Development and Application of a Mechanistic Nutrient-Based Model for Precision Fish Farming

Abstract

This manuscript describes and evaluates the FEEDNETICS model, a detailed mechanistic nutrient-based model that has been developed to be used as a data interpretation and decision-support tool by fish farmers, aquafeed producers, aquaculture consultants and researchers. The modelling framework comprises two main components: (i) fish model, that simulates at the individual level the fish growth, composition, and nutrient utilization, following basic physical principles and prior information on the organization and control of biochemical/metabolic processes; and (ii) farm model, that upscales all information to the population level. The model was calibrated and validated for five commercially relevant farmed fish species, i.e., gilthead seabream (Sparus aurata), European seabass (Dicentrarchus labrax), Atlantic salmon (Salmo salar), rainbow trout (Oncorhynchus mykiss), and Nile tilapia (Oreochromis niloticus), using data sets covering a wide range of rearing and feeding conditions. The results of the validation of the model for fish growth are consistent between species, presenting a mean absolute percentage error (MAPE) between 11.7 and 13.8%. Several uses cases are presented, illustrating how this tool can be used to complement experimental trial design and interpretation, and to evaluate nutritional and environmental effects at the farm level. FEEDNETICS provides a means of transforming data into useful information, thus contributing to more efficient fish farming.

1. Introduction

As fish farming is becoming more digitized, data analytics tools are required to transform data into useful information. Mathematical models that are able to describe and predict the dynamics of fish farming systems are candidate tools in this regard, with great potential to contribute as data interpretation and decision support tools [1,2].

Over the last decades, several mathematical models that describe the state of fish have been developed to support fish farming operations [3,4,5,6,7,8,9]. Some models describe the growth of fish based on simple empirical equations that assume that feeding is a non-limiting factor, and is therefore not included as a model input [8,10,11,12,13,14,15]. Others, based on differential equations, describe the growth and, in some cases, the composition of fish (among other indicators), given time-dependent information about environmental parameters (e.g., water temperature) and feeding operations (e.g., ration size, feed composition) [3,6,16,17,18]. For a more detailed description of different models that describe fish growth and body composition, see the reviews from Dumas et al. [7] and Chary et al. [9].

Usually, the choice of one type of model over another strongly depends on the availability of data and the purpose for which the model is intended. Within the framework of precision fish farming [1], where highly detailed information systems are required to support a knowledge-based decision-making process, the use of more detailed mathematical models as complementary tools to simpler models may open the door to new insight. For example, detailed mathematical models that account for the effects of ration size and feed composition (e.g., bioenergetic or nutrient-based models) can be used to quantify the impact of different feed formulations or feeding strategies on fish performance, allowing to pre-screen the most promising solutions before practical application [19]. Besides, this type of models can also be used to monitor and predict on a higher time resolution basis (e.g., hourly or daily basis) the state of production using data from past observations and data defining the expected environmental and feeding conditions of the farming unit. Having access to this additional information allow fish farmers to make more informed decisions based on the specific conditions of their farms, and on the current and future state of production; e.g., estimate feed requirements, manage feed stocks, detect disease outbreaks, and plan sales and harvesting dates [16,17,18,20,21,22].

Nutrient-based models—defined here as models that explicitly describe the nutritional state of fish based on nutrient intake and utilization—enable a quantitative prediction of the effects of feed composition, in terms of macronutrients, on fish performance [19,23]. This is a particularly relevant aspect, as the range of commercially available feeds for each growth stage can differ in composition. Besides, since feeds with the same energy content may present different nutrient composition, consequently inducing different fish performance [24,25,26], the use of nutrient-based models may be preferable to bioenergetic models. Within the domain of nutrient-based models, a relevant aspect to be considered is the level of detail included in the modelling approach. Simpler nutrient-based models (also known as “nutrient mass-balance models”) that only consider the protein and energy content of the feed are not able, for example, to predict the impact of imbalanced dietary amino acid profiles on fish performance [18]. To work with greater precision, it is therefore necessary to use mechanistic nutrient-based models (also known as “metabolic-flux models”) that account for the effects of the feed amino acid profile and that describe the main physiological and metabolic processes of fish (e.g., protein synthesis and degradation, amino acid oxidation and conversion, glycolysis, gluconeogenesis, glycogenesis, and glycogenolysis).

