Modeling the weight gain of freshwater-reared rainbow trout (Oncorhynchus mykiss) during the grow-out phase

We used five nonlinear models to calculate the weight gain of rainbow trout (122.11±15.6 g) during the final grow-out phase of 98 days under three different feed types (two commercials diets, A and B, and one experimental diet, C) in triplicate groups. We fitted the von Bertalanffy growth function with allometric and isometric scaling coefficient, Gompertz, Logistic, and Brody functions to weight (g) at age data of 900 fish, distributed in nine tanks. The equations were fitted to the data based on the least squares method using the Marquardt iterative algorithm. The accuracy of the fitted models was evaluated using a model performance metrics, combining mean squared residuals (MSR), mean absolute error (MAE), and Akaike’s Information Criterion corrected for small sample sizes (AICc). All models converged in all cases tested. The evaluation criteria for the Logistic model indicated the best overall fit (0.704) under all different feed types, followed by the Gompertz model (0.148), and the von Bertalanffy-I and von Bertalanffy-A with 0.074 each. The obtained asymptotic values are in agreement with the biological attributes of the species, except for the Brody model, whose values were massively exceeding the biologic traits of rainbow trout in 0.556 of tested cases. Additionally, ∆AICc results identify the Brody model as the only model not substantially supported by the data in any case. All other models are capable of reflecting the effects of various feed types; these results are directly applicable in farm management decisions.


Introduction
Rainbow trout (Oncorhynchus mykiss) aquaculture is one of the most widespread aquaculture activities in the world. Its contribution to the worldwide aquaculture production (excluding aquatic plants) is 0.01 and 0.04 for global harvest and value, respectively. With a world production of 814.000 metric tons in 2016, rainbow trout aquaculture generated an income of 3.4 billion United States Dollars (USD) (FAO, 2018).
The demand for rainbow trout is high, because it contains all the essential amino acids (Sabetian et al., 2012) and has a lower saturated fat content, which is recommended for the prevention of human cardiovascular diseases (Scherr et al., 2014). In addition, it has a wide acceptance in the consumer market because of its soft and delicate texture, white to pink color, and mild flavor (Singh et al., 2016).
In aquaculture production, the growth performance of the reared organisms is the most important factor regarding economic benefit (Baer et al., 2011). Fish growth is the positive result between anabolic and catabolic interaction, and when a more positive balance is desired, feed supply in terms of quality and quantity can be used as a strategy (Cho and Bureau, 2001).
For rainbow trout, like most carnivorous species, high quality and quantities of protein (45-50 g crude protein for 100 g of diet) and energy (around 4,500 kcal for kg of diet) in diets are recommended for optimal growth (Rodrigues et al., 2013). However, diets with this type of requirements increase production cost (Badillo et al., 2014). Therefore, improving the efficiency of fish production is always desired and can be achieved by studying growth curves, which offer an empirical support during the management of growth traits (Dumas et al., 2007;Lugert et al., 2019).
Models are mathematical equations, meeting the specific attributes needed, to represent animal growth. Predominantly, non-linear equations are used to model population and/or individual growth of fish in their natural environment. However, research on modeling the growth of fish in aquaculture has proven great suitability and has presented relevant contributions within the last years Lugert et al., 2017;Powel et al., 2019). Therefore, the objective of this study was to evaluate the fit of five non-linear three-parametric growth models (von Bertalanffy with allometric and isometric scaling coefficients, Gompertz, Logistic, and Brody) to weight gain data of rainbow trout during the grow-out phase under aquaculture conditions and three different feed types.

