Summary

Regarding the context of animal production, monogastric livestock systems have to be more sustainable. For this, an increase in the efficiency of feed utilization is necessary. This requires a better understanding and prediction of the nutritional value of the feeds. Many experimental data provide the digestibility of nutrients in various conditions. However, very few studies describe the digestive mechanisms along the digestive tract and for all the nutrients contained in the diet. However, it seems very important to be able to identify the reasons why a diet is digested more efficiently than another and what the causes of differences in digestibility are in the digestive system. Consequently, a modeling approach seems a good way to integrate information available, be able to represent the digestive processes and predict the digestibility. The work performed aimed at developing a model representing the transit, hydrolysis, fermentation and absorption/excretion of the main nutrients along the digestive tract. A similar structure was adopted for pig and poultry and only selected model parameters were adapted for each species. Parameters for pigs come from existing models and were then modified for the broiler based on biological knowledge and literature data. The transposition of a digestive model for pigs to a broiler has been done step by step to preserve the genericity and make only those modifications that are really needed. Modifications were driven either for biological reasons or to improve the predictive capacity of the model. For now, a generic model has been developed for growing pigs and broilers.

Objectives: The aim of the project was to develop a generic, dynamic and mechanistic model to represent transit and digestion in pigs and poultry and to integrate factors of variation of the nutritional value of feed in complex diets fed to animals at different physiological stages.

Rationale:

Modelling is an appropriate tool to integrate knowledge, to represent biological mechanisms and to test hypotheses. It is also a very useful approach to understand how a biological system works and to predict responses of animals in various conditions (e.g. species, physiological status, diet composition). Several models have been developed to represent the digestive process in pigs but this type of model is scarce in poultry. Considering many similarities in feeds used for both species and that the main digestive functions can be represented in a same way, it seemed logical to use the same generic model concept to be able to carry out comparative studies on the digestive use of feeds and nutrients.

Teams involved:

Feed-a-Gene partner involved:

-INRA, AFZ contact person for model request: Emilie Recoules (emilie.recoules@inra.fr)

Species and production systems considered:

The conventional pig and broiler production systems were considered.

Introduction

Regarding the context of animal production (e.g., increase in meat demand, international market, climate change, rarefaction of resources, competition between food, feed and biofuel production, consumers' expectations) monogastric livestock systems have to be more sustainable while maintaining or increasing their production efficiency. For this, it is necessary to better understand and predict the digestive use of feeds and nutrients by the animals. In pig and poultry production systems, feed formulation is mainly based on models in which the nutritional values of feedstuffs are considered fixed and additives. However, interactions exist between feed components within a diet and between the feed and the animal [START_REF] Bedford | Mechanism of action and potential environmental benefits from the use of feed enzymes[END_REF][START_REF] Jiménez-Moreno | Oat hulls and sugar beet pulp in diets for broilers 1. Effects on growth performance and nutrient digestibility[END_REF][START_REF] Quinsac | Comparison of yellow seed trait and dehulling effects on the chemical composition and nutritional value of rapeseed meal[END_REF]. Considering this interaction in the estimation of the nutritional value of the diet could be a way to better predict multiple responses of the animal to a diet (e.g., growth performance, environmental impact through nutrient excretion) and thus better adapt the diet to the productive goals.

Modelling is an appropriate tool to integrate knowledge, to represent biological mechanisms and to test hypotheses. It is also a very useful approach to understand how a biological system works and to predict responses of animals in various conditions (e.g., specie, physiological status, diet composition). Several models have been developed to represent the digestive process in pigs but this type of model is scarce in poultry. In addition, the model developed and the approach used highly depends on the objective of the work. However, no existing models have fully responded to the goals of the current project.

The aim of the project was thus to develop a generic, dynamic and mechanistic model capable to predict the digestive utilization of feeds and nutrients. To ensure the predictive potential, the model should be able to predict nutrient digestibility of feedstuffs listed in common nutritional tables (e.g., INRA, CVB, or NRC), account for differences among and within livestock species in terms of digestive capacity, and have the potential to accommodate aspects of feed technology, including the use of byproducts and advances in enzymology as addressed in WP1. The approach is mechanistic, but emphasis is given to the representation of processes that can be adequately parameterized, which is a significant improvement from current empirical approaches.

Results

The digestive model described here is based on concepts used in existing models (Bastianelli et al., 1996;[START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF][START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. It represents the transit, hydrolysis, fermentation and absorption of the different nutrients ingested in a dynamic way. The model was developed in Matlab (MATLAB R2015b). The integration time step is one minute and parameters have been those used in existing models, representative for a pig of 40 kg.

Structure of the model

The model contains four compartments that represent different anatomical sections of the digestive tract:

 Stomach  Proximal small intestine (duodenum and proximal jejunum)  Distal small intestine (distal jejunum and ileum)  Large intestine

The separation of the small intestine is the same as that described by Bastianelli et al. (1996); [START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF] and [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. The first part of the small intestine is supposed to account for 9% of the mean residence time of the small intestine (Bastianelli, 1996). This part was separated from the rest of the small intestine because of its particular importance in hydrolysis and absorption (Bastianelli et al., 1996). Also [START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF] considered the first part of the small intestine of particular importance because of endogenous secretions and the high rate of hydrolysis. The second part of the small intestine is anatomically a long tube. It was represented in the model as a single compartment but with the introduction of a delay to account for digesta flow (Bastianelli et al., 1996). It is the primary site for absorption [START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF]. Regarding minerals, the separation of the small intestine into two parts is also of importance. The first part of the small intestine is a section of intense Ca and P absorption, especially due to the low pH that allows minerals to be solubilized. In the distal part, non-phytic phosphorus (NPP) and Ca can also be absorbed but because of the increase in the pH in this section, NPP and Ca are not fully solubilized and therefore cannot be absorbed [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. This representation is a simplification that seems the best compromise between biological complexity and model simplification. Indeed, it was important to consider the small intestine as two distinct parts to represent biological phenomena that change along the small intestine. Representing these phenomena through more compartments would result in a more complicated model requiring more parameters without the insurance to improve the predictive ability of the model.

The diagram of the model is represented in Figure 1 and is explained in following sections. The flow chart of the model shows that nutrients are hydrolyzed (or fermented) in the different compartments. The rates of hydrolysis and absorption are different, and the outflow of a compartment (of non-absorbed nutrients) is the major inflow for the following compartment.

Inputs of the model

The inputs of the model are the quantitative and qualitative description of nutrient feed intake, representing daily dry matter and nutrient intake.

Qualitative description of the feed

The dietary nutrients considered in the model are proteins, free amino acids, lipids, starch, soluble sugars, fiber, calcium and phosphorus. These come from the concepts of the three existing models on which this model is based. The model is able to simulate the effect of plant and microbial phytase.

undegradable fiber (Bastianelli et al., 1996;[START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF]. The potentially degradable fiber represents 70% (Bastianelli et al., 1996) or 80% [START_REF] Strathe | A dynamic model of digestion and absorption in pigs[END_REF] of the fiber fraction. The fiber fraction was assumed to be represented by the NDF content in Bastianelli et al. (1996). However, as reported in Figure 2, the NDF fraction does not represent the total fiber content of the diet. It was assumed that the potentially degradable fiber fraction is 70% of the total dietary fiber (TDF). This is calculated by equation 1 and the potentially degradable and undegradable fiber contents are then calculated by equations 2 and 3 respectively. Regarding the minerals:

In the model, absorption of two minerals, Ca and P is simulated. In the feed, calcium can be of plant, animal or mineral origin. Plant calcium is considered to be non-soluble whereas calcium from animal and mineral origin can be found in solubilized or non-solubilized forms [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. In this version of the model, it is assumed that the diet does not contain any feedstuffs of animal origin. Regarding the mineral source of calcium, a coefficient of solubilisation in the stomach of 1 was used [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. Indeed, in nutritional value tables, calcium carbonate is considered as the reference and has a biological value of 100%. Due to little available information on the other sources of calcium, only one solubilisation coefficient was used. Consequently, the non-soluble part of mineral calcium is considered to be totally solubilized in the stomach (Létourneau-Montminy 2009). For the models inputs:

-PCAamnsFEED = 0 -PCAvnsFEED = calcium from plant origin -PCAsFEED = calcium from mineral origin Similarly, phosphorus can come from plant, animal or mineral origin and can be found as phytic or non-phytic phosphorus. Once the nutrient composition and other preliminary calculations have been performed in Excel, inputs were transferred to the Matlab model.