A few detailed mechanistic nutrient-based models of fish have been previously developed by other authors [3,4,5,6,19,23,27,28]. For example, Bar et al. [23] developed a highly detailed model of nutrient-pathways, growth, and body composition of fish, calibrated for Atlantic salmon (Salmo salar), that considers the effects of limiting amino acids. In turn, Hua et al. [19] adapted a non-ruminant nutrient-based model for rainbow trout (Oncorhynchus mykiss), which explicitly describes the utilization of energy-yielding nutrients and metabolites for protein and lipid deposition in the fish body, also considering the effects of limiting amino acids. These previous works set solid foundations that can be further explored to advance on the development of more detailed mechanistic nutrient-based models of fish growth and composition.

This manuscript presents the FEEDNETICS model, a detailed mechanistic nutrient-based model that has been developed to be used as a data interpretation and decision-support tool in the context of precision fish farming. This tool has a wide range of potential applications and it is aimed at fish farmers, aquafeed producers, aquaculture consultants and researchers. The main objectives of this work were:

• To develop a nutrient-based model that considers the main physiological and metabolic processes of fish;
• To calibrate and validate the model for five relevant farmed fish species, i.e., gilthead seabream (Sparus aurata), European seabass (Dicentrarchus labrax), Atlantic salmon (Salmo salar), rainbow trout (Oncorhynchus mykiss), and Nile tilapia (Oreochromis niloticus);
• To demonstrate the use of the model as a data interpretation and decision-support tool through several use cases.

2. Materials and Methods

2.1. Model Description

FEEDNETICS is a dynamic mechanistic nutrient-based model that simulates fish growth, composition and environmental impact, within the context of fish farming operations, while following basic physical principles (e.g., energy and mass conservation) and prior information on the organization and control of biochemical/metabolic processes. It deterministically simulates the time-dependent physiology and metabolism of a single fish (the “average fish”; fish model), being driven by temperature data, feed properties and quantity, which is then upscaled to a population of fish (farm model). Figure 1 shows an overview of the input data needed to run the model and the output data it provides.

In particular, the model considers the most relevant physiological and metabolic processes, including feed intake, digestion, nutrient absorption and transport, central metabolism (i.e., glycolysis, gluconeogenesis, glycogenesis, glycogenolysis, beta-oxidation of fatty acids, Krebs cycle) and nitrogen metabolism (i.e., protein synthesis, protein degradation, synthesis of non-essential amino acids, amino acid oxidation, amino acid conversion) in order to maximize its capability to effectively predict fish body weight and composition along time for general scenarios. A detailed description of this model is provided in Appendix A.

The model has been implemented as a set of difference equations in the Powersim Studio 10 Expert software (Powersim Software AS, Nyborg, Norway), which are solved numerically through forward Euler integration with a fixed timestep of 0.01 days.

2.2. Model Calibration and Validation

The model was calibrated and validated for five fish species: gilthead seabream, European seabass, Atlantic salmon, rainbow trout, and Nile tilapia. The data sets used to calibrate and validate the model, as well as the calibration and validation procedures, are described below.

2.2.1. Data Processing and Data Sets Description

All data sets used in this work include data about several in vivo growth trials, covering a wide range of rearing and feeding conditions (see

Table 1. Overview of the data used to calibrate and validate the model for each species.

Attributes	Unit	Gilthead Seabream	European Seabass	Atlantic Salmon	Rainbow Trout	Nile Tilapia
Nr. of data sources	-	19	37	61	33	44
Nr. of observational units	-	118	126	398	110	186
Nr. of diets	-	30	66	350	58	175
Body weight range	g	1–478	5–482	1–6645	2–2080	1–559
Temperature range	°C	11–28	18–26	4–20	4–19	18–30
Diet composition range
Crude protein	% as fed	37–58	37–56	29–54	26–58	23–46
Crude lipids	% as fed	9–23	10–31	10–47	6–31	3–15
Gross energy	MJ/kg	19–23	18–25	18–29	17–26	13–21
DP/DE	g/MJ	21–26	19–30	12–26	11–28	14–26

). Those were collected from the scientific literature and from data generated by Sparos Lda., and its partners, in R&D projects, some of which were specifically designed to support model calibration and validation.

