New aspects of the consideration of feedback mechanisms in...

New aspects of the consideration of feedback mechanisms in an ecological model of the pelagic zone

Jiirgen Benndorf, Dietrich Uhlmann and Frieder Recknagel Dresden, GDR

Abstract. In order to achieve a more realistic model of the pelagic zone of lakes and reservoirs three feedback mechanisms have been investigated. (1) A 'mixing model' involving higher light utilization by the phytoplankton at intensive vertical mixing of the water body, a high light attenuation coefficient and a low mixing depth. This combination of variables results in higher primary production rates compared with the conventional (static) model. As the high light attenuation coefficient is self-generated by the high phytoplankton biomass this phenomenon can be taken as a positive feedback mechanism. (2) In connection with the vertical mixing in water with mass growths of phytoplankton a further positive feedback loop plays an essential role: the self-regenerated carbon dioxide flux within the algal-bacterial association. (3) The negative feed-back between the biomass and specific activity (e.g. rate of phytosynthesis per unit biomass) leads to a more sophisticated version of the Monod kinetics and enables the model to respond realistically to external disturbances.

Nouveaux aspects sur la considération de mécanismes de 'feedback' pour un modèle écologique de la zone pélagique Résumé. Dans le but de bâtir un modèle plus réaliste de la zone pélagique des lacs et réservoirs, trois mécanismes de 'feedback' doivent être étudiés dans un modèle écologique d'une zone pélagique. (1) Un modèle de 'mélanges' comprenant l'utilisation plus intensive de la lumière par le phytoplancton quand l'eau se mélange plus rapidement dans la direction verticale, un coefficient d'atténuation de la lumière élevé et une faible profondeur de mélange. De cette constellation de variables résultent des taux de production primaire plus élevés que dans le cas du modèle conventionnel (statique). Comme le coefficient d'atténuation élevé de la lumière résulte de l'abondance de la biomasse de phytoplancton ce phénomène peut être considéré comme un mécanisme de 'feedback' positif. (2) Une autre source de 'feedback' positif, liée au mélange vertical des eaux avec la croissance de la masse de phytoplancton, joue un rôle essentiel: le flux auto-régénéré de dioxide de carbone par l'association algues-bactéries. (3) Le 'feedback' négatif entre la biomasse et l'activité spécifique (par example taux de photosynthèse par unité de biomasse) permet d'aboutir à une version plus sophistiquée de la cinétique de Monod et permet au modèle de répondre de manière plus réaliste aux perturbations externes.

NOTATION

E excess primary production (or growth) rate calculated as the relative difference between the primary production rate (PPR) of the mixing and the static model, respectively [per cent];

G grazing rate of the zooplankton on phytoplankton [day-1 ] ; C'max maximum grazing rate [day-1]; I_ light intensity (400 .. .700 nm) [J cm"2 day"1]; / mean light intensity within the mixed layer [J cm"2 day -1]; I0 incident light intensity at the surface [J cm"

2 day"1]; Iz light intensity at depth z [J cm"

2 day"1]; Ki,K2,K3,K4 parameters of the light-growth rate relations according to equations

(2)-(5)[Jcm"2day"1]; KM Michaelis constant of an enzymatic reaction [JJM/ 1.] ; Ks half saturation constant of the Monod growth kinetics [ugl 1.] ; Kx half saturation constant of the inverse relation between phytoplankton

growth rate and biomass [mm3/l.] ;

170

Feedback mechanisms in an ecological model of the pelagic zone 171

KxG half saturated constant of the relation between grazing rate and phytoplankton biomass [mm3/l.];

Kz half saturation constant of the inverse relation between grazing rate and zooplankton biomass [mm3/l.];

LX, MX parameters of the nutrient dependence of the half saturation con-stant Kx ;

PP primary production [mg 171 day -1]; PPR = n primary production or growth rate [day - 1 ] ; PPR vertical mean primary production or growth rate [day - 1 ] ; PPRmstx =nmax maximum primary production or growth rate at optimum conditions

[day"1]; S substrate or nutrient concentration [//g/1.]; T water temperature [°C]; v velocity of an enzymatic reaction [MM/S] ; Kmax maximum velocity of an enzymatic reaction [MM/S] ; W individual body mass of one zooplankter [mm3]; x phytoplankton biomass [mm3/l.]; Z zooplankton biomass [mm3/!.]; zmix depth of the mixed layer [m]; z depth of the euphotic zone [m]; e light attenuation coefficient (400 . . . 700 nm) [ITT1 ] ; li see PPR;

Mmax See " K m a x > n primary production [g m - 2 day - 1 ] .

