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Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?
- R. Muñoz-Tamayo, L. Puillet, J. B. Daniel, D. Sauvant, O. Martin, M. Taghipoor, P. Blavy
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What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.
Review: Deciphering animal robustness. A synthesis to facilitate its use in livestock breeding and management
- N. C. Friggens, F. Blanc, D. P. Berry, L. Puillet
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As the environments in which livestock are reared become more variable, animal robustness becomes an increasingly valuable attribute. Consequently, there is increasing focus on managing and breeding for it. However, robustness is a difficult phenotype to properly characterise because it is a complex trait composed of multiple components, including dynamic elements such as the rates of response to, and recovery from, environmental perturbations. In this review, the following definition of robustness is used: the ability, in the face of environmental constraints, to carry on doing the various things that the animal needs to do to favour its future ability to reproduce. The different elements of this definition are discussed to provide a clearer understanding of the components of robustness. The implications for quantifying robustness are that there is no single measure of robustness but rather that it is the combination of multiple and interacting component mechanisms whose relative value is context dependent. This context encompasses both the prevailing environment and the prevailing selection pressure. One key issue for measuring robustness is to be clear on the use to which the robustness measurements will employed. If the purpose is to identify biomarkers that may be useful for molecular phenotyping or genotyping, the measurements should focus on the physiological mechanisms underlying robustness. However, if the purpose of measuring robustness is to quantify the extent to which animals can adapt to limiting conditions then the measurements should focus on the life functions, the trade-offs between them and the animal’s capacity to increase resource acquisition. The time-related aspect of robustness also has important implications. Single time-point measurements are of limited value because they do not permit measurement of responses to (and recovery from) environmental perturbations. The exception being single measurements of the accumulated consequence of a good (or bad) adaptive capacity, such as productive longevity and lifetime efficiency. In contrast, repeated measurements over time have a high potential for quantification of the animal’s ability to cope with environmental challenges. Thus, we should be able to quantify differences in adaptive capacity from the data that are increasingly becoming available with the deployment of automated monitoring technology on farm. The challenge for future management and breeding will be how to combine various proxy measures to obtain reliable estimates of robustness components in large populations. A key aspect for achieving this is to define phenotypes from consideration of their biological properties and not just from available measures.
Introducing efficiency into the analysis of individual lifetime performance variability: a key to assess herd management
- L. Puillet, O. Martin, D. Sauvant, M. Tichit
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Lifetime performance variability is a powerful tool for evaluating herd management. Although efficiency is a key aspect of performance, it has not been integrated into existing studies on the variability of lifetime performance. The goal of the present article is to analyse the effects of various herd management options on the variability of lifetime performance by integrating criteria relative to feed efficiency. A herd model developed for dairy goat systems was used in three virtual experiments to test the effects of the diet energy level, the segmentation of the feeding plan and the mean production potential of the herd on the variability of lifetime performance. Principal component analysis showed that the variability of lifetime performance was structured around the first axis related to longevity and production and the second related to the variables used in feed efficiency calculation. The intra-management variability was expressed on the first axis (longevity and production), whereas the inter-management variability was expressed on the second axis (feed efficiency) and was mainly influenced by the combination of the diet energy level and the mean production potential. Similar feed efficiencies were attained with different management options. Still, such combinations relied on different biological bases and, at the level of the individual, contrasting results were observed in the relationship between the obtained pattern of performance (in response to diet energy) and the reference pattern of performance (defined by the production potential). Indeed, our results showed that over-feeding interacted with the feeding plan segmentation: a high level of feeding plan segmentation generated a low proportion of individuals at equilibrium with their production potential, whereas a single ration generated a larger proportion. At the herd level, the diet energy level and the herd production potential had marked effects on production and efficiency due to dilution of fixed production costs (i.e. maintenance requirements). Management options led to similar production and feed efficiencies at the herd level while giving large contrasts in the proportions of individuals at equilibrium with their production potential. These results suggested that analysing individual variability on the basis of criteria related to production processes could improve the assessment of herd management. The herd model opens promising perspectives in studying whether individual variability represents an advantage for herd performance.
