LIST OF SYMBOLS by udr54629

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									A dynamic model of the chemical nutrient system for research of the
ecological consequences in natural water


Alexander V . Leonov             Laxenburg, Austria


Abstract.     This paper presents a mathematical model of the ecological-biochemical system.
The model is intended for investigation of the interaction of chemical components with the living
matter of natural water and of the transformation of organic and inorganic substances and of bio-
chemical oxygen uptake. The transformation of dissolved organic matter, of organic and inorganic
phosphorus compounds, and oxygen uptake are discussed on the basis of computer simulation
results. The phosphorus compounds turnover intensity along different transformation pathways is
estimated.

Modèle dynamique du système des éléments nutritifs chimiques pour la recherche des consé-
quences écologiques dans les eaux naturelles
Résumé.       Dans cet article, nous proposons un modèle mathématique pour l'étude des processus
biochimiques et écologiques en milieu aquatique naturel. Le modèle décrit les interactions entre
les composants chimiques et la matière vivante, la transformation des substances organiques et
inorganiques, et l'utilisation biochimique de l'oxygène. Nous discutons les résultats des simulations
numériques en ce qui concerne la transformation de la matière organique dissoute, des composés
phosphores organiques et inorganiques, ainsi que l'utilisation de l'oxygène. Le temps de recyclage
des différents composés phosphores selon différentes voies de transformation est estimé.

LIST OF SYMBOLS
DOP, DIP dissolved organic and inorganic phosphorus;
Dp           detrite phosphorus;
02           oxygen;
 Y           biochemical oxygen consumption;
Ci, CM       labile and metabolic carbonous organic matter;
B, Pr, Z      the biomass of bacteria, protozoa and zooplankton (in units of phosphorus);
Nmin         mineral nitrogen;
PoolCB, PoolNB, PoolPB pools of carbon, nitrogen and phosphorus, utilized by
             bacteria;
Pool Cpr, PoolNpn PoolPPr pools of carbon, nitrogen and phosphorus, utilized by
             protozoa;
PB, PPr, Pz specific rate of substrate uptake by bacteria, protozoa and zooplankton ;
PDIP>PDOP>PDP>PC\ specific rate of bacterial uptake of DIP, DOP, Dp and d ;
P'DIP-> P'DOP> P'opr P'B specific rate of protozoa uptake of DIP, DOP, Dp, B;
P'DP> Ppr     specific rate of zooplankton uptake of Dp, Pr;
LB, Lpr, Lz specific rate of metabolic liberation by bacteria, protozoa and zooplankton;
r
  B> rPr, rz metabolic activities of bacteria, protozoa and zooplankton ;
SB, Spy, Sz specific rate of bacteria, protozoa and zooplankton elimination;
dbKi, Vj coefficients.

INTRODUCTION
The study of the problem of pollution and its effect on quantitative compounds of
natural water puts forward the necessity of elaborating mathematical models. It is
possible to use these models in the forecasting of natural water regime changes, in the
                                                24
                                  A dynamic model of the chemical nutrient system   25
study of the dynamics of anthropogenic chemical elements, and also in solutions to the
problem of natural water quality maintenance and control. Up to now the study of all
these problems has been based on a strict scientific orientation. For example, when
evaluating the quantity of biochemical oxygen demand (BOD), only two elements -
oxygen and organic substances — are taken into consideration. BOD is known as the
most widespread quantitative water index. Quantitative estimation of BOD is carried
out using the Streeter-Phelps highly specialized analytical model (Streeter and Phelps,
1925). This model includes only one of the chemical elements. As a rule, the trans-
formations of the chemical elements are not taken into consideration in wide-range
analytical and numerical models, or their usage is limited by the principal transformers
of substances in a body of water, that is, the population of organisms at the primary
trophic levels. The ecology-oriented models are based on the scientific trend and that
is why the transformation of the chemical combinations is not reflected at all (for
instance, in the 'prey—predator' type model), or, at best, only one parameter limiting
the development of the micro-organism population is used. Some hydrochemical,
 sanitary-microbiological and ecological problems were solved with the help of such
models. Studies of the antropogen and chemical anomalies and the processes connected
 with the transformation of natural organic and inorganic components are required for
 the complex method of approach to that problem.