INTRODUCTION

There is a need in modelling fresh water ecosystems to concentrate on the most essential processes because a comprehensive analysis of all processes is, at present, quite impossible, due to the enormous structural complexity and the multitude of stochastic disturbances influencing such systems. In this connection one of the most difficult problems is to separate the essential processes from the less essential ones. Furthermore a sufficient amount of information on essential processes is necessary before their inclusion into a model is possible and appropriate.

Without a doubt the ecological feedback mechanisms belong to the most important processes contributing to a great extent to the stability of a given state of an eco-system. Models neglecting these mechanisms reveal a too great a sensitivity to dis-turbances. This unrealistic sensitivity is at present still a great disadvantage even for such models for which the coefficients have been estimated very exactly (Hobbie and Tiwari, 1978). As a prerequisite of the completion and improvement of an ecological model of the pelagic zone of lakes, reservoirs or ponds, three feedback mechanisms have been investigated :

(1) a combined negative and positive feedback between primary production (PP) rate and light utilization;

(2) a positive feedback between PP rate and carbon dioxide availability, (3) a negative feedback between the plankton biomass and specific activity

(such as PP rate or grazing rate).

LIGHT UTILIZATION AS A FUNCTION OF VERTICAL MIXING INTENSITY

Usually a depth-averaged value of the PP rate is obtained by determining all rates at different depths and computing the arithmetic mean. The related experimental

172 Jùrgen Benndorf, Dietrich Uhlmann and Frieder Recknagel

approach is the vertical arrangement of bottles for the measurement of primary pro-duction in a fixed position (e.g. Vollenweider, 1974). On the other hand some authors have pointed out that with this procedure the passive vertical transport of the phyto-plankton by turbulent mixing of the water body and therefore also the corresponding changes in the illumination of the cells are neglected (Ruttner, 1952; Riley, 1957; Benndorf, 1973; Baumert, 1974, 1976a and 1976b; Jewson and Wood, 1975; Stephan etal, 1976; Benndorf and Werner, 1978; Benndorf and Baumert, 1978).

Benndorf and Baumert (1978) revealed that under certain conditions (high tur-bulence, high attenuation coefficient and. shallow depth of the .water.body) the use. of a 'mixing model' yields much higher and perhaps more realistic PP rates than the conventional (static) photosynthesis model. This mixing model starts with the assump-tion that the algae change their position in the vertical light gradient with a velocity exceeding the ability of the photosynthetic mechanism to respond to the change in light intensity. This essential premise can only be fulfilled in shallow water with a high attenuation coefficient and in times of strong wind action. In this case the PP rates at the several depths are no longer controlled by the light intensities at these depths, but only by the mean vertical light intensity. This is calculated according to equation (1) (Riley, 1957; Stephan etal, 1976; Benndorf and Baumert, 1978):

/-= _ _ î _ (1 _ e-«mix) (1) ez mix

In the mixing model the light intensity Iz at a given depth in the well-known light//*/5

rate equations is replaced by / :

FFR=PPRmax(S,T,X)—-= (2) Ai + /

PPR = PPRmax (S, T, x) ~^== (3)

PPR = PPRmax (S, T, x) V^3 = (4) A 3 +1

PPR=PPRmaK{S,T,x)Kje-K"r (5)

The PP rates calculated on the basis of equations (2)—(5) (mixing model version) in all cases exceed the rates obtained by the corresponding static model (Fig. 1). Under natural conditions 'excess PP rates' of more than 100 per cent can be expected. The differences between the results of the mixing and the static model substantially increase with the increasing magnitude of the product of mixing depth and attenuation coefficient.

These mixing-induced 'excess PP rates' may substantially contribute to the marked maintenance tendency of a highly eutrophic state of a water body. Since nutrient limitation (except inorganic carbon; see following section) barely plays a role in such waters, light availability represents a very important limiting factor for primary pro-duction. But just at the high attenuation coefficients characteristic of highly eutrophic waters high mixing-caused 'excess' production rates can be expected if the water body is well mixed. This mechanism leads to an increase of the light availability and has therefore a positive feedback effect other than the simple self-shading effect in the static case (Fig. 2).

