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Annex 3
Fish growth model

This section summarizes the basic features of the fish growth model used in the current study, together with calibration and validation results. The model structure is discussed in detail elsewhere (Bolte, Nath and Ernst, 1995).


In general, fish growth is primarily a function of fish size (or weight), food availability, photoperiod, temperature, dissolved oxygen and unionized ammonia concentrations (Brett et al., 1969; Cuenco et al., 1985). These variables appear to affect fish growth through their influence on food intake (Brett, 1979). The growth model developed by Bolte, Nath and Ernst (1995) provides capabilities for considering the effects on fish growth of all these variables.

For the current study, it was assumed that dissolved oxygen (DO) and ammonia are present at concentrations that do not limit fish growth. Further, although reproductive development also influences growth, it was assumed that such influences are negligible (e.g., as might be the case in intensive carp ponds or in a monosex culture of Nile tilapia).

Effects of Size

It is generally accepted that the growth rate of fish increases at a declining rate with size or weight (W); further, anabolic and catabolic processes may be paced at different rates in relation to fish weight, with subsequent effects on fish growth (von Bertalanffy, 1938). This is captured in the following equation for fish growth rate which has separate exponents for anabolism and catabolism (m and n respectively) (von Bertalanffy, 1938; Ursin, 1967):


where:     H = coefficient of anabolism (d1-m); and
k = coefficient of anabolism (d1-n).

Equation A3.1 is integrated over the simulation period to obtain predicted fish weights.

Effects of Food Availability

The parameter H in Equation A3.1 can be expanded as follows to consider daily ration, feeding catabolism and digestibility of the food consumed (Ursin, 1967):


where:     a = fraction of the food assimilated that is used for feeding catabolism (0–1);
b = the efficiency of food assimilation (0–1); and
R = daily ration (g d-1), which is the sum of daily intake of endogenous or natural food (Rn) and supplemental feed (Rs).

In energetic terms, b is the proportion of gross energy or food intake that is available as metabolizable energy. The term b(1 - a) represents energy that is available for growth and fasting catabolism.

Based upon previous work (e.g., Winberg, 1960; Ursin, 1967), daily ration can be calculated as a function of fish size, food availability (f; 0–1) and environmental conditions (E):

R = h f E Wm(A3.3)

where:     h = coefficient of food consumption.

The parameter f is the ratio of the actual food intake rate (R) to the food intake rate at complete satiation (Rmax; obtained by setting f = 1 in Equation A3.3). In fertilized ponds, f = fn (the proportion of natural food resources in the diet). In ponds that receive supplemental feed, f = fn + fs, where fs is the proportion of artificial feed in the diet. For ponds where only feed is used to enhance fish growth, f = fs..

Intake rates of natural food and/or artificial feed can be obtained as the product of their respective proportions and Rmax (e.g., Rn = fn Rmax).

It is difficult to predict the parameter fn because natural food production varies substantially in response to climatic, soil and water conditions as well as management practices. Moreover, the biomass of fish in a pond also determines natural food availability. Indeed, work by Hepher (1978) suggests that that this variable may be the principal factor controlling natural food availability for a given set of culture conditions. During the initial phase of fish production in fertilized ponds, adequate natural food appears to be produced, thereby sustaining fish at satiation feeding levels (i.e., f = 1), but once the fish biomass (FB) or standing crop in the pond exceeds the “critical standing crop” (CFB), natural food availability declines until the carrying capacity of the pond is reached (Hepher, 1978). Although endogenous food availability can potentially decline to zero, this situation does not appear to occur in fertilized ponds, perhaps because there is adequate food to meet maintenance requirements of fish even at relatively high biomass levels as a result of adaptation to conditions of poor food availability (Hepher et al., 1983; Hepher 1988). A simple expression to approximate the relationship between natural food availability and fish biomass (kg m-3) is as follows (Bolte, Nath and Ernst, 1995):

fn = 1.0if FB < CFB 
CFB/FB,if FB ≥ CFB(A3.4)

This expression appears to capture the pattern of decreasing natural food availability when fish biomass exceeds CFB and approaches the carrying capacity of a pond. Use of Equation A3.4 requires estimates of the CFB, which can be approximated from observed fish growth data as the biomass at the point when individual growth rates reach a maximum.

Apart from fish growth predictions, estimation of feed requirements is an important model output. For such purposes, it is convenient to define a satiation or target feeding level parameter ft (0 ≤ft≤ 1). This control parameter enables prediction of fish growth and feed requirements under different intensities of culture. The daily ration associated with the target feeding level in ponds that receive only artificial feed (i.e., fn is assumed to be zero) is obtained as the product of ft and Rmax.