MODEL DESCRIPTION
This paper gives information about the interaction of community population organisms
with the chemical components of the water and the creation of a mathematical model
intended for the study of transformation processes of organic and inorganic substances
by micro-organisms. Ideas used in model development of BOD and the transformation
of carbon (Leonov and Ajzatullin, 1975c), nitrogen (Ajzatullin and Leonov, 1975;
Leonov and Ajzatullin, 1975b), and phosphorus (Ajzatullin and Leonov, 1975) com-
ponents were taken into account during elaboration of the model of such an ecological-
biochemical system*. This model is more ecologically complex because it includes
organisms of different trophic levels: bacteria, protozoa, zooplankton.
   A diagram showing the associations of the chemical components and micro-
organisms of the given system is illustrated in Fig. 1.
   The tendency of biological objects to reproduce at the maximum rate and hence
increase to a maximum the trophodynamical activity, was taken into consideration in
the model and moreover, any one of the given chemical components — the dissolved
organic matter and components of nitrogen and phosphorus — was assumed to be a
potential limitation of micro-organism development. The ratio of the elements C : N : P
accumulated in the biomass during micro-organism growth was assumed to be constant
and equal to 100 : 20:1 (atomic ratio).
   The following terms, PB,Ppr,Pz, define the specific rate of substratum consumption
by micro-organisms (PB = bacteria,PPr = protozoa,P z = zooplankton) in the given
equations (Ajzatullin, 1974):
             dm                           Ks                        Kn
     p -                  p       -                 p
        B                                               Z
            1+MB'             *       1+MpPr'               l+Z/(d13Pr + dlsDP)
where
             C/P          N/P                  P
     M=              +                +
            PoolCB       PoolNB            PoolPB
*Term by K. M. Khailov (1974).
26         Alexander V. Leonov
and
                              C/P            N/P           P
                   MP =               +              +
                           PoolCp,,        P00lNPr       PoolPPr
          PoolCB = dMCx + C/P(d2DOP + d3DP)
      PoolNB = (N/C)di4Ci + (N/P)d3A>
          PoolPB = d1DIP + d2DOP + d3DP
      PoolCPr = C/P(d9B+d8DOP + dl2Dp)
      PoolNpr =               (N/P)(d9B+dl2Dp)
      PoolPpr = d9B+d1DIP + dsDOP + d12DP
   The environment reserves of nutrients, such as carbon (Pool Cg, Pool Cpr), nitrogen
(PoolNB, PoolNPr) and phosphorus (PoolPB, PoolPPr) combinations, indicate the
general specific rate of substratum consumption by bacteria and protozoa. These com-
pounds are concentrated in micro-organisms and are defined by the proportional
ratio C :N:P. The values of coefficients, dh reflect the different trophic importance of
interchangeable substratum and hence micro-organism preferences in their consump-
tion. The total specific rate of zooplankton substrate consumption is defined as the
sum content of available food sources per unit biomass of zooplankton.
   Appropriate equations for specific rates of individual substratum consumption may
be derived from equations of general specific rates of substratum consumption. There-
fore, the following equations characterize specific consumption rates of substrate by
bacteria and protozoa, and are written as follows:
for bacteria:
                         dl0d{DIP PoolCB PoolNB
       r
          p   —
          DIP                         MI