Material and Methods
The data were collected on a commercial rainbow trout farm, which is located in the municipality of Nova Friburgo, a mountainous region of the state of Rio de Janeiro, Brazil (22°23'36" S, 42°29'12" W, 1.032 m altitude).
The fish, without distinction of sex, were acquired from the farms-own breeding program. Nine hundred fish with an age of 273 days post-hatch (dph), mean weight of 122.11±15.6 g, and mean length of 22.42±0.71 cm, were selected. They were randomly distributed into nine masonry tanks with a volume of 40 m 3 each. Table 1 shows the proximal composition profile of the three diets (two commercials diets, A and B, and one experimental diet, C), which were offered in three triplicate groups [(A/1, A/2, A/3) (B/1, B/2, B/3) (C/1, C/2, C/3)]. Rations were offered twice a day until apparent saturation during 98 days. Table 2 shows the weight and length at the beginning and the end of the trial for each feed type. Water quality was maintained within optimal rearing levels and monitored on a daily basis at: dissolved Modeling the weight gain of freshwater-reared rainbow trout (Oncorhynchus mykiss) during the grow-out phase Janampa-Sarmiento et al.
Models were fitted using the Marquardt algorithm (Marquardt, 1963) through the NLIN computational process of SAS (Statistical Analysis System, version 9.4). This process uses the nonlinear least squares (nls) method. The default convergence conditions used include measure of Bates and Watts (10 -5 ), inverse Hessian (1e-12), and number of iterations (100).
The accuracy of the fitted models was evaluated using a model performance metrics. The performance criteria to evaluate the goodness of fit are: The mean squared residuals (MSR = RSS * [n -p] -1 ), in which RSS is the residual sum of squares, n is the number of observations, and p is the number of parameters of the model (Rawlings et al., 1998).
The Akaike Information Criterion (AIC) corrected for small sample sizes (AICc), AIC = 2k -2ln(L), in which k is the number of estimated parameters in the model, L̂ is the maximum value of the likelihood function for the model, and ln is the natural logarithm (Akaike, 1973); , in which n is the sample size and k is the number of parameters.  Bertalanffy Gompertz Y = A * exp (-exp (-B * (t -T))) Tjørve and Tjørve, 2017 Brody, 1945 We calculated the difference in AICc (∆AICc) values to test the support of inferior models by the data. ∆AICc is calculated as: AICc (AICc i -AICc min ) (Katsanevakis and Maravelias, 2008). Models with ∆AICc >10 have no support by the data, while models with ∆AICc <2 have substantial support (Burnham and Anderson, 2002). Models with ∆AICc between 4-7 are somewhat supported by the data and might be considered.
The Mean Absolute Error (MAE) is the average absolute difference between observed and predicted outcomes and is calculated as: MAE = mean (|observed -predicted|).
The MSR, AICc, and MAE were calculated using the SAS software.
Finally, the results from MSR, AICc, and MAE were analyzed using a scoring system in which each best fit accounted for one score. The model that had the best fit in most tested cases achieved the highest score.
In addition, we interpreted the estimated regression parameters of each model in regards to the biological attributes of the species.

Results
Values of parameter b (Table 4), the allometric constant of the length-weight relationship equation, ranged between 2.879 and 3.239 in all tested groups. As b never equaled 3.0, it indicates a strong allometric relationship between length and weight.
All models met convergence in all (nine out of nine evaluations) tested cases through Marquardt's iterative method and their parameters were obtained (Table 5). All models needed a comparably low number of iterations, and convergence was generally met within 17 and 95 iterations.
Parameter A values ranged between 563.14 and 31333.6. Within each group, the lowest value was always obtained by the Logistic model, while the highest value was always estimated by the Brody model. In contrast, parameter B values ranged between 0.0086 and 4.9661 with lowest values obtained by the Gompertz model, and highest by both Bertalanffy models.
Parameter T ranged between 303.2 and 372.3 in the Gompertz model and between 311.8 and 341.5 in the Logistic model.
All models, except Brody, displayed sigmoid curves ( Figure 1). Bertalanffy-I and Brody are highlighted as they predicted negative weight prior to the data.
The lowest MSR values were produced by the Logistic model in 0.67 of tested cases, followed by the Gompertz (0.22) and the von Bertalanffy-A (0.11) models (Table 6). Von Bertalanffy-I and Brody did not perform the lowest MSR in any case. MAE was lowest in the Logistic model in seven out of nine tested groups, 0.78. The von Bertalanffy-I model produced the lowest MAE twice (0.22) ( Table 6). The lowest Modeling the weight gain of freshwater-reared rainbow trout (Oncorhynchus mykiss) during the grow-out phase Janampa-Sarmiento et al.
5 Undisputedly, the Logistic model achieved the best overall scoring with 19 out of 27 best fits (0.704) ( Table 6). The Gompertz model achieved the best overall fit in 4 of 27 cases. The von Bertalanffy-I and von Bertalanffy-A models scored only 2 out of 27, and the Brody model did not archive the best fit in any tested cases and criteria. ∆AICc values ranged between 0.004 as the lowest and 14.9 as the highest.
The Logistic model had substantial support by the data in all cases (Table 6); the Gompertz model, in six cases; and the von Bertalanffy-A and von Bertanlanffy-I, in five cases each. The ∆AICc values of the Brody model ranged between 2.2 and 14.9. Accordingly, the model was never substantially supported by the data.