If both natural food and supplemental feed are available, the growth model assumes that fish preferentially consume the former resource before feed is used. Thus, feed is consumed only if fn (calculated from Equation A3.4) is less than ft.. Supplemental feed consumption is then calculated as the product of (ft - fn) and Rmax.

Effects of Environmental Variables

Environmental variables considered in the growth model are limited to photoperiod and temperature, and the parameter E for use in Equation A3.3 is assumed to be the product of the photoperiod (π) and temperature (τ) scalars, additional parameters that are calculated as described below.


Most cultured fish tend to feed only during daylight hours. Estimates of daily photoperiod for different sites can be obtained from sunrise and sunset hour angle calculations (Hsieh, 1986). The growth model uses the scalar π based on such calculations to adjust daily food intake. For example, a photoperiod of 12 h would result in π = 0.5.


Food consumption tends to increase with temperature (T) from a lower limit below which fish will not feed (Tmin) until the optimum temperature (Topt) for the given fish species is reached; beyond Topt, consumption decreases rapidly to zero until an upper limit (Tmax) is reached, above which fish will not feed (Brett et al., 1969). Cuenco et al. (1985) used a triangular function to describe this relationship. However, many fish species, such as tilapias (Caulton, 1978), tend to have a maximum food consumption rate within a temperature range rather than at a single optimum temperature. Therefore the dimensionless function used by Svirezhev et al. (1984), which is rather flat around a known optimal temperature, is more appropriate for describing the effects of temperature on food intake, and therefore anabolism. This function is as follows:

τ = exp{-4.6 [(Topt - T)/(Topt - Tmin)]4},if T < Topt(A3.5)
exp{-4.6 [(T - Topt)/(Tmax - Topt)]4},if T ≥ Topt

However, catabolism increases exponentially with temperature within the tolerance limits for a given species (Ursin 1967; see also Cuenco et al., 1985). In the growth model, the exponential function suggested by Ursin has been modified to include the lower temperature tolerance limit for the given species (assumed to be equivalent to Tmin) as follows:

k = kmin exp [s (T - Tmin)](A3.6)

where:     k = coefficient of fasting catabolism (g1-n/day);
kmin = coefficient of fasting catabolism at Tmin (g1-n/day); and
s = a constant to describe temperature effects on catabolism (°C-1).


The growth model has been calibrated (parameterized) for four fish species, namely Nile tilapia (Oreochromis. Niloticus), tambaqui (Colossoma macropomum), pacu (Piaractus mesopotamicus), and common carp (Cyprinus carpio). Ten parameters are estimated for each species. These include anabolism parameters (b, h, and m), catabolism parameters (a, kmin, s and n) and temperature scalars (Tmin, Tmax, and Topt).

Parameter estimation was accomplished by the use of an iterative, non-linear, adaptive search algorithm implemented in the POND software (Bolte and Nath, 1996). A listing of the parameters estimated for the four fish species is provided in Table A3.1. Data sources used for model calibration and parameter estimation results for each of these species are discussed below. Feeding rates in the data sources were typically reported in terms of % body weight (BW) per day, and were used to estimate the amount of feed available for fish intake. Feed in excess of fish satiation was assumed to be wasted.

Nile Tilapia

Data from an experiment conducted at the El Carao research station in Honduras (Teichert-Coddington et al., 1991) were used for model calibration. Experimental details reported by these authors are briefly summarized here.

The experiment involved the following four treatments (replicated three times in 0.1 ha earthen ponds):

For all treatments, chicken litter and feed were supplied at 1000 kg ha-1 wk-1 (dry matter basis) and 3% of the estimated fish biomass respectively. Sex-reversed Nile tilapia (mean weight of 28.5 g) were stocked at 1 fish m-2 and harvested after 147 days.

Experimental data for the above study were retrieved from the aquaculture database maintained by PD/A CRSP. The parameter estimation procedure involved the simultaneous prediction of water temperature (using daily weather data reported in the CRSP database) and fish growth. Only growth data from the CL1 and CL3 treatments were used for model calibration. Data from the other two treatments were used for validation purposes (see Model Validation section below). Mean CFBs for the CL1 and CL3 treatments were estimated from observed growth data to be 0.156 and 0.143 kg m-3 respectively. The simulation runs assumed that feed consumption would commence only once the CFB was exceeded.