                         d10d2DOPPoolCB         PoolNB
      "DOP                            Mi

                         d10d3Dp PoolCB PoolNB
           P         =
            Dp
                                      My

               p         dl0dlAC\PoolNB       PoolPB
                cr                    Mi

for protozoa:
                         Ks d1DIP Pool CPr PoolNpr
          P'DIP ~
                                      M2
                         Ksd8DOP PoolCp, PoolNp,.
      P              =
       DOP
                                      M2
                         Ksdl2DP    PoolCpr PoolNp,
           P'r,
           r         -
           Dp                         M2
                         Ksd9B PoolCpy PoolNPr
               P'B =
                                      Mn
                                                   A dynamic model of the chemical nutrient system               27

                          0,                       c
                                                       i
                           /
                               1


              ' ...,j               •   •   \                  '                                    1
        r->       B            Pr                          z            U
                                                                            P
                                                                                          DOP      DIP

                      1            i                       ,
                                                                                                    1

              '                                                                     ' '
                                                                                N    .
              CM




        FIGURE 1.     The model element associations.
        C\, Cy[      the labile and the metabolic carbonous organic matter [jugC/1.]
        02           oxygen [mgOj/1.];
        B, Pr, Z      the biomass of the bacteria, protozoa and zooplankton [/xgP/L]
        DIP, DOP, Dj> the dissolved inorganic, organic and detrite phosphorus [iigP/l.
        N„           the mineral nitrogen [/ugN/1.].



where
        Mi = [PoolCB PoolNB PoolPB + B PoolCB                                             PoolNB
                  + B (C/P) PoolPB PoolNB + B (N/P) Pool CB PoolPB] (I + K4                              Nmin)

and
        M2 = Pool CPr PoolNpr PoolPPr + Pr Pool CPr PoolNpr
                  + (C/P) Pr PoolNpr PoolPpr + (N/P) Pr PoolPPr Pool Cpr
   Specific consumption rates of substratum (detrite and protozoa) by zooplankton
are reflected in the following equations:
                      -Kl3^15-Dp                           Ki.d^Pr
        p"        =                             Ppr-
                          M,                                       M3
where
        M3 = d13Dï,+dl5Pr                        +Z

Thus the vital functions of micro-organisms provide the uptake of chemical organic
and inorganic substances in water, and realize the processes of these transformations.
Metabolic products of micro-organisms are exhausted into the environment at the rate
of utilization of initial products and they undergo subsequent transformation. The
following equations characterize specific rates of metabolic liberations by micro-
organisms - bacteria (LB), protozoa (Lp,-) and zooplankton (Lz) - and they are
written as follows:
        LB    ~ rB PB>                  LPr - rPr PAPn                          rZPz
where metabolic activities of bacteria (rB) and protozoa (rPr) depend on the total
specific rate of substratum consumption (Leonov and Ajzatullin, 1975a) and are
28      Alexander V. Leonov
described by the following equations:
                  d4PB        I     d4\                d6PPr    I        d6\
      r    =             +i                 rp    =           + h
                l+d5PB        \     dsJ               1+duPpr   \         dj
where rz is considered to be constant and equal to 0.1 for zooplankton.
   Some formulae were used to show the specific rate of elimination of bacteria (SB),
protozoa (Sp,,) and zooplankton (Sz):
      SB = Vl+V2rB,               SPr=V3^{VAPr)IPPn             Sz = V5 + (V6Z)IPZ
   The mathematical model which describes the change of concentration in the
ecological-biochemical system elements is given in the following form:
             dS
             —            =(PB-LB-SB)B-PBPr
              At
          d(DIP)
                    -• K6LBB -     KJDIPB   -    P'DIPPT   + Kï4{DOP) + (1 - K16)LPrPr
           dt
      d(DOP)
                   = KnDP - K2PDOPB -       P'DOPPT -      K14(DOP) + Kl6LPrPr
           dt

            —      = SBB + (Sz +LZ)Z+       SprPr - K1DV ~~ PDpB - P'DpPr - P'£pZ

            dCM _
            —      K8LBB - K9CM
             dt
             dY
             —         =Kl0LBB+KlsPPrPr+KllZ
              dt
             dC,
            — = KnKnDv - K3PC.B - 0.376 (K10LBB + KlsPp, + K„Z)
              dt
             dPr _
            ~~.      (Ppr — Lpr - Sp^)Pr — PprZ
             dt
             dZ
             —           =(PZ~LZ~SZ)Z
             dt
          dN,mm
                   = 0.376 (N/C) (K10LBB + Kl5Pp,,Pr + K„Z) -           KlsNmin
           dt
where Y is the biochemical oxygen consumption and dt, Kb Vt are coefficients.


