The development from kinetic coefficients of a predictive model for the growth of Eichhornia crassipes in the field . II . Testing and refining the model under field conditions

Kinctic cocfficicnts derived for Eichhornia crassipes (M art.) Solm s under culture conditions o f N and P limita­ tion were used in the Monod model to identify the limiting nutrient and to predict specific growth rates under conditions o f varying water nutrient concentration and air temperature. Predicted data were validated by compari­ son with specific growth rates measured for plants growing in loosely and densely crowded populations at tw o field sites. T he use o f culture-derived maximum specific growth rates (U m ax) in the model resulted in maccurate predic­ tions o f plant growth rates in loosely and densely crowdcd field populations. The use o f field-derived Umax values in the m odel, however, resulted in adequate predictions o f plant growth rates in loosely crowded field populations. T he incorporation o f radiant flux density (diffuse com ponent o f the radiant flux) and relative humidity into the m odel considerably improved its accuracy o f prediction. In all cases, specific growth rates were more accurately predicted from the limiting total N or total P concentrations, than from other N or P fractions, in the water.


IN TRO DUC TIO N
Harvesting Eichhornia crassipes (M art.)Solms (water hyacinth) growing in eutrophied aquatic sys tems may constitute an effective means o f removing nutrients and controlling excessive growth of plants ( Boyd, 1970;Yount & Crossman, 1970).However, to achievc maximum nutrient removal efficiency by E. crassipes in a nutrient removal schem e, it is necessary to establish the size of the population re quired to maintain desirable nutrient concentrations in the w ater under varying conditions o f nutrient loading and climate, and the amounts and frequen cies of harvest required to control the population size.
From the kinetic standpoint, it is theoretically feasible to construct a mathematical model for E. crassipes from which population sizes, yields, growth rates and frequencies and amounts of harvest, under varying conditions o f nutrient loading and climate, can be predicted to control nutrient inputs and ex cessive growth in eutrophied aquatic systems (Toe rien, 1972;Musil & Breen, 1977).Musil & Breen (1985) m easured the kinctic coefficients, maximum specific growth rate (Um ax), half saturation coeffi cient (Ks) and yield coefficient (Yc) for E. crassipes growing in N and P tlim ited batch cultures in a greenhouse with the objective of developing a pre dictive m odel.This investigation was designed to test the validity of and refine these culture deter mined kinetic coefficients for predicting growth rates of E. crassipes under field conditions.In testing the model, two assumptions were made: (i) T hat the maximum specific growth rates (U m ax), derived under culture conditions o f N and P limitation (Musil & Breen, 1985), followed the van't Hoff rule, i.e. they approximately doubled for each 10°C rise in the tem perature, as has been de m onstrated for various species o f marine and fresh water algae ( Goldman, 1972;Goldman & C ar penter, 1974).
(ii) That the specific growth rate (U ) was not limi ted in a multiplicative o r additive m anner, but in a threshold mode by the single nutrient in shorter sup ply.This principle, based on 'Liebig's law of the mi nimum' that the maximum population size o r maxi mum yield in plant material is controlled by the sin gle factor in shorter supply (Blackm an, 1905), has been observed to apply in the regulation of phyto plankton organic production by soluble nutrients (B randt, cited in G ran, 1912).M ore recently, this has been extended to include the regulation or con trol o f phytoplankton growth rate by the limiting nu trient (O ' Brien, 1972Brien, ). D roop (1974)), for example, has shown that the growth rate of Monochrysis lutheri growing under P and B 12 limitation in culture is not limited in a multiplicative pattern, but by the sin gle nutrient in shorter supply. Rhee (1978) also sur mized that the growth rate of Scenedesmus sp. is limited in a threshold pattern.
A part from air tem perature, seasonal variations in radiant flux density and relative humidity may influ ence E. crassipes growth rate under field conditions.In view o f this, two hypothetical, multiplicative ex pressions were investigated for correcting the pre dicted growth rates for the effects o f radiant flux density and relative humidity respectively.These are defined as follows: Umax = A rE'R ITx i)................................(i) where Umax = maximum specific growth rate g fresh mass g~* d-*; T = absolute m ean daily air tem perature °K; I = radiant flux density (diffuse compo nent o f the radiant flux) MJ m 2 h-'; A = constant d a y 1; E = activation energy cal.mole*1; R = uni versal gas constant cal.mole-' "K-*.
The first multiplicative expression assumed that the effect of radiant flux density on E. crassipes growth rate could not be considered independent of the effect of tem perature.This is in accordance with the observed interaction between tem perature and light intensity in influencing both growth rates and photosynthetic responses o f algae (Sorokin, I960; Sorokin & Krauss, 1962;Maddux & Jones, 1964;Smayda, 1969;Eppley, 1972;Harris & Lott, 1973) and higher plant species (Pisek et al ., 1973;Billings, 1974).Several environmental modellers (Di Toro ei al ., 1971;Chen & O rlob, 1972;Park el al., 1975;Kieffer & Enns, 1976) have used a multiplication of independent light and tem perature functions in phy toplankton population/productivity models, in many instances without experimental evidence.Rodhe (1948Rodhe ( , 1978) ) suggested that the combined effects of factors such as tem perature, light and daylength on the population dynamics o f phytoplankton may be more important than the effect of any single one fac tor.
The second multiplicative expression assumed that the observed effect of relative humidity on E. crassipes growth rate (Freidel el al ., 1978) could not be considered independent of the effect o f tem pera ture and radiant flux density.This is because of the interaction between tem perature, light intensity and relative humidity in influencing transpiration rates (Crafts et al ., 1949) which in turn may indirectly in fluence growth rate, possibly by altering the water potential (Slayter, 1967;M eidner& Sheriff, 1976) of leaf cells.Cell and leaf growth are highly sensitive to a reduced water potential, particularly as cell expan sion is caused by the action o f turgor pressure upon 'softened' cell walls (Greacen & O h, 1972).In fact, Hsiao et al. (1976) show that even mild water stress in mesophytic leaves, i.e.where the water potential of leaf cells is reduced by only a few bars, can result in a reduction in growth rate and the disruption of several metabolic processes, including protein and chlorophyll biosynthesis.

