This contribution focuses on applying mathematical models based on systems of ordinary first-order differential equations to synthesize and interpret data from egg production experiments. Models based on linear systems of differential equations are contrasted with those based on nonlinear systems. Regression equations arising from analytical solutions to linear compartmental schemes are considered as candidate functions for describing egg production curves, together with aspects of parameter estimation. Extant candidate functions are reviewed, a role for growth functions such as the Gompertz equation suggested, and a function based on a simple new model outlined. Structurally, the new model comprises a single pool with an inflow and an outflow. Compartmental simulation models based on nonlinear systems of differential equations, and thus requiring numerical solution, are next discussed, and aspects of parameter estimation considered. This type of model is illustrated in relation to development and evaluation of a dynamic model of calcium and phosphorus flows in layers. The model consists of 8 state variables representing calcium and phosphorus pools in the crop, stomachs, plasma, and bone. The flow equations are described by Michaelis-Menten or mass action forms. Experiments that measure Ca and P uptake in layers fed different calcium concentrations during shell-forming days are used to evaluate the model. In addition to providing a useful management tool, such a simulation model also provides a means to evaluate feeding strategies aimed at reducing excretion of potential pollutants in poultry manure to the environment.

The objectives of the study were to develop a multivariate framework for analyzing energy balance data from lactating cows and investigate potential changes in maintenance requirements and partial efficiencies of energy utilization by lactating cows over the years. The proposed model accounted for the fact that metabolizable energy intake, milk energy output, and tissue energy balance are random variables that interact mutually. The model was specified through structural equations implemented in a Bayesian framework. The structural equations, along with a model traditionally used to estimate energetic parameters, were fitted to a large database of indirect calorimetry records from lactating cows. Maintenance requirements and partial efficiencies for both models were similar to values reported in the literature. In particular, the estimated parameters (with 95% credible interval in parentheses) for the proposed model were: net energy requirement for maintenance equal to 0.36 (0.34, 0.38) MJ/kg of metabolic body weight·day; the efficiency of utilizing dietary energy for milk production and tissue gain were 0.63 (0.61, 0.64) and 0.70 (0.68, 0.72), respectively; the efficiency of utilizing body stores for milk production was 0.89 (0.87, 0.91). Furthermore, additional analyses were conducted for which energetic parameters were allowed to depend on the decade in which studies were conducted. These models investigated potential changes in maintenance requirements and partial efficiencies over the years. Canonical correlation analysis was used to investigate the association between changes in energetic parameters with additional dietary and animal characteristics available in the database. For both models, net energy requirement for maintenance and the efficiency of utilizing dietary energy for milk production and tissue gain increased in the more recent decades, whereas the efficiency of utilizing body stores for milk production remained unchanged. The increase in maintenance requirements in modern milk production systems is consistent with the literature that describes increased fasting heat production in cows of higher genetic merit. The increase in utilization of dietary energy for milk production and tissue gain was partially attributed to the changes in dietary composition, in particular to the increase in dietary ether extract to levels closer to currently observed in modern milk production systems. Therefore, the estimated energetic parameters from this study can be used to update maintenance requirements and partial efficiencies of energy utilization in North American feeding systems for lactating cows.

