Review: The prevailing mathematical modeling classifications and paradigms to support the advancement of sustainable animal production

Mathematical modeling is typically framed as the art of reductionism of scientific knowledge into an arithmetical layout. However, most untrained people get the art of modeling wrong and end up neglecting it because modeling is not simply about writing equations and generating numbers through simulations. Models tell not only about a story; they are spoken to by the circumstances under which they are envisioned. They guide apprentice and experienced modelers to build better models by preventing known pitfalls and invalid assumptions in the virtual world and, most importantly, learn from them through simulation and identify gaps in pushing scientific knowledge further. The power of the human mind is well-documented for idealizing concepts and creating virtual reality models, and as our hypotheses grow more complicated and more complex data become available, modeling earns more noticeable footing in biological sciences. The fundamental modeling paradigms include discrete-events, dynamic systems, agent-based (AB), and system dynamics (SD). The source of knowledge is the most critical step in the model-building process regardless of the paradigm, and the necessary expertise includes (a) clear and concise mental concepts acquired through different ways that provide the fundamental structure and expected behaviors of the model and (b) numerical data necessary for statistical analysis, not for building the model. The unreasonable effectiveness of models to grow scientific learning and knowledge in sciences arise because different researchers would model the same problem differently, given their knowledge and experiential background, leading to choosing different variables and model structures. Secondly, different researchers might use different paradigms and even unalike mathematics to resolve the same problem; thus, model needs are intrinsic to their perceived assumptions and structures. Thirdly, models evolve as the scientific community knowledge accumulates and matures over time, hopefully resulting in improved modeling efforts; thus, the perfect model is fictional. Some paradigms are most appropriate for macro, high abstraction with less detailed-oriented scenarios, while others are most suitable for micro, low abstraction with higher detailed-oriented strategies. Modern hybridization aggregating artificial intelligence (AI) to mathematical models can become the next technological wave in modeling. AI can be an integral part of the SD/AB models and, before long, write the model code by itself. Success and failures in model building are more related to the ability of the researcher to interpret the data and understand the underlying principles and mechanisms to formulate the correct relationship among variables rather than profound mathematical knowledge.

PhenoBR: a model to phenotype body condition dynamics in meat sheep

In situations of negative energy balance (NEB) due to feed scarcity or high physiological demands, body energy reserves (BRs), mainly stored in adipose tissues, become the main sources of energy for ruminants. The capacity to mobilise and restore such BRs in response to different challenges is of major concern in the current context of breeding for resilience. Body condition score (BCS) is a common, practical indicator of BR variations throughout successive productive cycles, and quantitative tools for characterising such dynamics at the individual level are still lacking. The main objective of this work was to characterise body condition dynamics in terms of BR mobilisation and accretion capacities of meat sheep during their productive lifespan through a modelling approach, using BCS measurements. The animal model used in this work was the reproductive meat ewe (n = 1 478) reared in extensive rangeland. Regular measurements of BCS for each productive cycle were used as the indicator of BR variations. A hybrid mathematical model and a web interface, called PhenoBR, were developed to characterise ewes' BCS variations through four synthetic and biologically meaningful parameters for each productive cycle i: BR accretion rate (k_bⁱ), BR mobilisation rate (k_pⁱ), plus the time of onset and the duration of the BR mobilisation, t_bⁱ and ΔTⁱ, respectively. The model PhenoBR converged for all the ewes included in the analysis. Estimation of the parameters indicated the inter-individual variability for BR accretion and mobilisation rates, and the length of the mobilisation period. The present study is a proof of concept that the combination of data-driven and concept-driven models is required for the estimation of biologically meaningful parameters that describe body reserve dynamics through consecutive productive cycles. Individual characterisation of animals by these parameters makes it possible to rank them for their efficiency in the use of body reserves when facing NEB challenges. Such parameters could contribute to better management and decision-making by farmers and advisors, e.g. by adapting feeding systems to the individual characteristics of BR dynamics, or by geneticists as criteria to develop future animal breeding programmes including BR dynamics for more robust and resilient animals.

