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Stochastic model of carp fingerling growth
Aquaculture, 9 1 ( 1990) 87-99 Elsevier Science Publishers B.V., Amsterdam
87
Stochastic model of carp fingerling growth Maria Anna Szumiec PolishAcademy OfSciences, Experimental Fish CultureStation. Go&z, 43-422 Chybie, Poland (Accepted 12 April 1990)
ABSTRACT Szumiec, M.A., 1990. Stochastic model of carp fingerling growth. Aquaculture, 9 1: 87-99. A model of unit body mass growth of carp fingerlings in ponds was set up and calibrated using multiple regression equations between the fish mass stated empirically every fortnight and the cumulative sums of the water temperature effective for carp growth, the final fish densities and the food values. The model shows high accuracy and flexibility.
INTRODUCTION
The goal of this work was to quantify the combined effect of water temperature, final fish density and feed value on the unit growth of carp fingerlings and the yield obtained from the unit surface area of ponds when different methods of tish production intensification were introduced. Water temperature, being the resultant value of local climatic conditions, and thus fully out of control, is a factor limiting carp production in temperature climates throughout a considerable part of the year. The quantification of its impact on fingerling growth, considered together with the effect of fish density and food values, allows farming methods to be adapted to the local fish culture possibilities and to the actual economic demands, applying a proper method of production intensification. MATERIAL
AND METHODS
OF DATA COLLECTION
AND ESTABLISHING
THE
MODEL
Experiments concerning different methods of fish production intensification were carried out in 1982-1985 at Golysz Station (49”52’N; 18”48’E; 273 m altitude) in 24 ponds each of 1500 m2 area and average depth 1 m. In the consecutive years the ponds were divided into groups differing in the origin of the stocking material and fish density, in the kind of fertilization applied and in the amount of protein in feeds. In 1982-l 983 the work was aimed
0044-8486/90/$03.50
0 1990 -
Elsevier Science Publishers
B.V.
88
M.A. SZUMIEC
at determining optimal pond fertilization; four variants of fertilization were used with two stock densities and uniform fish feeding (Szumiec, J., 1988). The goal of the experiments in 1984- 1985 was to define the effectiveness of feeding the carp fingerlings on feeds of different protein content with uniform pond fertilization (Table 1) . Ponds were stocked with fingerlings originating from carp hybrids (third line of Polish carp) in May or in the first days of June. Fish were harvested in mid October. The survival rate varied between ponds and seasons from lo20% to over 80%. The biometric measurements of fish, taken mostly every fortnight throughout the farming season, and the mean die1 water temperature, calculated from measurements taken in ponds three times a day, were the basis for the regression calculations between the fish body mass, G, and the cumulative sums of temperature &9, effective for carp growth (above 14’ C, Szumiec and Szumiec, 1985 ) . Thus, the fingerling growth rate is not considered in this work as a function of time but of C&. As fish mortality occurs mostly during the first 3-4 weeks after stocking, only the biometric measurements from July onwards were used in computing the results. To simplify the model it was assumed that fish density was constant later on. Only ponds with a final fish density higher than 1.6 x 1 O4 ha- I season- ’ were considered, as in ponds in which the survival rates are lower than a few percent, fish mortality is caused by diseases or unfavourable chemical conditions and occurs mostly in the later part of the season. Dimensionless coefficients characterizing the food value v were assumed to be equal to 10, 20, 30 and 40, respectively, for the protein contents in feeds, and to the following types of pond fertilization: P,- without, 0 - with organic, NP- with mineral, and M - with mineral-organic fertilization (Table 1). Thus, in the fertilization experiment ( 1982- 1983 ) different food values mean different natural food resources in ponds stimulated by the above-mentioned types of fertilization when fish were fed pellets containing 20% protein. TABLE 1 Schedule of the variants of the experiment on carp fingerling culture Year Stock density (x 104fish ha-‘) A: Fertilization
B: Protein content in feeds (O/o) Food conversion ratio
1982
1983
12,15 9, 12 0 - non-fertilized 0 - organic fertilized NP - mineral fertilized M - organic-mineral 30 1.6-1.9