Quantitative description of the feed intake

The quantitative description of feed intake is first the amount of the daily dry matter intake (g) and then how it is ingested along the day. In the model, the feed intake pattern within a day is set up as an input. For this, several criteria have to be defined:

-Hour of the morning meal (H_meal_morning) -Hour of the evening meal (H_meal_evening) -Meals in between are defined by a regular interval between meals (meal frequency) (H_meal_day_interval) -Duration of the meal The model assumes that all meals have the same duration. A value of 15 min was used as done by Bastianelli et al. (1996). Meals occurring between morning and evening meals are defined by a regular interval which is not always very well represented when the time interval between morning and evening meals cannot be divided by the interval defined in input (see following examples no. 3 and 4).

In a classic pig production system as well as in digestibility studies, animals are usually fed restrictively using two meals per day. To represent this, H_meal_morning and H_meal_evening are the times (in minutes) at which the morning and evening meals occur. However, to introduce some flexibility in the feeding pattern, more than two meals can be represented. For this, H_meal_day_interval has to be defined. It calculates the regular interval between morning and evening meals. Some examples are given below: This works well when the interval defined fits with the difference between evening and morning meals. However, in cases where the interval between evening and morning meals cannot be divided by the interval given in input, meals are not evenly spaced (Examples no.3 and 4). The interval between 08:00 am and 04:00 pm (8 hours cannot be divided by 6). In this way, the model defines the meals at 08:00, 02:00 pm (6 hours after the morning meal) and 04:00 pm (defined in input). In this case meals are not evenly spaced.

Example no. 4: Two meals at 08:00 am and 04:00 pm and meals every 3h in between  H_meal_morning = 480  H_meal_evening = 960  H_meal_day_interval = 180

The model defines meals at 08:00 am, 11:00 am, 02:00 pm and 04:00 pm. As in the previous example, the 8 hours between morning and evening meals cannot be divided by 3 (interval defined in input). In this way, the model applies the request whenever this is possible (until 02:00 pm) but meals are not evenly spaced. This approach might be further improved, however, the representation of the feed intake pattern as it is now in the model can be sufficient for the digestive model.

Regarding the amount of feed intake, three inputs have to be defined:

 DailyDMInt  P_meal_morning  P_meal_evening
DailyDMInt corresponds to the amount of dry matter intake during the day. Then P_meal_morning and P_meal_evening are the proportions of the DailyDMInt that are ingested during the morning and evening meals respectively.

Once these inputs are defined, the model calculates the DM intake that is not ingested during morning and evening meals and share it equally between the other meals performed during the day.

Example with feeding pattern no. 4 (meals occurring at 08:00 am, 11:00 am, 02:00 pm and 04:00 pm).

 DailyDMInt = 1600 g  P_meal_morning = 30%  P_meal_evening = 20% 50% of the DailyDMInt are not ingested in morning and evening meals and will be shared: 25% at 11:00 am and 25% at 02:00 pm.

Finally the animal eats:

 08:00 -30% DailyDMInt = 480 g (32 g / min as the meal duration is 15 min)  11:00 -25% DailyDMInt = 400 g (26.7 g/min)  02:00 -25% DailyDMInt = 400 g (26.7 g /min)  04:00 -20% DailyDMInt = 320 g (21.3 g /min)

This gives the amount of dry matter intake at each time step of the simulation.

Model: representations and calculations

Feed intake calculations

At each time step, the model calculates the amount of nutrients ingested (Equation 6).

𝑁𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑖𝑛𝑡𝑎𝑘𝑒 (𝑔) = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑖𝑒𝑡 𝑜𝑛 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑏𝑎𝑠𝑖𝑠 (%) * 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛𝑡𝑎𝑘𝑒 (𝑔) (𝟔)

Representation of the digestive phenomena

For growing pigs, all equations and parameters come from Bastianelli (1996), Bastianelli et al. (1996), [START_REF] Létourneau-Montminy | Modélisation du métabolisme phosphocalcique chez le porc et le poulet en croissance[END_REF] and [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF].

Digestion is represented by four main functions that are transit of the digesta, hydrolysis, fermentation and the absorption of nutrients. All nutrients that are not absorbed are excreted in the feces. To represent the main biological phenomena and have a global overview of the process of digestion from ingestion to excretion it was necessary to keep a simple representation and not to represent the mechanisms in too much detail. The objective was to be able to:

-represent the amount of each nutrient in each compartment and at each time step -predict the ileal and fecal digestibilities for each nutrient -predict the amount of each nutrient absorbed at each time step (inputs for the metabolic model, WP3, task 3.2)

Representation of the transit

The transit or the flow of nutrient from one compartment to another is represented by a mass action law considering that the transit at time t depends on the amount of the nutrient in the compartment at time t-1 (Equation 7). For simplification, it was assumed that all nutrients follow the same transit as dry matter (Bastianelli et al., 1996). The constant value represents the fractional rate of dry matter flow and equals:

-0.0058%/min from stomach to small intestine 1 -0.05%/min from small intestine 1 to small intestine 2 -0.005%/min from small intestine 2 to large intestine -0.0005%/min from large intestine to feces These coefficients are the inverse of the mean retention time (MRT) of dry matter in the compartment (Fractional rate = 1/MRT).

In the distal small intestine, a delay of 30% of the retention time in that compartment (60 min for growing pigs, Bastianelli et al., 1996;[START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF] was added to smooth the passage of nutrients. It is represented by a flow through three virtual compartments in the distal small intestine.

In the large intestine, the output flow of nutrient (transit from large intestine to feces) is represented by equation 8 because a higher ileal dry matter flow decreases the residence time in the large intestine. The fractional rate of the large intestine emptying is an exponential function rather than a linear function of the dry matter content in the large intestine. With a = 450 and b = 250 (Bastianelli, 1996) Representation of the hydrolysis

The hydrolysis of a nutrient by endogenous enzymes at each time t is considered to be related to the nutrient quantity considered at the time t-1 and this holds for all the digestive compartments. It is defined by Equation 9.

𝐻𝑦𝑑𝑟𝑜𝑙𝑦𝑠𝑖𝑠 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡 -1) (𝟗)

One constant value is defined for each nutrient and for each digestive compartment (Bastianelli et al., 1996).

Equation 9 represents the hydrolysis by endogenous digestive enzymes. For the minerals, the hydrolysis results from endogenous enzymes but also from plant and microbial phytase. Hydrolysis due to plant and microbial phytase is represented by a Michaelis-Menten law (Létourneau-Montminy et al., 2011, equation 10).

𝐻𝑦𝑑𝑟𝑜𝑙𝑦𝑠𝑖𝑠 𝑜𝑓 𝑚𝑖𝑛𝑒𝑟𝑎𝑙𝑠 𝑏𝑦 𝑝𝑙𝑎𝑛𝑡/𝑚𝑖𝑐𝑟𝑜𝑏𝑖𝑎𝑙 𝑝ℎ𝑦𝑡𝑎𝑠𝑒 (𝑡) = (𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡 -1) * 𝑉𝑚𝑎𝑥) (𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡 -1) + 𝐾𝑚) (𝟏𝟎)

With Vmax and Km parameters [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF].

Solubilization and insolubilization of minerals

In the stomach and in the distal part of the small intestine, solubilization and insolubilization of minerals occur (see Figure 1). This is represented by considering the effect of the pH on the solubility of minerals. The equations and parameters used come from [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF].

Endogenous secretions

Endogenous secretions are represented (see Figure 1) and are assumed to be a function of the dry matter. It is calculated by multiplying the flow of dry matter entering the compartment by a constant (Equations 11 and 12). In this way, any increase in the dry matter (either by an increase in feed intake or by a decrease in digestibility) induces an increase in the endogenous secretion (Bastianelli et al., 1996).

For the endogenous secretions in the stomach:

𝐸𝑛𝑑𝑜𝑔𝑒𝑛𝑜𝑢𝑠 𝑠𝑒𝑐𝑟𝑒𝑡𝑖𝑜𝑛𝑠 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛𝑡𝑎𝑘𝑒 (𝑡 -1) (11)

For the other compartments: With constant values from Bastianelli et al. (1996) and [START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF].

Representation of fermentation

The fermentation process is represented in a very simple way with a microbial sub-compartment in the large intestine. The composition of the microflora is one defined as given by Bastianelli et al. (1996). The growth and maintenance of the microbes requires uptake of soluble sugars, fatty acids, amino acids and non-protein nitrogen. The uptake of each one of these four nutrients is represented by equation 13.

