used to predict the effect of lowering nutrient levels in an eutrophicated lake. When eq. A is used in such a model, any lowering of, say, the phosphate level will, through the decreasing Lp, result in a lower algal growth. As will be clear from Fig. 2, in models containing eq. C the algal growth rates will not respond to the phosphate level as long as any other nutrient (e.g. nitrate) is more limiting than phosphate; even below the threshold concentration the predicted ecosystem's reaction will be less. Equations A and C differentiate
.3"
~o~
.25
~~
.50
~'
.75
LI
~.o
Fig. 2. Cross-section of Fig. 1, at Lj = 0.50. Visible is the continuous influence of L, in eq. A, the threshold effect in eq. C and the intermediate and shifting character of eq. B.
106
in the eutrophic region of L~,j. Because of this feature, a choice in favour of eq. A or C might be arrived at by means of calibrations. This calibration method is less feasable for a decision between eqs. A and B, since eqs. A and B behave essentially the same in the eutrophic region. Yet, eq. B will predict consistently smaller effects of phosphate a n d / o r nitrate loading reductions. This difference becomes even more evident when the rates of loss of algae (sinking, grazing, washout etc.) are subtracted from the predicted rates of growth. EXPERIMENTAL TESTS OF THE EXPRESSIONS
The expressions derived above can be experimentally tested. The design of such experiments would be: (i) establish L, and Lj of an algal species by measuring growth rates under jand/-saturated conditions; (ii) measure concentrations and growth rates under conditions of simultaneous limitation by i and j; (iii) calculate the L i's and L f s corresponding to the concentrations and then the predicted growth rates according to the three equations A, B and C, to test which one best predicts the measured growth rates. In this paragraph, a visualized inspection of the empirical merits of the three equations will precede a statistical treatment. In the literature, three experiments have been found which can be analysed by this method. Since their authors' aims and tools were, in varying degrees, different from those which are relevant here, a re-interpretation of their data was necessary. This is the subject of Annexes II, Ill and IV. In the discussion, two other experiments are briefly mentioned. Droop (1975) investigated the behaviour of Microcystis lutheri in four batch cultures, with phosphate and Vitamin B12 as simultaneously limiting nutrients. From these data, L e, LvIT and the measured multiple nutrient limitation Le, vl a- can be calculated for 17 combinations of [P] and [VIT]. Lp and L v r r can be combined according to eqs. A, B and C, yielding the predicted Lp,vi-r. The complete procedure is given in Annex II. As it turns out, in 13 combinations Lp or Lv~ x is very close to either 0 or 1. In these cases, the three equations predict the same growth rate. These non-discriminating cases can be left out of the analysis. In Fig. 3, measured and predicted growth rates are compared for the three equations and the four remaining discriminating cases. As can be seen, the multiple nutrient limitations predicted by eq. C come closest to those measured, while eq. A gives the worst predictions. Feuillade and Feuillade (1975) experimented with Oscillatoria rubescens in chemostat cultures limited by phosphate and nitrate simultaneously. The N-
107 o ,.: . . . . . . . . . . . . . . . . . . . . . . .
o
o
-2],
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Fig. 3. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the single species experiments of Droop (1975). The discriminating cases are represented by the heavy dots. On the diagonal, predicted values equal those measured•
and P-inflows (fluxes) are treated as the independent variables, and the resulting rate of production is measured. The relation between the N- and P-inflows and L N and Lp is not given; they have to be inferred from the dataset before L N and Lp can be determined. In this procedure, fully explained in Annex III, three of the eight experimental cases are used, leaving five cases for which L N and L p can be independently calculated and, through these, the LN, p'S predicted by the three equations. In Fig. 4 these are compared with the limitations measured in the corresponding cases. None of the equations accounts for the zero-growth measured in one case. Accepting this common discrepancy, the visual result is the same as for Droop's experiment: the limitations predicted by eq. C give the best overall fit, while eq. A predicts worst, with eq. B between the two.
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0.42
EQ. C
Fig. 4. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the single species experiments of Feuillade and Feuillade (1975). Small crosses indicate experiments used to infer the growth constants, for which M and PR are automatically equal.
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Fig. 5. Measured (M) and predicted (PR) multiple nutrient limitations, according to the three equations, as derived from the experiments of Shapiro and Glass (1975), with natural algal assemblies, Small crosses indicate experiments used to infer the growth constants, for which M and PR are automatically equal.