Prediction of carcass composition through measurements in vivo and measurements of the carcass of growing Santa Inês sheep

In vivo and carcass measurements were evaluated to predict carcass physical and chemical composition and to list the measurements that best fit the prediction of the composition of growing Santa Inês sheep carcasses. Thirty-three animals were used to measure the loin eye area by ultrasound in vivo (LEAu) and in the carcass. We used 39 animals for biometric measurement in vivo and 42 sheep for morphometric measurement in the carcass. For the physical and chemical compositions of carcasses, dissection of the half left carcass was carried out in 42 animals. The data were submitted to Pearson’s correlation analysis and t test. Simple and multiple linear regressions were performed using a stepwise procedure. All correlations between in vivo measurements and the physical and chemical compositions of carcasses (in kg) were significant, except for LEAu. Biometric measurements and hot (HCW) and cold (CCW) carcass weights were considered as predictors of the carcasses’ physical and chemical compositions. Slaughter body weight (SBW) was the variable that most influenced the equations in the assessment of in vivo measurements and HCW and CCW most influenced the equations for measurements on carcasses. Biometric measurements of Santa Inês sheep can be used together with the SBW to estimate the physical and chemical compositions of carcasses, with emphasis on body compactness index, breast width, wither height, and croup height. The morphometric measurements can be used together with carcass weight to estimate the physical and chemical compositions of carcasses, with emphasis on croup width, carcass compactness index, croup perimeter, external and internal carcass lengths, chest width, and leg length and perimeter. The HCW can be used to predict the physical and chemical composition of carcasses without affecting the accuracy of the prediction model.

Prediction of Carcass Traits of Hair Sheep Lambs Using Body Measurements

Simple Summary

Some authors have reported that the use of body measurements (BMs) could be a useful tool for predicting carcass characteristics in sheep. Hair sheep breeds have been adopted for lamb production in the tropical regions of Latin America. Among these, Pelibuey and Katahdin breeds and their crosses have shown great reproductive capacity and adaptation, contributing to improving the productive efficiency of flocks in tropical production systems. However, few studies have been carried out on this breeds to define its BMs correctly, and little work has been found using BMs to predict the carcass characteristics in different physiological stages.

Abstract

The present study was designed to evaluate the relationship between the body measurements (BMs) and carcass characteristics of hair sheep lambs. Twenty hours before slaughter, the shrunk body weight (SBW) and BMs were recorded. The BMs involved were height at withers (HW), rib depth (RD), body diagonal length (BDL), body length (BL), pelvic girdle length (PGL), rump depth (RuD), rump height (RH), pin-bone width (PBW), hook-bone width (HBW), abdomen width (AW), girth (GC), and abdomen circumference (AC). After slaughter, the carcasses were weighed and chilled for 24 h at 1 °C, and then were split by the dorsal midline. The left-half was dissected into total soft tissues (muscle + fat; TST) and bone (BON), which were weighed separately. The weights of viscera and organs (VIS), internal fat (IF), and offals (OFF—skin, head, feet, tail, and blood) were also recorded. The equations obtained for predicting SBW, HCW, and CCW had an r² ranging from 0.89 to 0.99, and those for predicting the TST and BON had an r² ranging from 0.74 to 0.91, demonstrating satisfactory accuracy. Our results indicated that use of BMs could accurately and precisely be used as a useful tool for predicting carcass characteristics of hair sheep lambs.