RESULTS OF MODELLING AND DISCUSSION
Differential equations were solved on the 'Minsk-22M' computer with the help of the
Runge-Kutta-4 method, using a constant time step consisting of 0.01 day. The beha-
viour of a non-photosynthetic ecological-biochemical system in a series of computer
experiments, with initial values of chemical and biological components which are
often observed in natural water, was simulated in two sets of mathematical runs.
Values of rate constants were estimated when considering individual parts of the model
                             A dynamic model of the chemical nutrient system         29
for the study of biochemical oxygen uptake and the transformation processes of the
carbon, nitrogen and phosphorus compounds on actual experimental data (Ajzatullin
and Leonov, 1975; Leonov and Ajzatullin, 1975b,c).
   With different input concentrations of the chemical components, the change of
concentrations of all components shown in Fig. 1 was traced in the computer's experi-
ments on the limitation by one form of substrate, while all other necessary substrates
were available in sufficient quantity. Also estimated were the specific rates of consump-
tion of individual substrates, metabolic liberation and dying organisms, uptake of prey
by predators, formation and detrite destruction, and consumption of oxygen (total
and individual populations of organisms).
   The result of the first simulation is presented in Fig. 2 and shows the dynamics of
the main system elements. This variant is obtained with the following values of input
data and rate constants:
     DIP = 20;DOP = 5; C, = 4.000; CM = 0.0001 ; B = 0.05; Y = 0;£>P = 0;
     Pr = 0.06; Z = 0.008;^ = 0.012; d2 = 0.0032; d3 = 0.003; d4= 0.06;
     ds = 0.1;d6= 0.09; dn= d8 = 0.1 ;d9 = 50;d 10 = 70; dn = 0.1 ;
     d12 = 0.5;<i13 = l;d 1 4 = 10;£?1S = 0.1;K1=K2= 1;K3 = 40;
     K* = 0 ; K s = 2 ; K 6 = l ; K y = 2.8; Ks = 0.01 ; K9 = K10 = 0.1 ; Kn = 40;
     Kl2 = 0.4; Kl3 = 1.25; K14 = 0.2; Kls = 0.01 ; Kl6 = 0; Kin = 0.01 ;
     Kl8 = 0; K19 = 0.7; Vl =0.1 ; V2 = 0.15; V3 = 0.07;F 4 = 0.001 ;VS = 0.03;
     F 6 = 0.02.
    Some regularity is traced in the dynamics of biomass micro-organisms dependent
upon trophodynamic interrelations, including the relations of the 'prey—predator'
type. The change in the micro-organism biomasses exercises a significant influence on
the dynamics of the water phosphorus components [Fig. 2(b)], and also on charac-
teristics used as water quality criteria [Fig. 2(c)]. The first essential change in micro-
organism biomasses occurs during the first 12 days; the bacteria first begin to increase
the biomass and later on the protozoa, then finally the zooplankton. In the initial
period of bacterial activity the concentration of water chemical components under-
goes significant changes: the content oîtheDIP,DOP and C\ decreases 3, 2 and 1.5
times respectively. The oxygen uptake increases to 2.9mg0 2 /l., and the sum content
of the mineral nitrogen to 0.2mgN/L, at the expense of the bacterial activity. During
the relaxation of the bacterial activity (5-8 days), when there is development of the
organisms in higher trophic levels, the initial reserves of phosphorus substances, especi-
ally organic phosphorus, reconstruct almost completely and partial regeneration of the
labile organic matter takes place. The oxygen uptake develops slowly at the expense of
the protozoa and zooplankton. When the zooplankton utilizes the main part of the
protozoa — the direct bacterial predator — the latter activate and their biomass regene-
rates. This takes place between the eighth and the fifteenth day. The active bacterial
behaviour at that time causes the second decrease of chemical compound concen-
trations used by bacteria as food sources — DIP, DOP and C\ - and it contributes to
the accumulation of detrite, the increase of the oxygen uptake and the mineral nitrogen.
The further development of bacterial predators causes regeneration of the nutrient
supply in the environment between the sixteenth and twenty-fourth day. The equi-
librium of the micro-organism biomasses defined fully by the interrelation of the
'prey—predator' type takes shape in the system after 30 days.
    The dynamics of the phosphorus components show that the DIP begins operating
in the turnover quickly enough: its content may be decreased about 4 times and
regenerated quickly to the initial values. The concentration of DOP is subjected to
30     Alexander V. Leonov
                                        2       '            1            i                 i
                                                                                                   (a)
              pgp/l                 1