Sites
Two sites, characterized by different nutrient con centrations in the water, were selected for field esti mations of specific growth rates.These rates, meas ured periodically throughout the year, were com pared with those predicted by the use o f culture-de rived kinetic coefficients (Musil & Breen, 1985) in the Monod model.The sites are in the Durban Dis trict of Natal (Fig. 1) in the climatic region described by Schulze (1965) as warm to hot and humid, subtro pical.The M aturation Pond 3 (MP3) site is enriched by secondary treated waste-water effluent dis charged from the northern sewage treatm ent works and the Botanic Gardens Lake (BGL) site by ferti lizer run-off.

Measurement o f growth
Specific growth rates were measured by tagging plants and introducing them for 12 to 14 day periods into field populations (Bock, 1966;1969).These rates were measured in both loosely and densely crowded field populations. Mitsch (1977) found that the two different growth forms, viz marginal and central forms (Musil, 1982), associated with these two different population densities (R ao, 1920;Mc-Clean, 1922;Lansdell, 1925;Bruhl & G upta, 1927;La G arde, 1930;W eber, 1950) exhibit different net carbon uptakes and photosynthesis/respiration ratios which suggests that they may have different growth rates under similar field conditions.
Vcgctatively propagated offsets (daughter plants) possessing three pseudolaminae of the marginal (fresh mass: ca 7 to 22 g) and central (fresh mass: ca 47 to 111 g) forms were collected at 12 to 14 day in tervals from loosely and densely crowded popula tions at each site.Plants were washed in site water, to remove al) extraneous particles, tagged and al lowed to drain for two minutes.They were shaken to dislodge adhering w ater, their fresh masses recorded on an electric, top-loading balance and introduced into enclosures.Two sizes of enclosures were used.Both were constructed o f plastic-coated wire mesh held in place by metal fencing posts driven into the sediment.To minimize disturbance by wind and wave action, they were located on the leeward side of each water body in water ca 1,0 m deep.
Cylindrical enclosures with a diam eter of ca 1 m and a height o f ca 1,5 m were used for containing plants of the marginal form at each site.The water area, ca 0,8 m2, contained within each enclosure was adequate lo accommodate an increase in the size of the introduced population, over a 12 to 14 day period, without causing the plants to become unduly crowded.Forty plants of the marginal form collected at each site were introduced into four enclosures (10 plants per enclosure) located at each site.
Two densely crowded populations (ca 6 m2) of the central form were enclosed by wire mesh, ca 1,5 m high, at the MP3 site.This ensured that the plants within were kept in densely crowded situations and maintained in the central form.Thirty plants of the central form collected at this site were inserted at random (15 plants per enclosure) into the two en closed populations.
After 12 to 14 days, marginal and central forms and their offsets were harvested from the enclosures at each site.Care was taken not to separate the off sets from their respective parents, Plants were pick ed free of debris, washed and reweighed as de scribed above.Specific growth rates were calculated for marginal and central forms, over each 12 to 14 day interval, using the general growth equation (Ma lek & Fencl, 1966;Radford, 1967): _ In Xt -In Xo t where Xo = fresh mass at time = tj(g); Xt = fresh mass at time = t 2(g); U = specific growth rate (g fresh mass g-> d~!); t = time period between time t2 and t, (days); £n = log,, (natural logarithm).

Chemical analyses
W ater samples were collected between llhOO and 14h00 from within the loosely and densely crowded populations enclosed at each site, at the commence ment and termination of each growing interval.W ater samples were collected ca 20 cm below the water surface, to avoid surface contamination, in 500 ml plastic bottles (Golterm an, 1969) previously cleaned with conc.HCI and rinsed thoroughly in deionised-distilled water, Bottles were sealed with Parafilm (American Can Company, Greenwich, Connecticut) and immediately transported in an in sulated container to the laboratory.
The following N and P fractions were analysed in the water samples using published methods (E n vironmental Protection Agency, 1974; American Public Health Ass.: Standard methods, !975).In fil tered samples (Rigler, 1964;Olsen, 1967;G olter man, 1969), nitrate-nitrogen (N O j-N ) by colorime try after reduction to nitrite and soluble reactive phosphorus (SRP) (Twinch & Breen. 1980) by colo rimetry using the molybdenum blue method.In un filtered samples, Kjeldahl nitrogen as ammonium ( N H 4 -N ) after digestion o f the samples by conc.H2SO4 in the presence of a mercury catalyst and total phosphorus (total P) as SRP after digestion of the samples with H2S 0 4 and persulphate.Total ni trogen (total N) was calculated as the sum of Kjeldahl nitrogen and nitrate plus nitrite (NH4-N + N O j-N + NO2-N ).
The pH o f the water was recorded daily between llh(X) and 14h00, beneath the loosely and densely crowded populations enclosed at each site, with an electrolytic probe connected to a portable pH meter.

