A centenary review presents an opportunity to ponder over the processes of concept development and give thought to future directions. The current review aims to ascertain the ontogeny of current concepts, underline the connection between ideas and people and pay tribute to those pioneers who have contributed significantly to modelling in animal nutrition. Firstly, the paper draws a brief portrait of the use of mathematics in agriculture and animal nutrition prior to 1925. Thereafter, attention turns towards the historical development of growth modelling, feed evaluation systems and animal response models. Introduction of the factorial and compartmental approaches into animal nutrition is noted along with the particular branches of mathematics encountered in various models. Furthermore, certain concepts, especially bioenergetics or the heat doctrine, are challenged and alternatives are reviewed. The current state of knowledge of animal nutrition modelling results mostly from the discernment and unceasing efforts of our predecessors rather than serendipitous discoveries. The current review may stimulate those who wish for greater understanding and appreciation.

A model of reproductive performance was developed to study the influences of breeding management decisions and animal characteristics on the reproductive performance and the calving distribution in a beef herd. In the model, reproductive performance is formalized as a sequence of events (parturition, ovulation, conception), each of which modifies the reproductive status of the simulated cow. With respect to reproduction, a cow can be in one of three possible states: open-not-cycling, open-cycling or pregnant. The length of the different intervals that are included between two successive reproductive events (calving to first cycle interval, length of oestrous cycles, calving–conception and calving intervals) is formalized using stochastic or empirical laws that may be influenced by numerous animal or environmental factors or by management decisions (feeding strategy, breeding season and length of the breeding period). Within the herd, cows are considered to differ from each other by their parity, calving date, body condition at calving and their bull exposure.

Calving to first oestrous interval (postpartum anoestrous interval (PPAI)) is expressed as the sum of three equations which formalize the respective effects of calving date, body condition score at calving (BCScalving) and the response to early bull exposure in interaction with BCScalving. The influences of these variables on reproductive performance were quantified by analysing data sets (three bibliographical and two experimental) or by expertise. Special attention was paid to the influence of calving date on PPAI and a biological interpretation of this effect is proposed. Probabilities of natural insemination success were estimated according to the number of oestrus and the number of matings.

The model was fitted to data from primiparous Charolais cows (n=139) bred at the experimental station Laqueuille (French National Institute for Agricultural Research (INRA)). Its ability to simulate PPAI was tested using an independent data set of primiparous Charolais cows (n=188) from the experimental farm Le Pin. The model only accounts for 39% of the observed inter-individual variability. However, the analysis of the mean square deviation components led to validation of the structure of the model. In particular, the assumption that the influence of calving date on PPAI can be attributed to a sensitivity of the reproductive function to the variation of the photoperiod during the month preceding parturition was confirmed. Simulations also revealed that fat cows could have similar anoestrus to thin cows when they are exposed early to a bull. Such a result emphasized the necessity to investigate further and better calibrate the combined effects of BCS at calving and bull exposure on PPAI.

The results from three types of study with broilers, namely nitrogen (N) balance, bioassays and growth experiments, provided the data used herein. Sets of data on N balance and protein accretion (bioassay studies) were used to assess the ability of the monomolecular equation to describe the relationship between (i) N balance and amino acid (AA) intake and (ii) protein accretion and AA intake. The model estimated the levels of isoleucine, lysine, valine, threonine, methionine, total sulphur AAs and tryptophan resulting in zero balance to be 58, 59, 80, 96, 23, 85 and 32 mg/kg live weight (LW)/day, respectively. These estimates show good agreement with those obtained in previous studies. For the growth experiments, four models, specifically re-parameterized for analysing energy balance data, were evaluated for their ability to determine crude protein (CP) intake at maintenance and efficiency of utilization of CP intake for producing gain. They were: a straight line, two equations representing diminishing returns behaviour (monomolecular and rectangular hyperbola) and one equation describing smooth sigmoidal behaviour with a fixed point of inflexion (Gompertz). The estimates of CP requirement for maintenance and efficiency of utilization of CP intake for producing gain varied from 5·4 to 5·9 g/kg LW/day and 0·60 to 0·76, respectively, depending on the models.