                      8

                          -(}               l/"\ 3
                                                                                                          -




                     0                                               -,                   =
                                                1            1            1                 l

                 /l
                                   1

                 15 -                                                         /



                 10                                              /



                     5                                                            \   v

                               ^ 2

                                        3
                  n            -                I            i



                          i                     1            1        1                    I
                                                                                                                  DOM,
          Oxygen                                                                                   (c)
                              \ 1                                                                                 N       .
                                                                                                                      r.i i n
     Consumption
            mg/1                                                                                                  mg/1

                 12                                                   ?




                  8                                                                                       - 2
                                                     / S ^ ^          3


                 4                                                    4
                                                     V                                                    -



                 0        tr^^              1            i            i
                                                                                                              .
                                            8            16          24                   32 t ,   days

     FIGURE 2.        Dynamics of ecosystem elements (mathematical modelling), (a) biological
     elements, 1 -B;2-Pr,3      -Z; (b) phosphorus elements, 1 - DIP; 2 - DOP; 3 - Dp;
     (c) water quality parameters, 1 - dissolved organic matter (DOM); 2 - sum biochemical
     oxygen consumption; 3 - bacterial oxygen consumption; 4 - mineral nitrogen.
                                  A dynamic model of the chemical nutrient system     31
oscillation also, but it is fully marked in comparison with the change of the DIP
content in the system. The detrite phosphorus changes in the range 0—1 MgP/1- and it
transforms to DOP adequately fast.
   The alteration in the concentration of the labile organic matter and the oxygen
uptake demonstrates that these two components are criteria of bacterial activity: the
time of the decrease of C\ content and the increase of oxygen uptake coincides with
the commencement of bacterial biomass growth.
   Thus, the results show that the model may be used for the dynamic study of
natural ecological-biochemical systems. It should be noted however that values of
some constants require correction, particularly those parameters which reflect the
behaviour of biological components. The model's orientation to experimental data
will make it possible to correct the values of some constants.
   For the second simulation, the experimental data used were those of Canale and
Cheng (1974), obtained in a closed water system with micro-organisms often found in
natural conditions, in order to correct the oxygen uptake activity by the protozoa.