Physical analyses
Over each growing interval, hourly measurements o f radiant flux density (diffuse component of the ra diant flux) and daily maximum, minimum and mean air temperatures and relative humidities were ob tained from the nearby meteorological station at Louis Botha A irport, Durban.
The diffuse component o f the radiant flux was chosen as a measure of the light as this includes a greater proportion o f the photosynthetically active radiation.About one-third of the direct solar radia tion, often referred to as the global component, is photosynthetically active compared with over twothirds for the diffuse component (Ross, 1975;Fitter & Hay, 1981).Theoretical calculations have shown that even under cloudless skies, the diffuse radiation (D) may account for between one-third and threequarters of the total irradiance (T), and in a series of m easurem ents Szelez (1974) showed that the ratio D/T was always greater than 0,5.

Physical factors
A ir tem peratures, relative humidities and diffuse radiant fluxes at the two sites are depicted in Figs 2, 3 & 4.They showed a seasonal pattern decreasing progressively after summ er (Septem ber to M arch) through to w inter (May to August).Mean daily air tem peratures and relative humidities at the two sites ranged from 16,1 to 25,2°C and from 64 to 85% re spectively.Diffuse radiant fluxes ranged from 0,30 to 0,72 MJ nr* h-'.trations in the water were much higher, The latter was attributed partly to enrichment by fertilizer run off and to a higher rate o f nitrification at the BGL site (Musil, 1982).There were no differences in the N and P concentrations in the water beneath the loosely and densely crowded populations enclosed at the MP3 site (Musil, 1982).At both sites, the water pH values were in close proximity to pH 7,0.Only minor variations in the water pH (pH 7,0 to 7,3 and pH 7,2 lo 7,7 at the BGL and MP3 sites respectively) occurred at these two sites during the year.There were no differences in the water pH beneath the loosely and densely crowded populations enclosed at the MP3 site (Mu sil, 1982).

Specific growth rates
Specific growth rates of marginal and central forms, growing in loosely and densely crowded pop ulations respectively, at the two sites are illustrated in Fig. 8.
At both sites, specific growth rates of marginal forms followed a distinct seasonal pattern with val ues decreasing progressively after summer (Septem ber to March) through to winter (May to August).During 1978, the highest specific growth rates, 0,1698 and 0,1227 g fresh mass g-> d-> (16,98 and 12,27% d-1 at the MP3 and BGL sites respectively, were measured during summer, in February, with the lowest specific growth rates, 0,0526 and 0,0305 g fresh mass g-1 d-1 (5,26 and 3,05% d -1) al these two sites respectively, being measured for marginal forms during midwinter, in June.Throughout 1978, specific growth rates of marginal forms at the MP3 site were significantly higher (P=S 0,01), than those at the BG L site (Musil, 1982) and reflected the higher total N and total P concentrations in the water at the MP3 site (Figs 5 & 6).In general, speci fic growth rates o f marginal forms at both sites fell in the range of specific growth rates (3,0 to 12,5% d -1) reported by various authors (Seaman & Porterfield, 1964;Bock, 1969;Knipling et al., 1970;M orris, 1974;Boyd, 1976) for E. crassipes growing under subtropical to tropical climates in other parts of the world.
In contrast to marginal forms, specific growth rates of central forms at the MP3 site did not show any distinct seasonal pattern, since no plants o f the central growth form were produced during the mid w inter m onths o f June, July and August.The highest specific growth rate, 0,0659 g fresh mass g~! d : (6,59% d-*), was m easured for central forms during sum m er, in Decem ber, 1977, with the lowest specific growth rate, 0,0202 g fresh mass g-> d-* (2 ,02% d-i), being m easured during winter, in May, 1978. Throughout 1977and 1978, specific growth rates of central forms at the MP3 site were significantly lower (17 to 35% , P=£ 0,001) than those of marginal forms (M usil, 1982).
The significantly lower specific growth rates meas ured in densely crowded field populations may be partly related to the adverse effects of self shading and intraspecific competition arising through over crowding in such populations, as well as to the intrin sic morphological limitations of plants of the central form occurring in such populations. C enter & Spencer (1981) have shown that the production of E. crassipes plants with elongate petioles (central forms) results in a decline in the lamina area ratio (L A R ), i.e. in relatively less photosynthetic area per unit of plant weight.The ratio o f the lamina area to plant weight is similar to the leaf area ratio of other authors (e.g.Beevers & Cooper, 1964;Radford, 1967).Consequently, if it is assumed that photosyn thesis is proportional to the lamina area and respira tion to weight, the LAR should be an index o f the P/R ratio and an indicator o f the growth potential, i.e.net photosynthesis.U nder these assumptions, central forms with elongate petioles should have the smallest potential for growth and consequently the lowest specific growth rate,