𝑈𝑝𝑡𝑎𝑘𝑒 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑓𝑜𝑟 𝑚𝑖𝑐𝑟𝑜𝑏𝑖𝑎𝑙 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑠𝑜𝑙𝑢𝑏𝑙𝑒 𝑠𝑢𝑔𝑎𝑟𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑟𝑔𝑒 𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑛𝑒 (𝑡 -1)(𝟏𝟑)

The microbial compartment produces volatile fatty acids (equation 14).

𝑉𝐹𝐴 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑠𝑜𝑙𝑢𝑏𝑙𝑒 𝑠𝑢𝑔𝑎𝑟𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑟𝑔𝑒 𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑛𝑒 (𝑡 -1) (𝟏𝟒)

Representation of the absorption

The absorption process is supposed to be saturable and for this a Michaelis-Menten representation is used. Absorption in the stomach is considered to be negligible and was not represented except for phosphorus. In the other compartments, it is represented by equation 15 for amino acids, soluble sugars, fatty acids.

𝐴𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 (𝑡) =

(𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡-1) * 𝑉𝑚𝑎𝑥)

(𝐾𝑚+𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡-1)) (15)
Parameters Vmax and Km are those proposed by Bastianelli et al. (1996) for each nutrient and each compartment. For minerals, active and passive absorption are represented by equations and parameters defined by Létourneau-Montminy et al. ( 2011).

Outputs of the model

Amount of nutrients in the compartment at each time step

The amount of nutrient in each digestive compartment and at each time step is the difference between the inflow (intake, transit, endogenous secretions and hydrolysis) and the outflow from the compartment (hydrolysis, transit, absorption, excretion). This can be generalized by equation 16.

𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 (𝑡) = 𝑎𝑚𝑜𝑢𝑛𝑡 (𝑡 -1) + 𝑎𝑚𝑜𝑢𝑛𝑡 𝑖𝑛 (𝑡) -𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑢𝑡 (𝑡) (𝟏𝟔)

Amount of nutrients absorbed at each time step

The amount of nutrient absorbed at each time step (for the total digestive tract) is calculated by summing the amount of nutrient absorbed in each compartment. This information is useful for the metabolic model. The figure 4 illustrates the amount of nutrients absorbed in function of time.

Digestibility

For the digestibility calculations, the transit flow from large intestine to feces was calculated for different classes of nutrients:

 crude protein = protein + amino acids + protein from microbes  total nitrogen = crude protein + non-protein nitrogen + nitrogen from microbes  lipids = lipids + fatty acids + lipids from microbes  sugars = starch + soluble sugars + VFA + sugars from microbes

The digestibility is calculated only for the last day of simulation and the right value to consider is the one of the last minute (when steady-state is achieved). The amount of nutrient excreted is obtained by summing the flow on the 24h. It is represented by equation 17.

𝐴𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝐷𝑖𝑔𝑒𝑠𝑡𝑖𝑏𝑖𝑙𝑖𝑡𝑦 = 1 -𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑚𝑜𝑢𝑛𝑡 𝑒𝑥𝑐𝑟𝑒𝑡𝑒𝑑 (𝑔)

𝑑𝑎𝑖𝑙𝑦 𝑎𝑚𝑜𝑢𝑛𝑡 𝑖𝑛𝑔𝑒𝑠𝑡𝑒𝑑 (𝑔)

(17)

For the ileal digestibility, the amount excreted is the flow of nutrient from distal small intestine to large intestine and for the fecal digestibility, it is the flow from large intestine to feces.

The figure 5 illustrates the apparent ileal digestibility obtained with five different diets.

Transposition from pigs to broilers

The aim of the project was to develop a generic model that could be suitable for pigs and poultry. The work described above is a model representing digestion in pigs and is largely based on existing concepts. The second step of the work was to transpose this model to broiler chickens. For this and in an objective of genericity, a step-by-step method was adopted. The general structure of the model remains the same. Despite some anatomical differences between pigs and broilers, it was assumed that digestive functions described for the pig model were suitable for the broiler. The objective was thus to quantify the digestive mechanisms represented in the model to be able to simulate digestion in growing broilers by changing as few parameters as possible (and needed).

The first step was to change the daily dry matter intake (on average 82 g/d). Results indicated an overestimation of the apparent ileal and fecal dry matter digestibility by 7 and 8 points respectively. This result suggests that parameters have to be changed to be able to represent a broiler, keeping the structure of the pig model.

Changing parameters requires the quantification of these for the broiler. For this, data from literature were used. Articles investigating ileal and fecal digestibility of dry matter, starch, crude protein, ether extract, calcium and phosphorus in broilers as a function of diet composition (amount of nutrients) from years 2000 to 2015 were selected. The database included 128 experimental treatments in 20 published articles. Chickens ranged from 18 to 29 days of age and from 445 to 1365 g of body weight. Daily dry matter intake ranged from 45 to 148 g/d. Diet composition was: starch from 0 to 52.3% of DM, soluble sugars from 0 to 51.9% of DM, crude protein from 7.3 to 29.6% of DM, total dietary fibres from 2.8 to 17.4% of DM, lipids from 2.9 to 13.9% of DM, calcium from 0 to 6.7% of DM, and phosphorus from 0 to 3.6% of DM.

As indicated above, simulating dry matter intake of chicken with pig parameters induced an overestimation of the apparent ileal and fecal digestibility. As the digestibility depends on the ingested and excreted amount of dry matter and because dry matter intake was measured in literature articles, this overestimation was hypothesized to be due to an underestimation of output transit flows. The only parameters used to calculate the output flow is the constant (fractional rate, Equation 7). As the fractional rate is the inverse of the mean retention time (fractional rate = 1/MRT), a decrease in MRT would increase the output flow. In addition, it is known that the total tract mean retention time is about 8 times shorter in broilers compared to growing pigs (5 h vs 39 h; [START_REF] Ouhida | The effects of sepiolite in broiler chicken diets of high, medium and low viscosity. Productive performance and nutritive value[END_REF]Wilfart et al., 2007). This confirms the need to change the mean retention time for broilers.

Step 1 : Quantification of the mean retention time for broilers Keeping in mind that the objective of the model is to represent the animal along its productive life and to represent pigs and broilers in a generic way, the mean retention time was changed by making it a function of body weight (BW).

The mean retention time is the ratio of the indigestible dry matter in transit and the indigestible dry matter intake [START_REF] Van Der Klis | Absorption of Minerals and Retention Time of Dry Matter in the Gastrointestinal Tract of Broilers[END_REF].

𝑀𝑒𝑎𝑛 𝑟𝑒𝑡𝑒𝑛𝑡𝑖𝑜𝑛 𝑡𝑖𝑚𝑒 (ℎ) = 𝑈𝑛𝑑𝑖𝑔𝑒𝑠𝑡𝑖𝑏𝑙𝑒 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 (𝑘𝑔 ℎ ⁄ ) 𝑈𝑛𝑑𝑖𝑔𝑒𝑠𝑡𝑖𝑏𝑙𝑒 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛𝑡𝑎𝑘𝑒 (𝑘𝑔) (𝟏𝟗)
The indigestible dry matter in transit linearly increases with gut capacity [START_REF] Clauss | A case of non-scaling in mammalian physiology? Body size, digestive capacity, food intake, and ingesta passage in mammalian herbivores[END_REF][START_REF] Steuer | Is there an influence of body mass on digesta mean retention time in herbivores? A comparative study on ungulates[END_REF].

𝑈𝑛𝑑𝑖𝑔𝑒𝑠𝑡𝑖𝑏𝑙𝑒 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 (𝑘𝑔) = 𝑎 * 𝑔𝑢𝑡 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 (𝑘𝑔) (𝟐𝟎)

The gut capacity increases linearly with body weight [START_REF] Parra | Comparison of foregut and hindgut fermentation in herbivores[END_REF][START_REF] Demment | A nutritional explanation for body-size patterns of ruminant and nonruminant herbivores[END_REF].

𝐺𝑢𝑡 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 (𝑘𝑔) = 𝑏 * 𝐵𝑊 (𝑘𝑔) (𝟐𝟏)

Consequently, the indigestible dry matter in transit can be expressed in function of the body weight.