Discussion
Convergence is met when the iterative process successfully estimates parameters for the function within the given maximum number of iterations set in the fitting algorithm (Rodrigues et al., 2010). In this study, all models met convergence in all tested cases using the Marquardt algorithm. This algorithm is described as more robust than others offered in statistical software (Elzhov et al., 2013;Lugert et al., 2017). This is especially important, as non-convergence situations of models for aquaculture data are described by several authors (Costa et al., 2009;Mansano et al., 2012;Allaman et al., 2013;Sousa Júnior et al., 2014).
The allometric scaling coefficient b values obtained by weight-length relationship equation is similar to values described by Dumas et al. (2007) on 20 to 500 g size rainbow trout. This is in agreement with the assumption that isometric-growing fish can grow more allometrically in terms of weight-length relationship under aquaculture conditions (Jobling, 2003). Furthermore, cultured rainbow trout are known to have a largely varying parameter b, ranging from as low as 1.3 to as high as 6.6 (Dumas et al., 2007); in contrast to values of 2.88 to 3.39 for wild rainbow trout (Cilbiz and Yalim, 2017).
Parameter A describes the infinite size of an organism (in this case, weight), and can be interpreted as the possibility of the model to reflect the biological properties of the species. O. mykiss is known to exceed 120 cm in length (Eaton et al., 1995) and weight of 25 kg (Robins and Ray, 1986). Despite these massive documented maximal sizes, the species commonly does not exceed 60 cm in length (Bristow, 1992) and a responding weight of 5 kg (Davidson et al., 2014). Accordingly, all models tested in this study, except for the Brody model, estimated A within the biological range of the species. Besides, the obtained values are in accordance with observed values from other aquaculture studies of this species (Dumas et al., 2007;Davidson et al., 2014). It is important to note that the obtained     ∆AICc values indicate support of the model by the data. * Best adjustment criterion achieved among all models. ** Each best fit accounts for one score for the regarding model.
Modeling the weight gain of freshwater-reared rainbow trout (Oncorhynchus mykiss) during the grow-out phase Janampa-Sarmiento et al.