Identifying the limiting nutrient
At each site, the nutrient limiting E. crassipes growth rate was estimated from the average total N and total P concentrations in the water using the mean half saturation (Ks) concentrations of 976 ug N and 94.1 ^g P f-i, derived under culture conditions of N and P limitation (Musil & Breen 1985), in the Monod model.For example, the aver age total N and total P concentrations determined in the water at the MP3 site during 1978 were 20 746 (Ag N €-• and 6 569 jig P €-i ( Musil, 1982).The per centage o f the maximum specific growth rate (Umax) that E. crassipes would achieve at (i) the average total N, (ii) the average total P concentrations in the water at this site were estimated using the Monod model as follows: T he results show that at the MP3 site E. crassipes would achieve a lower percentage o f the Umax at the average total N than at the average total P con centrations in the water which indicates that N was the limiting nutrient.
At the BGL site, on the other hand, the average total N and total P concentrations determined in the water during 1978 were 10 206 pg N £-> and 150 pg P (Musil, 1982).Using the Monod model, it was estimated that E. crassipes would achieve 91,3% and 61,4% o f the Umax at the average total N and total P concentrations in the water respectively at this site, indicating that P was the limiting nutrient.

Predicting Umax fo r different temperatures
The Umax values derived for E. crassipes under culture conditions of N and P limitation, at mean daily air tem peratures o f 24°C and 28°C respectively, were 0,0886 g fresh mass g-' d~l for N and 0,1089 g fresh mass g-> d~! for P (Musil & Breen, 1985).Using these values, the Umax of E. crassipes, under condi tions of N or P limitation, may be predicted for other tem peratures according to the Van't Hoff rule (Tables 1 & 2) from the Arrhenius equations for the exponential relationships between Umax(n) and Umax(p) and tem perature.Assuming a Qio of 2,0, these relationships are: where Umax(p) = maximum specific growth rate for P (g fresh mass g-1 d~'); T = absolute m ean daily air tem perature (°K).

Comparison o f predicted and measured specific growth rates
Incorporating these equations and the Ks concen trations o f 976 pg N and 94,1 pg P into the M onod model, specific growth rates were predicted for E. crassipes, over each growing interval at the MP3 and BGL sites, from the limiting N o r P con centrations in the water and mean daily air tem pera tures.These were then compared with measured specific growth rates.A t the MP3 site, where N was estimated to be the limiting nutrient, specific growth rates were predicted from the various N (NO3-N , NH4-N and total N) fractions in the water using Equation 1 in the Monod model.At the BGL site, where P was estimated to be the limiting nutrient, specific growth rates were predicted from the vari ous P (SRP and total P) fractions in the water using Equation 2 in the Monod model.For example, the mean daily air tem perature and total N concentra tion in the water at the MP3 site over the growing interval 1/2 to 16/2/78 were 24,9°C and 15 970 jug N €_I respectively (Musil, 1982).The specific growth rate (U ) was predicted for E. crassipes for this set o f conditions as follows: U = 3,9151 x 107e -5916/24,9 + 273,2 X ___-------------976 4-15 970 = 0,0887 g fresh mass g-i d-i (8,87% d ■).
Predicted specific growth rates and those meas ured for marginal and central forms, growing in loosely and densely crowded populations respect ively, at the two sites are illustrated in Figs 9 and 10.
At the MP3 site, specific growth rates predicted from total N, NH4-N and to a lesser extent from NO3-N concentrations in the water followed a fairly similar seasonal pattern to those measured for m ar ginal forms with values decreasing progressively after summer (September to March) through to win ter (May to August), A similar relationship between m easured specific growth rates and those predicted from total P concentrations in the water was ob tained at the BGL site, Specific growth rates pre dicted from SRP concentrations in the water at the BGL site, however, were extremely variable and did not follow any recognizable seasonal pattern.
In general, specific growth rates predicted from the various N or P fractions in the water at the two sites were significantly lower than those measured for marginal forms.For example, o f the 29 specific growth rates predicted from total N concentrations in the water at the MP3 site, only 3 (during April and June) fell within the standard deviations of meas ured values.O f the 22 specific growth rates pre dicted from total P concentrations in the water at the BGL site, only 6 (during M arch, April, June and July) fell within the standard deviations o f m easured values.T he differences betw een m easured specific growth rates and those predicted from o th er N (NH4-N and N O 3 -N ) o r P (SRP) fractions in the w ater at the two sites, however, were generally larger than the differences between measured speci fic growth rates and those predicted from total N or total P concentrations in the water.This suggests that specific growth rates of E .crassipes in the field may be m ore accurately predicted from total N or total P concentrations, than from other N o r P frac tions, in the water.At both sites, the smallest differences betw een m easured specific growth rates and those predicted from total N o r total P concentra tions in the water occurred during midwinter in June when relative humidities and diffuse radiant fluxes were at their lowest levels (Figs 3 & 4).These results suggest that the Umax values derived for E .cras sipes, under culture conditions of N and P limitation (Musi! & B reen, 1985), may have been depressed by lowered relative humidities and radiant flux densi ties in the greenhouse, since specific growth rates predicted from total N or total P concentrations in the w ater at the two sites were significantly lower than those measured for marginal forms, except when diffuse radiant fluxes and relative humidities in the field were low.
Relative humidities recorded in the greenhouse during the experimental determ ination o f kinetic co efficients for E. crassipes (Musil & Breen, 1985) were lower than those recorded during summer for the two field sites in the Durban area (Fig. 3).Mean daily relative humidities in the greenhouse ranged from 61 to 67% , whereas those recorded during summ er for the two field sites ranged from 74 to 85% . Since Freidel et al. (1978) have shown that E. crassipes growth rate decreases with a decrease in relative humidity, it would appear that the Umax values derived for E. crassipes in culture were de pressed by lower relative humidities in the green house.In addition, it is possible that radiant flux densities in the greenhouse limited E. crassipes growth rate during the experimental determination of kinetic coefficients for this plant.Measurements o f light intensity at midday in full sunlight inside and outside the greenhouse showed that light intensity in the greenhouse was attenuated by ca 37% ( Musil. 1982).rates varied considerably during the year.For ex ample, at the MP3 site the ratios between specific growth rates measured for marginal forms and those predicted from total N concentrations in the water ranged from ca 2,5 during summer to ca 1,1 during winter.Similarly, at the BGL site, the ratios be tween measured specific growth rates and those pre dicted from total P concentrations in the water ranged from ca 2,6 during summer to ca 1 ,1 during winter.
In contrast to specific growth rates of marginal forms, those of central forms growing in densely crowded populations at the MP3 site were often sig nificantly lower than those predicted from the vari ous N fractions in the water (Fig. 9).For example, only 2 of the 18 specific growth rates predicted from total N concentrations in the water at this site fell within the standard deviations o f measured values.No significance, however, was attached to the obser vation that the differences between specific growth rates measured for central forms and those predicted from total N concentrations in the water were slightly larger than the differences between meas ured specific growth rates and those predicted from other N (NH4-N and N 0 3-N ) fractions in the water.The poor affinity obtained between predicted speci fic growth rates and those measured for central forms may be attributed to the specific growth rate of E. crassipes being depressed by intraspecific com petition and self shading in densely crowdcd field populations.This suggests that, unless a correction factor is introduced into the model to amend the Umax for the density of the plant population, Umax values derived for plants o f the marginal growth form in culture cannot be used in the Monod model to predict specific growth rates o f central forms growing in densely crowded field populations.
Although Umax values derived for £ .crassipes under culture conditions of N and P limitation were of little value in the Monod model for accurately predicting specific growth rates of plants growing in loosely o r densely crowded field populations, it is possible that more reliable estimates o f Umax may be derived for £ .crassipes from the field data.At the MP3 site, where N and P concentrations in the water were very high, it was estimated from the mean Ks concentrations, derived under culture con ditions of N and P limitation, that E. crassipes would achieve ca 95,5 and 98,6% of its Umax at the aver age total N and total P concentrations in the water respectively.It is evident, therefore, that, even though N was considered to be the limiting nutrient at this site, N and P concentrations in the water ap proached those saturating to E. crassipes growth rate.Consequently, if it is assumed that specific growth rates measured for plants o f the marginal and central form s at specific tem peratures at the MP3 site closely approxim ate their respective Umax val ues, it should be possible to express the relation ships, if exponential, between these estimated Umax values of marginal and central forms and air tem peratures in the form o f Arrhenius equations.These equations can then be incorporated into the Monod m odel to predict specific growth rates of marginal and central forms, from the limiting N o r P concen trations in the water and m ean daily air tem pera tures, at other field sites.This should improve the m odel's accuracy o f prediction for plants growing in both loosely and densely crowded field populations.T he mean Ks concentrations derived for E. crassipes under culture conditions o f N and P limitation, on the o th er hand, appear fairly reliable, since pre dicted specific growth rates generally followed a similar seasonal pattern to those measured.