The Cornell-Penn-Miner (CPM) Dairy is an applied mathematical nutrition model that computes dairy cattle requirements and the supply of energy and nutrients based on characteristics of the animal, the environment and the physicochemical composition of the feeds under diverse production scenarios. The CPM Dairy was designed as a steady-state model to use rates of degradation of feed carbohydrate and protein and the rate of passage to estimate the extent of ruminal fermentation, microbial growth, and intestinal digestibility of carbohydrate and protein fractions in computing energy and protein post-rumen absorption, and the supply of metabolizable energy and protein to the animal. The CPM Dairy version 3.0 (CPM Dairy 3.0) includes an expanded carbohydrate fractionation scheme to facilitate the characterization of individual feeds and a sub-model to predict ruminal metabolism and intestinal absorption of long chain fatty acids. The CPM Dairy includes a non-linear optimization algorithm that allows for least-cost formulation of diets while meeting animal performance, feed availability and environmental restrictions of modern dairy cattle production. When the CPM Dairy 3.0 was evaluated with data of 228 individual lactating dairy cows containing appropriate information including observed dry matter intake, the linear regression between observed and model-predicted milk production values indicated the model was able to account for 79·8% of the variation. The concordance correlation coefficient (CCC) was high (rc=0·89) without a significant mean bias (0·52 kg/d; P=0·12). The accuracy estimated by the CCC was 0·997. The root of mean square error of prediction (MSEP) was 5·14 kg/d (0·16 of the observed mean) and 87·3% of the MSEP was due to random errors, suggesting little systematic bias in predicting milk production of high-producing dairy cattle. Based upon these evaluations, it was concluded the CPM Dairy 3.0 model adequately predicts milk production at the farm level when appropriate animal characterization, feed composition and feed intake are provided; however, further improvements are needed to account for individual animal variation.

A deterministic model for the spread of an important animal disease, bovine spongiform encephalopathy (BSE), is described. It is a sparse three-pool model, the pools corresponding to susceptible, infected and sick animals. Simulations illustrate the biological meaning of the parameters, and pinpoint how parameters affect epidemic characteristics: infectivity impacts on the leading edge of the epidemic and its intensity. The times when infection is introduced and when control is exercised determine the length and possibly the intensity of the epidemic. Finally, the (single) incubation rate influences the trailing edge of the epidemic. It is applied to data describing the recent UK BSE outbreak, with reasonable success. Apart from a scaling factor, only these four quantities were adjusted to achieve this. The contributions from greater modelling complexity are discussed. It is concluded that the simple model is well suited for grasping the essentials, whereas further detail is needed for a more mechanistic representation, in particular concerning incubation delays in developing infectiousness and clinical sickness.

Currently, society awareness, legislations and competing markets demand dairy farming systems which are sustainable. In the near future, farm management and animal genetics will be key elements in developing such sustainability. Although the effect of farm management on some attributes of sustainability has already been studied, the impacts and scope for realizing goals of agricultural multifunctionality through genetic changes are still to be tested. Sustainable and Integrated Management Systems for Dairy Production (SIMSDAIRY) is a new farm level modelling framework which integrates these concepts to practical actions and brings all of this complexity into an operational and scientific modus operandi.

The current paper provides a brief description of the structure of SIMSDAIRY and an example of how it can be used to compare the scope for improving the overall sustainability of a dairy farm by: (i) future system changes aimed at improving genetic characteristics of plants and animals with (ii) current system structural changes aimed at improving nutrient management efficiency. In order to do this comparison, management factors and new genetic traits from plants or/and animals, acting singly or in combination, are evaluated against a baseline dairy farm scenario. Sustainability is measured in terms of targets associated with: (i) the Nitrates Directive, (ii) phosphorus (P) threshold for eutrophication, (iii) the Kyoto Protocol, (iv) the Gothenburg Protocol, (v) an adequate net farm income for standard of living and acceptable standards of (vi) quality of milk, (vii) animal welfare, (viii) level of biodiversity, (ix) landscape aesthetics and (x) soil quality.

Results suggest that genetic-based changes offer greater scope than management-based ones to improve sustainability up to an acceptable level. Costs associated with management changes are often too high within current socio-economics circumstances. Optimizing nitrogen (N) mineral fertilizer rate and timing was the only management-based measure that, while improving most of the environmental and biodiversity indices, resulted in improved economic results. Some genetic-based changes offered substantial scope for reducing environmental losses while having economic benefits. However, only those decreasing the crude protein (CP) of the plant and increasing the diet N cow partition into milk seemed to result in non-significant pollution swapping and be achievable in the nearby future.

Methane, in addition to being a significant source of energy loss to the animal that can range from 0·02 to 0·12 of gross energy intake, is one of the major greenhouse gases being targeted for reduction by the Kyoto protocol. Thus, one of the focuses of recent research in animal science has been to develop or improve existing methane prediction models in order to increase overall understanding of the system and to evaluate mitigation strategies for methane reduction. Several dynamic mechanistic models of rumen function have been developed which contain hydrogen gas balance sub-models from which methane production can be predicted. These models predict methane production with varying levels of success and in many cases could benefit from further development. Central to methane prediction is accurate volatile fatty acid prediction, representation of the competition for substrate usage within the rumen, as well as descriptions of protozoal dynamics and pH. Most methane models could also largely benefit from an expanded description of lipid metabolism and hindgut fermentation. The purpose of the current review is to identify key aspects of rumen microbiology that could be incorporated into, or have improved representation within, a model of ruminant digestion and environmental emissions.