      FIGURE 3.        The computer simulation of the DOM transformation and oxygen uptake
      in bacterial presence (curves) for the experimental data (points) of Canale and Cheng
      (1974). The input data: DIP = 100; DOP = 700; Cx = 60.000; C M = 0; B = 0.05; Y = 0;
      Pr = 0.06; Z = 0;DP = 0; dl = 0.012; d2 = 0.0032; d3 = 0.003; rf„ = 0.06; ds = 0.1; d6 = 0.06;
      d, = ds = 0.0001; d9 = 0.0001 ;dl0= 2500; dn = 0.1; rf,s = d I 3 = 0.0001;d 14 = 0.006;
      dls = 0.0001; K t = Kt= 1;K3 = 500;KA = 0;K s = 5;K6= 1; K7 =2.S;KS= 0.025 ;K, = 0.05:
      # 1 O = 0 . 1 ; A ' 1 1 = 5 0 ; . K ' 1 2 = 0 . 4 ; A : 1 3 = 1.25; A"„ = 0.2; KIS   = 0.4; Kl6   = 0;Kn   = 0.01;
      A: 18 =0;A: 19 = 0.7; I', = 0.1; K 2 = 0 . 1 5 ; K3 = 0.07; K4 = 0.07; V5 = 0.1.
      1 - log (5); 2 -DOP; 3 -DIP; 4 -DOM; 5 - bacterial oxygen consumption.
32       Alexander V. Leonov
         ig(ii) I              i
        lK(l'r)            1          2




             i7^


                               2.5           5           1.5            10 t , days
      FIGURE 4.         The computer simulation of the DOM transformation and the oxygen
      uptake in the presence of bacteria-protozoa (curves) for the experimental data (points)
      of Canale and Cheng (1974). The input data: DIP = 50;DOP = 200; Q = 30.000; CM = 0;
      5 = 0.05; K = 0;/,/- = 0.06;Z = 0;Z)p = 0;d, = 0.012;d2 =0.0032;d3 =0.003;d 4 =0.06;
      rf5 = 0.1 ;d 6 = 0.06 ;rf, =d8 = 0.01;d, = 5;d10 = 2500;d u = 0.1 ;dl2 =0.5 ;dl3 =di5 = 0.0001;
      d1A= 0.006; K1 = K2 = 1;K3=500;K4=0;K5=5;K6              = 1;K1=2.&;KS = 0.025;K9=0;
      K,0=0.1;Kn = 50;Kl2=0A;Kl3 = 1.25;Klt=0.2;Kls = OA;Ku = 0;K17 = 0.01;
      £ 1 S =0.7; K, = 0.1; K2= 0.15; V3 = K 4 =0.07; VS = 0.05; K„ = 0.1.
      1 - log (£); 2 - log (Pc); 3 -OOP; 4 -£>//>; 5 -DOM; 6 - sum biochemical oxygen
      consumption; 7 - bacterial oxygen consumption.

The destruction of the carbohydrate* and oxygen uptake by bacteria Aerobacter
aerogenes and by protozoa Tetrahymena pyriformis (strain D) were investigated in
experiments at 25°C.
   Figures 3 and 4 illustrate the computer simulation results (curves) and the experi-
mental data (points) achieved by Canale and Cheng in an environment, with and
without bacterial predators. Figures 3(a) and 4(a) show the changes on a logarithmic
scale in micro-organism biomasses, estimated in computer experiments in terms of
/xgP/1. The dynamics of the phosphorus components (in terms of/igP/1.), used by
micro-organisms as a substratum, is presented in Fig. 3(b) and 4(b), but the concen-
tration changes of the dissolved organic matter (DOM = C\ + C M ) in terms of mgC/1.
and the oxygen uptake — by bacteria (BOD B ) and the sum (BOD^) — in terms of
mg0 2 /l- are demonstrated in Figs 3(c) and 4(c).
   The value of the dw coefficient defines the intensity of utilization of the nutrient.
In particular, Ch in the period 0—0.5 days, was corrected by the initial values of DIP
and DOP. The first values of the latter were selected from adequate conditions for the
experimental oxygen uptake data caused by micro-organism activity; that is to say,
*The sucrose and the cerophyl extract with high nutrient content.
                                A dynamic model of the chemical nutrient system               33
      TABLE 1.       The transformation of carbonaceous compounds by bacteria
                                                             In the presence of
                    In the ]presence of bacteria             bacteria- -protozoa
      Time
      [days]       DOM          Cy                 C
                                                       M     DOM             Ci          C
                                                                                          M