Shapiro and Glass (1975) measured growth rates of a natural algal assembly from Lake Superior in batch cultures, with phosphate and taconite (a manganese containing mining waste product) as limiting nutrients. The dataset consists of nine cases of P-tac-combinations. As explained at length in Annex IV, five cases are needed to allow testing of the equations by means of the then remaining four cases. Fig. 5 is the result of the Annex calculations. As in the preceding experiments, it is found that eq. A gives the worst predictions, while eqs. B and C both predict satisfactorily, with errors one quarter of those of eq. A. It can be seen that eq. B's predictions have a better overall direction. F r o m the above analysis it can be hypothesized that eq. A, the multiplicative expression, consistently yields the worst growth rate predictions, while eqs. B and C are close to each other in quality, with eq. C probably slightly better. This hypothesis can be tested statistically. The first step is to calculate the differences between the predicted and measured values of Li. j with regard to the three equations, for the 4 + 5 + 4 experimental cases. Then in order to treat the 13 cases as one dataset, these absolute errors must be made mutually comparable. This is done by making them relative to the largest Li. j, measured in the experiment they are derived from. This scale factor is 1.01 for the errors from Droop (1975), 0.28, 0.060 and 0.47 for the errors in the analysis of the Feuillade and Feuillade data (1975), concerning eqs. A, B and C respectively, and 0.37 for Shapiro and Glass (1975) (ref. Tables II, IV and VI). Thus, the absolute differences become relative differences on the same scale, with n = 13 for all three equations. The averages and standard devia-
109
tions of these series are as follows: eq.A:
d A=0.10;
sA = 0 . 2 0
(n=13)
eq. B:
d B = 0.06;
s~ = 0.12
(n = 13)
eq.C:
dc=0.04;
sc = 0 . 1 2
(n=13)
Here, d is the average relative prediction error, while s is the standard deviation, indicating the degree to which prediction errors vary around their mean. The closer both values approach zero, the better is the predicting equation. As expected, the results of eqs. B and C closely resemble each other. The slight advantage of eq. C can be shown not to be statistically significant in Student's t-test. Equation A seems inferior in terms of both j and s. The significance of the difference in d can only be tested under the assumption that no difference in the standard deviation exists between the populations from which the cases are drawn. Hence, sh must be inspected first against s B and s c. In order to test this standard deviation difference, Cochran's test (Dixon and Massey, 1969) is most suitable. The null-hypothesis (the equations generate relative error populations with the same standard deviation) is tested against the alternative hypothesis that eq. A generates relative errors with a larger standard deviation than both eqs. B and C. As it turns out, Cochran's statistic C ---
s~ 2 + 2+s ~ SA S B
is significant at the 5% level. Since also d A is inferior to both d B and dc, it is inferred that eq. A is the statistically significant worst growth predictor. DISCUSSION
As Tailing (1979) has pointed out, the use of fixed limitation parameters and interaction formulae still has a weak empirical basis. A recent experiment may serve to illustrate this. Young and King (1980) measured growth rates of A n a c y s t i s nidulans under various light, carbon dioxide and phosphate conditions. Under constant light conditions, they found that the impact of a low but constant external phosphate concentration increases at decreasing external carbon dioxide levels. Obviously, none of the three alternative expressions can account for such a phenomenon; at most, a constant influence of phosphate is predicted. The internal nutrient concentration ratio, r, assumed to be constant in the derivation of the alternatives, might be the key factor in explaining the observed facts. Young and King show that the internal C/P-ratio may increase by a factor of 10 when the external
110