Parameters estimated from this experiment (Table A3.1) resulted in good predictions of Nile tilapia growth (Figure A3.1; Table A3.2). Reduced growth rates for the CL3 treatment presumably resulted from food limitation because supplemental feeding commenced well after the CFB was reached. Periods of slow growth predicted by the model for both treatments were apparently the result of low water temperatures (in the range of 19° to 22°C).


Data from a 255-day experiment conducted at the Centro de Pesquisa e Treinamento em Aquicultura (CEPTA) in Itiquira, Brazil, (Merola and Pagan-Font, 1988) were used for calibrating the fish growth model for tambaqui. Fish of mean weight 11.7 g were stocked at a density of 1.7 fish m-2 in a 0.71 ha earthen pond. Feed was applied at rates varying from 3.5% to 2% BW d-1 depending on fish size. Time-series water temperature data reported by Merola and Pagan-Font (1988) were used as input to the fish growth model. Parameter estimation for this species assumed that feed was the only source of nutrition.

The best-fit parameters for tambaqui (Table A3.1) result in the growth profile shown in Figure A3.2 (see also Table A3.2). Simulated fish weights tended to deviate slightly from observed weights, perhaps as a result of the extreme sensitivity of this species to low water temperatures (< 20°C), a characteristic that may not be adequately represented in the temperature function for anabolism (Equation A3.6). It would perhaps have been possible to generate a more appropriate parameter set for tambaqui had additional replicates been available for the above experiment.


The data used for calibrating the growth model for pacu were obtained from Lima et al., (1988). This study involved two phases in 0.1 ha ponds, namely fingerling culture for 230 days (two replicates), followed by a grow-out phase that lasted 89 days (three replicates). Stocking rates were 2 and 0.6–0.8 fish m-2 respectively for the two phases. Feeding rates ranged from 1.5–3% BW d-1; model calibration assumed that the rates decreased with increasing biomass. Because the growth model does not account for the effects of stocking density in fed ponds, data from both phases were combined for model calibration. This enabled analysis of growth over a longer duration of culture. Time-series water-temperature data reported by Lima et al. (1988) were used as input to the fish growth model.

Simulations with the calibrated model (Figure A3.3; Table A3.2) indicate that the best-fit parameter set (Table A3.1) results in good correspondence with observed data. The model also accurately predicted periods of poor growth associated with low water temperatures during the initial phase of culture.

Common Carp

Reports that document fish growth profiles, water temperature, and feeding rates for common carp were not available for Latin America. Therefore, data from experiments conducted during 1969–1974 at the Golysz experimental station in Poland were used for model calibration. Mean monthly water temperatures for this station were obtained from Szumiec (1979a), feeding rates from Szumiec (1979b) and growth data from Szumiec and Szumiec (1985).

For parameter estimation, growth data for 1–2 year old fish (C1-2) were used. Model predictions with the best-fit parameter set (Table A3.1) correspond very well with observed growth data (Figure A3.4; Table A3.2).


Nile Tilapia

As indicated earlier, data from two of the treatments (CL and CL2) in the experiment conducted by Teichert-Coddington et al. (1991) were used for model validation. Mean CFBs for these two treatments were estimated from observed growth data to be 0.144 and 0.204 kg m-3 respectively.

Growth predictions were comparable to observed data (Figure A3.5; Table A3.2), although somewhat higher for the CL2 treatment. This discrepancy may have resulted from the higher CFB value estimated for the CL2 ponds compared to the range of 0.143–0.156 kg m-3 for the other ponds in this experiment. Once again, low growth phases in the simulated growth profiles were apparently the result of sub-optimal water temperatures. It is also interesting to note that fish weights up to day 60 in all the four experimental treatments (Figures A3.1 and A3.5) were not substantially different, suggesting that supplemental feed addition in well-fertilized ponds is not warranted until the CFB is reached.

Additional validations (although the growth profiles are not shown here) were conducted to evaluate model predictions at CRSP sites in Honduras (El Carao) and Thailand (AIT). The relevant experiments are described in Teichert-Coddington et al. (1990) and Diana et al. (1993) respectively. For the El Carao site, the final predicted weight for a 132-day trial in fertilized ponds was comparable to the mean of the observed weights (Table A3.2). For the AIT site, where fish were fed to complete satiation, the final simulated weight was 13.5% higher than the observed mean (Table A3.2). This result is not entirely unexpected because Diana et al. (1993) reported a correlation between fish growth and periods of low DO concentrations in the AIT ponds. As indicated earlier, such effects were not accounted for in the simulation runs with the fish growth model.