𝑼𝒏𝒅𝒊𝒈𝒆𝒔𝒕𝒊𝒃𝒍𝒆 𝒅𝒓𝒚 𝒎𝒂𝒕𝒕𝒆𝒓 𝒊𝒏 𝒕𝒓𝒂𝒏𝒔𝒊𝒕 (𝒌𝒈) = 𝒂 * 𝒃 * 𝑩𝑾 (𝒌𝒈) (𝟐𝟐)

Feed intake is considered to be related to the energy requirement, which can be scaled to the metabolic body weight. Equation 25 allowed estimating the mean retention time according to the body weight. For each simulation, the mean retention time and consequently the fractional rate of the output of nutrients are better adapted to the animal simulated. However, this modification resulted in an underestimation of the apparent ileal and fecal digestibility of dry matter ranging from 29 to 56 points and from 19 to 28 points respectively. This underestimation may result from the distribution of the mean retention time between the different parts of the gastro-intestinal tract (as a percentage of total retention time) which also differs between pigs and chickens [START_REF] Rougière | Étude comparée des paramètres digestifs des poulets issus des lignées génétiques D+ et D-sélectionnées pour une efficacité digestive divergente Université François Rabelais de Tours[END_REF].

Step 2 : Modification of the distribution of the total tract mean retention time

As a percentage of the total tract MRT, the MRT in the stomach and small intestine is higher for chickens than for growing pigs. This result in increased hydrolysis and absorption of nutrients for the broiler compared to the growing pig and also leads to a lower fermentation of dietary fiber. For broilers, the distribution of the total MRT was estimated as 22% in the stomach (compared to 11% in pigs), 53% in the small intestine (9% in pigs) and 25% in the large intestine (80% in pigs; Dänicke et al., 1999;[START_REF] Shires | Rate of Passage of Corn-Canola Meal and Corn-Soybean Meal Diets Through the Gastrointestinal Tract of Broiler and White Leghorn Chickens[END_REF]. The distribution of the mean retention time in the small intestine was not changed (9% in the proximal part and 91 % in the distal part). With this modification the apparent ileal and fecal digestibility of dry matter were underestimated from 0 to 17 points and from 5 to 13 points respectively.

Considering that the output flows were quantified better, it was hypothesized that this underestimation may be due to other digestive processes (i.e. hydrolysis, fermentation, absorption).

To identify which parameters have the greatest impact on the dry matter ileal and fecal digestibility, a sensitivity analysis was carried out. Parameters involved in hydrolysis, fermentation and absorption for each nutrient and each compartment were increased or decreased by 25, 50 or 75% of their default values (Bastianelli et al., 1996;[START_REF] Létourneau-Montminy | Modeling the fate of dietary phosphorus in the digestive tract of growing pigs[END_REF]. Results indicated that the ileal and fecal digestibility of dry matter are most sensitive to the hydrolysis rate. Consequently, this parameter was modified.

Step 3 : Modification of the hydrolysis rate

Endogenous secretions are more important in broilers than in growing pigs. Endogenous secretions include digestive enzymes which further justifies the choice to increase the hydrolysis rate in broilers compared to the default value of pigs. Firstly, to simplify, the same coefficient of correction was applied to hydrolysis rates for all nutrients (multiplying by 2). With this modification, the predicted ileal digestibility of dry matter varies from an underestimation of 5 points to an overestimation of 3 points. After having obtained a reasonably good prediction of dry matter ileal and fecal digestibility, the digestibility of each nutrient is currently tested.

Conclusions

Based on existing digestion models for pigs, the first part of the work consisted in combining information from these pig models. This pig model has then been transposed to broilers. The developed model is an interesting research tool to describe the digestive steps for the main nutrients in growing pigs and broilers, and it can be used to determine the nutritive value of feedstuffs and mixed feeds. The digestive model is available for the DSS development in T3.5.

  1. Summary ........................................................................................... 2. Introduction ........................................................................................ 3. Results .............................................................................................. 3.1 Structure of the model ........................................................................... 3.2 Inputs of the model ............................................................................... 3.3 Model: representations and calculations ............................................. 3.4 Outputs of the model ........................................................................... 3.5 Transposition from pigs to broilers ...................................................... 4. Conclusions ..................................................................................... 5. References ...................................................................................... 6. Annex 1: Information used for simulations (Figures 4 and 5) ...........

Figure 1 .

 1 Figure 1. Diagram of the model. Stomach (STO), Proximal small intestine (SI1), Distal small intestine (SI2), Large intestine (LIC). Proteins (PRO), Amino Acids (AA), Non-protein nitrogen (NPN), Starch (ST), Soluble sugars (SS), Potentially degradable (DF) and undegradable fibres (UF), Lipids (LIP), Fatty acids (FA), Volatile fatty acids (VFA), non-solubilized calcium of animal and mineral origins (CAamns), non-solubilized calcium of plant origin (CAvns), non-solubilized calcium of all origins (Cans), solubilized calcium (CAs), non-solubilized phytic phosphorus (PPns), solubilized phytic phosphorus (PPs), solubilized non-phytic phosphorus (NPPs), non-solubilized non-phytic phosphorus (NPPns), microorganisms (MIC) and residuals (RES). Transit flows are represented by solid arrows between compartments. Hydrolysis flows are represented by solid arrows between two different nutrients. Solubilization and insolubilization flows are represented by solid arrows between solubilized and unsolubilized calcium or phosphorus. Fermentation in the large intestine is represented by solid lines with white arrows. Endogenous flows are represented by dotted lines which came from "Endo" boxes. Absorption flows are represented by dotted lines which reached "Abs" boxes.

Figure 2 :

 2 Figure 2 : Classification of the fiber

  TDF (%) = 1.03 * NDF(%) + 0.05 (𝟏) (Le[START_REF] Gall | Prédiction de la teneur en fibres totales et insolubles de matières premières courantes dans l'alimentation du porc à partir de leur composition chimique[END_REF] Degrdable fiber (PDFFEED, %) = 0.7 * TDF (%) (2) Undegradable fiber (PUFFEED, %) = 0.3 * TDF (%) (3)

  Phytic phosphorus (g kg ⁄ ) = Total phosphorus (g kg ⁄ ) * Phytic phosphorus Total phosphorus (%) (𝟒) ⁄  Non-phytic phosphorus comes from plant, mineral and animal origin. Coefficients of solubilization defined by Létourneau-Montminy et al. (2011) are used (Figure3).

Figure 3 :

 3 Figure 3 : Different forms of phosphorus in the diet. * phytic phosphorus in feedstuffs is calculated by equation 4 For the model inputs: -PPPnsFEED = phytic phosphorus calculated by equation 4 for each feedstuff -PNPPnsFEED = non-phytic phosphorus non-solubilized from plant, animal or mineral origin -PNPPsFEED = non-phytic phosphorus solubilized from plant, animal or mineral origin Residuals represent the dry matter that is not represented explicitly by nutrients in the model. Thus, PRESFEED is calculated by equation 5. PRESFEED (%) = 100 -(PAAFEED + PPROFEED + PLIPFEED + PSTFEED + PSSFEED + PUFFEED + PDFFEED + PCAamnsFEED + PCAvnsFEED + PCAsFEED + PPPnsFEED + PNPPnsFEED + PNPPsFEED)(𝟓)

  Example no. 1: Two meals per day at 08:00 am and 04:00 pm  H_meal_morning = 480 (08*60 to represent 08:00 am in minutes)  H_meal_evening = 960 (16*60 to represent 04:00 pm in minutes)  H_meal_day_interval = 0 Example no. 2: One meal per hour between 08:00 am and 04:00 pm  H_meal_morning = 480  H_meal_evening = 960  H_meal_day_interval = 60

  Example no. 3: Two meals at 08:00 am and 04:00 pm and meals every 6h in between  H_meal_morning = 480  H_meal_evening = 960  H_meal_day_interval = 360 (6*60)

  𝑂𝑢𝑡𝑝𝑢𝑡 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 𝐴 𝑡𝑜 𝐵 (𝑡) = 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 (𝑡 -1)𝑖𝑛 𝐴 * 𝑓𝑙𝑜𝑤 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑓𝑟𝑜𝑚 𝐴 𝑡𝑜 𝐵 (𝑡) = (𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑛𝑢𝑡𝑟𝑖𝑒𝑛𝑡 (𝑡 -1) 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 (𝑡 -1)) 𝑖𝑛 𝐴 * 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 (𝑡 -1) 𝑖𝑛 𝐴 ⁄ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒐𝒇 𝒏𝒖𝒕𝒓𝒊𝒆𝒏𝒕 (𝒕 -𝟏) 𝒊𝒏 𝑨 (𝟕)

  𝑂𝑢𝑡𝑝𝑢𝑡 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝐿𝑎𝑟𝑔𝑒 𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑛𝑒 𝑡𝑜 𝐹𝑒𝑐𝑒𝑠 (𝑡)= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑟𝑔𝑒 𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑛𝑒 (𝑡 -1) * exp ((𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑟𝑔𝑒 𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑛𝑒 (𝑡 -1) -𝑎)/𝑏) (𝟖)

  𝐸𝑛𝑑𝑜𝑔𝑒𝑛𝑜𝑢𝑠 𝑠𝑒𝑐𝑟𝑒𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝐵 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝑓𝑟𝑜𝑚 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 𝐴 𝑡𝑜 𝐵 (𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 * 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑑𝑟𝑦 𝑚𝑎𝑡𝑡𝑒𝑟 𝑖𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 𝐴 (𝑡 -1))(𝟏𝟐)

Figure 4 :

 4 Figure 4 : Dynamic of the quantity of nutrients absorbed along the day for a growing pig. The daily dry matter intake was 1924 g shared in two meals (start times = 1 and 721 min, duration = 15 min) represented by the red lines. Diet composition was wheat (87.25%), soybean meal (10%), dicalcium phosphate (0.7%), calcium carbonate (1.1%), salt (0.45%) and other vitamins and minerals (0.5%) as described in Cozannet et al., 2010.