For purposes of uniformization and data handling, all data sets were processed into a standard format and stored in a database structured according to ‘tidy data’ principles [29]. Three types of data are associated to each data set (Figure 2):

• Raw data: data in the exact same format/structure as it was collected from the data source;
• Processed data: processed data stored in a standard format, where each observational unit is a table (representing a tank or cage), each attribute is a column (including responses and covariates), and each observation is a row (numerical data that describe the state of the observational unit over time, on a daily resolution basis);
• Metadata: data that describe the main characteristics of each data set.

During data processing, different methods were applied to convert the raw data into processed data, including the use of simple conversion factors (e.g., to convert whole-body composition and diet composition from dry weight to wet weight, and to calculate the energy content of diets), linear and non-linear interpolation methods (e.g., to distribute feed intake and temperature data over time, on a daily resolution basis, whenever the data source provided aggregated information), and additive models (e.g., to estimate the dietary amino acid profile and apparent digestibility coefficients based on ingredient composition). In addition, in cases where it was not possible to apply these methods, default values per species were used to ensure that all the minimum required attributes were defined.

After data processing, all processed data sets were subjected to quality analysis (e.g., comparison of diet proximate composition and fish whole-body composition data against realistic bounds; analysis of zootechnical indicators, namely nutrient retention efficiencies) in order to identify potential flaws in the data and remove outliers. The different data sets were then partitioned into calibration and validation data sets (roughly 80% for calibration and 20% for validation), based on metadata attributes. This process was performed ensuring that the calibration and validation data sets included data covering similar ranges of experimental conditions.

2.2.2. Model Calibration

For each species, the model was calibrated using the data sets allocated for calibration purposes, in a two-step approach.

In a first step, a subset of model parameters was statically calibrated. This process included the use of linear regression techniques (e.g., least squares, Huber loss minimization, quantile regression) to calibrate body composition and feed intake models, and the assignment of published values to constant or quasi-constant biochemical attributes (e.g., fatty acid molecular weights, energy conversion factors, whole-body amino acid profile).

In a second step, the remaining model parameters were dynamically calibrated in the Powersim Studio 10 Expert software (Powersim Software AS, Nyborg, Norway), using an evolutionary optimization algorithm based on the CMA-ES (covariance matrix adaptation evolutionary strategy) approach [30]. This process comprised the specification of general settings related to the evolutionary algorithm (i.e., number of generations, parents and offspring), the selection of a set of decisions (i.e., model parameters to be calibrated), the adjustment of the calibration space (i.e., minimum and maximum values allowed for each parameter), and the definition of the target set of objectives (weighted loss functions; i.e., ${MAP{E}_{cal}}_{bw}$ weighing 10.00, ${CAE}_{bw}$ weighing 0.01, and ${WE}_{CL}$ weighing 0.10). The following Equations (1)–(3) describe the loss functions:

$${MAP{E}_{cal}}_{bw}\left(\mathrm{\%}\right)=\frac{100}{n}\sum _{i=1}^n\left|\frac{{P_{bw}}_i-{O_{bw}}_i}{{O_{bw}}_i}\right|,$$

(1)

$${CAE}_{bw}\left(g\right)=\sum _{i=1}^n\left|{P_{bw}}_i-{O_{bw}}_i\right|,$$

(2)

$${WE}_{CL}=\frac{1}{m}\sum _{j=1}^m{\left(C{L}_{ref}-{CL}_{predicted}\right)}^2\times 0.1,$$

(3)

where ${MAP{E}_{cal}}_{bw}$ is the body weight mean absolute percentage error, ${CAE}_{bw}$ is the body weight cumulative absolute error, $n$ is the number of predicted-observed value pairs, ${P_{bw}}_i$ is the predicted body weight value (g), ${O_{bw}}_i$ is the observed body weight value (g), ${WE}_{CL}$ is the crude lipids weighted error, ${CL}_{predicted}$ is the predicted whole-body crude lipids content (%), ${CL}_{ref}$ is a reference value for whole-body crude lipids content (%) derived from quantile regression of body composition data, and $m$ is the number of time steps.

Model calibration was run until the evolutionary optimization algorithm reached the minimum convergence rate or limit of generations specified, in order to find a plausible parameterization that minimizes the prediction error for the calibration data set.