A mechanistic model (COWPOLL) was used to estimate enteric methane (CH4) emissions from beef production systems in Chile. The results expressed as a proportion of gross energy intake (GEI) were compared with enteric fermentation data reported in the last Chilean greenhouse gases inventory, which utilized an earlier the Intergovernmental Panel on Climate Change Tier 2 approach. The simulation analysis was based on information from feedstuffs, dry matter intake (DMI), body weight (BW) and average daily gain (ADG) of steers raised and finished at two research facilities located in Central and Southern Chile, as well as three simulated scenarios for grass-based finishing systems in Southern Chile. Data for feedlot production systems in the central region were assessed by considering steers fed a forage : concentrate ratio of 23 : 77 using maize silage and wheat straw as roughage sources during the stages of backgrounding and fattening. Average DMI were 7.3±0.62 and 9.2±0.55 kg/day per steer for backgrounding and fattening, respectively, whereas ADG were 1.1±0.22 and 1.3±0.37 kg/day for backgrounding and fattening. For the Southern Chilean fattening production systems, the forage : concentrate ratio was 56 : 44 with ryegrass pasture as the sole forage source. In this case, average DMI was 9.97±0.51 and ADG was 1.1±0.24 kg/day per steer. Two of the grass-based scenarios used the same initial BW information as that used for the Central and Southern Chilean systems, but feedlot diets were replaced by ryegrass pasture. The third grass-based scenario used an initial BW of 390 kg. In all the grass-based scenarios an ADG of 0.90 kg/day, with maximum DMI estimated as a proportion of BW (0.01 of NDF, kg/kg BW), was assumed. The results of the simulation analysis showed that emission factors (Ym; fraction of GEI) ranged from 0.062 to 0.079 of GEI. Smaller values were associated with finishing systems that included a lower proportion of forage in the diet due to higher propionate production, which serves as a sink for hydrogen in the rumen. Cattle finished in feedlot systems had an average of 0.062 of GEI lost as CH4, whereas grass-based cattle had losses of 0.079 of GEI. Enteric CH4 emissions for the systems using grass-based and concentrate diets were 261 and 159 g/kg weight gain, respectively. The Chilean CH4 inventory employs a fixed Ym of 0.060 to estimate enteric fermentation for all cattle. This value is lower than the average Ym obtained in the current simulation analysis (0.071 of GEI), which results in underestimation of enteric CH4 emissions from beef cattle. However, these results need to be checked against field measurements of CH4 emissions. Implementation of mechanistic models in the preparation of national greenhouse gas inventories is feasible if appropriate information is provided, allowing dietary characteristics and regional particularities to be taken into consideration.

Phytase, a widely used feed additive in poultry diets, increases P availability and subsequently reduces inorganic-P supplementation and P-excretion. Phytase supplementation effect on P-retention in poultry has been investigated, but the effect sizes were highly variable. The present study's objective was to conduct several meta-analyses to quantitatively summarize the phytase effect on P-retention in broilers and layers. Data from 103 and 26 controlled experiments testing the phytase effect on P-retention were included in 2 separate meta-analyses for broilers and layers, respectively. The mean difference calculated by subtracting the means of P-retention for the control group from the phytase-supplemented group was chosen as an effect size estimate. Between-study variability (heterogeneity) of mean difference was estimated using random-effect models and had a significant effect (P < 0.01) in both broilers and layers. Therefore, random-effect models were extended to mixed-effect models to explain heterogeneity and obtain final phytase effect size estimates. Available dietary and bird variables were included as fixed effects in the mixed-effect models. The final broiler mixed-effect model included phytase dose and Ca-to-total-P ratio (Ca:tP), explaining 15.6% of the heterogeneity. Other variables such as breed might further explain between-study variance. Broilers consuming control diets were associated with 48.4% P-retention. Exogenous phytase supplementation at 1,039 FTU/kg of diet increased P-retention by 8.6 percentage units on average. A unit increase of phytase dose and Ca:tP from their means further increased P-retention. For layers, the final mixed-effect models included dietary Ca, age, and experimental period length. The variables explained 65.9% of the heterogeneity. Layers receiving exogenous phytase at 371 FTU/kg were associated with a 5.02 percentage unit increase in P-retention. A unit increase in dietary Ca from its mean increased P-retention, whereas an increase in the experiment length and layer's age decreased P-retention. Phytase supplementation had a significant positive effect on P-retention in both broilers and layers, but effect sizes across studies were significantly heterogeneous due to differences in Ca contents, experiment length, bird age, and phytase dose.

Linear and non-linear models have been extensively utilised for the estimation of net and metabolisable energy requirements and for the estimation of the efficiencies of utilising dietary energy for maintenance and tissue gain. In growing animals, biological principles imply that energy retention rate is non-linearly related to the energy intake level because successive increments in energy intake above maintenance result in diminishing returns for tissue energy accretion. Heat production in growing cattle has been traditionally described by logarithmic regression and exponential models. The objective of the present study was to develop Bayesian models of energy retention and heat production in growing cattle using parametric and non-parametric techniques. Parametric models were used to represent models traditionally employed to describe energy use in growing steers and heifers whereas the non-parametric approach was introduced to describe energy utilisation while accounting for non-linearities without specifying a particular functional form. The Bayesian framework was used to incorporate prior knowledge of bioenergetics on tissue retention and heat production and to estimate net and metabolisable energy requirements (NEM and MEM, respectively), and the partial efficiencies of utilising dietary metabolisable energy for maintenance (km) and tissue energy gain (kg). The database used for the study consisted of 719 records of indirect calorimetry on steers and non-pregnant, non-lactating heifers. The NEM was substantially larger in energy retention models (ranged from 0.40 to 0.50 MJ/kg BW0.75.day) than were NEM estimates from heat-production models (ranged from 0.29 to 0.49 MJ/kg BW0.75.day). Similarly, km was also larger in energy retention models than in heat production models. These differences are explained by the nature of y-intercepts (NEM) in these two models. Energy retention models estimate fasting catabolism as the y-intercept, while heat production models estimate fasting heat production. Conversely, MEM was virtually identical in all models and approximately equal to 0.53 MJ/kg BW0.75.day in this database.