[P] decreases, at a constant CO2-1evel. It seems reasonable to suppose that this also takes place at constant [P] and decreasing [CO2]; in other words, the algae need relatively more P when [CO2] decreases. Hence, the P-influence may be stronger, the lower [CO2] becomes. If r is allowed to fluctuate with the external CO2-P-circumstances, all three expressions could be mathematically reformulated and tested. Young and King do not provide enough data for such an analysis. Also the pH, which, through its influence on the growth rate (Golterman, 1975), might be an alternative explanation, is not given. The problems which can be encountered in analysing experimental data can be illustrated in the case of G. Ahlgren (1980). In a chemostat experiment with Oscillatoria agardii it was tried to find a best L N, Lp and LN, p-expression simultaneously. The analysis does not allow for the concluded "limited multiplicative effect". For example, N is supposed to be the (most) limiting factor in cases where clearly L N > Lp; half-value constants found by assuming one Li-formula are applied in another; and very low external N-concentrations which, according to the conclusions, would yield an impossibly low LN, are ignored. One conclusion can be drawn from Ahlgren's data: the measured growth rates are explained by the internal P-concentrations, whatever the other circumstances are. This falls in line with Droop's (1975) findings, but unfortunately is not mentioned by the author. Besides the equations A, B and C, an infinite number of alternative expressions for the multiple nutrient limitation can be formulated. Apart from their empirical value, the L,,/-formulae should meet two "logical" criteria: (i) if any nutrient's concentration is zero (e.g., Lj = O) then the growth should be zero (Li. j = 0), independent of the other nutrients, (ii) if any nutrient concentration is saturated, it should exert no influence; e.g., if Lj = 1, all expressions should yield Li, j = L i. These criteria define the outcomes on the extremes in Figs. 1 and 2, where the lines of eqs. A, B and C meet. All three meet the criteria--as will, in fact, any expression derived from a reasonable biochemical growth model. Jorgensen et al. (1981) apply five formulae in a calibration research, among which are eqs. A, C and two alternatives: 1
L,,j = 1/L, + 1/Lj
(D)
L~.j= ( L~ + L/)/2
(E)
while Scavia and Park (1976), in their model CLEANER, use: 2 L i ' j = 1 / L i q- I / L j
(F)
111
The generalised forms are given in Annex I. None of these expressions meets the second criterion. If nutrient j is in unlimited supply (Lj--- 1), eqs. E and F always predict Li,j's higher than L i, while eq. D always predicts them lower than L~. Moreover, eq. E does not meet the first criterion. If nutrient j is not present at all (Lj = 0), the growth rate is still predicted as positive. Given the many degrees of freedom in most calibration processes, it cannot be excluded that "illogical" equations may yield acceptable results in the calibrated ranges of L~ and Lj. However, for predicting growth rates outside these ranges, e.g. predicting the effect of proposed phosphate control measures, they are inherently weaker tools. Finally, it must be considered along which lines further research might be undertaken. Both laboratory experiments and ecosYstem model calibration procedures may be fruitful in solving the existing discrepancies concerning the multiple nutrient limitation expressions, shown in this paper. Possibly one condition for success might be that one should not try to find all growth influencing factors and growth formulae simultaneously. First, the single nutrient limitation functions must be firmly established, e.g. by measuring them under truly saturated conditions of all nutrients except one. Then, using "logical" formulae only, the multiple nutrient expression can come under test. The direct practical importance in water management justifies a search for an improved empirical grounding of the modelling of nutrient limitation. ACKNOWLEDGEMENTS
Thanks are due to M. Donze and E. Meelis for their general and statistical support. ANNEX
Limitations in terms of concentrations; generalized expressions. Equation 2, for one nutrient only, reads: Li
bti bt, + tt i
The building-in time can be assumed constant, while tt might depend on a constant transport velocity, v, and the average transport distance, d; assuming this distance to be inversely proportional to the concentration (d = a / [ i ] ) , the. nutrient limitation becomes: L,-
bt
a bt + - -
v.[i]
112
Dividing numerator and denominator by bt/[i] it is found that: Li =
[i] a [ i ] +v- •- bt
This is the Michaelis-Menten expression, in which the denominator's second term is measured as the half-value constant, ki. If this type of growth curve is substituted for L i and Lj, the two equations A and B read as follows:
[;].[j] eq.A:
Li,j=Li,LJ-[i].[j]+[i].kj+[j].ki+ki.k
eq.B"
Li,j=l/Li+l/Lj_l=[i].[j]+[i].kj+[j].ki
1
j
[i]. [j]
Note that in terms of concentrations, eq. B shows as a simplification of eq. A. The generalised multi-nutrient forms of the equations, mentioned in this paper, are: eq. A:
L,,j ...... = L~ x Lj x ... X L ,
eq.B"
L,,j ..... " = 1 / L i+ 1 / L j + ... + 1 / L , - ( n -
eq.C:
Li, j ..... , = M I N ( L i , L
eq. D:
Li, j ......
eq.E:
L,,j ...... = ( L , + L j + . . . L , ) / , .