2.2.3. Model Validation

For each species, the model was validated using the independent data sets allocated for validation purposes (i.e., processed data files not used for model calibration). The validation results were evaluated qualitatively, through visual inspection of the model behavior over time in comparison with point observations, and quantitatively, by estimating the body weight mean absolute percentage error for the validation data set (${MAP{E}_{val}}_{bw}$) (4), as follows:

$${MAP{E}_{val}}_{bw}\left(\mathrm{\%}\right)=\frac{100}{n}\sum _{i=1}^n\left|\frac{{P_{bw}}_i-{O_{bw}}_i}{{O_{bw}}_i}\right|,$$

(4)

where $n$ is the number of predicted-observed value pairs, ${P_{bw}}_i$ is the predicted body weight value (g), and ${O_{bw}}_i$ is the observed body weight value (g).

2.3. Model Application

Various applications are presented to illustrate the use of the model to complement experimental trial design and interpretation, and to support the aquaculture industry in evaluating the impact of nutritional and environmental effects at the farm level.

2.3.1. Complement Trial Design and Interpretation

The experiment conducted by dos Santos et al. [31] aimed to evaluate the performance of different Nile tilapia strains (i.e., GIFT-1, GST-24, GIFT-2, GST-14) reared under similar conditions. The authors used the Gompertz model (sigmoid function) as a tool to support the analysis of the results, namely, to characterize the growth pattern and growth rate of the different strains evaluated. The nutrient-based model presented here can be used to further complement the analysis of the results. In particular, and because the model includes feeding as an input, it provides a method to ascertain if the differences in weight gain are due to differences in feed intake. To this end, we ran the model considering as inputs data made available by dos Santos et al. [31] in the published version of the article (i.e., initial weight, temperature, FCR), as well as other additional information provided by the main author (personal communication; i.e., feeding rates and diet proximate composition). The results of the model were compared with the experimental results to make further inferences about the performance of the aforementioned Nile tilapia strains.

Additionally, we considered the experiment presented by Farinha et al. [32], which focused on the evaluation of European seabass juveniles that were fed diets with varying methionine levels. To formulate the diets and define the methionine levels to be considered, the authors relied on the findings of a previous study carried out with similar diets and with fish of about the same size. The nutrient-based model presented here was applied to this study to show how it could have contributed as a complementary tool to support the design of the experiment, namely, in defining the different methionine levels to be tested. This illustrates the usefulness of the model for cases where there are no previous scientific findings available to support the design of the experiment. To this end, we ran the model considering as inputs the detailed records of the experiment presented by Farinha et al. [32]. The results of the model were compared with experimental data to illustrate the model’s ability to predict the growth of European seabass when fed diets with varying methionine levels.

2.3.2. Evaluate Nutritional and Environmental Effects at the Farm Level

To demonstrate further uses of the model by the industry, three distinct use cases are made available in Appendix B:

• Use case 1: Evaluate the impact of different commercial feeds on trout production performance;
• Use case 2: Evaluate the impact of different temperature profiles on post-smolt production performance;
• Use case 3: Predict the long-term effects of marginal changes in diet digestibility on bream production performance.

These model applications were carried out considering typical fish farming production scenarios, generally using information at the level that is available to the typical user (e.g., diet composition and feeding rates were defined based on information available in official technical sheets provided by feed producers). In each application, the different scenarios were evaluated by comparing several key-performance indicators provided by the model (e.g., time to reach harvest weight, growth rate, feed conversion ratio—FCR, economic conversion ratio—ECR).

3. Results

3.1. Model Calibration and Validation

3.1.1. Model Performance for Calibration Data Sets

Table 2. Error metrics for the calibration data set for each species.

Loss Functions/Error Metrics	Gilthead Seabream	European Seabass	Atlantic Salmon	Rainbow Trout	Nile Tilapia
${MAP{E}_{cal}}_{bw}$ (%)	6.82	5.15	6.33	8.28	18.18
${CAE}_{bw}$ (g)	91.29	75.70	588.83	162.12	166.04
${WE}_{CL}$ (−)	0.24	0.16	0.17	0.27	0.20

presents the final estimates of the three loss functions (error metrics) used to dynamically calibrate the model for each species (i.e., ${MAP{E}_{cal}}_{bw}$, ${CAE}_{bw}$ and ${WE}_{CL}$; see Equations (1)–(3) for formulas).

Overall, the performance of the model in predicting the body weight of fish for the calibration data sets is similar for most fish species in terms of ${MAP{E}_{cal}}_{bw}$. However, it must be pointed out that the model presents worse performance for Nile tilapia (${MAP{E}_{cal}}_{bw}$ = 18.18%) compared to other species (${MAP{E}_{cal}}_{bw}$ between 5.15% and 8.28%).