Six classical growth functions (monomolecular, Schumacher, Gompertz, logistic, Richards, and Morgan) were fitted to individual and average (by parity) cumulative milk production curves of Canadian Holstein dairy cows. The data analyzed consisted of approximately 91,000 daily milk yield records corresponding to 122 first, 99 second, and 92 third parity individual lactation curves. The functions were fitted using nonlinear regression procedures, and their performance was assessed using goodness-of-fit statistics (coefficient of determination, residual mean squares, Akaike information criterion, and the correlation and concordance coefficients between observed and adjusted milk yields at several days in milk). Overall, all the growth functions evaluated showed an acceptable fit to the cumulative milk production curves, with the Richards equation ranking first (smallest Akaike information criterion) followed by the Morgan equation. Differences among the functions in their goodness-of-fit were enlarged when fitted to average curves by parity, where the sigmoidal functions with a variable point of inflection (Richards and Morgan) outperformed the other 4 equations. All the functions provided satisfactory predictions of milk yield (calculated from the first derivative of the functions) at different lactation stages, from early to late lactation. The Richards and Morgan equations provided the most accurate estimates of peak yield and total milk production per 305-d lactation, whereas the least accurate estimates were obtained with the logistic equation. In conclusion, classical growth functions (especially sigmoidal functions with a variable point of inflection) proved to be feasible alternatives to fit cumulative milk production curves of dairy cows, resulting in suitable statistical performance and accurate estimates of lactation traits.

A meta-analysis investigation based on literature data was conducted to estimate the effect size of nutritional and animal factors on phosphorus (P) excretion in feces and concentrations of P in milk. Two data sets were created for statistical analysis: One to derive prediction equations for P in feces (25 studies; 130 treatments) and another for P in milk (19 studies; 94 treatments). Prediction equations were derived using mixed model regression analysis with a random effect for study, and equations were evaluated based on values for Bayesian information criterion (BIC), root mean square prediction error (RMSPE), and concordance correlation coefficient (CCC) statistics. In terms of RMSPE and CCC values, fecal P excretion was best predicted by P intake, where P in feces (g/d)=-3.8(±3.45) + 0.64(±0.038) × P intake (g/d) (RMSPE: 18.3%, CCC: 0.869). However, significant effects of crude protein [g/kg of dry matter (DM)], neutral detergent fiber (g/kg of DM), and milk yield (kg/d) on fecal P excretion were also found. Despite a lack of improvement in terms of RMSPE and CCC values, these parameters may still explain part of the variation in fecal P excretion. For milk P, expressed as a fraction of P intake, the following equation had the highest CCC and the lowest RMSPE value: P in milk as a fraction of P intake (g/g)=0.42(±0.065) + 0.23(±0.018) × feed efficiency (i.e., fat- and protein-corrected milk yield/dry matter intake) - 0.11(±0.0199) × P in feed (g/kg of DM) (RMSPE: 19.7%; CCC: 0.761). Equations derived to predict fecal P as a fraction of P intake (g/g) or milk P content (g/kg) could not adequately explain the observed variation and did not perform well in terms of RMSPE and CCC values. Examination of the residuals showed that P balance was a seemingly confounding factor in some of the models. The results presented here can be used to estimate P in feces and milk based on commonly measured dietary and milk variables, but could also be used to guide development of mechanistic models on P metabolism in lactating dairy cattle. Factors to consider in future research and modeling efforts regarding efficiency of P use include the effects of dietary neutral detergent fiber, crude protein, starch, variation in P content of milk, and effects of P resorption from bone and body tissues during early lactation.