An individual-based model simulating goat response variability and long-term herd performance
- L. Puillet, O. Martin, D. Sauvant, M. Tichit
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Finding ways of increasing the efficiency of production systems is a key issue of sustainability. System efficiency is based on long-term individual efficiency, which is highly variable and management driven. To study the effects of management on herd and individual efficiency, we developed the model simulation of goat herd management (SIGHMA). This dynamic model is individual-based and represents the interactions between technical operations (relative to replacement, reproduction and feeding) and individual biological processes (performance dynamics based on energy partitioning and production potential). It simulates outputs at both herd and goat levels over 20 years. A farmer’s production project (i.e. a targeted milk production pattern) is represented by configuring the herd into female groups reflecting the organisation of kidding periods. Each group is managed by discrete events applying decision rules to simulate the carrying out of technical operations. The animal level is represented by a set of individual goat models. Each model simulates a goat’s biological dynamics through its productive life. It integrates the variability of biological responses driven by genetic scaling parameters (milk production potential and mature body weight), by the regulations of energy partitioning among physiological functions and by responses to diet energy defined by the feeding strategy. A sensitivity analysis shows that herd efficiency was mainly affected by feeding management and to a lesser extent by the herd production potential. The same effects were observed on herd milk feed costs with an even lower difference between production potential and feeding management. SIGHMA was used in a virtual experiment to observe the effects of feeding strategies on herd and individual performances. We found that overfeeding led to a herd production increase and a feed cost decrease. However, this apparent increase in efficiency at the herd level (as feed cost decreased) was related to goats that had directed energy towards body reserves. Such a process is not efficient as far as feed conversion is concerned. The underfeeding strategy led to production decrease and to a slight feed cost decrease. This apparent increase in efficiency was related to goats that had mobilised their reserves to sustain production. Our results highlight the interest of using SIGHMA to study the underlying processes affecting herd performance and analyse the role of individual variability regarding herd response to management. It opens perspectives to further quantify the link between individual variability, herd performance and management and thus further our understanding of livestock farming systems.
Simple representation of physiological regulations in a model of lactating female: application to the dairy goat
- L. Puillet, O. Martin, M. Tichit, D. Sauvant
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A dynamic model of the lactating dairy goat, combining a minimum of mechanistic representations of homeorhetic regulations and a long-term approach, was developed. It describes (i) the main changes in body weight, dry-matter intake, milk production and composition of a dairy goat; (ii) the succession of pregnancy and lactation throughout the productive life; and (iii) the major changes in dynamics induced by the female profile (production potential and body weight at maturity). The model adopts a ‘pull’ approach including a systematic expression of the production potential and not representing any feed limitation. It involves three sub-systems. The reproductive events sub-system drives the dynamics through time with three major events: service, kidding and drying off. It also accounts for the effect of production potential (kg of milk at the peak of lactation) and lactation number (potential reached at the fourth lactation). The regulating sub-system represents the homeorhetic mechanisms during pregnancy and lactation with two sets of theoretical hormones, one representing gestation and the other lactation. The operating sub-system describes the main physiological flows and the energetic requirements linked to these functions through a compartmental structure. Simulations were run in order to test (i) the behaviour of the model at the scale of the productive life for an average profile of female (60 kg at maturity and 4 kg of milk at peak); (ii) the sensitivity of the simulated dynamics (mainly milk production and body reserves) to the production potential and body weight at maturity; (iii) external validation with comparison of model outputs to data from the experimental flock of Grignon and data from the French milk record organization (French organism in charge of animal recording for dairy farmers). The results at the scale of one productive life show the model simulates a relevant set of dynamics. The sensitivity analysis suggests that the model fairly well simulates the link between a female’s ability to produce and mobilise reserves. Finally, external validation confirms the model’s ability to simulate a relevant set of physiological dynamics while pointing out some limits of the model (simulation of milk fat and protein content dynamics, for example). The results illustrate the relevance of the model in simulating biological dynamics and confirm the possibility of including minimum representations of homeorhetic regulations with a simple structure. This simplicity gives an opportunity to integrate this basic element in a herd simulator and test interactions between females’ regulations and management rules.