2.0-2.3
1984
1985
9
12 M
10,20,30,40 3.9-4.6
3.2-3.1
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
89
The growth of the unit carp fingerling body mass G (g fish-’ season-’ ) was calculated as the product of the increasing cumulative sum of effective water temperature CS, (“C) throughout the farming season, the final fish density d (fish ha- * season- ’ ) in ponds and the food value zx G=f(C&)
f(d) f(v)
The model was calibrated using the multiple regression procedure (StraSkraba and Gnauck, 1983 ). Among the multiplicative, exponential and reciprocal regression equations, those characterized by the highest correlation coefficients were chosen. It was verified by the Kolmogorov-Smirnov test (Taubenheim, 1969; Kaczmarek, 1970). The prediction and confidence limits of the regression curves between the empirical and theoretical G values as well as between the relative residuals ( Gem,,- Gtheor/Gtheor ) and theoretical G were presented. For model validation the data obtained in the farming seasons 1987 and 1988 were used. In these years the fish stock density was higher (24x lo4 fish ha-’ ) in some ponds than in the years 1982-l 985, and further, some of the ponds were covered with Lemna minor which decreased the water temperature. To correct its shading effect (Szumiec, M.A., 1984), in those ponds where the plant covered more than 50% of the water surface, the sum of the effective temperature (measured only in ponds with a free water surface) was lowered by 1 “C day-‘. The accuracy of the predicted values was also estimated by the Kolmogorov-Smirnov test and on the regression curves between the theoretical and empirical fish body masses. The above equation presents the submodel of the general model of the functioning of the pond ecosystem (Jorgensen, 1983; Ginot, 1985). RESULTSAND
DISCUSSION
The correlation coefficients R of most of the regression equations where G was the multiplicative function of CS, or lnZr9~and the exponential function of d were distinctly higher than the coefficients R of the other equations (Table 2). The highest R and the high significance level, SL, equal to 0.999 were obtained for the equations: G=7.054~
10-g. (l~~~)~‘~~~~~.e~p-~~~~~~~~~,~~~~~~~
(1)
in the case of the fertilization variant (Table 2A) and G=3_723x
10-9. (lnC~~)‘2.3488.exp-O.O00012d. (lnv)O.5695
in the case of the feeding variant found for some other regression the shape (C29,)” or (lti& ) n and 2) show that they can be applied
(2)
(Table 2B). The same high SL and high R equations where the temperature term has the d term has the exponential form (Table for defining growth of carp fingerling body
90
M.A. SZUMIEC
TABLE 2 Significance levels SL (calculated by the Kolmogorov-Smirnov test ) of the coincidence between the theoretical and empirical unit body mass G of carp fingerlings and correlation coeffkient R of the particular multiple regression equations between G and the cumulative sum of water temperature Cr9, effective for carp growth, the final fish density d and food value 11 A: Fertilization SL=O.108 SL=O.371 SL=O.431 SL=O.lOO SLZO.517 SLCO.517 SL=O.108 SLZO.517 SLZO.517 SL=O.102 SLZO.517 SL=O.999 SL=O.l08 SL=O.441 SL=O.517 SL=O.108 SL=O.441 SL=O.441 SL=O.108 SL=O.999 SL=O.999 SL=O.108 SL=O.517 SL=O.517 SL=O.l08 SL=O.517 SL=O.441
R=0.9370 R=0.9561 R =0.9568 R=0.9337 R=0.9505 R~0.9514 R=0.9329 R~0.9493 R=0.9502 R=0.9358 R=0.9549 R=0.9561 R=0.9330 R~0.9501 R=0.9512 R=0.9321 R=0.9489 R=0.9499 R=0.9354 R=0.9543 R=0.9555 R=0.9326 R=0.9497 R=0.9508 R=0.9317 R=0.9485 R=0.9495
B: Protein content in feeds SL=O.355 SL=O.418 SL=O.418 SL=O.298 SL=O.999 SL=O.999 SL=O.167 SL=O.999 SL=O.487 SL=O.248 SL=O.O35 SL=O.999 SL=O.355 SL=O.999
R=0.9337 R=0.9452 R=0.9454 R=0.9300 R=0.9414 R=0.9415 R=0.9274 R=0.9404 R=0.9405 R=0.9352 R=0.9472 R=0.9450 R=0.9319 R=0.9439
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
91
TABLE 2 (continued) B: Protein content in feeds R=0.9441 R=0.9309 R=0.9339 R=0.9428 R=0.9430 R=0.9500 R=0.9502 R=0.9325 R~0.9446 R~0.9448 R=0.9315 R~0.9345 R~0.9437
SLZO.999 SL=O.355 SL~0.487 SL~0.487 SL~0.248 SLZO.999 SL=O.999 SL=O.355 SLZO.999 SL=O.999 SL=O.355 SL=O.999 SL=O.999
fOO] 601
4m1
Relative
residuals (%)
mb!
Empirical
P
1G (g fist?)
Relative I ..
residuals
(% 1
* .
.
I
-f&7--_,,,, 0
20
40
Theoretical 60
60
*. L
G _ 100
0
---7 20
Theoretical 40
I
r
60
60
Fig. 1. Regression curves between the empirical and theoretical fish masses and between the relative residuals and theoretical ones; the prediction (farther) and confidence (closer bands) limits are shown. A - fertilization, B - feeding variant of the experiment.