ASN-ASAS SYMPOSIUM: FUTURE OF DATA ANALYTICS IN NUTRITION: Mathematical modeling in ruminant nutrition: approaches and paradigms, extant models, and thoughts for upcoming predictive analytics1,2

This paper outlines typical terminology for modeling and highlights key historical and forthcoming aspects of mathematical modeling. Mathematical models (MM) are mental conceptualizations, enclosed in a virtual domain, whose purpose is to translate real-life situations into mathematical formulations to describe existing patterns or forecast future behaviors in real-life situations. The appropriateness of the virtual representation of real-life situations through MM depends on the modeler's ability to synthesize essential concepts and associate their interrelationships with measured data. The development of MM paralleled the evolution of digital computing. The scientific community has only slightly accepted and used MM, in part because scientists are trained in experimental research and not systems thinking. The scientific advancements in ruminant production have been tangible but incipient because we are still learning how to connect experimental research data and concepts through MM, a process that is still obscure to many scientists. Our inability to ask the right questions and to define the boundaries of our problem when developing models might have limited the breadth and depth of MM in agriculture. Artificial intelligence (AI) has been developed in tandem with the need to analyze big data using high-performance computing. However, the emergence of AI, a computational technology that is data-intensive and requires less systems thinking of how things are interrelated, may further reduce the interest in mechanistic, conceptual MM. Artificial intelligence might provide, however, a paradigm shift in MM, including nutrition modeling, by creating novel opportunities to understand the underlying mechanisms when integrating large amounts of quantifiable data. Associating AI with mechanistic models may eventually lead to the development of hybrid mechanistic machine-learning modeling. Modelers must learn how to integrate powerful data-driven tools and knowledge-driven approaches into functional models that are sustainable and resilient. The successful future of MM might rely on the development of redesigned models that can integrate existing technological advancements in data analytics to take advantage of accumulated scientific knowledge. However, the next evolution may require the creation of novel technologies for data gathering and analyses and the rethinking of innovative MM concepts rather than spending resources in collecting futile data or amending old technologies.

The assessment of supplementation requirements of grazing ruminants using nutrition models

This paper was aimed to summarize known concepts needed to comprehend the intricate interface between the ruminant animal and the pasture when predicting animal performance, acknowledge current efforts in the mathematical modeling domain of grazing ruminants, and highlight current thinking and technologies that can guide the development of advanced mathematical modeling tools for grazing ruminants. The scientific knowledge of factors that affect intake of ruminants is broad and rich, and decision-support tools (DST) for modeling energy expenditure and feed intake of grazing animals abound in the literature but the adequate predictability of forage intake is still lacking, remaining a major challenge that has been deceiving at times. Despite the mathematical advancements in translating experimental research of grazing ruminants into DST, numerous shortages have been identified in current models designed to predict intake of forages by grazing ruminants. Many of which are mechanistic models that rely heavily on preceding mathematical constructions that were developed to predict energy and nutrient requirements and feed intake of confined animals. The data collection of grazing (forage selection, grazing behavior, pasture growth/regrowth, pasture quality) and animal (nutrient digestion and absorption, volatile fatty acids production and profile, energy requirement) components remains a critical bottleneck for adequate modeling of forage intake by ruminants. An unresolved question that has impeded DST is how to assess the quantity and quality, ideally simultaneously, of pasture forages given that ruminant animals can be selective. The inadequate assessment of quantity and quality has been a hindrance in assessing energy expenditure of grazing animals for physical activities such as walking, grazing, and forage selection of grazing animals. The advancement of sensors might provide some insights that will likely enhance our understanding and assist in determining key variables that control forage intake and animal activity. Sensors might provide additional insights to improve the quantification of individual animal variation as the sensor data are collected on each subject over time. As a group of scientists, however, despite many obstacles in animal and forage science research, we have thrived, and progress has been made. The scientific community may need to change the angle of which the problem has been attacked, and focus more on holistic approaches.

A dynamic model as a tool to describe the variability of lifetime body weight trajectories in livestock females