Deriving Umax in ihe field
A where U = specific growth rate (estimated maxi mum specific growth rate) g fresh mass g-> d->; T = absolute mean daily tem perature °K.
A Q 10 value of 2,14, in the tem perature range 15 to 25°C, was calculated from the above expression.This compares favourably with the Cho value of 2,12 reported by Goldman (1972) for the exponential re lationship between the Umax values of various species of fresh water algae and tem perature, in the range 19 to 39°C.It demonstrates that the effect of air tem perature on the Umax of E. crassipes (margi nal forms) follows the Van't Hoff rule and confirms the initial hypothesis made.In addition, the acti vation energy o f 12,978 calories mole-1 calculated for marginal forms from the above expression com pares favourably with the activation energy o f 13,356 calories mole-1 reported by Goldman (1972) for al gae.
The regression equation exponentially relating specific growth rates (estimated Umax values) of central forms at the MP3 site to the reciprocals of absolute mean daily air tem peratures was: U = 1,9932 x 10SíH661.t ..4 where U = specific growth rate (estimated maxi mum specific growth rate) g fresh mass g-1 d _1; T = absolute mean daily air tem perature °K.A O ]0 value of only 1,71 in the tem perature range 15 to 25°C, was calculated from the above expres sion.This demonstrates that central forms growing in densely crowded field populations show a propor tionally smaller increase in their specific growth rate (estimated Umax) with a 10°C rise in the mean daily air tem perature than do marginal forms growing in loosely crowded field populations.The lower Q I() value obtained for central forms was attributed to their specific growth rate being depressed in densely crowded field populations.

< / > S
With respect to the above exponential expres sions, it should be pointed out, however, that recent investigations have shown that plant growth in the field can respond linearly rather than exponentially to environmental tem perature, which suggests that the asymetric bell-shaped response to tem perature may not be as widely applicable as was formerly con sidered.Gallagher & Biscoe (1979), for example, have demonstrated that, in the absence of water stress, the expansion rate of barley leaves is directly proportional to the temperature of the stem apex.It remains to be seen, however, what the theoretical interpretation of such linear responses can be.