The consequences of the consideration of this / concept have been investigated by means of an analytical model of the pelagic region of a reservoir (Benndorf and Recknagel, unpublished). In this model the following ecosystem elements are taken


E[%1PPR[%] 100-

E[°/o],PPR[%l 100

Ef%]PPf?[%] 10a

E[%],PPRM no-

FIGURE 1. 'Excess' primary production rates (E) calculated as the difference between the mixing and the static model; the average primary production rate (JPPR) of both models corresponds (a) to equation (2), (b) to equation (3), (c) to equation (4) and (d) to equation (5); abscissa: normalized light intensity ; d = product of light attenuation coefficient and mixing depth (from Benndorf and Baumert, 1978).

•

(-)

I

Iz

I

k /M

PPR X

FIGURE 2. Schematic representation of both the ways in which self shading by phyto-plankton (x) influences the primary production rate (PPR) in shallow turbulent waters; /= use of the / concept (mixing model) ; Iz - Iz concept (static model).

into account: light, temperature, orthophosphate, phytoplankton, zooplankton and fish. For the special purpose of checking the influence of the mixing-induced increase of light utilization, the model has been simplified by equating the grazing rate of the zooplankton to zero in order to get high phytoplankton biomass and attenuation coefficients. The model outputs for PP rate and phytoplankton biomass according to both model concepts (mixing and static) are compared in Figs. 3 and 4. The higher PP


0.1001

®

FIGURE 3. Growth rates of the phytoplankton calculated by the static model (a) and by the mixing model (b); input data of the Saidenbach Reservoir, 1975; grazing rate = 0; note the difference in the ordinate scales.

rates of the mixing model [Fig. 3(b)] cause only a slightly higher biomass during spring and summer [Fig. 4(b)] compared with the static model [Figs. 3(a) and 4(a)]. On the other hand, during the autumn the biomass of the mixing model is consider-ably higher than in the static model. The absolute values of both model runs (Figs. 3 and 4) are of course unrealistic because of the absence of the zooplankton grazing. The only purpose of these runs is the comparison of both models which indicates that the mixing model indeed shows the supposed positive feedback mechanism.

There are also empirical results which demonstrate that mixing involves higher PP rates. Some authors have compared measurements of PP rates using vertically arranged light/dark bottles with calculations of the PP rates resulting from diurnal oxygen changes recorded directly in the water body (Winberg and Jarovitzina, 1939; Uhlmann, 1966; Kalbe, 1972). Only the latter method takes into consideration the mixing effects. Consequently very much higher PP rates have been found. Unfortunately, it remains doubtful to what extent the higher PP rates both calculated by the mixing model and measured by the direct oxygen method really are caused by the assumed relation between mixing intensity and increasing light utilization. This is emphasized because the possibly better light utilization as a consequence of intensive vertical mixing in a highly eutrophic water is in each case also combined with a better supply of carbon dioxide.


I A N . FCI. WW. APRIL HOT JUKE JJLI RUE. HPT. OCT. K». K C .

FIGURE 4. Phytoplankton biomass calculated by the static model (a) and by the mixing model (b); input data and grazing rate as in Fig. 3; note the difference in the ordinate scales.

THE UPPER LIMIT OF PHYTOPLANKTON PRODUCTION AS CONTROLLED BY CARBON TURNOVER

A very high level of phytoplankton production (per unit area and time) presupposes a maximum areal density of the phytoplankton biomass in connection with an optimum supply of light energy. As Fig. 5 indicates, the necessary conditions obviously are only given in hyper-eutrophic water bodies with an excess supply of dissolved nitrogen and phosphorus. It also becomes evident from this figure that the maximum production estimates in terms of carbon fixation per unit area substantially exceed the highest possible values for inorganic carbon flux from the atmosphere as calculated by Schindler and Fee (1973). On the other hand, a permanent release of carbon dioxide from bottom sediments an order of magnitude exceeding 5 gCm-2day_1 also is improb-able. Even in the case of highly polluted water bodies such as sewage ponds, corre-sponding estimates of oxygen consumption by bottom sediments obviously have not been recorded (Uhlmann, 1970). As a summation of the carbon fluxes from the atmosphere and from the bottom sediment is insufficient to support a primary pro-duction in excess of 20 gC m"2 day-1 in waters more than 0.1 m deep, the most probable source to meet the corresponding demand is a fast recirculation of the carbon dioxide released by microbial respiration in the water body. That such a positive feedback loop between net primary production and dark respiration of phytoplankton


R

100-

10-

1:

m-

Primary Production

* " \ -* • data used for calculation of the envelope curve Jl(zO

\ * Supplementary data vfï documenting a high level of Jl

Water hyacinths C Eichhornia J ' ^Sewage ponds in sewage ^~">i. \ lagoons * • * : ^ O r „ ,