        0.0         1.000       1.000              0.000     1.000           1.000       0.000
        0.2         0.809       0.792              0.017     0.608           0.578       0.030
        0.5         0.241       0.151              0.090     0.128           0.035       0.093
        1.0         0.156       0.013              0.143     0.115           0.006       0.109
        1.5         0.155       0.006              0.149     0.114           0.003       0.112
        2.0         0.155       0.004              0.151     0.113           0.002       0.111
        3.0         0.157       0.004              0.153     0.113           0.002       0.111
        4.0         0.160       0.004              0.156     0.116           0.002       0.114
        5.0         0.164       0.004              0.160     0.125           0.002       0.123
        7.5         0.179       0.004              0.175     0.135           0.001       0.134
       10.0         0.201       0.004              0.197     0.137           0.001       0.136
       12.5         0.212       0.004              0.208     0.137           0.001       0.136


their ability to uptake and to liberate the phosphorus components and also to increase
the biomasses. The initial content of C\ amounted to 60mgC/l. and 30mgC/l. in
experimental systems with and without protozoa respectively. We chose the following
initial phosphorus component concentrations, which reflected satisfactorily the
experimental picture of the DOM transformation and the BOD at fixed values of
<f10 = 2500* and changeable initial concentrations of phosphorus compounds — DIP =
 100;DOP = 700/xgP/l- and DIP = 50; DOP = 200 jugP/l. in environments without and
with protozoa respectively.
    It should be noted that the initial concentrations of DIP and DOP were corrected
by the liberation rate of CM in the environment, which depends on the bacterial bio-
mass and the lower level of DOM in the system after 0.6 days. It was assumed that the
all-bacterial metabolic phosphorus liberation at these initial ratios of DIP and DOP
was quickly transforming to phosphate (orK6 = 1).
    Table 1 shows the computer simulation results which allowed us to estimate (1) to
what extent the non-refractory materials were decomposed; (2) the accumulation of
metabolic organic matter at different initial values of C\. Concentrations of DOM, C\
and CM are presented in Table 1 in unsealed units, that is to say in respect of the
initial content of C\ (in terms of jugC/1.). It follows from Table 1 that the DOM trans-
formation develops quickly at the lower content of C\, but the accumulation of CM
evidences more intense DOM transformations during the period 0-0.5 days in the
bacterial-protozoa system. The model also makes possible the estimation of the C\
turnover in the DOM biochemical destruction. The complete utilization of the initial
amount of C\ is carried out over 0.25 days in the bacterial-protozoa system and 0.4
days in the bacterial system. The regenerated organic matter by the destruction of
detrite will be utilized further in the environment. It should be noted that the change
of DOM concentration [Figs. 3(c) and 4(c)] from the model results, and compatible
with the experimental data, does not reflect the dynamics of its biochemical trans-
formation, because the equilibrium between the DOM utilization and the intake
by the destruction of the detrite and the CM accumulation is established in the system
almost within 0.5 days. At the same time the oxygen uptake curves give more infor-
mation about the bacterial transformation of DOM dynamics in the systems.
    The ^second stage of BOD [Fig. 4(c)] is caused by the protozoa activity and it is
reflected by the model in detail (without consideration of the oxygen consumption on
the nitrification).
*The comparative high value of dl0 obviously testifies that fermentation takes place in the DOM
destruction.
34      Alexander V. Leonov
   Thus, the model makes it possible to reveal the specific features of the oxygen
uptake regime in different experiments and to give a qualitative explanation for BOD
kinetic curve types in the process of the DOM transformation. From computer simula-
tion it is also possible to estimate the intensity of the phosphorus component trans-
formation.
   The analysis of the DOP balance suggests that the consumption of DOP by bacteria
into the bacterial—protozoa system is 1.1 to 1.4 times higher in the period 0—10 days
than during the absence of bacterial predators. The main part of DOP (about 23-25
per cent) is utilized by the bacteria in the period 0—2 days; however, the basic pathway
of transformation is by direct decay to DIP after the second day. The latter increases
from 1.5 to 2.3 times in comparison with the bacterial consumption of DOP in the
period of 3—12.5 days, with the existence in the system of bacteria only, and from
1.34—2.6 times with the existence still of protozoa. Part of the transformation DOP
flow to the DIP has an essential significance in the total balance of DIP at the same