G 1
400
M.A. SZUMIEC
92
mass, maybe under slightly differentiated environmental conditions or using a different technology of carp farming. Including or withdrawing from the model some G values changes the shape of the v term and slightly alters the model parameters. The verification of the ‘model by the regression between the empirical G and its relative residuals plotted against the theoretical fish masses (Fig. 1) also confirmed high significance of the predicted values. The theoretical fingerling growth throughout the season showed a high agreement with the empirical one when G was calculated as the function of Cfie for constant d and v (Fig. 2). Higher deviations appeared when G was considered as a function of d for constant C$ and v (Fig. 3 ). This resulted from the assumption that d is constant during the whole farming season. The two dimensional plots of G as a function of v, calculated for the particular d and for average, extremely cold and extremely warm seasons (determined by the long-term average, minimum and maximum B?e), present the combined effect of these three state variables on fingerling growth (Fig. 4). The highest growth rate as well as the highest differentiation of carp fingerling mass appeared in the warmest season when G varied from ca 30 g fish-’ sea-
A
G(g fist?) 8o
d=58000(fish hB’ season)
74 000
0
NP
c
M
0
2
4
6
8
10
Z9eX
103(‘C)
89 000
STOCHASTIC MODEL OF CARP FINGERLING
GROWTH
93
30%
40%
0
2
4
6
8
10 ~L3ex103(oC)
Fig. 2 Simulated and measured (dots) growth of the unit body mass G of carp fingerlings throughout the farming season plotted against the sums of the effective temperature 1~9~for the stated survival rates d and food values v. A - fertilization, B - feeding variant of the experiment.
son-’ (in ponds where d= 10x lo4 fish ha-’ and v= lo), up to 135 g fish-’ season-’ (in ponds where d=Zx lo4 fish ha-’ and v=40). In the coldest season this differentiation covered the limits ca 9-38 g fish-’ season-’ in ponds with the respective d and v values. The results presented in Fig. 4 also show that the decreasing stock density from 10 x 1O4to 2 x 1O4fish ha- ’ season- I caused a much higher increase of G than that caused by the increasing food value from 10 to 40. Further, they show a distinctly higher growth rate of G in the feeding variant of the experiment than in the fertilization variant. The carp fingerling body mass calculated from equations ( 1) and (2 ) as the continuous three-dimensional function of C&, d and v (Fig. 5) allows G to be determined at any moment of the farming season when all the three state variables are known. As G can be determined experimentally, equations ( 1) and (2) can be used for fish density evaluation in ponds during the farming season, thus for a more precise calculation of feed rations. The accuracy of the d value, depending on the representativeness of the fish samples, i.e.
94
M.A. SZUMIEC
8 G(g fisK’seaso&
0
A NP
)
2 4 6 8 10X104 d(fish hri’seasori’) 8
30%
20%
10%
40%
G(g fish-’ seasori’)
I$&&&_~
h/11111
-0
2 4 6 8 d(fish h&eason-‘)
10x10~
Fig. 3. Simulated and measured (dots) final fingerling body mass G plotted against final fish density d for the considered thermal conditions and food values. A - fertilization variant, (I) Ble=990,(2).X&.=970: B- feeding variant. (1 ) H&=670, (2) Z&=600.
G(g fisti’season? 150
’ ’ J ’ ’ - rae 500 OC
I
,
.
,
,
,
1
750 OC
120-
90co-
_ /CL60-
1 10
20
30
I
/
I
,
40 v
Fig. 4. Simulated final fingerling body mass G plotted against food value v for different fish densities d and for cold (Ze= 500), average (Z’r9,= 750) and warm (zi9,= 1000) seasons. Fertilization, solid line; feeding variant, broken line.
STOCHASTIC
MODEL OF CARP FINGERLING
GROWTH
95
A
IO"
d x10 (fish hi’ seame’)
Fig. 5. Three-dimensional plots of the theoretical G against d and 219~for defined v. A - fertilization, B - feeding variant of the experiment.
biometric measurements of G, seems to be higher than those taken from some theoretical formulae (Backiel and Zamojska, 1969; Wlodek, 197 I). The model equations multiplied by fish density d allow the fish yield from 1 ha to be estimated (Fig. 6). In the considered climatic conditions (C&= 750°C), applying either fertilization or feeding technology of carp rearing, the increase of d up to ca 6 x 1O4fish ha- ’ season- ’ caused a consid-
M.A.
96
SZUMIEC
dtkg
4
6
a
10
dxlO?fish h&.easorV)
2 Y
O
2
dxlO?fish
4 c 8 IO hd’sea! soti')
Fig. 6. Three-dimensional plots of the simulated fingerling yield G’ against d and 28, for fined v. A - fertilization, B - feeding variant of the experiment.
de-
91
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
60 4
G (g fish-‘)
60
IOOl
Relative
4001 Relative residuals f% )
residuals (% )
400
Theoretical d
I
20
I
40
I
60
I
80
G
I
100
-400
Theoretical 0
I 20
I 40
I 60
6b
Fig. 7. Regression curves between the empirical and theoretical fish masses and between the relative residuals and theoretical ones. For further explanation see Fig. 1.