\ K i • *\\Hyper-eutrophic \ x : • •^^coastat o. inland lakes \ ••Vc*

i * " X n. 35 I *"* ' X ^ (z'+0.5)°-9

\ ; x ^\JiQrine \ : \^waters

\ . \ i x\4 -X

- •• N : \ ; \ : \

!2v

001 0.1 m 100[m] Euphotic Zone Depth z'

FIGURE 5. The highest levels of phytoplankton production (n) as documented in the literature as a function of light penetration depth (z ' ) ; / , = tangent to the lower part the envelope curve; z'w = depth corresponding to the vertex of the envelope curve; f2 = highest possible level of the C0 2 invasion from the atmosphere as calculated by Schindler and Fee (1973); data adopted and, in part, converted from Wolverton and McDonald (1976; Eichhornia), McGarry and Tong Kasame (1971), Odum and Wilson (1962), Odum et al. (1963), Sreenivasan (1976), Hubel (1968), Ryther (1963).

associated with bacteria really exists, can be demonstrated by means of continuous-flow laboratory models of sewage ponds (Fig. 6). In these models, dark respiration is proportional to the net primary production of the previous photo-period. On the other hand, the net primary production is carbon-limited and thus essentially depends upon the carbon dioxide stock accumulated during the previous dark period. About one hour after the beginning of the light period the carbon supply is usually exhausted, and a further increase in photosynthetic activity can only be achieved by injection of carbon dioxide.

In a water body with an extremely dense phytoplankton population, the photo-synthesis in the upper layer is usually also carbon-limited, but from the horizons below the light penetration depth a carbon dioxide flux sufficiently high to maintain a high production rate can be provided by wind-induced agitation. Thus the most plausible explanation of the extremely high primary production estimates plotted in Fig. 5 seems to be the self-regenerated carbon dioxide flux within the algal-bacterial association. The magnitude of this flux, however, is primarily physically controlled, with the discontinuous wind action as the principal driving force for the vertical dispersion of phytoplankton, bacteria, products of phytosyn thesis, and carbon dioxide, within the light/dark gradient. By these means 'bottle effects' such as photo-inhibition, accumulation of toxic (free) ammonia resulting from an increase in pH and limitation of photosynthesis by too high an oxygen pressure are prevented as far as vertical mixing attains a sufficiently high intensity or frequency, respectively. The highest estimates included in the plot relate to tropical sewage ponds which were artificially turned over twice a day (McGarry and Tong Kasame, 1971). The data for Eichhornia are only included for completion and in order to get a corresponding estimation for a floating-leaved system in which both carbon dioxide regeneration and transport are obviously

Feedback mechanisms in an ecological model of the pelagic zone 177 dark tight dark light

02 [mg/I.J

20-

10

0 A A A A A A A A A

d02 dT

5-

04

[mg/l-h] net production

n^vUi^VVV respiration

PPCOJ[mg/t-dJ

20 \ ©

FIGURE 6. (a) Diurnal changes of oxygen concentration in a continuous-flow laboratory model of a sewage pond with a 8-h light and a 16-h dark period; organic load = 25 mg BODs l"1 day -1; (b) primary production and dark respiration calculated from the rate of change; (c) level of net production and dark respiration per day as calculated from (b).

subject to a highly efficient internal feedback mechanism. In some cases the estimation had to be scaled up from bottle experiments by using a multiplication factor of 2.35 (Kalbe, 1972). This still can be an underestimation. In some other cases, estimation of z was only possible on the basis of secondary data such as the productivity/depth curve of bottle experiments or the secchi transparency, and thus somewhat uncertain. Nevertheless the total set of data seems to be sufficient as a basis for a computation of the enveloping curve designating the upper limit of primary production II for different euphotic zone depths. This curve exhibits a vertex at a euphotic zone depth of 0.5 m. Thus the further increase in II with decreasing z is not as steep as would be the case with the function/j. There is a high probability that wind action converted into Langmuir circulation provides optimal conditions for vertical dispersion of both biotic and abiotic components of the carbon turnover. At depths less than 0.5 m this circulation is limited by frictional resistance; usually the diameter of Langmuir spirals amounts to about 1-3 m. On the other hand, with increasing z the probability that carbon dioxide can efficiently be recycled from the tropholytic layers substantially decreases. Thus only in situations with extremely dense phytoplankton growths does a wind-induced rapid turnover of the inorganic carbon pool seem to be able to attain the level necessary to maintain a correspondingly high primary production. Since the described carbon dioxide, limitation and regeneration are at present not incorporated into our model, no results of model experiments can be given.