Figure 5 :

 5 Figure 5 : Apparent ileal digestibility of dry matter, crude protein, calcium and phosphorus for different diets. Simulations were performed with two meals per day occurring à 1 and 721 min (duration of the meals = 15 min) with 50% of the daily feed intake ingested at each meal. Daily dry matter intake and diet compositions are available in Annex 1.

Figure 6 :

 6 Figure 6 : Linear relationship between the mean retention time and the metabolic body weight in broiler.

Table of contents

 of



  Phytic phosphorus comes from plants only and is calculated from INRA table values by equation 4. It is non-soluble.

To represent the action of phytase, the approach of Letourneau-Montminy et al. (2011) was used.

The amount of each nutrient is expressed as a percentage of the dry matter intake. In this way it was not necessary to consider the moisture content of the diet and water consumption through the day. According to Bastianelli et al. (1996), traits such as transit time are affected by the water content of the diet. However, for simplicity reasons it was assumed that the water content was constant for conventional diets, thereby having no impact on the transit time and digestibility (Bastianelli et al., 1996).

The nutrient content of the feed is taken from table values (INRA-AFZ, 2004). Once the amount of proteins, amino acids, lipids, starch, sugars, fiber, calcium and phosphorus have been calculated, some additional calculations are required to fit the inputs required by the Matlab model (Table 1). Regarding the fiber part of the diet, chemical analysis or information available in nutritional tables usually gives NDF, ADF, ADL (lignin). However, in the model, inputs are potentially degradable and


Summary

Objectives

The purpose of the model is to quantify the variability in individual trait responses within a livestock population whose individuals share a common factor, e.g., are of the same breed, live in the same farm, or are offered the same feed. In addition to assessing trait variability within a given a population, the approach will also allow comparisons between populations, e.g., that differ in species, breed, environment, or feed, accounting for both their average and their inherent variability in trait responses. The Deliverable has the following objectives towards these aims:

1) To characterise and estimate trait variability from a dataset on a herd, flock or breed. 2) To predict and summarise the influence of variation in model-derived traits on trait variation in a production system. 3) To demonstrate the approach on different monogastric species and on performance and reproduction traits.

Rationale:

We developed a modelling approach that has multiple applications in precision livestock farming, nutrition, and selective breeding. The approach is demonstrated through three case studies on monogastric livestock: growth in pigs, and reproduction in sows and in laying hens. Comparison of trait variability across these species and traits shows common aspects that they share as well as their distinctive features.

This methodology comprises a data-driven (top-down) approach, where models are fitted to phenotypic trait data obtained from multiple individual animals; and a simulation (bottom-up) approach, where population phenotypic variation is derived and summarised by the average and deviation (i.e., median and confidence interval) for each modelled trait. The approach has the following benefits in relation to current alternatives: 1) Making no prior assumptions about the distributions of traits and their correlations within the population; specifically, it is assumed that the population traits are distributed according to the trait distribution in the group of sampled animals (nonparametric approach). 2) Being computationally faster than current parametric approaches; specifically, the distribution of traits in the wider population is inferred from that in the sample through a process of individual resampling. 3) Having no specific requirements on the size and quality of the datasets input.

Teams involved:

UNEW, INRA, Topigs

Species and production systems considered:

Commercial pig breeds, experimental hen lines, simulated sows. The results are applicable to all monogastric species raised across Europe.

Introduction

Motivation

It is necessary to introduce stochasticity (i.e., variation) in predictive models of livestock production to account for trait variation across individuals in a herd, flock or breed. The introduced stochasticity should preferably be based on empirical data, because different forms of modelling stochasticity in individual traits can lead to disparate conclusions about variation in populations.

There are important benefits from accounting for phenotypic variation in model predictions rather than using predictions solely for the 'average' animal [START_REF] Kyriazakis | Opportunities to improve nutrient efficiency in pigs and poultry through breeding[END_REF][START_REF] Van Milgen | Precision pork production: Predicting the impact of nutritional strategies on carcass quality[END_REF][START_REF] Saintilan | Genetics of residual feed intake in growing pigs: Relationships with production traits, and nitrogen and phosphorus excretion traits[END_REF]Filipe et al., 2018a;[START_REF] Pomar | Meeting individual nutrient requirements to improve nutrient efficiency and the sustainability of growing pig production systems[END_REF][START_REF] Filipe | Bayesian, likelihood-free modelling of phenotype plasticity and variability in individuals and populations[END_REF][START_REF] Tullo | Review: Environmental impact of livestock farming and Precision Livestock Farming as a mitigation strategy[END_REF]; these include the following:

1) Knowledge of the range and distribution of individual output within a group provides the user with information about deviations from the average individual output; moreover, it offers users a more accurate prediction of the group's total output, i.e. the total is usually different from (average productivity) x (number of animals). 2) This knowledge also allows prediction of the level of uniformity in the group (e.g., how many individuals conform to market requirements) and thus allows prediction of the current market value of a given batch, herd or flock. 3) Knowledge of the dynamic performance of each individual or each pen, allows for real-time implementation of individually-or pen-targeted feeding and healthimprovement schedules for optimal allocation of resources. 4) Prediction of the variation in model-derived traits within a population can assist the achievement of breeding goals when genetic selection utilises modelderived traits. 5) A more robust comparison between populations (e.g., different breeding lines) in different environmental conditions, or offered different diets.

These benefits will help breeders and producers, improve animal welfare, and reduce environmental impact by contributing to the implementation of modern precision livestock farming practices (PLF), and, therefore, will contribute to the broader aim of delivering greater sustainability and robustness in intensive livestock production systems.

Problem

Estimating and modelling variation in traits requires much more computation than estimating and modelling an average animal. In other words, instead of doing one calculation, we need to perform many calculations, which is has extra computational cost. The approach that we have developed and implemented here is a trade-off between this extra computational cost and the benefits from having information on the deviations from the average.

Progress has been made in animal science to predict the distribution of animal productivity within populations [START_REF] Ferguson | Predicting the effects of animal variation on growth and food intake in growing pigs using simulation modelling[END_REF][START_REF] Knap | Characterization of pig genotypes for growth modeling[END_REF][START_REF] Pomar | Modeling stochasticity: Dealing with populations rather than individual pigs[END_REF][START_REF] Vautier | Accounting for variability among individual pigs in deterministic growth models[END_REF]. This methodology relies on two steps. First, the assumption and calibration of a certain plausible distribution of model trait parameters in the population; typically, a multivariate-normal distribution of parameters calibrated via an empirical or expert-based variance-covariance matrix (top-down approach). Second, the simulation of the trait variation in the population under the assumed distribution (bottom-up approach). There are several disadvantages in assuming explicit population trait distributions, such potential inaccuracy and difficulty of calibration.