erable increase of fish yield. For higher d the yield increased much slower and when d exceeded ca 10 x 1O4 (in the case of fertilization) and/or ca 8.4 x 1O4 fish ha-’ season-’ (in the case of feeding), the yield decreased as functions exp -“dud for n=O.OOOOlO and 0.000012 achieve their maximum when d takes the above-mentioned values. The accuracy of the model, proved by calculations of the regression equations between the theoretical and empirical fingerling mass growth in 1987 and 1988 (Fig. 7), seems to be good, especially the slope of the curves (probability level 0.000); the intercept of the regression curves is shifted by ca - 1 g fish- ’ . The approximate significance level ( Kolmogorov-Smimov test ) is equal to 0.999. The flexibility of the model can also be seen from the good coincidence between the theoretical and empirical fingerling mass growth throughout the farming season in several ponds where different fish survival rates occurred (Fig. 8). When the survival rate was very low, fish mortality occurred not only at the beginning of the farming season but also later on. Comparing the theoretical and empirical G, the moment can be found at which
G I 100
98
M.A. SZUMIEC
60-
47407
Fig. 8. Theoretical growth of the unit fingerling body mass G in ponds in I987 and 1988 (curves) and the measured G (columns, the black ones present the final G calculated after pond harvesting). The percentages indicate the survival rates; 60% and 100% - the assumed ones.
mortality appears; when it occurs at the end of the season, i.e., when the temperature is too low for fish growth, final G is determined by the higher fish density reduced shortly before pond harvesting (Fig. 8, last graph). The assumption that fish density is constant from about the fourth week after pond stocking was fulfilled in over 70% of the observed 24 ponds and in all the ponds where the fish survival rate was higher than 30%. Despite other factors limiting carp growth in ponds, like low oxygen concentration in the water (Huisman, 1974) and the many feed-back effects existing in the pond ecosystem, the model can be used as a deterministic one for temperate climates. It needs validation for different climatic conditions and for other carp hybrids.
STOCHASTIC
MODEL OF CARP FINGERLING
GROWTH
99
CONCLUSIONS
The model equations allow individual carp fingerling body mass to be determined as well as the yield at any moment of the farming season when the fish density, sum of temperature effective for carp growth, and the food value are known and when no other limiting factors, like fish diseases or lack of dissolved oxygen in pond water, are involved. Among the three state variables considered, temperature and fish density play the decisive role in carp fingerling growth. The quantity and quality of the feed applied do not compensate for the decrease of individual fingerling mass caused by increasing density, even when density changes only from 1.6 x 1O4to 2.0 x 1O4fish ha- ’ season- ‘. The model equations show that in warm seasons the fingerling yield increases distinctly with increasing fish density up to ca 6 x 1O4fish ha- ’ season-’ and with protein content in feed from 10 to 20% or when organic fertilization is replaced by mineral-organic. The yield decreases when fish density exceeds ca 10 x 1O4 (in fertilization) or ca 8.4 x 1O4fish ha- ’ season- ’ (in the feeding variant). In a cold season all these differences are distinctly reduced.
REFERENCES Backiel, T. and Zamojska, B., 1969. Obliczanie produkcji karpi i spoiycia przez nie paszy w stawach. [ Berechnung der Karpfenproduktion und der Nahrungsmenge, welche durch die Karpfen in Teichen verzahrt wird. ] Zesz. Nauk. SGGW. Ser. Zootechnika-Rybactwo. No. 3. pp. 95-105 (in Polish, with German summary). Ginot, V., 1985. Modelisation of the pond ecosystem. Summary. Invited paper at Symposium on the Aquaculture of Carp and Related Species, 2-5 September 1985. INRA, Evry. Huisman, E.A., 1974. Optimalisering van de groei bij de karper (Cyprinus curpio L. ). [A study on optimal rearing conditions for carp ( Cyprinus curpio L. ). ] OVB, Wageningen, 95 pp. Jorgensen, S.E. (Editor), 1983. Application of Ecological Modelling in Environmental Management, Part A. Elsevier, Amsterdam, viii + 735 pp. Kaczmarek, Z., 1970. Metody statystyczne w hydrologii i meteorologii. Wydaw. Komunikacji i Lacznosci, Warsaw, 3 12 pp. StraSkraba, M. and Gnauck, A., 1983. Aquatische Ekosystemen. Gustav Fischer, Jena, 279 pp. Szumiec, J., 1988. Investigation on intensification of carp fingerling production. 1. Optimization of rearing biotechniques. Acta Hydrobiol., 29: 275-289. Szumiec, M.A., 1984. Termika stawow karpiowych. [Thermal properties of fish ponds. ] PWN, Warsaw, 136 pp. Szumiec, M.A. and Szumiec, J., 1985. Studies on intensification of carp farming. 2. Effect of temperature on carp growth. Acta Hydrobiol., 27: 147-158. Taubenheim, J., 1969. Statistische Auswertung geophysikalischer und meteorologischer Daten. Akad. Verlag Geest und Portig, K.G., Leipzig, 386 pp. Wiodek, J.M., 197 1. Biomass and production of carp fry in differently fertilized fingerling ponds. Acta Hydrobiol., 18(2): 247-264.