THE INTERACTION BETWEEN SPECIFIC ACTIVITY, LIMITING 'RESOURCE' AND BIOMASS

Usually equations according to the Monod kinetics [equation (6)] are used as expres-sions of a 'resource' based limitation of a specific activity (e.g. growth) including parameters characteristic of the particular species such as the half saturation constant


and the maximum rate of the activity considered:

M = / W ^ (6)

This approach has been derived from the Michaelis-Menten enzyme kinetics:

v = Vmax (7) KM + S

There is, however, an essential distinction between both these approaches: whereas the Michaelis-Menten enzyme kinetics considers a velocity (that means an effective per-formance), the Monod kinetics describes the relation between a specific performance (that means a performance per unit biomass) and the substrate (or nutrient) concen-tration. Therefore the velocity v in equation (7) increases with increasing enzyme concentration but the growth rate ^ in equation (6) decreases with increasing biomass. Hitherto this fact has not been taken into consideration by almost all authors although the inverse relationship between specific activity and biomass is well known (Micheewa, 1970; Uhlmann, 1971; Spodniewska, 1971; Findenegg, 1971; Benndorf and Stelzer, 1973; Javorni£ky, 1974; Klapwijk et al., 1974; Koschel, 1977). Proposals for the con-sideration of this inverse relationship in the calculation of the growth rate have only been made by Contois (1959) and more recently by StrasTorabâ (1978) in connection with his hypothesis of multiple resource kinetics. From these reflections and from the works cited it is evident that the 'classic' Monod relation cannot be applied to models of ecosystems which are quite different from a chemostate, because biomass — and therefore /! — are not only controlled by the nutrient concentration but also by other factors (e.g. grazing and sinking). [In the chemostate an equilibrium between the growth limiting nutrient and the biomass is automatically achieved, and consequently only in this case can /x be related to the nutrient concentration according to equation (6).]

The use of a kinetic model which takes into account the relation between the specific intracellular concentration of the limiting nutrient and the growth rate, is much more realistic. This approach leads to two-step growth models, involving nutrient uptake and cell growth as separate processes (Fuhs, 1969; Kozerski and Benndorf, 1972;Kozerski, 1974; Bierman etal, 1974; Wassiljew etal, 1975; Bierman, 1976; Straslcraba and Dvorakovâ, 1977). But in this connection a substantial difficulty remains: the determination of exact and realistic uptake constants. This point has to be emphasized because the commonly used tracer methods do not produce realistic values if the concentration of the limiting nutrient lies in the concentration range of the added tracer (Hobbie, 1969). Under these circumstances the measured uptake values are an expression of the degree of the intracellular nutrient deficiency (Koschel, 1974).

In order to overcome the above-mentioned difficulties a synthesis of the conven-tional Monod relation and the inverse relationship between activity and biomass seems to be useful. This synthesis starts from the fact that in batch-like systems (with opti-mum light, temperature and pH conditions) the correct kinetic parameters Ks and Mmax °f t n e Monod relation [equation (6)] can only be derived by means of experi-ments with a low biomass (Fig. 7). Only the kinetic parameters determined in such a way are comparable with parameters obtained from continuous culture experiments. As a next step the following inverse biomass growth rate relationship (Fig. 8) is for-mulated :

M = Mmax (/, T, S) 777-4^777", (8) (!/**) +(1/x)


- and x Biomass < 1mm3//.

o " < 3mm3/l.

• » > 3mm3//.

20 40 60

Orthophosphate [mgP/m3]

100

FIGURE 7. The relation between dissolved orthophosphate, biomass and growth rate; data of batch cultures {Asterionella formosa); the kinetic parameters for minimum biomass: Ks = 1.7 MgP/1-and Mm a x = 1.2 day

-1.

1 ra

te

Gro

wh

V

/"max

^—-~""^ '

• ~ s v

OSpmar

~~-~~—— s, • • — s f

Biomass

Kx-f(S)

FIGURE 8. The inverse relationship between biomass and growth rate; Sl and S2 = low nutrient concentration (S lies in the range of Ks); S3 and 5 4 = high nutrient concentration (S>KS).