Quantification of dairy cattle nitrogen (N) excretion and secretion is necessary to improve the efficiency with which feed N is converted to milk N (ENU). Faecal and urinary N excretion and milk N secretion are correlated with each other and thus are more accurately described by a multivariate model that can accommodate the covariance between the three observations than by three separate univariate models. Further, by simultaneously predicting the three routes of excretion and taking advantage of the mass balance relationships between them, covariate effects on N partitioning from feed to faeces and absorbed N and from absorbed N to milk and urine N and animal ENU can be estimated. A database containing 1094 lactating dairy cow observations collated from indirect calorimetry experiments was used for model development. Dietary metabolisable energy content (ME, MJ/kg DM) increased ENU at a decreasing rate, increased the efficiency with which feed N was converted to absorbed N and decreased the efficiency with which absorbed N was converted to milk N. However, the parameter estimate of the effect of ME on post-absorption efficiency was not different from zero when the model was fitted to a data subset in which net energy and metabolisable protein were at or above requirement. This suggests the effect of ME on post-absorption N use is dependent on the energy status of the animal.

In vitro fermentation techniques (IVFT) have been widely used to evaluate the nutritive value of feeds for ruminants and in the last decade to assess the effect of different nutritional strategies on methane (CH4) production. However, many technical factors may influence the results obtained. The present review has been prepared by the ‘Global Network’ FACCE-JPI international research consortium to provide a critical evaluation of the main factors that need to be considered when designing, conducting and interpreting IVFT experiments that investigate nutritional strategies to mitigate CH4 emission from ruminants. Given the increasing and wide-scale use of IVFT, there is a need to critically review reports in the literature and establish what criteria are essential to the establishment and implementation of in vitro techniques. Key aspects considered include: i) donor animal species and number of animal used, ii) diet fed to donor animals, iii) collection and processing of rumen fluid as inoculum, iv) choice of substrate and incubation buffer, v) incubation procedures and CH4 measurements, vi) headspace gas composition and vii) comparability of in vitro and in vivo measurements. Based on an evaluation of experimental evidence, a set of technical recommendations are presented to harmonize IVFT for feed evaluation, assessment of rumen function and CH4 production.

Monensin is a widely used feed additive with the potential to minimize methane (CH4) emissions from cattle. Several studies have investigated the effects of monensin on CH4, but findings have been inconsistent. The objective of the present study was to conduct meta-analyses to quantitatively summarize the effect of monensin on CH4 production (g/d) and the percentage of dietary gross energy lost as CH4 (Ym) in dairy cows and beef steers. Data from 22 controlled studies were used. Heterogeneity of the monensin effects were estimated using random effect models. Due to significant heterogeneity (>68%) in both dairy and beef studies, the random effect models were then extended to mixed effect models by including fixed effects of DMI, dietary nutrient contents, monensin dose, and length of monensin treatment period. Monensin reduced Ym from 5.97 to 5.43% and diets with greater neutral detergent fiber contents (g/kg of dry matter) tended to enhance the monensin effect on CH4 in beef steers. When adjusted for the neutral detergent fiber effect, monensin supplementation [average 32 mg/kg of dry matter intake (DMI)] reduced CH4 emissions from beef steers by 19±4 g/d. Dietary ether extract content and DMI had a positive and a negative effect on monensin in dairy cows, respectively. When adjusted for these 2 effects in the final mixed-effect model, monensin feeding (average 21 mg/kg of DMI) was associated with a 6±3 g/d reduction in CH4 emissions in dairy cows. When analyzed across dairy and beef cattle studies, DMI or monensin dose (mg/kg of DMI) tended to decrease or increase the effect of monensin in reducing methane emissions, respectively. Methane mitigation effects of monensin in dairy cows (-12±6 g/d) and beef steers (-14±6 g/d) became similar when adjusted for the monensin dose differences between dairy cow and beef steer studies. When adjusted for DMI differences, monensin reduced Ym in dairy cows (-0.23±0.14) and beef steers (-0.33±0.16). Monensin treatment period length did not significantly modify the monensin effects in dairy cow or beef steer studies. Overall, monensin had stronger antimethanogenic effects in beef steers than dairy cows, but the effects in dairy cows could potentially be improved by dietary composition modifications and increasing the monensin dose.