                                   1           1            1
       igoy                  (a)
      is (iv

                     - \                           2


                                                   3
                     ~
                                                   1



                                 I             1            1


                                   I           I            1
               lg          (b}
                                                       l
                                           /"""
                1




                0




                                          N.           2


               -:2                                              -^_^__-
                                1              1               1
               0               2.5             5             7.5     t , days
     FIGURE 5. Comparison of bacterial functions with and without the presence of protozoa
     (mathematical modelling), (a) A change of specific rates of phosphorus compound consump-
     tion: 1 and 2 - by bacteria with and without protozoa presence; 3 - by protozoa, (b) Altera-
     tion of bacterial activity with and without the presence of protozoa: 1 - using the specific
     rate of compound uptake; 2 - using the values of bacterial biomasses.
                                A dynamic model of the chemical nutrient system                 35
time. The inorganic phosphorus intake through transformation during the entire study
period (0-12.5 days) is 11.2 times better than in the initial moment, when this value
decreases 3.14 times in the system still supporting the protozoa population. The
general contribution gives part of the bacterial liberation in the balance of the DIP
which increases the direct flow of DIP from DOP more than twice during 0.2 days in
the system with bacteria only, as with the bacteria—protozoa system. At the establish-
ment in the system of such phosphorus component concentrations, when the products
of dx DIP and d2DOP become comparable, these components begin to be utilized by
the bacteria at the same rates. Such conditions are formed in the system with bacteria
only in almost 7.5 days [Fig. 3(b)] and then the DIP begins to decrease noticeably.
In the bacteria—protozoa system in approximately 10 days [Fig. 4(b)] the state of
equilibrium is almost reached and it conforms to the conditions present during 5-7.5
days in the system with bacteria only.
    The comparison of the phosphorus balance ways allows us to deduce that the total
bacterial consumption of DIP and DOP is more than an order higher than protozoa
consumption. It also should be noted that the inorganic phosphorus utilization is
about an order higher for the organic phosphorus uptake by bacteria than by protozoa.
It is noteworthy that the general specific rate of the phosphorus substrate consump-
tion to the unity of the bacteria biomass is 2.5—3 orders higher in the bacteria-
protozoa system than in the system with bacteria only (Fig. 5).

CONCLUSION
Thus, the results of the practical usage of the model provide not only an explanation
for the dynamic peculiarities of the component consumption, but also allow us to
study the individual transformations of compounds at different conditions and to check
the total accepted kinetic outlines of the process. The calculations of the quantity of
transforming matter show that significant mistakes may be received in the evaluation
of some included metabolism transforming compounds (organic matter, DIP, DOP)
directly from experimental data, if the formation of secondary matter (liberations by
organisms and detrite destruction) is not taken into account. The conclusions must be
drawn with great caution on the basis of these estimations. For example, formation of
the secondary organic matter can give the illusion of stability of the initial organic
matter to biochemical destruction in the period of minimal biomass of the bacteria
 and maximal biomass of the predators. The consumption of oxygen is the best indi-
cation of biochemical activity of the non-photosynthetic system in this case, with the
total combination of, for example, the data on the accumulation of C0 2 and mineral
nitrogen compounds.
   The mathematical model considered may be used for analysis of the following:
(1) complex observed results in dark laboratory conditions; (2) data on the dynamics
of components under the photosynthetic layer of simple hydrodynamic models; (3) as
the basis for a sub-model of more complex models of real natural productivity systems;
(4) for the investigation of self-purification processes in water.

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