The function MmaxC^) in equation (8) can be substituted by equation (6):

S/x M = ^max:0

(KJKX) + (S/Kx) + (Ks/x) + (S/x) (9)

As Fig. 8 indicates, the numerical value of the parameter Kx is not expected to be constant but is variable dependent on the nutrient concentration. An example of the relationship between Kx and nutrient concentration is given in Fig. 9 and equation (10):

Kx = LX S" (10)

The numerical values of the parameters LX and MX (for orthophosphate) are almost equal for large diatoms (Asterionella formosa), small diatoms (Stephanodiscus hantz-schii) and green algae (Chlorococcales).


20 \

Î? 75-

20 40 SO 80 Orthophosphatë [mg P/m3]

FIGURE 9. The relation between the dissolved orthophosphate and the parameter Kx; each point represents the arithmetic mean of four batch experiments with Asterionella formosa.

"Q

9J

I e CO

12

1.0

0.8

06-

04

0.2 \

5, o CD M

ale

so •ft n u. i= * &

1.2-

w-0.8-

0.6-

04-

0.2^

5 - - '

/ n r

99% y ,- '

= 0,02 = 26 = 0,92

y=X

99%

+ 0,93x

...x=y

- 99%

y = 0,29+ 0.73 K r> = 26 r = 0,73

02 0.4 0.6 0.8 1.0 1.2

Growth rate (experimental) [d~'J

FIGURE 10. Comparison between the experimental determined growth rate of Asterionella formosa and Fragilaria crotonensis and growth rates calculated (a) by means of equation (9) and (b) by means of equation (6).


The reliability of the proposed growth rate equation (9) has been tested by means of an independent experimental data set (a data set which had not been used for the estimation of Ks, nmax,LX and MX). The comparison of the experimentally deter-mined growth rates with the calculated growth rates [equation (9)] reveals a satis-factory degree of conformity [Fig. 10(a)], whereas the comparison of the same experi-mental rates with the rates calculated by the conventional equation (6) shows quite unsatisfactory results [Fig. 10(b)].

The concept described of the combined control of a specific activity by the limiting nutrient and the biomass is regarded as an alternative to the two-step concept includ-ing separate nutrient uptake and growth. The new concept exhibits the advantage that all kinetic constants can easily be determined.

This concept has been applied also to the calculation of the grazing rate of the zoo-plankton:

G = Gmax(T,W) x/z

(11) {KxGlKz) + (x/Kz) + (KxGlz) + (x/z)

where £ z = 5.76.x;0-41.

The consideration of the discussed negative feedback mechanism between biomass and specific activity in the construction of our model of the pelagic zone leads to a higher degree of stability of this model. This is shown in Fig. 11(a) which represents

& 0.075'

| 0-050-

è 0025-

0.800

0.600

0A00

g 0.200

®

'Ms s...

que. «ti •. ocr.

FIGURE 11. Computed growth rate of the phytoplankton at high phytoplankton biomass (G = 0) (a), and at low phytoplankton biomass (as a consequence of zooplankton grazing) (b); note the difference in the ordinate scales.


the same model run as in Figs. 3(a) and 4(a) with the theoretical assumption of a grazing rate equal to zero. Because of the low losses (sinking only) the biomass of the phyto-plankton from the spring up to the end of the year is very high in this theoretical case [see Fig. 4(a)]. But the growth rate exhibits extremely low values even during the summer [Fig. 11(a)]. On the other hand a much higher growth rate is obtained [Fig. 11(b)] if grazing of the zooplankton is involved in the model, thus causing a low phytoplankton biomass (maximum value for the year = 4 mm3/L). As a consequence of this model behaviour, the ecologically most important output (primary production) is buffered to a large extent.

REFERENCES

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Baumert, H. (1976a) Mathematisches Modell zur Deutung der durch intermittierende Belichtung von Phytoplankton hervorgerufenen Mehrleistung der Photosynthèse (Mathematical model for the explanation of the increased phytoplankton production caused by intermittent light), for. Revue ges. Hydrobiol. 61 , 517-527.

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Contois, D. E. (1959) Kinetics of bacterial growth relationship between population density and specific growth rate of continuous cultures. J. Gen. Microbiol. 2 1 , 4 0 - 5 0 .

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Fuhs, G. W. (1969) Phosphorus-limited growth of plankton diatoms. Verh. Internat. Verein. Limnol. 17 ,784-786.

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Feedback mechanisms in an ecological model of the pelagic zone 183 Hiibel, H, (1968) Die Bestimmung der Primàrproduktion des Phytoplanktons der Nord-Riigenschen

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