Comparison o f predicted and measured specific growth rates
The usefulness o f Equation 4 in the Monod model could not be assessed, since no measurements of specific growth rates of central forms were obtained at the BGL site.
Substituting Equation 3 for Equation 1 in the Monod model, specific growth rates were repre dicted for marginal forms at the BGL site from the limiting P (SRP and total P) concentrations in the water and mean daily air temperatures.Predicted specific growth rates and those measured for margi nal forms are illustrated in Fig. 13.O f the 22 specific growth rates predicted from total P concentrations in the water at the BGL site, 14 (ca 64% ) fell within the standard deviations of measured specific growth rates.Predicted specific growth rates generally followed the same seasonal pattern as measured specific growth rates with va lues decreasing progressively after summer (Septem ber to March) through to winter (May to August).Bell (1981) pointed out that, by analogy with the correlation coefficient (r), a coefficient of variation (R 2) may be computed to test model outputs com pared with data defined as: , sum of squares of residuals -i - ---------------------------------------n SD2y where n is the number of data points and SD2y is the variance in predicted values.For values of R2 that are high and approximate to 1 , the fit is good; for values of R2 that arc low and approximate to 0, the fit is poor.For values of R2 that are inbetween and approxim ate to 0,5, the situation is uncertain.Dent & Blackie (1979) recom mended a simple regression analysis between model outputs and data as paired observations which produces the same value of R2 as defined.T he coefficient o f variation (R 2) calculated between m easured specific growth rates and those predicted from total P concentrations in the w ater at the BGL site had a value of R2 = 0,5321.Conse quently, the observed affinity between predicted and measured values could not be regarded as significant over the entire growing season.Specific growth rates predicted from SRP concentrations in the w ater at this site were extremely variable and did not follow any recognizable seasonal pattern.Only one o f the predicted values fell within the standard deviations o f m easured values.
T he differences betw een m easured specific growth rates and those predicted from total P concentra tions in the water at the BGL site were considerably smaller than the differences between measured specific growth rates and those predicted from other P (SRP) fractions in the water (Fig. 13).Similar re sults were obtained at sites where N was estimated to be the limiting nutrient (M usil, 1982).These results indicate that specific growth rates o f E. crassipes in the field are more accurately predicted from the limiting total N o r total P concentrations, than from other N o r P fractions, in the w ater which suggests that the plant may utilize both inorganic and organic forms o f N and P for growth.
With respect to P , Jeschke & Simonis (1965) re ported that the main source of P for growth o f aqua tic plants is in the form o f inorganic phosphates.However, specific growth rates of marginal forms at the BGL site were more accurately predicted from total P than from SRP concentrations in th e water.Consequently, it would appear that total P concen trations in the water at this site better reflected the total amount of P available to plants during growth, some o f the P for growth of plants possibly being provided by release of that bound to sedim ents as well as to other soluble and insoluble fractions in the water.For example, when P is added to lakes, it is rapidly removed from solution by adsorption onto sediments (H epher, 1958;Hayes & Phillips. 1968).This P is not rendered entirely unavailable since sediment P and dissolved P exist in equilibrium (H e pher, 1958;Pomeroy et al., 1965).T he equilibrium concentration increases with increased P content in the sediment (Pomeroy et al., 1965).Consequently, removal of P from the water by E. crassipes during growth would displace the P equilibrium allowing additional P to be released from the sedim ents into the overlying water.Alternatively, it has been shown that many zooplankton and phytoplankton species excrete alkaline phosphatases which may ac celerate phytoplankton growth by supplying ortho phosphate from suitable organic esters (Berm an, 1969;1970;Jansson, 1976;Wynne & Gophen, 1981).T he excretion of alkaline phosphatases by higher aquatic plant species has not been reported in the literature, though Wetzel (1969aliterature, though Wetzel ( , 1969b) ) has shown that some species do excrete various dis solved organic compounds.O ne may, therefore, speculate that species such as E. crassipes may ex crete alkaline phosphatases which would allow them to utilize normally unavailable organic forms of P for growth.
With respect to N, Sculthorpe (1967) suggested that NH4-N does not serve as an N source for growth of aquatic plants.However, several authors (Von Schwoerbel & Tillmans, 1972;Toetz, 1971;1973) have subsequently shown a preference by aquatic plants for NH4-N as an N source for growth.Best (1980) demonstrated that, although NH4-N supplied at low concentrations for a short period stimulated the growth o f Ceratophyllum demersum L. in cul ture, higher concentrations (in excess o f 45 x 103 fig NH4 -N €-■) applied for a prolonged period were toxic.The ammonium-induced inhibition o f growth has been reported for several other aquatic macrophyte species (Mulligan el al ., 1976).High NH4-N concentrations in the water may repress the in duction of nitrate reductase as shown by Joy (1969) and O rebam jo & Stewart (1975a, 1975b) in Lemna minor L. Reduced nitrate reductase activity would mean reduced N uptake and consequently a reduc tion in growth rate.
The largest differences between measured specific growth rates and those predicted from total P con centrations in the water at the BGL site (Fig. 13) occurred during the summer months (September to March) when radiant flux densities (diffuse compo nent of the radiant flux) were high (Fig. 4).This sug gested that if the effects of radiant flux density plus air tem perature were incorporated into the model, it might improve its accuracy of prediction.