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A values are generally higher than harvest weights required by the average Brazilian market (350 to 400 g). Accordingly, parameter A can be considered a useful attribute in growth prediction for farming management purposes. Similar conclusions about parameter A are indicated by Gomiero et al. (2009) for Brycon orbignyanus, Mansano et al. (2012) for Lithobates catesbeianus, and Lugert et al. (2017) for Scophthalmus maximus, with all of the above in aquaculture grown species.
Parameter B denotes the precocity index. This means that the larger the numeric value, the quicker the fish will reach the asymptotic or infinite weight (Malhado et al., 2009). Estimated B values in this study (between 0.008 to 4.9661) have the tendency to be greater than those values (between 0.061 to 1.76) obtained from wild rainbow trout (Blair et al., 2013;Sloat and Reeves, 2014;Cilbiz and Yalim, 2017). Similarly, Lugert et al. (2016) found similar differences in parameter B between cultured and wild Scophthalmus maximus, relating these differences to the positive effect of controlled environmental conditions in recirculation production systems.  Sloat and Reeves (2014) for wild rainbow trout. However, our estimated weights at the POI (between 203.2 g and 372.2 g) were higher compared with wild rainbow trout (91.06 to 117.28 g) (Sloat and Reeves, 2014), being evidence of genetic improvement of breeding strains, optimal feed supply, and good rearing conditions. Furthermore, in aquaculture operations, parameter T can be useful in the empiric adjustment of management strategies, as it is proven to correlate with other husbandry information. For instance, parameter T has significant meaning on cultured Carassius auratus gibelio because it positively correlates with dietary protein level (Yun et al., 2015). Likewise, Oreochromis niloticus shows significant influence of water temperature on weight gain and on the age at the inflexion point .
When analyzing growth, model selection is a proven and reliable method to find the best suitable model for a given data set. The von Bertalanffy-I as the most widely, and a priori, used model has widely been disputed (Katsanevakis and Maravaelias, 2008), and Burnham and Anderson (2002) pointed out that the chances of a priori choosing the best suitable model for a given dataset are minor. Accordingly, model selection studies, as the current one, are of great importance in understanding the underlying biological properties of a reared species, by evaluating the most suitable model. Our results indicate that the von Bertalnaffy-I model is, besides the Brody model, the least suitable one to use on aquaculture data of O. mykiss during the grow-out phase.
In model selection, the goodness of fit should generally not be based on a single criterion. Correspondingly, it has become common practice to evaluate the most suitable model based on an evaluation metrics of mostly three statistical parameters of different properties (e.g., Yun et al., 2015;Lugert et al., 2017;Powell et al., 2019). The first parameter should be based on the residuals from fitting the model. The second parameter is often based on information theory, either AIC, AICc, or BIC. A third parameter is mostly somehow based on the deviation between estimated and sampled data. For these three categories of evaluation parameters, several different statistical parameters are available.
The author needs to decide which parameter is most suitable for the study.
In our study, we used MSR, AICc, and MAE. The non-linear least squares method aims to achieve non-linear equation parameter by minimizing the Residual Sum of Squares (RSS). The smaller RSS, the smaller the MSR and the better the fit (Rawlings et al., 1998). In this study, the Logistic, von Bertalanffy-A, and Gompertz models achieved the smallest RMS values. Similar results were obtained by Mansano et al. (2012) and Costa et al. (2009) in growth studies of Lithobates catesbeianus and Orechormis niloticus under aquaculture conditions.
We used ∆AICc to identify whether our datasets were supported by more than one model. This was necessary, as the outcome from the analysis revealed very close numeric results between different models within tested groups. ∆AICc <2 indicates substantial support of a model by the data (Burnham and Anderson, 2002). Indeed, in five out of nine analyses, four out of five tested models were supported by the data, namely von Bertalanffy-A, von Bertalanffy-I, Gompertz, and Logistic. This might be due to the specific pattern of our recorded data (grow-out phase), which are distributed around the POI of the growth curve. Accordingly, several models of sigmoidal behavior can reflect this segment of the curve.
Primarily, we observed that the different non-linear models adjusted their fit individually to the various growth trajectories expressed by rainbow trout caused by different diet treatments. Araneda et al. (2013) observed similar results when fitting models on various growth data of Penaeus vannamei. This specific application has huge potential in predicting the effects of new feed formulations, harvest size, and production period in all aquaculture species. However, it is necessary to verify and validate this potential through studies with rigorous control of diet quality and quantity as recorded in carp (Yun et al., 2015).

Conclusions
The Logistic, Gompertz, and both versions of the von Bertalanffy models show capacity to fit the weight-at-age data of cultured rainbow trout during the grow-out phase. However, in the current study, the Logistic model achieved the highest accuracy in fit. Therefore, the Logistic model is the best model to represent growth of cultured rainbow trout during growth-out phase and is useful to predict long-term growth and harvest size in fish aquaculture management.