eq. F:
Le, j ...... - 1 / L , + 1 / L j + ... 1 / L ,
1 1)
j ..... L,)
1 1/L~ + 1 / L j + ... 1 / L n
11
in which n is the number of nutrients. ANNEX
I1
An interpretation of the Droop (1975) data. Droop (1975) experimented with Monochrysis lutheri in four batch cultures, with phosphate and Vitamin B12 as simultaneously limiting nutrients. No relation exists between the growth rates and external concentrations, but the internal nutrient concentrations do exert a clear influence. As Droop points out, this influence can be satisfactorily explained by the assumption that only the most limiting nutrient (in terms of internal concentrations) determines the growth rate, that is, by equation C. For the purpose of this
113
article, it makes sense to investigate if the other hypotheses are not better. This is achieved as follows. From the given biomass levels B, the growth rates g (in fraction per day) can be calculated through: ln(1
+ g,) =
In Bt+at- In Bt_at 2 At
in which t and At denote sampling day numbers and their differences, respectively. Also, from the given internal nutrient concentrations, L p and Lvl x can be found, using the expression and parameters mentioned by Droop. Then, setting gmax at 1.5, a dataset is arrived at in which are Lp, Lvi v and the measured Lp.vl r, consisting of 17 cases. The latter can be compared with the Lp,vla-'s, predicted by the eqs. A, B and C, to find a best predicting equation. The data and predictions are given in Table II. In the majority of cases, Lp a n d / o r L v v r turns out to be so close to zero or unity that the three equations do not differentiate (see Table I). In fact, only the cases 11, 12, 15 and 16 offer an opportunity to evaluate the equations. In Fig. 3, these cases are represented by the large dots. Although not relevant for the purpose of
T A B L E II M e a s u r e d a n d predicted growth rates, derived from the four one-species b a t c h cultures of D r o o p (1975), in which p h o s p h a t e a n d vitamin B12 can be simultaneously limiting
Expt. I
Expt. II
Expt. III
Expt. IV
Case
Day
Le
Lvv r
Lp, vl T Lp, vlT, predicted by: measured eq. A eq. B eq. C
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
4 6 8 11 4 6 8 11 15 2 3 5 7 3 5 7 10
0.84 0.39 0.04 - 0.05 0.96 0.96 0.94 0.92 0.88 0.93 0.75 0.40 0.01 0.95 0.82 0.31 0.04
0.98 0.92 0.88 0.85 0.96 0.71 0.56 0.36 -0.03 0.96 0.91 0.76 0.59 0.96 0.83 0.23 -0
1.01 0.39 0.07 - 0 0.92 0.41 0.09 0.07 0.01 0.84 0.66 0.30 0.07 0.99 0.84 0.27 0.03
0.82 0.36 0.04 - 0.04 0.92 0.68 0.53 0.33 -0.03 0.89 0.68 0.30 0.01 0.91 0.68 0.07 -0
0.83 0.38 0.04 - 0.05 0.92 0.69 0.54 0.35 -0.03 0.90 0.70 0.36 0.01 0.91 0.70 0.15 -0
0.84 0.39 0.04 - 0.05 0.96 0.71 0.56 0.36 -0.03 0.93 0.75 0.40 0.01 0.95 0.82 0.23 -0
114
this paper, it may be noted that the growth rates of cases 6, 7 and 8 are grossly overestimated by all three equations. Those cases were characterized by a high total-P concentration, giving rise to high phosphate cell quota. These phosphates evidently act more as "burden-P" than as "luxury-P", and Droop's way of calculating Lp does not account for this. ANNEX III
An interpretation of the Feuillade and Feuillade (1975) data. Feuillade and Feuillade (1975) experimented with Oscillatoria rubescens in chemostat cultures, with N and P as limiting nutrients. Independent variables were the N and P input rates (fluxes); the dependent variable was the rate of production, g. This dataset, comprising eight experimental cases, is represented in Table III. The phosphorus and nitrogen limitations (Lp and LN) are functions of the respective input rates, P-flux and N-flux. In Feuillade and Feuillade, contrary to Droop (1975), these functions are not presented. This implies that the two half-value constants k P and k N (see Annex I) have to be found before the data can be analysed. The same holds for gmax, the maximum growth rate, which determines how a measured growth rate g is translated into a measured Ly, p. These three unknowns have to be inferred from the Feuillade and Feuillade dataset. The following procedure is applied: (i) assume one of eq. A, B or C to hold, (ii) write out this equation in terms of fluxes (Annex I), (iii) substitute the P-flux, N-flux and measured g for three arbitrary experi-
TABLE III The Feuillade and Feuillade (1975) dataset: measured production rates (g), in mg dry matter per liter per hour, and phosphate and nitrate inputs (fluxes) in 8 one-species chemostat experiments Case
P-flux, #g P/1 h
1
7
2 3 4 5 6 7 8
12 22 7 14 24 7 12
N-flux,/xg N / 1 h
g mg/1 h
27 27 27 53 68 64 119 128
0.0 0.80 0.65 i.10 1.30 1.35 1.80 2.40
1 2 3 4 5 6 7 8
Case
0.48 (0.61 0.71 0.48 0.64 0.75 (0.48 (0.61
0.15 0.15 0.15 0.26 0.31 0.30 0.45 0.46 0
0.09) 0.08 0.13 0.15 0.16
0.21) 0.28)
0.07
0.09
0.11 0.125 0.20 0.225
0.21 0.28
LN, P (meas.)