Substituting Equation 5 dieted for marginal forms at the BGL site from the limiting P (SRP and total P) concentrations in the water, mean daily air tem peratures and diffuse ra diant fluxes.
Predicted specific growth rates and those meas ured for marginal forms are illustrated in Fig. 15.
In general, specific growth rates were more accu rately predicted from the limiting total P concentra tions in the w ater at the BGL site using Equation 5in the M onod model than Equation 3. In the form er exam ple, 15 o f the 22 specific growth rates predicted from total P concentrations in the water (ca 68% ) fell within the standard deviations o f m easured growth rates (Fig. 15), compared with 14 (ca 64% ) in the latter (Fig. 13).However, the coefficient of va riation (R 2) calculated between measured and pre dicted values was considerably higher in the form er example (R 2 = 0,7973) than in the latter (R 2 = 0,5321).Furtherm ore, the differences between m easured and predicted values were also generally much smaller in the form er example than in the lat ter (Table 3).Similar results were obtained at sites where N was estimated to be the limiting nutrient (Musil, 1982).In both examples, specific growth rates were m ore accurately predicted from total N or total P concentrations than from other N o r P frac tions in the water.
The largest differences between m easured specific growth rates and those predicted from total P con centrations in the water at the BGL site (Fig. 15) occurred during midsummer (November to Feb ruary) when relative humidities were highest (Fig. 3).This suggested that if the effects o f relative hu midity, in addition to air tem perature plus radiant flux density, were incorporated into the model, it might further improve its accuracy of prediction.
In general, specific growth rates were predicted with similar accuracy from the limiting total P con centrations in the water at the BGL site using Equa tion 6 in the Monod model compared with Equation 5, In the former example, 17 of the 22 specific growth rates predicted from total P concentrations in the water (ca 77% ) fell within the standard devia tions of measured specific growth rates (Fig, 17), compared with 15 (ca 68% ) in the latter (Fig. 15).However, the coefficients of variation (R 2), and the differences calculated between measured and pre dicted values (Table 4), were not very much differ ent in the former example (R2 -0,7870) compared with the latter (R2 = 0,7973).Similar results were obtained at sites where N was estimated to be the limiting nutrient (Musil, 1982).In both examples, specific growth rates were more accurately predicted from total N or total P concentrations, than from other N o r P fractions, in the water.T he largest differences between measured specific growth rates and those predicted from total P con centrations in the w ater at the BGL site (Fig. 17), however, still occurrcd during midsummer (Novem ber to February).It appeared unlikely that the water pH significantly influenced E. crassipes growth rate at this site o r at the MP3 site from where the Umax values o f marginal forms were derived.The re corded variation in the water pH at these two sites (Fig. 7) being considerably smaller than the varia tion in pH o f ca 1,2 pH units o r greater, in the range pH 3,0 to 8,2 shown by Chadwick & Obeid (1966) as significantly influencing E, crassipes growth in cul ture.In addition, the water pH values at both sites were in close proximity to pH 7,0 at which maximum growth o f E. crassipes occurs (Chadwick & Obeid, 1966).In view of this, two possible reasons are given to explain why predicted specific growth rates were significantly lower than those measured during mid summer: (i) That the Ks concentrations derived for E. cras sipes under culture conditions o f N and P limitation (Musil & Breen, 1985) were tem perature depen dent-In Chlorella pyrenoidosa and Oscillatoria agardhii, Shelef et al. (1970) andAhlgren (1978) have shown that Ks varies with tem perature.Simi larly, in Aerobacter aerogenes and Escherichia coli, Topiwala & Sinclair (1971) and Sawada et al. (1978) have also dem onstrated that Ks changes with tem perature and an Arrhenius plot o f the change is lin ear.Lower Ks concentrations in the w ater for the specific limiting nutrient during midsummer, when tem peratures were highest, would have brought the predicted specific growth rates closer to those meas ured.Consequently, it may be possible to describe more accurately the effect o f a limiting nutrient on E. crassipes growth rate by expressing Ks as a func tion o f tem perature. (ii) T hat the specific growth rates (estimated Umax values) measured for marginal forms at the MP3 site, particularly during midsummer, may have been depressed by some toxic factor in the water.High phytoplankton population densities in the water at the MP3 site, particularly during midsum m er, may have resulted in the excretion by phyto plankton of some toxic factor in sufficiently high concentrations to be inhibitory to E. crassipes growth rate.Algae in some instances can severely interfere with the growth of higher plants ( Hewitt, 1966), possibly through the production of antibiotic substances (Jorgensen, 1962) or toxic am ino and carboxylic acids (Steinberg, 1947a;1947b;Woltz & Jackson, 1961;Woltz, 1963).Furtherm ore, the pres ence of phenolic acids, common by-products of de composition, in the secondary treated waste-water effluent at the MP3 site may also have inhibited E. crassipes growth rate.Glass (1973Glass ( , 1974)), for ex am ple, has shown that phenolic acids severely inhibit This is what was observed during midsummer (Fig. 17).Finally, it should be pointed out that at the BGL site relative humidities were not substantially differ ent from the MP3 site, at which the Umax values of E. crassipes were derived.However, at sites where relative humidities were substantially different from the MP3 site, Musil (1982) showed that specific growth rates of marginal forms were generally more accurately predicted from the limiting nutrient con centrations in the water using Equation 6 in the Monod model than Equation 5.