87
Stochastic model of carp fingerling growth Maria Anna Szumiec PolishAcademy OfSciences, Experimental Fish CultureStation. Go&z, 43-422 Chybie, Poland (Accepted 12 April 1990)
ABSTRACT Szumiec, M.A., 1990. Stochastic model of carp fingerling growth. Aquaculture, 9 1: 87-99. A model of unit body mass growth of carp fingerlings in ponds was set up and calibrated using multiple regression equations between the fish mass stated empirically every fortnight and the cumulative sums of the water temperature effective for carp growth, the final fish densities and the food values. The model shows high accuracy and flexibility.
INTRODUCTION
The goal of this work was to quantify the combined effect of water temperature, final fish density and feed value on the unit growth of carp fingerlings and the yield obtained from the unit surface area of ponds when different methods of tish production intensification were introduced. Water temperature, being the resultant value of local climatic conditions, and thus fully out of control, is a factor limiting carp production in temperature climates throughout a considerable part of the year. The quantification of its impact on fingerling growth, considered together with the effect of fish density and food values, allows farming methods to be adapted to the local fish culture possibilities and to the actual economic demands, applying a proper method of production intensification. MATERIAL
AND METHODS
OF DATA COLLECTION
AND ESTABLISHING
THE
MODEL
Experiments concerning different methods of fish production intensification were carried out in 1982-1985 at Golysz Station (49”52’N; 18”48’E; 273 m altitude) in 24 ponds each of 1500 m2 area and average depth 1 m. In the consecutive years the ponds were divided into groups differing in the origin of the stocking material and fish density, in the kind of fertilization applied and in the amount of protein in feeds. In 1982-l 983 the work was aimed
0044-8486/90/$03.50
0 1990 -
Elsevier Science Publishers
B.V.
88
M.A. SZUMIEC
at determining optimal pond fertilization; four variants of fertilization were used with two stock densities and uniform fish feeding (Szumiec, J., 1988). The goal of the experiments in 1984- 1985 was to define the effectiveness of feeding the carp fingerlings on feeds of different protein content with uniform pond fertilization (Table 1) . Ponds were stocked with fingerlings originating from carp hybrids (third line of Polish carp) in May or in the first days of June. Fish were harvested in mid October. The survival rate varied between ponds and seasons from lo20% to over 80%. The biometric measurements of fish, taken mostly every fortnight throughout the farming season, and the mean die1 water temperature, calculated from measurements taken in ponds three times a day, were the basis for the regression calculations between the fish body mass, G, and the cumulative sums of temperature &9, effective for carp growth (above 14’ C, Szumiec and Szumiec, 1985 ) . Thus, the fingerling growth rate is not considered in this work as a function of time but of C&. As fish mortality occurs mostly during the first 3-4 weeks after stocking, only the biometric measurements from July onwards were used in computing the results. To simplify the model it was assumed that fish density was constant later on. Only ponds with a final fish density higher than 1.6 x 1 O4 ha- I season- ’ were considered, as in ponds in which the survival rates are lower than a few percent, fish mortality is caused by diseases or unfavourable chemical conditions and occurs mostly in the later part of the season. Dimensionless coefficients characterizing the food value v were assumed to be equal to 10, 20, 30 and 40, respectively, for the protein contents in feeds, and to the following types of pond fertilization: P,- without, 0 - with organic, NP- with mineral, and M - with mineral-organic fertilization (Table 1). Thus, in the fertilization experiment ( 1982- 1983 ) different food values mean different natural food resources in ponds stimulated by the above-mentioned types of fertilization when fish were fed pellets containing 20% protein. TABLE 1 Schedule of the variants of the experiment on carp fingerling culture Year Stock density (x 104fish ha-‘) A: Fertilization
B: Protein content in feeds (O/o) Food conversion ratio
1982
1983
12,15 9, 12 0 - non-fertilized 0 - organic fertilized NP - mineral fertilized M - organic-mineral 30 1.6-1.9
2.0-2.3
1984
1985
9
12 M
10,20,30,40 3.9-4.6
3.2-3.1