Until now, the development of precision livestock farming has been largely based on data acquisition automation. The future challenge is to develop interpretative tools to capitalize on high-throughput raw data and to generate benchmarks for phenotypic traits. We developed a dynamic model of female BW that converts BW time series into a vector of biologically meaningful parameters. The model is based on a first submodel that split a female's weight into elementary mass changes related to biological functions: growth (G component), reserves balance (R component), uterine load (U component), and maternal investment (M component). These elementary weight components are linked to the second submodel, which represents the litter developmental stages (oocyte, fetus, neonate, and juvenile) that drive elementary components of dam weight over each reproductive cycle. The so-called GRUM model is based on ordinary differential equations and laws of mass action. Input data are BW measures, age, and litter weight at birth for each parturition. Outputs of the fitting procedure are a vector of parameters related to each GRUM component and indexed by reproductive cycle. We illustrated the potential application of the model with a case study including growth and successive lactations ( = 202) from 45 dairy goats from the Alpine ( = 27) and Saanen ( = 18) breeds. The fitting procedure converged for all individuals, including goats that went through extended lactations. We analyzed the fitted parameters to quantify breed and parity effects over 4 reproductive cycles. We found significant differences between breeds regarding gestation components (fetal growth and reserves balance). We also found significant differences among reproductive cycles for reserves balance. Although these findings are based on a small sample, they illustrate how use the model can be to adapt herd management and implement grouping strategies to account for individual variability.

Development of equations, based on milk intake, to predict starter feed intake of preweaned dairy calves

There is a lack of studies that provide models or equations capable of predicting starter feed intake (SFI) for milk-fed dairy calves. Therefore, a multi-study analysis was conducted to identify variables that influence SFI, and to develop equations to predict SFI in milk-fed dairy calves up to 64 days of age. The database was composed of individual data of 176 calves from eight experiments, totaling 6426 daily observations of intake. The information collected from the studies were: birth BW (kg), SFI (kg/day), fluid milk or milk replacer intake (MI; l/day), sex (male or female), breed (Holstein or Holstein×Gyr crossbred) and age (days). Correlations between SFI and the quantitative variables MI, birth BW, metabolic birth BW, fat intake, CP intake, metabolizable energy intake, and age were calculated. Subsequently, data were graphed, and based on a visual appraisal of the pattern of the data, an exponential function was chosen. Data were evaluated using a meta-analysis approach to estimate fixed and random effects of the experiments using nonlinear mixed coefficient statistical models. A negative correlation between SFI and MI was observed (r=-0.39), but age was positively correlated with SFI (r=0.66). No effect of liquid feed source (milk or milk replacer) was observed in developing the equation. Two equations, significantly different for all parameters, were fit to predict SFI for calves that consume less than 5 (SFI5) l/day of milk or milk replacer: ${\rm SFI}_{{\,\lt\,5}} {\equals}0.1839_{{\,\pm\,0.0581}} {\times}{\rm MI}{\times}{\rm exp}^{{\left( {\left( {0.0333_{{\,\pm\,0.0021 }} {\minus}0.0040_{{\,\pm\,0.0011}} {\times}{\rm MI}} \right){\times}\left( {{\rm A}{\minus}{\rm }\left( {0.8302_{{\,\pm\,0.5092}} {\plus}6.0332_{{\,\pm\,0.3583}} {\times}{\rm MI}} \right)} \right)} \right)}} {\minus}\left( {0.12{\times}{\rm MI}} \right)$ ; ${\rm SFI}_{{\,\gt\,5}} {\equals}0.1225_{{\,\pm\,0.0005 }} {\times}{\rm MI}{\times}{\rm exp}^{{\left( {\left( {0.0217_{{\,\pm\,0.0006 }} {\minus}0.0015_{{\,\pm\,0.0001}} {\times}{\rm MI}} \right){\times}\left( {{\rm A}{\minus}\left( {3.5382_{{\,\pm\,1.3140 }} {\plus}1.9508_{{\,\pm\,0.1710}} {\times}{\rm MI}} \right)} \right)} \right)}} {\minus}\left( {0.12{\times}{\rm MI}} \right)$ where MI is the milk or milk replacer intake (l/day) and A the age (days). Cross-validation and bootstrap analyses demonstrated that these equations had high accuracy and moderate precision. In conclusion, the use of milk or milk replacer as liquid feed did not affect SFI, or development of SFI over time, which increased exponentially with calf age. Because SFI of calves receiving more than 5 l/day of milk/milk replacer had a different pattern over time than those receiving <5 l/day, separate prediction equations are recommended.