CO NCLUSIONS
Specific growth rates o f E. crassipes growing in loosely crowded field populations were adequately predicted from the limiting total N or total P concen trations in the water using culture-derived half satu ration coefficients (Ks) and field-derived maximum specific growth rates (U m ax), expressed as a func tion of air tem perature, diffuse radiant flux and rela tive humidity, in the Monod model.The possible tem perature dependency o f Ks requires further in vestigation.This, however, will require precise measurements of Ks, under conditions of N o r P limitation, at different tem peratures in some type of continuous flow culture system, as suggested by Mu sil & Breen (1985).The relationship between Ks and tem perature will then need to be mathematically for mulated and incorporated into the model-W hether such refinements will improve the model's accuracy of prediction needs verification.Nature, Lond. (New Biol.)

V lT T R E K S tL
tion w ere used in the M onod model to identify the limiting nutrient and to predict specific growth rates under conditions o f varying w ater nutrient concentration and air temperature.Predicted data w ere validated by compari son with specific growth rates measured for plants growing in loosely and densely crowded populations at tw o field sites.T h e u se o f culture-derived maximum specific growth rates (U m ax) in the m odel resulted in maccurate predic tions o f plant growth rates in loosely and densely crowdcd field populations.The use o f field-derived Umax values in the m odel, how ever, resulted in adequate predictions o f plant growth rates in loosely crowded field populations.T h e incorporation o f radiant flux density (diffuse com ponent o f the radiant flux) and relative humidity into the m odel considerably improved its accuracy o f prediction.In all cases, specific growth rates w ere more accurately predicted from the limiting total N o r total P concentrations, than from other N or P fractions, in the water.
FIG, l. -Location o f field sites.I, maturation pond, northern sewage treatment works; 2, Botanic Gar dens Lake.

FIG. 2 ,-F
FIG. 2, -Air temperatures, as daily averages over each growing interval, at tw o field sites.
FIG . 5. -Nitrogen (N O ,-N , N H .-N and total N ) concentrations in the w ater, averaged over each growing interval, beneath loosely crowded populations enclosed at tw o field sites.

FIG. 8
FIG. 8. -Specific growth rates o f E. crassipes over each growing interval, at tw o field sites.Solid line = marginal forms growing in loosely crowded populations (means o f 4 0 replicates).Broken line = central forms, growing in densely crowded populations (m eans o f 30 replicates).N o plants o f the central form were produced during June, July and A u gust.Standard deviations o f measured specific growth rates are shown by bars.

FIG . 9
FIG .9. -Specific growth rates predicted for E. crassipes from 2 , total N ; 3, N H j-N and 4 , N O j-N concentrations in th e w ater, over each growing interval, at the maturation pond 3 site com pared with those measured for 1, marginal forms, growing in loosely crowded populations and 5 , central forms, growing in densely crowdcd populations.Standard deviations o f m easured specific growth rates are shown by bars.Um ax values used in the M onod m odel derived under culture conditions o f N limitation and expressed as a function o f air temperature.
FIG. 10. -Specific growth rates predicted for E. crassipes from 2 , total P and 3, SR P concentrations in the water, over each growing interval, at the Botanic Gardens Lake site compared with those measured for 1, marginal forms, growing in loosely crowded populations.Standard deviations o f measured specific growth rates are shown by bars.Um ax values used in the M onod model derived under culture conditions o f P limitation and expressed as a function o f air temperature.
rrhenius plots of specific growth rates (estim ated Umax values) o f marginal and central forms, ex pressed as natural logarithms (Loge), against the re ciprocals o f absolute m ean daily air tem peratures at the MP3 site (Figs 11 & 12), over the period Feb ruary to Decem ber, 1978, yielded linear relation ships with the correlation coefficients being high and significant at P ^ 0,01.T he regression equation exponentially relating specific growth rates (estim ated Umax values) of marginal forms at the MP3 site to the reciprocals of absolute m ean daily air tem peratures was: U = 5,2631 x 108e^540/r.........................3 FIG .11. -A n Arrhenius plot o f specific growth rates (estim ated U m ax values) o f marginal forms (Loge) against the reciprocals o f absolute mean daily air temperatures at the maturation pond 3 site.
FIG .13. -Specific growth rates predicted for marginal forms from 2 , total P and 3 , SR P concendations in th e w ater, over each growing interval, at the Botanic Gardens Lake site compared w ith 1, measured specific growth rates.Standard deviations o f measured specific growth rates are shown by bars.U m ax values used in the M onod m odel derived under field conditions and expressed as a function o f air temperature.
FIG .14. -A n Arrhenius plot o f specific growth rates (estimated Um ax values) o f marginal forms (Logf) against the products o f the reciprocals o f absolute mean daily air temperatures and diffuse radiant fluxes at the maturation pond 3 site.
the uptake of P and K by barley.Consequently, Umax values derived for E. crassipes at the MP3 site, under conditions o f growth rate inhibition by some toxic factor in the w ater, when incorporated into the Monod model would have underestimated the specific growth rates o f plants at the BGL site.

FIT
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