Approach taken

We develop a combination of top-down and bottom-up approaches that has two important features (Filipe et al, 2018a;[START_REF] Filipe | Bayesian, likelihood-free modelling of phenotype plasticity and variability in individuals and populations[END_REF]. First, the methodology is non-parametric, i.e., it draws the trait distribution and correlations implicit in the empirical data without making explicit and potentially inaccurate assumptions about such distributions and correlations. Second, the methodology requires much less computation than using the latter assumptions, because it does not require the construction of and simulation from a hypothetical trait distribution. In the top-down step, a data-driven approach is applied where models are fitted to phenotypic trait data obtained from multiple individual animals. In the bottom-up step, the following approach is applied for estimating population phenotypic variation. Instead of assuming a theoretical distribution or expert-informed parameter ranges, it is assumed that the population traits are distributed according to the trait distribution in the group of sampled animals. This assumption is common to a broad range of non-parametric approaches. In addition, the distribution of traits in the wider population is inferred from that in the sample through a process of random individual resampling. The approach is completed by a statistical evaluation of the average and deviation (i.e., median and confidence interval) of each modelled trait in the population. Regarding the literature, this method contains assumptions that relate (but are not identical) to those made in bootstrap [START_REF] Efron | The jackknife, the bootstrap and the other resampling plans[END_REF] and quantile regression [START_REF] Koenker | Regression quantiles[END_REF]) and combines them with other modelling components.

Scope

This modelling approach applies to the prediction and quantification of phenotypic variation within populations of different species, breeds, and herds of livestock. The phenotypic variation can comprise any number of traits that are measured or modelderived, and the correlations among traits are inherently preserved. The approach can be used for multiple purposes. Here, we focus on a selection of non-ruminant livestock species and performance and reproduction traits relevant to applications in farming, nutrition, and selective breeding.

Method

A first requirement in modelling population phenotypic-variation is the availability of longitudinal trait observations from multiple individual animals. These individuals are a sample from the relevant large population or are the full population. The first stage in the method (Filipe et al., 2018a;[START_REF] Filipe | Bayesian, likelihood-free modelling of phenotype plasticity and variability in individuals and populations[END_REF] is to estimate the stochasticity in trait parameters across the population (top-down approach); here, model curves are fitted to each individual dataset and their associated trait parameters are estimated. The second stage is to quantify the variability in the relevant traits that is a consequence of this stochasticity (bottom-up approach). The detailed components of the method are the following.

1-Assumption.

The distribution of a given set of animal traits within the large population is represented by the joint variation of these traits across the individuals in the sample taken from that population. As a result, the variation in other, model-derived traits will be naturally associated to the variation in observed traits among the individuals in the sample.

2-Variability in trait parameters.

Trait parameters for each individual in the sample are estimated by fitting a model to the observed traits. There are different techniques for fitting data and estimating parameters. Given that the models are non-linear, we used non-linear least-squares point estimation; other estimation techniques would also be suitable for this purpose [START_REF] Filipe | Bayesian, likelihood-free modelling of phenotype plasticity and variability in individuals and populations[END_REF]. The outcome is a distribution of model-based trait parameters across the sample. According to the assumption, this distribution is representative of that in the large population.

3-Resampling.

The variation in model-derived traits in the large population is inferred by resampling individuals randomly from the sample, repeatedly up to a chosen population size. This step has the effect of making the statistical value of the quantiles (evaluated below) converge to a stable value as the size of the resample becomes large compared to the size of the sample (unless the sample is very large, in which case both values should agree). As, according to the assumption, the individuals in the resample are repeated from the sample, no further computation of the model traits is required, only a recalculation of the statistics of these model traits. The resampling step is optional. The alternative is to use the sample itself, in which case the inferences do not extrapolate to a large population. The choice of whether to focus on trait distributions within the sample or in a larger population depends on the specific application and on the relevant question addressed (see applications in Discussion).

4-Quantiles of the trait distribution.

For dynamic traits, a set of time points (or points on another covariate) are chosen. The set can be within or, in the case of forecast or hind-cast, beyond the longitudinal range of the data. For each modelled trait, quantiles of its distribution (in the resample or sample) are evaluated at each time point. Here, the 5%, 50%, and 95% quantiles were used to represent the trait average (via the median) and confidence interval (area between the 5% and 95% quantile curves, which contains 90% of the individuals in the population). Examples of dynamic traits are live weight and protein weight. Non-dynamic traits associated to the latter are the live weight and protein weight at a given stage (e.g., maturity), or the average weight gain or protein deposition over a given period. For non-dynamic traits, quantiles are evaluated only once. A modelled trait can be a model representation of an observed trait (e.g., live weight or live weight gain) or a model-derived trait that is not observed but is related to observed traits via biological assumptions (e.g., the intrinsic growth rate, dynamic or mature protein weight, or mature live weight) if not observed.

Diagnostics of model fitting and data outliers. The user of the method may wish to check the quality of the model fitting or to identify and tackle potential outlier individuals in the population sample. A detailed account of the diagnostics on fitting and outliers and example diagnostic results (case study 1) is given in Annex 1.

Case studies

Growth of live weight in boars

 Data was provided by Topigs.

 Datasets: Individual data on live weight of boars of a single breed during the finishing growth: 119 animals; weight range 25-165 kg. The animals are a contemporary cohort offered the same dietary schedule ad libitum.

 Production model: Gompertz curve fitted to live weight (Filipe et al., 2018a). Other growth models could be used without altering the general outcomes of the method.

Egg production in hens

 Data was provided by INRA.

 Datasets: Individual data on hen egg laying and egg weight in two experimental lines divergently selected for residual food intake: 29 animals; age range while laying was 19-38 weeks; weighing of the eggs took place during 28-31 weeks of age. The lines are part of a long-term experimental programme [START_REF] Bordas | Direct and correlated responses to divergent selection for residual food-intake in rhode-island red laying hens[END_REF]. The data from the two lines (one with 16, another with 13 animals) were merged into a single population sample to increase the sample size. While it is expected that reproduction traits are not significantly affected by this selection, some differences between lines are still likely. For our purpose of illustrating the approach, it is adequate to consider a genetically heterogeneous group, which tests the approach in the characterisation of potentially large phenotypic variation. Future applications of the approach could focus on the differences between these lines.

 Production model: Gompertz curve fitted to egg output; power-law curve fitted to cumulative egg mass. Again, other curves could be used here.

Gestation and lactation in sows

Sow datasets at individual level and with longitudinal observations, as in the other case studies, were currently not available. As an alternative, we used simulated data, i.e. phenotypically-diverse outputs from a model where the variation is not informed by observation data. This case study allows us to demonstrate that: 1) gestation and lactation traits can be characterised using this approach in the same way that other traits can; and 2) the approach can be used for analysing artificial trait variability as well as trait variability based on observations. In addition, this case study shows that when it is advantageous to use simulated rather than observation-based population data, trait variability within the simulated population can still be adequately characterised.

Assumption: To simulate artificial phenotypic variability, trait parameters that are inputs of the model were varied under the assumption that their range of variation is representative of the parameter distribution that would be obtained if the model were fitted to data from multiple individual sows. Such approach to generating phenotypic variability in a model population relates to those used in animal science [START_REF] Knap | Characterization of pig genotypes for growth modeling[END_REF][START_REF] Pomar | Modeling stochasticity: Dealing with populations rather than individual pigs[END_REF][START_REF] Vautier | Accounting for variability among individual pigs in deterministic growth models[END_REF]. in each of these individuals no more than one trait was changed with respect to the basic sow profile because the trait correlations across sows are not known.



Results

Growth of live weight in boars

The live weight observations of the 119 boars (Figure 1A) show that each individual has a distinct growth trajectory and a specific age range during which weight was recorded. The X-axis ranges from the median first age of recording (76 d, 33 kg) to the maximum age of recording across animals (196 d, 167 kg); it does not start at the minimum first age of recording because there are too few individuals at the earlier ages of recording to estimate population quantiles.

The average (median) and confidence interval (CI, area between the 5% and 95% quantiles) characterise adequately the distribution of live weight across the population over the recorded finishing period (Figure 1B): approximately 10% of the individuals are outside the CI. The quantiles are based on resampling 1000 individuals out of the sample of 119 animals.

The median live weight has an inflection point around age 175 d (Figures 1C andE), which is very close to the average age of maximum growth for males of this breed (Egbert Knol, personal communication). An estimate of the inflection point based on the median is generally more robust and thus preferable to one based on the mean, because the median is not sensitive to outlier animals. The predicted daily gain, which is above 1 kg/d for only a short period (Figure 1E), is also in agreement with empirical expectation for this breed.

Prediction beyond the observed range shows the expected population average and CI up to 250 d (Figure 1D). The future width of the CI indicates how variability in weight might increase as the animals in the herd grow further. The CI width quantifies both predicted variability and uncertainty about future growth. The degree of uncertainty depends on the amount of information and noise in the past data; therefore, it is possible and plausible that a CI is wider than expected in practice, especially if the expectation were based on average observations (i.e., without deviations). Real-time prediction, where performance in the following day or period is predicted based on performance in a previously recorded period, could be tested by fitting the model to a narrower age range than in Figure 1D and predicting performance based on the parameters estimated using the narrower range. It is likely that the average would not be significantly affected, but that the CI would be wider than that based on the full data as in Figure 1D.