0.08 (0.13 0.21 0.08 0.15 0.23 (0.08 (0.13
0.023 0.023 0.023 0.044 0.056 0.053 0.095 0.101
LN
Lp
LN, P (pred.)
Lp
LN
Testing eq. B
Testing eq. A
0.045 0.060
0.021 0.029 0.042 0.045
0.020
0.018
LN. P (pred.)
0.045) 0.060)
0.016 0.027 0.032 0.034
0.020)
0
LN, P (meas.)
0.35 (0.48 0.62 0.35 0.51 0.65 (0.35 (0.48
Lp
0.15 0.15 0.15 0.26 0.31 0.30 0.45 0.47
L~
Testing eq. C
0.35 0.47
0.15 0.26 0.31 0.30
0.15
0.15
LN. p (pred.)
0.35) 0.47)
0.125 0.21 0.25 0.26
0.15)
0
LN. P (meas.)
Testing eqs. A, B and C through the Feuillade and Feuillade (1975) data. Method explained in Annex III. The cases used to calculate the parameters k N, kp and gma~, for which the predicted and measured production rates are automatically equal, are put between brackets. Values used for testing are printed in italics
TABLE IV
116
mental cases of Feuillade and Feuillade, thus creating three equations to solve the three unknowns. With the now known kN, kP and gmax, it is possible to calculate LN, Lp and the measured LN, P for the remaining five Feuillade and Feuillade cases, as well as the errors in the predicted LN, P generated by eq. A, B or C. E.g., eq. A, in terms of fluxes, reads: gJgmax =
P-flux P-flux W k P
N-flux N-flux + k N
Substituting the P-flux, N-flux and g of the experimental case 2, one finds: 12
27
0 " 8 / g m a x ----- 12 + k P " 27 + k N
This can be repeated for two other cases, e.g., cases 7 and 8, and kN, k P and gmax are found to be 148, 7.8 and 8.5, respectively. This is used to calculate LN, Lp and the measured LN, P as well as the LN, P predicted by eq. A. The results are shown in Table IV. Repeating this procedure for eqs. B and C, Table IV is completed. F r o m the Table and Fig. 4, it is obvious that none of the equations can account for the zero growth in case 1. Disregarding this c o m m o n discrepancy, it can be seen that in this analysis eq. A comes third in prediction accuracy, while eq. C comes out best. ANNEX IV
A n interpretation of the Shapiro and Glass (1975) data. Shapiro and Glass (1975) noticed a synergistic effect of phosphate and manganese enrichment on algal growth in Lake Superior. This was investi-
TABLE V The Shapiro and Glass (1975) dataset: radiocarbon uptake after 20 days, in thousands Cpm, under various conditions of taconite (mg/1) and phosphate ( # g / l ) additions. Natural algal populations of Lake Superior Additions of phosphate
+0 +5 + 25
Additions of Taconite +0
+0.4
+4
3 7 8
3 8 16
3 10 37
117
gated in batch cultures with natural algal assemblies. Three samples were taken of Lake Superior water, and various amounts of phosphate and taconite, a manganese containing mining waste product, were added. Sufficient specific growth rates are only given for their Sample III. The experiments with this sample will be analysed here. Table V shows the results of the combined effects of the addition of phosphate and taconite, in terms of radiocarbon uptake after 20 days, for nine experiments. The experiments are still in their exponential growth phase after 20 days; hence, the carbon uptakes are indicative for the growth rates (uptake/day), and will be treated as such. The additions will be regarded as indicative for concentrations. As in Annex III, the single nutrient limitations ( L e and LTAC) are not given for the P and taconite additions; neither is g . . . . the maximum growth rate. Again, they have to be inferred from the dataset, using some experimental cases, before eqs. A, B and C can be tested by means of the remaining ones. In Shapiro and Glass, taconite additions over 4 m g / l do not increase growth rates. Hence, over the last column of Table V, the experiment is taconite saturated, implying that, assuming Michaels-Menten kinetics, it can be written, for TAC = 4 mg/1, g -Lpgmax