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
89
The growth of the unit carp fingerling body mass G (g fish-’ season-’ ) was calculated as the product of the increasing cumulative sum of effective water temperature CS, (“C) throughout the farming season, the final fish density d (fish ha- * season- ’ ) in ponds and the food value zx G=f(C&)
f(d) f(v)
The model was calibrated using the multiple regression procedure (StraSkraba and Gnauck, 1983 ). Among the multiplicative, exponential and reciprocal regression equations, those characterized by the highest correlation coefficients were chosen. It was verified by the Kolmogorov-Smirnov test (Taubenheim, 1969; Kaczmarek, 1970). The prediction and confidence limits of the regression curves between the empirical and theoretical G values as well as between the relative residuals ( Gem,,- Gtheor/Gtheor ) and theoretical G were presented. For model validation the data obtained in the farming seasons 1987 and 1988 were used. In these years the fish stock density was higher (24x lo4 fish ha-’ ) in some ponds than in the years 1982-l 985, and further, some of the ponds were covered with Lemna minor which decreased the water temperature. To correct its shading effect (Szumiec, M.A., 1984), in those ponds where the plant covered more than 50% of the water surface, the sum of the effective temperature (measured only in ponds with a free water surface) was lowered by 1 “C day-‘. The accuracy of the predicted values was also estimated by the Kolmogorov-Smirnov test and on the regression curves between the theoretical and empirical fish body masses. The above equation presents the submodel of the general model of the functioning of the pond ecosystem (Jorgensen, 1983; Ginot, 1985). RESULTSAND
DISCUSSION
The correlation coefficients R of most of the regression equations where G was the multiplicative function of CS, or lnZr9~and the exponential function of d were distinctly higher than the coefficients R of the other equations (Table 2). The highest R and the high significance level, SL, equal to 0.999 were obtained for the equations: G=7.054~
10-g. (l~~~)~‘~~~~~.e~p-~~~~~~~~~,~~~~~~~
(1)
in the case of the fertilization variant (Table 2A) and G=3_723x
10-9. (lnC~~)‘2.3488.exp-O.O00012d. (lnv)O.5695
in the case of the feeding variant found for some other regression the shape (C29,)” or (lti& ) n and 2) show that they can be applied
(2)
(Table 2B). The same high SL and high R equations where the temperature term has the d term has the exponential form (Table for defining growth of carp fingerling body
90
M.A. SZUMIEC
TABLE 2 Significance levels SL (calculated by the Kolmogorov-Smirnov test ) of the coincidence between the theoretical and empirical unit body mass G of carp fingerlings and correlation coeffkient R of the particular multiple regression equations between G and the cumulative sum of water temperature Cr9, effective for carp growth, the final fish density d and food value 11 A: Fertilization SL=O.108 SL=O.371 SL=O.431 SL=O.lOO SLZO.517 SLCO.517 SL=O.108 SLZO.517 SLZO.517 SL=O.102 SLZO.517 SL=O.999 SL=O.l08 SL=O.441 SL=O.517 SL=O.108 SL=O.441 SL=O.441 SL=O.108 SL=O.999 SL=O.999 SL=O.108 SL=O.517 SL=O.517 SL=O.l08 SL=O.517 SL=O.441
R=0.9370 R=0.9561 R =0.9568 R=0.9337 R=0.9505 R~0.9514 R=0.9329 R~0.9493 R=0.9502 R=0.9358 R=0.9549 R=0.9561 R=0.9330 R~0.9501 R=0.9512 R=0.9321 R=0.9489 R=0.9499 R=0.9354 R=0.9543 R=0.9555 R=0.9326 R=0.9497 R=0.9508 R=0.9317 R=0.9485 R=0.9495
B: Protein content in feeds SL=O.355 SL=O.418 SL=O.418 SL=O.298 SL=O.999 SL=O.999 SL=O.167 SL=O.999 SL=O.487 SL=O.248 SL=O.O35 SL=O.999 SL=O.355 SL=O.999
R=0.9337 R=0.9452 R=0.9454 R=0.9300 R=0.9414 R=0.9415 R=0.9274 R=0.9404 R=0.9405 R=0.9352 R=0.9472 R=0.9450 R=0.9319 R=0.9439
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
91
TABLE 2 (continued) B: Protein content in feeds R=0.9441 R=0.9309 R=0.9339 R=0.9428 R=0.9430 R=0.9500 R=0.9502 R=0.9325 R~0.9446 R~0.9448 R=0.9315 R~0.9345 R~0.9437
SLZO.999 SL=O.355 SL~0.487 SL~0.487 SL~0.248 SLZO.999 SL=O.999 SL=O.355 SLZO.999 SL=O.999 SL=O.355 SL=O.999 SL=O.999
fOO] 601
4m1
Relative
residuals (%)
mb!
Empirical
P
1G (g fist?)
Relative I ..
residuals
(% 1
* .
.
I
-f&7--_,,,, 0
20
40
Theoretical 60
60
*. L
G _ 100
0
---7 20
Theoretical 40
I
r
60
60
Fig. 1. Regression curves between the empirical and theoretical fish masses and between the relative residuals and theoretical ones; the prediction (farther) and confidence (closer bands) limits are shown. A - fertilization, B - feeding variant of the experiment.