The worse performance for Nile tilapia may be related to several reasons linked to the data sets of this species, two of which are worth mentioning: (i) the different publications on the experiments carried out with Nile tilapia appear to show a much greater variability in terms of rearing conditions (e.g., rearing density, water quality) than the other species, which may explain part of the deviation between model predictions and observations; and (ii) the high variability of Nile tilapia strains used in different studies can also contribute to the higher calibration error in Nile tilapia, since, if the differences between strains are manifested in different metabolic rates or nutrient utilization efficiencies, those differences may not be fully replicated by the current model parametrization as the model calibration was done at the species level and not at the strain level.

The ${CAE}_{bw}$ and ${WE}_{CL}$ directly depend on the size, structure, and content of the calibration data set; therefore, they are presented here for reference only and not used to make any considerations about the model performance.

Given the outcomes of the calibration procedure, the resulting parameter sets were accepted and used for the model validation stage.

3.1.2. Model Performance for Validation Data Sets

Figure 3 shows the model validation results for each species in terms of body weight prediction. In general, the model accuracy is consistent across species, presenting a mean absolute percentage error (${MAP{E}_{val}}_{bw}$) ranging between 11.7% and 13.8%. The relative error distribution shows a symmetrical pattern, meaning that the scale and sign of model deviations are homogeneous over the body weight range (i.e., there is no clear bias for under- or over-estimation). The higher density of points for lower body weight classes (more evident in the case of rainbow trout due to higher n), is explained by the fact that most of the data sets used to validate (and calibrate) the model for each species are related to experiments carried out with small fish.

Visual inspection of the model behavior over time indicates that the predictions are representative of the observed growth pattern of fish (see Section 3.2. Model Application for examples). However, as expected, for some validation data sets, the magnitude of the deviation between model predictions and observations tends to increase throughout the simulation, since simulation errors are cumulative and generally multiplicative, and the model is not re-initialized when there are intermediate sampling points.

It is also worth mentioning that, although ${MAP{E}_{val}}_{bw}$ can be seen as a proxy for the model generalization error, one cannot always expect an accuracy according to this error metric when extrapolating fish growth with this model. Model accuracy is not only dependent on the model itself, but also on the quality of the input data used to drive it. Thus, in order to obtain accurate predictions, we cannot fail to mention that the input data used to drive the model must be as representative as possible of the scenarios to be evaluated.

3.2. Model Application

3.2.1. Complement Trial Design and Interpretation

Figure 4 shows the time series of predicted and observed average body weight values for the dos Santos et al. [31] data set, which includes data about different Nile tilapia strains (i.e., GIFT-1, GST-24, GIFT-2, GST-14) reared under similar conditions. For all Nile tilapia strains, the model predictions for body weight are consistent with the observations, with a MAPE ranging between 11.4% and 16.9%. The predicted relative growth rate (RGR, %/day) is also in line with the observed values.

The ability of the model to predict the growth of these four strains suggests that differences in the growth performance of these strains are mainly related to feed intake, i.e., higher growth rates were achieved due to the capacity of some strains (i.e., GST-24 and GIFT-1) to ingest more feed. This is something that is not apparent when analyzing the experimental data or the results from simpler growth models, such as the Gompertz model, and that can complement the research presented by dos Santos et al. [31]. If differences in growth performance were manifested due to different metabolic efficiencies, the nutrient-based model should clearly reflect this, by predicting the growth of some strains expressively better than others, over the entire period. Information about the observed voluntary feed intake (VFI, % body weight/day; GIFT-1 = 5.92%/day, GST-24 = 6.45%/day, GIFT-2 = 5.33%/day, GST-14 = 5.64%/day) also indicate a direct positive relationship between feed consumption and growth rate. However, these values alone do not support the hypothesis that the main factor driving growth performance is feed intake capacity, since this indicator does not aggregate information on the nutrient utilization capacity. This application illustrates how this nutrient-based model can be a useful tool to support the interpretation of experimental results, contributing to new insights about the main factors that explain fish performance.