Diagnostics of model fitting and data outliers

The user of the method may verify the quality of the model fitting to the boar data or identify and tackle potential outlier animals in the sample. Results of the diagnostics on fitting and outliers are available (Figure 2 in Annex 1).

Egg production in hens

Egg output

The observed number of eggs laid by each of the 29 hens (Figure 3A) shows that each individual follows a distinct production trajectory since the age of start of laying and during the given period of observation. The age of start of laying differed among hens (19-23 wk), but here we focused on the production trajectory of each hen regardless of the age of start of production.

The average and CI characterise suitably the distribution of egg output across the population over the recorded laying period, as approximately 10% of the individuals are outside the CI (Figure 3B). The CIs are determined by the fitted model trajectories (Figure 3C) and are based on a 1000-large resample. The noise in some extreme observed trajectories makes them differ somewhat from the fitted trajectories (Figure 3C), leading to some difference between outliers in Figures 3B and3C. However, the average curve is robust to such variation.

The population average and CI of eggs output are prediction beyond the data range, up to 30 wk of laying (Figure 3D). In principle, it is possible to predict egg output up to any future point, and even to predict lifetime production individually and across the population. However, the shortness of the observed laying period (up to 18 wk) compared to the typical duration of laying (up to 2 years, depending on genotype), does not make for reliable long-term prediction; this is perhaps suggested by the shape of the average and CI beyond the data range.

The average daily rate of egg output declines consistently over time, although it is relatively constant up to 10 wk (Figure 3E). The decline at the end of the observations is probably excessive, which is likely a result of the short duration of the observations.

Egg mass

The cumulative egg mass produced during the laying sequence also shows variation among individuals, suggesting that some hens produce larger eggs (with the exception that a few hens lay multiple eggs on some days (Figure 3E), which does not imply larger eggs). A difference relative to egg output is that cumulative egg mass (along the egg sequence) is less variable within each individual's production history. The predictions up to 30 wk in laying are similar to but reflect the lesser variability than in egg output (Figures 4A-D). The average daily egg mass produced increased slightly during laying, but the corresponding population variation is nearly constant, (i.e., egg weight variation occurs among and not within hens; Figure 4E) as already suggested by Figure 4A. In the majority of occasions, the egg mass is that of a single egg laid; on a few occasions, some individuals laid >1 egg (i.e., 2 or 3; Figure 4E), in which cases the egg mass is that of the total eggs laid.

Gestation and lactation in sows

There is large variability in the four modelled traits of a gestating and lactating sow: live weight, protein weight, lipid weight, and mass of milk produced, both within the population and over time (i.e., across and within parities) (Figure 5A-B and E-F). The trait distributions, generated under the working assumptions (Case Study 3), are denser in the centre, which represent sows of similar phenotype expression (those that differed in body weight loss during lactation).

Above and below the centre, the distributions are relatively evenly dispersed, representing sows of differing litter size or feed intake level. The distributions are not symmetric (Figure 5A or B), and there is no reason they should be. Milk production (Figure 5F) occurs only during the lactation. In the model used [START_REF] Dourmad | InraPorc: A model and decision support tool for the nutrition of sows[END_REF], milk production is not sensitive to some of the defining traits varied as a result of key assumptions on which the model builds.

Clearly, these artificial-trait distributions are suitable for illustrating how the method characterises variability in sow populations. The median of each trait (Figure 5C-D and G-H) tracks the time trend within in the sampled population. The medians also happen to be close to the centre of the distributions (Figure 5A-B and E-F) because the sow population was built to be relatively symmetric, but could deviate from the centre if the distributions were more asymmetric.

The CIs follow the temporal trend of the averages and characterise the variability within the population, e.g. only a few of the 26 individuals are outside the 90% CIs. These CIs show that there is considerable deviation from the average, but to a degree that depends on the trait. Such differences are more clearly seen by examining the deviations relative to the average (DRtA = (Y(t)-Yav(t))/Yav(t) where Y is the response and t is time) (Figure 6). Population deviations from the average are larger during lactation as the sow's body resources are used to respond to the increased requirements of the piglets (which grow faster and are more physically active than during gestation). At its largest, the CI DRtA is 10% for live weight, <10% for body protein, and >20% for body lipid (Figure 6A-C).

Describing the population using the average curve would miss out important information: the amount of deviation from the average, how this deviation changes over time, and how the deviations differ among traits. On the other hand, describing such phenotypic characteristics would be difficult using serial observations across individuals (if such data were available) or multiple individually-fitted curves. That purpose is achieved adequately and simply by the CIs. The predicted CIs do not rely on assumptions about the population distribution and correlations of the traits and, therefore, are robust for applications to large numbers of simultaneous traits and to complex trait dynamics where the correlations may be unknown.

The relative deviations in protein and live weight within the population are comparable and appear plausible, while those for lipid weight are considerably larger. This fact results from the underlying assumptions of the sow model, where that all energy intake not used for protein deposition or maintenance is deposited as lipid [START_REF] Dourmad | InraPorc: A model and decision support tool for the nutrition of sows[END_REF]. In practical applications, the levels of maintenance energy and protein mobilisation during lactation are adjusted so that lipid and body mass remain stable across the multiple parities. This adjustment was not implemented here, which explains the variation in lipid. These technical aspects illustrate practical difficulties in modelling sows over multiple parities, but are not relevant to our purpose of illustrating the potential of the approach to characterise phenotypic variation in general. Moreover, while we could have focused on a single parity, we preferred to show the potential of the approach to capture population variation with complex changes over time.

Discussion

The modelling approach developed here quantifies and predicts variation in animal traits within populations using empirical data from livestock herds or flocks. The results demonstrated the approach on growth and reproduction traits in monogastric livestock, i.e. growing pigs, and reproducing sows and laying hens. Comparison of trait variability across the species and traits considered showed common aspects that they share, e.g. growth in pigs and egg output in hens, as well as their distinctive features, e.g. egg output and egg mass in hens and gestation and lactation in sows.

Approach benefits

The approach makes no prior assumptions about the distributions of traits and their correlations within the population (Filipe et al., 2018a;[START_REF] Filipe | Bayesian, likelihood-free modelling of phenotype plasticity and variability in individuals and populations[END_REF].

Additionally, the approach is computationally faster than parametric approaches by relying solely on model computations for the observed individuals, and thus avoiding model and statistical computations pertaining to hypothetical individuals drawn from an assumed trait distribution. Moreover, the approach has no specific requirements on the size and quality of the datasets input.

Data characteristics

The traits whose variation is predicted can be traits that are observed, i.e. at points during growth, or that are not observed such as model-derived traits, e.g. body lipid and protein in the sow example. The approach works both for empirical samples that have a small or a large number of individuals, and regardless of whether the longitudinal observations are smooth or noisy. In addition, the approach can quantify variation in datasets not from empirical samples but from model output generated under plausible assumptions, e.g. ranges or distributions of trait parameters, as in the sow case study.

Application areas

The approach can be used for multiple purposes in precision livestock farming, nutrition, and selective breeding. For example, if the animals in the dataset have been offered identical feed, the estimated variation can be attributed to genetic and environmental influences. Conversely, if the animals have been offered different diets or feeding schedules, the estimated variation includes the influence of nutrition. Moreover, if two or more datasets are considered from groups with distinct diets but similar variation in genetics and environment, then differences in the influence of these diets can be assessed and tested in a way similar to regression analysis but without the associated parametric assumptions. The same holds for comparisons among groups that are genetically distinct.

Such information about variability can help, for example, to quantify trait heritability, to implement targeted feeding and preventive treatments, or to make more realistic evaluations of livestock value, losses, and costs. For example, this approach was used to evaluate the economic gains from adopting alternative strategies for pig allocation to finishing pens [START_REF] Filipe | Pen-allocation strategies for uniform weights in finishing pigs[END_REF]. Applications can also be made to real-time prediction: given a trait observed over a window of time, its future average and population confidence interval can be predicted based on the earlier observations. For example, the prediction of the next-week's intake and performance of a group can be carried out in this way.