In general, model validation results for Nile tilapia suggest that the growth model will provide predictions of fish weights that are sufficiently accurate for regional-scale studies.


Published reports on tambaqui growth that include fish growth, water temperature and feeding data as well as stocking and harvest details are limited. The only report that provided complete details (Merola and Pagan-Font, 1988) was used for model calibration.

For the purpose of validation, two other reports that provided some experimental details of tambaqui culture were used. The first of these reports pertains to a 129-day experiment at Gualaca, Panama (Peralta and Teichert-Coddington, 1989). Fish were stocked at densities of 0.25 and 1 fish m-2; data from the latter treatment were used for model validation. Feeding rates reported by the authors were used as input to the growth model. Because stocking and harvest dates were not reported, we assumed a culture period from April to August, 1986. Only monthly fish weights and feeding rates for a 11 month culture period were available in the second report (Gomez et al., 1995). Fish were stocked at a density of 1 fish m-2. A culture period of January 1 to November 30 was assumed. For both experiments, water temperature data were not available and the weather model in POND was used to predict pond water temperatures.

Final fish weights predicted by the growth model for these experiments (Table A3.2) were about 12–14% lower than reported weights. These discrepancies may be the result of poor predictions of water temperatures by the POND weather model (which assumed constant wind, cloud cover and relative humidity conditions) or different culture periods in the actual experiment compared to the ones assumed. The discrepancies may also be due to less than ideal model parameters, as tambaqui in both of the reports used for validation were grown to a larger size than in the experiment used for model calibration (Merola and Pagan-Font, 1988; Table A3.2). Peralta and Teichert-Coddington (1989) reported that the growth rate of tambaqui increases after it has reached several hundred grams. However, in the absence of additional fish growth data and other grow-out culture details for tambaqui, we decided to use the parameter set (Table A3.1) for this species in the current study.


Model validation for pacu was accomplished by the use of data from experimental trials conducted at another CEPTA station in Pirrassununga, Brazil. The first set of growth, water temperature and feeding data were obtained from an experiment conducted during 1982–83 (Bernardino and Ferrari, 1989). The second set of data were obtained from a CEPTA researcher (Dr Newton Castagnolli, pers. comm.) and originate from an experiment conducted during 1983–84.

Validation results for this site (Figure A3.6; Table A3.2) suggest that the estimated growth parameters will result in reasonable growth profiles for pacu under grow-out conditions. The discrepancy between predicted and observed fish weights during the intermediate phase of culture for the 1983–84 experiment (Figure A3. 6) may have been caused by the use of inaccurate feeding rates. The original reports indicated that feed was supplied at 5% BW d-1 for the first month of culture followed by feed at the rate of 1–3% BW d-1. However, it was unclear as to how and when feeding rates were adjusted during the latter phase. Model validations assumed a decrease in feeding rate with increasing fish biomass, which may not have been the case in the actual experiment.

Common Carp

For model validation, growth data from the Golysz research station in Poland (Szumiec and Szumiec, 1985) for two-to-three-year-old fish (C2–3) were used. Temperature and feeding data were obtained from the sources listed in the Model Calibration section. Growth predictions (Figure A3.7; Table A3.2) were somewhat lower than observed values, and may be due to different temperature sensitivities of larger fish compared to smaller ones. It is also possible that water temperatures for the time period from which the growth data were obtained (Szumiec and Szumiec, 1985) were different from the average temperature data for the Golysz station (Szumiec, 1979a) used as input to the model.

Additional validations were conducted to compare model output to carp growth results reported from Israeli ponds (Rappaport and Sarig, 1979). Water temperature data were also obtained from this report. For carp stocked at 1 fish m-2 in May and harvested after 58 days, the model predicts a weight of 503.3 g compared to an observed weight of 482.0 g. Similarly, the final observed and simulated fish weights for fish stocked in July and harvested 52 days later were 533.0 and 540.6 g respectively. Finally, observed and simulated fish weights for fish stocked in September and harvested 48 days later were 335.0 and 347.9 g respectively.

The validation results obtained for Polish and Israeli ponds suggest that predictions using the parameter set for common carp (Table A3.1) are likely to be adequate for the purpose of growth forecasting relevant to planning applications.

Table A3.1 Best-fit growth parameters estimated for four fish species.