Figure 5 shows the time series of predicted and observed average body weight values for the Farinha et al. [32] data set, which includes experimental data about European seabass that were fed diets with different methionine levels (i.e., M0.65, M0.85, M1.25, M1.50). For all dietary treatments, the model predictions are consistent with the observations, with a MAPE ranging between 2.6% and 11.5%. This means that the model is able to predict the effects of different methionine levels on the growth performance of European seabass accurately.

As described by Farinha et al. [32], the analysis of experimental data through one-way ANOVA indicated significant differences between the dietary treatments M0.65 and M1.50 for final average body weight, pointing to methionine deficiency in fish fed with the M0.65 diet. The authors further carried out a linear broken-line regression analysis to find the breakpoint that defines the methionine level below which growth performance is affected (i.e., methionine requirement). Usually in this type of assessment (broken-line regression), it is recommended (not mandatory though) to have at least two observed points in each segment so as to ensure that the slope of each segment is estimated based on points directly related to observed data and that the breakpoint is then defined with better precision. However, this was something that did not happen, as M0.65 was the only diet that displayed a clear methionine deficiency, meaning that the broken-line was adjusted considering only one directly observed point below the breakpoint. The use of the nutrient-based model presented in this work could have been useful during the design of this experiment as it could have allowed a better definition of the methionine levels to be tested (prescreening of diets in silico), in order to ensure at least two points (methionine levels) below the breakpoint. This illustrates how this nutrient-based model can be a useful tool to support the design of experiments, helping to anticipate fish performance, and thus, make more informed decisions to ensure full effectiveness of the experiment to be performed. Furthermore, in nutrient requirements studies similar to the one presented by Farinha et al. [32], the model can also be used to extrapolate the performance of fish subjected to different rearing conditions (e.g., different temperatures) or diet compositions (e.g., different protein or energy content) to verify whether the nutrient requirements determined apply in those cases or not.

3.2.2. Evaluate Nutritional and Environmental Effects at the Farm Level

Additional model applications directed to industrial end users (e.g., fish farmers, aquafeed producers, and aquaculture consultants) are presented in Appendix B and include the following use cases: 1. evaluate the impact of different commercial feeds on trout production performance; 2. evaluate the impact of different temperature profiles on post-smolt production performance; and 3. predict the long-term effects of marginal changes in diet digestibility on bream production performance. These applications illustrate how the nutrient-based model presented here can be used as a decision-support tool, helping to define the best feeding and farming strategies for each specific case, through the analysis of the impact of nutritional and environmental effects at the farm level.

4. Discussion

This manuscript describes and evaluates the FEEDNETICS model, a mechanistic nutrient-based model intended to be used as a data interpretation and decision-support tool by different end users, i.e., fish farmers, aquafeed producers, aquaculture consultants, and researchers. The framework behind FEEDNETICS comprises two main components: (i) fish model, that simulates at the individual level the fish growth, composition, and nutrient utilization, and (ii) farm model, that upscales all information to the population level.

The fish model is based upon a mechanistic and deterministic modelling approach, following basic physical principles and prior information on the organization and control of biochemical/metabolic processes. It assumes a multi-compartmental structure describing the flow of nutrients and metabolites between the gut, blood, and body (single compartment representing the main body tissues, e.g., brain, liver, muscle, adipose tissue, bone). Similar mechanistic nutrient-based models have been previously developed by other authors for larval [6,28], juvenile, and adult fish stages [3,4,5,19,23,27]. However, not all of these models explicitly consider different compartments to describe the flow of nutrients and metabolites. Although not a requisite to predict the growth and body composition of fish, assuming a multi-compartmental structure enables a more realistic description of nutrient flow and utilization rates. In addition, its modular nature facilitates the continuous improvement of the model through the simplification or complexification of specific (sub-)processes. For example, in the current version of the model, the gut compartment does not mechanistically simulate the digestive process, since the apparent digestibility coefficients (ADCs) of nutrients are provided as input. However, as nutrient digestibility of feeds may vary with fish size, water temperature, and ration size [33,34,35,36,37], among other factors, it may be relevant to refine the model in order to account for such effects. As the gut is considered as a separate compartment in this model, future improvements aimed at incorporating a mechanistic description of the digestive process can be achieved without the need to completely reformulate the entire structure of the model. The same is applicable to other physiological and metabolic processes (e.g., appetite and ingestion).