While the results focused on specific species and traits, this modelling approach applies to any livestock species and genetic line and to any set of traits for which individual observations are available across a group. Therefore, on one hand, the approach allows the identification of commonalities and differences across species and lines. On the other hand, the approach allows consideration of other types of traits that affect production. For example, the degree to which production efficiency and robustness under disturbance are mutually exclusive or not, can be quantified by examining the joint variation and overlap of these traits in the same way that simultaneous traits of sows and hens were examined here.

Conclusions

A modelling approach was developed to quantify the phenotypic variation in animal traits within populations using empirical data from livestock herds or flocks, as well as to predict traits and their variation outside the range of empirical observation. The approach can also be used to quantify and predict trait variation in artificial data from simulation models. The approach has generic applications, including precision livestock farming, nutrition, and selective breeding in animal populations. First, the approach applies to any animal traits, although it was demonstrated here for growth and reproduction traits. Second, the approach applies to observable traits as well as to non-observable traits that can be related to the observed traits through a quantitative model. Third, the approach applies to monogastric livestock species as well as to any livestock species for which individual data are available. The advantages of the approach are its flexibility, i.e. no reliance on potentially-inaccurate assumptions about the form of variation and correlation of traits across individuals; its economy and speed of computation; and its applicability to small as well as to large animal sample sizes.

Given the broad range of potential uses of this approach in the implementation of precision livestock farming, and the resulting benefits of precision livestock farming for producers, for animal welfare and for the environment, the approach has the potential to contribute to the broader aim of delivering greater sustainability and robustness in intensive livestock production.

Annexes

Annex 1: Diagnostics of model fitting and data outliers

The predictions of trait variability in a population (Figure 1) are the ultimate purpose of the current analysis; in general, no further results need to be examined. However, the user of the method may wish to verify the quality of the model fitting or to identify and tackle potential outlier individuals in the population.

 Parameter distribution. The outcomes of fitting a Gompertz curve to the live weight of each individual (Figure 1A) are point estimates of the two parameters of this growth model (i.e., mature size and growth rate). The population distribution of these model-derived traits and the respective confidence intervals can be examined, either as a joint distribution or as separate marginal distributions (Figure A2.A-C).  Goodness-of-fit. Goodness-of-fit was quantified by the coefficient of determination R 2 and distribution of residuals (i.e., deviations of the data with respect to the model (Figure A1.E-F). The overall goodness-of-fit was high for most individuals, but a few individuals show a residual trend indicating that their specific growth conditions may have been disturbed and may not be describable by the proposed growth curve (Figure A1.D).  Outliers. Some individual growth trajectories stand out with respect to the majority (Figure 1A), which could have been caused by a variety of factors, including disease. For these individuals, growth conditions may have been affected by extreme perturbations or observational errors, in which case the growth model and the fitting approach may no longer be appropriate. Consequently, the parameter estimates may not be compatible with their biological interpretation, and the resulting fitted curve may be an outlier in the population distribution model curves.

We have used the option of tackling outliers by imposing presumed 'acceptable' bounds on the parameter ranges prior to their estimation; these ranges are reflected in the edges of Figure A1.A. Other approaches, such as the removal of identified outliers could be implemented.  Resampling. The effect of resampling individuals from the original sample has the effect of increasing the relative counts and stabilising some of the quantile estimates (Figure A1.G-H).

  1. Summary ........................................................................................... 2. Introduction ........................................................................................ 3. Method .............................................................................................. 4. Case studies ...................................................................................... 4.1 Growth of live weight in boars ............................................................... 4.2 Egg production in hens .......................................................................... 4.3 Gestation and lactation in sows ............................................................. 5. Results .............................................................................................. 5.1 Growth of live weight in boars ............................................................... 5.2 Egg production in hens ........................................................................ 5.3 Gestation and lactation in sows ........................................................... 6. Discussion ....................................................................................... 7. Conclusions ..................................................................................... 8. Annexes .......................................................................................... 8.1 Annex 1: Diagnostics of model fitting and data outliers ....................... 8.2 Annex 2: Bibliographic references ........................................................ 8.3 Annex 3: Publications and communications .........................................

o

  Data simulated by INRA.  Datasets: Individual data on four sow traits during gestation and lactation: 26 simulated animals, during eight parities (first service at 140 kg and mature body weight after farrowing of 270 kg).  Generation of data was via the model of Dourmad et al. (2008): Output traits: Live body weight, body protein, and body lipid during gestation and lactation; milk produced during lactation. o Variability in input traits: These model output traits were simulated for each of 26 sows with distinct artificially-defined phenotypic profiles. This phenotypicallydiverse population was constructed as follows. 1) A basic individual profile was defined by setting typical values of key sow and litter trait parameters in the model, such as sow body weight, backfat thickness, and feed intake at given critical stages of the production cycle, and piglet growth and survival. A facility of the model was used to ensure that the model output traits of the sow, such body weight and backfat thickness, did not deviate from the defining inputs throughout the long period of the simulation (eight parties, >1000 days); this involved calibrating the maintenance energy expenditure during gestation and lactation, and the minimum protein mobilisation during lactation. 2) Each of the other 25 individual sow profiles in the population was defined by changing one of the following traits: sow feed intake during lactation (basic value 5.8 kg/d, range 5.22 to 6.5 kg/d), sow body weight loss during lactation (basic value 25 kg, range 22 to 28 kg), and number of piglets born alive (basic value 12.5, range 10.5 to 14.5);

Figure 1 .

 1 Figure 1. Growth in boars. A) Observed individual trajectories. B) Predicted population average (median) and 90% CI, overlapped on the data to which the model was fitted to individual by individual. C) As B, but without showing the dataset; note the average age at maximum growth (inflection point) at 175 d. D) Prediction as in B, but beyond the data range (197 d) to 250 d. E) Daily rate of weight gain, population average and CI; the maximum gain occurs at 175 d; an alternative evaluation of the average daily rate of weight gain (blue line, obtained by differentiating the average curve in B) is close but is a less suitable estimation.

Figure 3 .

 3 Figure 3. Cumulative egg output of each hen over time since start of laying. A) Observed individual trajectories; the age of start of laying varies among hens (19-23 wk) but is not included here in order to focus on the period of production. B) Predicted population average (median) and 90% CI, overlapped on the data to which the model was fitted to individual by individual. C) As B, but showing individual model-fitted trajectories. D) Prediction as in B, but beyond the data range (18 wk) to 30 wk laying. E) Daily rate of egg output, population average and CI; the rate of egg laying decreases slowly but consistently (as also evident from A-D).

Figure 4 .

 4 Figure 4. Cumulative eggs mass laid during the laying sequence. Similar to Figure 3, but focusing on the mass of individual eggs (during 28-31 wk of age, i.e. 28 d) in terms of the ranking in laying sequence. A) Observed individual trajectories. B) Predicted population average (median) and 90% CI, overlapped on the data to which the model was fitted to individual by individual. C) As B, but showing individual model-fitted trajectories. D) Prediction as in B, but beyond the data range (29 eggs were laid) to 50 eggs laid. E) Daily egg mass laid, population average and CI; the egg mass increases very gently throughout.In the majority of occasions, the egg mass is that of a single egg laid; on a few occasions, some individuals laid >1 egg (i.e., 2 or 3; Figure4E), in which cases the egg mass is that of the total eggs laid.

Figure 5 .

 5 Figure 5. Gestation and lactation in sows. A-B, E-F) Artificial dataset of individual-sow trajectories for each of the traits considered: live weight, protein weight, lipid weight, and mass of milk produced during lactation. C-D, G-H) Average and 90% CI of population distribution of each of the traits.

Figure 6 .

 6 Figure 6. Gestation and lactation in sows. Deviation of the 90% CI with respect to the population average (DRtA). A, B, C, D correspond to C, D, G, H in Figure 2. Other details as in Figure 5.

Figure A1 .

 A1 Figure A1. Growth in boars -Diagnostics of model fitting and outliers. The outcomes of fitting the Gompertz curve, Y=a*(Y0/a)^{exp(-(X-X0)/b)), to each individual dataset (X,Y), where parameters (a,b) are the mature weight and time scale of growth (1/rate of growth), and vectors X and Y are the sequences of observed age and live weight: A) Population joint distribution of the point estimates of (a,b) and associated 95% confidence intervals. B-C) Marginal frequency distributions of parameters a and b. D) Fitted curved and datasets. E) Sequential residuals (deviations of the date from the model) for each individual. F) Frequency distribution of residuals for each individual. G-H) Population distribution of parameters in 1000 large random re-sample.

Table of contents

 of