G 1
400
M.A. SZUMIEC
92
mass, maybe under slightly differentiated environmental conditions or using a different technology of carp farming. Including or withdrawing from the model some G values changes the shape of the v term and slightly alters the model parameters. The verification of the ‘model by the regression between the empirical G and its relative residuals plotted against the theoretical fish masses (Fig. 1) also confirmed high significance of the predicted values. The theoretical fingerling growth throughout the season showed a high agreement with the empirical one when G was calculated as the function of Cfie for constant d and v (Fig. 2). Higher deviations appeared when G was considered as a function of d for constant C$ and v (Fig. 3 ). This resulted from the assumption that d is constant during the whole farming season. The two dimensional plots of G as a function of v, calculated for the particular d and for average, extremely cold and extremely warm seasons (determined by the long-term average, minimum and maximum B?e), present the combined effect of these three state variables on fingerling growth (Fig. 4). The highest growth rate as well as the highest differentiation of carp fingerling mass appeared in the warmest season when G varied from ca 30 g fish-’ sea-
A
G(g fist?) 8o
d=58000(fish hB’ season)
74 000
0
NP
c
M
0
2
4
6
8
10
Z9eX
103(‘C)
89 000
STOCHASTIC MODEL OF CARP FINGERLING
GROWTH
93
30%
40%
0
2
4
6
8
10 ~L3ex103(oC)
Fig. 2 Simulated and measured (dots) growth of the unit body mass G of carp fingerlings throughout the farming season plotted against the sums of the effective temperature 1~9~for the stated survival rates d and food values v. A - fertilization, B - feeding variant of the experiment.
son-’ (in ponds where d= 10x lo4 fish ha-’ and v= lo), up to 135 g fish-’ season-’ (in ponds where d=Zx lo4 fish ha-’ and v=40). In the coldest season this differentiation covered the limits ca 9-38 g fish-’ season-’ in ponds with the respective d and v values. The results presented in Fig. 4 also show that the decreasing stock density from 10 x 1O4to 2 x 1O4fish ha- ’ season- I caused a much higher increase of G than that caused by the increasing food value from 10 to 40. Further, they show a distinctly higher growth rate of G in the feeding variant of the experiment than in the fertilization variant. The carp fingerling body mass calculated from equations ( 1) and (2 ) as the continuous three-dimensional function of C&, d and v (Fig. 5) allows G to be determined at any moment of the farming season when all the three state variables are known. As G can be determined experimentally, equations ( 1) and (2) can be used for fish density evaluation in ponds during the farming season, thus for a more precise calculation of feed rations. The accuracy of the d value, depending on the representativeness of the fish samples, i.e.
94
M.A. SZUMIEC
8 G(g fisK’seaso&
0
A NP
)
2 4 6 8 10X104 d(fish hri’seasori’) 8
30%
20%
10%
40%
G(g fish-’ seasori’)
I$&&&_~
h/11111
-0
2 4 6 8 d(fish h&eason-‘)
10x10~
Fig. 3. Simulated and measured (dots) final fingerling body mass G plotted against final fish density d for the considered thermal conditions and food values. A - fertilization variant, (I) Ble=990,(2).X&.=970: B- feeding variant. (1 ) H&=670, (2) Z&=600.
G(g fisti’season? 150
’ ’ J ’ ’ - rae 500 OC
I
,
.
,
,
,
1
750 OC
120-
90co-
_ /CL60-
1 10
20
30
I
/
I
,
40 v
Fig. 4. Simulated final fingerling body mass G plotted against food value v for different fish densities d and for cold (Ze= 500), average (Z’r9,= 750) and warm (zi9,= 1000) seasons. Fertilization, solid line; feeding variant, broken line.
STOCHASTIC
MODEL OF CARP FINGERLING
GROWTH
95
A
IO"
d x10 (fish hi’ seame’)
Fig. 5. Three-dimensional plots of the theoretical G against d and 219~for defined v. A - fertilization, B - feeding variant of the experiment.
biometric measurements of G, seems to be higher than those taken from some theoretical formulae (Backiel and Zamojska, 1969; Wlodek, 197 I). The model equations multiplied by fish density d allow the fish yield from 1 ha to be estimated (Fig. 6). In the considered climatic conditions (C&= 750°C), applying either fertilization or feeding technology of carp rearing, the increase of d up to ca 6 x 1O4fish ha- ’ season- ’ caused a consid-
M.A.
96
SZUMIEC
dtkg
4
6
a
10
dxlO?fish h&.easorV)
2 Y
O
2
dxlO?fish
4 c 8 IO hd’sea! soti')
Fig. 6. Three-dimensional plots of the simulated fingerling yield G’ against d and 28, for fined v. A - fertilization, B - feeding variant of the experiment.
de-
91
STOCHASTIC MODEL OF CARP FINGERLING GROWTH
60 4
G (g fish-‘)
60
IOOl
Relative
4001 Relative residuals f% )
residuals (% )
400
Theoretical d
I
20
I
40
I
60
I
80
G
I
100
-400
Theoretical 0
I 20
I 40
I 60
6b
Fig. 7. Regression curves between the empirical and theoretical fish masses and between the relative residuals and theoretical ones. For further explanation see Fig. 1.