The farm model was implemented based on the assumption that the entire fish population can be described based on a linear scaling of the outputs provided by the fish model. Therefore, the existing inter-individual variability is not accounted for in the current model implementation. Although it is a strong assumption, this approach aims to simplify the required input data from the user. Furthermore, it must be pointed out that accurately modelling fish size distribution within a population is challenging, as it is not only affected by genetic variability among individuals, but may also be influenced by production practices (e.g., ration size and stocking density) [38,39,40]. Nevertheless, if relevant for future applications, some outputs provided by the farm model can be modified in order to scale from individual to population level based on fish size distribution functions, where the parameters of such functions would be considered as input.

In general terms, and despite the simplifications that were assumed to ensure greater usability, the model has a complex and detailed structure. Considering such a high level of detail has the advantage of making the model usable for a wider range of purposes. However, it also means that the input data needed to run the model are considerably more detailed than in the case of simpler models (e.g., bioenergetic models) [20]. While this may not be an issue for some end users (e.g., aquafeed producers, aquaculture consultants and researchers), it may compromise usability for others who may not have access to detailed information, namely in terms of the nutrient composition of feeds (e.g., fish farmers). This usability issue is highly relevant and should be considered when implementing this tool as a user-friendly application. To overcome it, it is suggested to define default values and implement auxiliary data processing methods to deal with missing input data. Default values can be defined under different assumptions, but it must be ensured that they meet the requirements of the species and that they are representative, as far as possible, of common practices applied in the industry. Auxiliary data processing methods can be implemented on top of different data structures, for example, to convert monthly resolution data to daily resolution data, or to estimate amino acids and apparent digestibility coefficients of feeds based on information about the composition of its ingredients.

The calibration and validation of the model was carried out at the species level, for five commercially relevant farmed fish species, i.e., gilthead seabream, European seabass, Atlantic salmon, rainbow trout, and Nile tilapia. In these procedures, we made use of data sets related to in vivo experimental trials covering a wide range of rearing and feeding conditions (see Section 2.2.1. Data Processing and Data Sets Description). Most of these data sets were collected from the scientific literature and include valuable knowledge effortfully generated over the last few decades by the academic community. Calibrating the model with these data sets means that part of this prior knowledge is being integrated through mathematical functions and made easily available to support new advances in fish farming and nutrition. The extension of the model to other species can be done following the same calibration and validation procedures, once sufficient data is collected and processed to do so. Furthermore, the model can also be calibrated in order to accommodate physiological and metabolic differences of different fish lineages/strains.

Overall, the results of the validation of the model for fish growth are consistent between species, presenting a mean absolute percentage error (MAPE) between 11.7 and 13.8%. An objective characterization of the model precision is something difficult to generalize, as the acceptable error range is often subjective and strongly depends on the application for which the model is intended to be used. Therefore, it is advisable for users to carry out more specific assessments in this regard, for example, by comparing the precision of this model for a particular application against other available methods that can be applied for the same purpose. While doing so, it is important to ensure that the input data used to run the model is representative and as accurate as possible, and that the complexity of the problem being solved is adequate for all methods being compared.

Various model applications are presented in Section 3.2. Model Application and in Appendix B. Those illustrate how the FEEDNETICS model can be used to complement trial design and interpretation and to support in the evaluation of nutritional and environmental effects at the farm level. However, the applications illustrated here represent only a small picture of the potential uses that can be made out of this model. Further potential applications include, for example, using the model to optimize finishing feeding strategies that aim to enrich fish with a high content of omega-3 fatty acids, or using it as a monitoring and forecasting tool to provide information about the current and future state of production on a daily basis.

Future model developments can be made at several levels, some of which have already been mentioned above. The multi-compartmental structure of some physiological and metabolic processes included in the fish model allows simplifying or complexifying the modeling approach to improve the model’s ability to describe fish performance. In addition, optimization algorithms can be implemented on top of the model to move from what-if scenario analysis to automated criteria-based optimization, where the algorithm would find the best solutions based on a set of criteria (e.g., key-performance indicators) pre-defined by the user.

5. Conclusions

In an era of increasing digitization, where it is important to transform data into useful information, the development and deployment of mathematical models as data interpretation and decision-support tools will contribute in the movement towards the implementation of precision fish farming practices. The ultimate goal is to increase production efficiency and productivity through greater knowledge and control of the systems. FEEDNETICS provides a means of transforming data into useful information, thus contributing to further advances in the fish farming industry and more efficient production.