P-add. P-add. + kP
independent of the equations to be tested. By substituting the P-addition values and the measured g, it is found that gm,x approximates to 100. If the values in Table V are divided by 100, they represent the measured Lp,a-AC (see Table VI). The phosphate limitation Lp equals Lp, TAc over the last column. Hence, for the addition of 0, 5 and 25 mg P / l , Lp equals 0.03, 0.10 and 0.37, respectively. No phosphate limitation exists in any row of Table V. Therefore, as in Annex III, the LTAc that can be found depends on which L P,'rAC equation is assumed to hold true. E.g. for the case (P = 5, TAC = 4), it is known that Lp - 0.10 and the measured Lp, a.AC = 0.08. Now, if eq. A is assumed to hold, LTAC must be 0.80. If eq. C is assumed to hold, LVAc must be 0.08. Using the whole row of P = 5 mg/1, --assuming eq. A to hold: LTAc = 0.7, 0.8 and 1.0 for the three taconite additions, --assuming eq. B to hold: La-AC= 0.19, 0.29 and 1.0, respectively, --assuming eq. C to hold: La-AC= 0.07, 0.08 and 1.0, respectively. For the row and column used, the predicted Lp,TAc will now automatically coincide with the measured Lp,a-AC, leaving four cases to test the equations, by comparing the predicted and measured L e,a-Ac's. This is represented in Table VI. To check the limitations stated above, the cases of the column TAC = + 4 and P = + 5 are given between brackets.
119
From Table VI, it can be concluded that in this analysis eqs. B and C both yield far better predictions than eq. A. The sum of the absolute errors of prediction by eq. A is 0.37, while this amounts to 0.09 for both eqs. B and C. REFERENCES Ahlgren, G., 1980. Effects on algal growth rates by multiple nutrient limitation. Arch. Hydrobiol., 89: 43-53. Baule, B., 1917. Zu Mitscherlichs Gesetz der physiologische Beziehungen. Landwirtsch. Jahrb., 51 : 363-385. Blackman, F.F., 1905. Optima and limiting factors. Ann. Bot., 19: 281-295. Dixon, W.J. and Massey, Jr., F.J., 1969. Introduction to statistical analysis, 3rd edn. McGraw Hill, New York, 638 pp. Droop, M.R., 1975. The nutrient status of algal cells in batch culture. J. Mar. Biol. Ass. U.K., 55: 541-555. Feuillade, J.B. and Feuillade, M.G., 1975. Etude des besoins en azote et en phosphore d'Oscillatoria rubescens D.C. h l'aide de cultures en chemostats. Verh. Int. Ver. Limnol., 19: 2698-2708. Golterman, H.L., 1975. Developments in Water Science, Vol. 2, Elsevier, Amsterdam, 489 pp. J4rgensen, S.E., J4rgensen, L.A., Kamp-Nielsen, L. and Meijer, H.F., 1981. Parameter estimation in eutrophication modelling. Ecol. Modelling, 13:111-129. Lederman, T.C. and Tett, P., 1981. Problems in modelling the photosynthesis-light relationship for phytoplankton. Bot. Mar., 24: 125-134. Rodhe, W., 1978. Algae in culture and nature. Mitt. Int. Ver. Limnol., 21: 7-20. Scavia, D. and Park, R.A., 1976. Documentation of selected constructs and parameter values in the aquatic model CLEANER. Ecol. Modelling, 2: 33-58. Shapiro, J. and Glass, G.E., 1975. Synergistic effects of phosphate and manganese on growth of Lake Superior algae. Verh. Int. Ver. Limnol., 19: 395-404. Tailing, J.F., 1979. Factor interactions and implications for the prediction of lake metabolism. Arch. Hydrobiol. Beih., 13: 96-109. Young, T.C. and King, D.L., 1980. Interacting limits to algal growth: light, phosphorus and carbon dioxide availability. Water Res., 14: 409-412.