erable increase of fish yield. For higher d the yield increased much slower and when d exceeded ca 10 x 1O4 (in the case of fertilization) and/or ca 8.4 x 1O4 fish ha-’ season-’ (in the case of feeding), the yield decreased as functions exp -“dud for n=O.OOOOlO and 0.000012 achieve their maximum when d takes the above-mentioned values. The accuracy of the model, proved by calculations of the regression equations between the theoretical and empirical fingerling mass growth in 1987 and 1988 (Fig. 7), seems to be good, especially the slope of the curves (probability level 0.000); the intercept of the regression curves is shifted by ca - 1 g fish- ’ . The approximate significance level ( Kolmogorov-Smimov test ) is equal to 0.999. The flexibility of the model can also be seen from the good coincidence between the theoretical and empirical fingerling mass growth throughout the farming season in several ponds where different fish survival rates occurred (Fig. 8). When the survival rate was very low, fish mortality occurred not only at the beginning of the farming season but also later on. Comparing the theoretical and empirical G, the moment can be found at which
G I 100
98
M.A. SZUMIEC
60-
47407
Fig. 8. Theoretical growth of the unit fingerling body mass G in ponds in I987 and 1988 (curves) and the measured G (columns, the black ones present the final G calculated after pond harvesting). The percentages indicate the survival rates; 60% and 100% - the assumed ones.
mortality appears; when it occurs at the end of the season, i.e., when the temperature is too low for fish growth, final G is determined by the higher fish density reduced shortly before pond harvesting (Fig. 8, last graph). The assumption that fish density is constant from about the fourth week after pond stocking was fulfilled in over 70% of the observed 24 ponds and in all the ponds where the fish survival rate was higher than 30%. Despite other factors limiting carp growth in ponds, like low oxygen concentration in the water (Huisman, 1974) and the many feed-back effects existing in the pond ecosystem, the model can be used as a deterministic one for temperate climates. It needs validation for different climatic conditions and for other carp hybrids.
STOCHASTIC
MODEL OF CARP FINGERLING
GROWTH
99
CONCLUSIONS
The model equations allow individual carp fingerling body mass to be determined as well as the yield at any moment of the farming season when the fish density, sum of temperature effective for carp growth, and the food value are known and when no other limiting factors, like fish diseases or lack of dissolved oxygen in pond water, are involved. Among the three state variables considered, temperature and fish density play the decisive role in carp fingerling growth. The quantity and quality of the feed applied do not compensate for the decrease of individual fingerling mass caused by increasing density, even when density changes only from 1.6 x 1O4to 2.0 x 1O4fish ha- ’ season- ‘. The model equations show that in warm seasons the fingerling yield increases distinctly with increasing fish density up to ca 6 x 1O4fish ha- ’ season-’ and with protein content in feed from 10 to 20% or when organic fertilization is replaced by mineral-organic. The yield decreases when fish density exceeds ca 10 x 1O4 (in fertilization) or ca 8.4 x 1O4fish ha- ’ season- ’ (in the feeding variant). In a cold season all these differences are distinctly reduced.
REFERENCES Backiel, T. and Zamojska, B., 1969. Obliczanie produkcji karpi i spoiycia przez nie paszy w stawach. [ Berechnung der Karpfenproduktion und der Nahrungsmenge, welche durch die Karpfen in Teichen verzahrt wird. ] Zesz. Nauk. SGGW. Ser. Zootechnika-Rybactwo. No. 3. pp. 95-105 (in Polish, with German summary). Ginot, V., 1985. Modelisation of the pond ecosystem. Summary. Invited paper at Symposium on the Aquaculture of Carp and Related Species, 2-5 September 1985. INRA, Evry. Huisman, E.A., 1974. Optimalisering van de groei bij de karper (Cyprinus curpio L. ). [A study on optimal rearing conditions for carp ( Cyprinus curpio L. ). ] OVB, Wageningen, 95 pp. Jorgensen, S.E. (Editor), 1983. Application of Ecological Modelling in Environmental Management, Part A. Elsevier, Amsterdam, viii + 735 pp. Kaczmarek, Z., 1970. Metody statystyczne w hydrologii i meteorologii. Wydaw. Komunikacji i Lacznosci, Warsaw, 3 12 pp. StraSkraba, M. and Gnauck, A., 1983. Aquatische Ekosystemen. Gustav Fischer, Jena, 279 pp. Szumiec, J., 1988. Investigation on intensification of carp fingerling production. 1. Optimization of rearing biotechniques. Acta Hydrobiol., 29: 275-289. Szumiec, M.A., 1984. Termika stawow karpiowych. [Thermal properties of fish ponds. ] PWN, Warsaw, 136 pp. Szumiec, M.A. and Szumiec, J., 1985. Studies on intensification of carp farming. 2. Effect of temperature on carp growth. Acta Hydrobiol., 27: 147-158. Taubenheim, J., 1969. Statistische Auswertung geophysikalischer und meteorologischer Daten. Akad. Verlag Geest und Portig, K.G., Leipzig, 386 pp. Wiodek, J.M., 197 1. Biomass and production of carp fry in differently fertilized fingerling ponds. Acta Hydrobiol., 18(2): 247-264.