ParameterNile tilapiaTambaquiPacu   Common carp
Anabolism Parameters    
Efficiency of assimilation (b)
Anabolism exponent (m)
Food consumption coefficient (h)
Catabolism Parameters    
Feeding catabolism coefficient (a)
Catabolism exponent (n)
Min. catabolism coefficient (kmin)
Temperature parameter (s)
Temperature Scalars    
Minimum (Tmin)
Maximum (Tmax)
Optimum (Topt)

Table A3.2. Summary model calibration and validation results for the four species chosen for analysis.

SpeciesSiteData sourcePeriod simulatedTreatmentaFinal fish weight (g)Relative error(b) (%)
Nile tilapiaEl Carao, Honduras127/7/89 to 21/12/89Feed + Fertilizer (CL1)276.4294.2+6.4
 127/7/89 to 21/12/89Feed + Fertilizer (CL3)258.4251.4-2.7
TambaquiPirassununga, Brazil21/10/84 to 13/6/85Feedc298.0304.6+2.2
PacuItiquira, Brazil326/4/86 to 11/03/87Feedd699.6727.9+4.0
Common carpGolysz, Poland4–610/6/72 to 10/10/72eFeed400.0401.8+0.5
Nile tilapiaEl Carao, Honduras127/7/89 to 21/12/89Fertilizer only (CL)206.4197.8-4.2
 127/7/89 to 21/12/89Feed + Fertilizer (CL2)256.5275.3+7.3
El Carao, Honduras711/8/88 to 20/12/88Fertilizer131.3124.7-5.0
AIT, Thailand89/10/91 to 19/3/92Feed325.7369.6+13.5h
TambaquiGualaca, Panamaf9no datesFeed426.0375.7-11.8h
 Colombiaf10no datesFeed1240.01068.1-13.9h
PacuPirassununga, Brazil1125/2/82 to 25/2/83Feed624.0649.2+4.0
Pirassununga, Brazilg1122/2/82 to 22/2/83Feed567.0589.7+4.0
Common carpGolysz, Poland4–610/6/72 to 10/10/72eFeed920.0889.4-3.3
 Haifa, Israel129/5/78 to 30/6/78Feed482.0503.3+4.4
 Haifa, Israel127/7/78 to 28/8/78Feed533.0540.6+1.4
 Haifa, Israel125/9/78 to 23/10/78Feed335.0347.9+3.9

Notes:   (a) Refer to text for explanation of treatment codes, if any;
(b) Calculated as [(P - O)/O] * 100, where P and O are the final predicted and observed weights (g) respectively;
(c)c Organic fertilizers (cattle or poultry manure) were added during the initial phase of the study, but suspended thereafter because a parallel study suggested that its contribution to fish growth was minimal;
(d) Cattle manure was added for the first eight months of the study, but its effects on growth were difficult to assess and therefore not considered in the current analysis;
(e) Stocking and harvest dates are approximate estimates because growth data were read off a graph;
(f) See text for additional assumptions that were made to accomplish model validation;
(g) Data for this experiment were obtained from Dr. Newton Castagnolli (personal comm.);
(h) Possible reasons for these relatively high error values are discussed in the text.

Data Sources:  1. Teichert-Coddington et al., 1991;
2. Merola and Pagan-Font, 1988;
3. Lima et al., 1988;
4. Szumiec, 1979a;
5. Szumiec, 1979b;
6. Szumiec and Szumiec, 1985;
7. Teichert-Coddington et al., 1990;
8. Diana et al., 1993;
9. Peralta and Teichert-Coddington, 1989;
10. Gomez et al., 1995;
11. Bernardino and Ferrari, 1989;
12. Rappaport and Sarig, 1979.

Figure A3.1.

Figure A3.1. Growth model calibration results for Nile tilapia at El Carao, Honduras.

Figure A3.2.

Figure A3.2. Growth model calibration results for tambaqui at Pirassununga, Brazil.

Figure A3.3.

Figure A3.3. Growth model calibration results for pacu at Itiquira, Brazil.

Figure A3.4.

Figure A3.4. Growth model calibration results for common carp at Golysz, Poland.

Figure A3.5.

Figure A3.5. Growth model validation results for Nile tilapia in fertilized (CL) and both fertilized and fed (CL2) ponds at El Carao, Honduras.

Figure A3.6.

Figure A3.6. Growth model validation results for pacu during 1982–83 and 1983–84 at Pirassununga, Brazil.

Figure A3.7.

Figure A3.7. Growth model validation results for common carp at Golysz, Poland.

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