The description of the temporal evolution of milk production in domestic rumi- nants is one of the most important applications of mathematical modelling in animal science. Lactation curve models are implemented in feeding management software for livestock species (Boeet al., 2005). They represent an essential com- ponent of random regression test day models, which are an upgraded version of
0 0.5 1.0 1.5 2.0 2.5
0 50 100 150 200
Days in milk
Daily milk yield (kg)
Fig. 2.1. Lactation pattern of a Sarda breed goat.
0 0.5 1.0 1.5 2.0
Regular and continuous Random perturbation 2.5
0 50 100 150 200
Days in milk
Daily milk yield (kg)
Fig. 2.2. The two main components of the lactation pattern of a Sarda breed goat:
regular and continuous component and random perturbation.
genetic models used to predict breeding values and to estimate variance compo- nents for milk production, in selection schemes for the genetic improvement of milk production traits in dairy animals (Schaeffer, 2004).
Data to be modelled usually consist of daily test day records of milk yield (or composition, e.g. fat or protein content or somatic cell count, SCC) measured at different times from parturition on the same animal, thus representing a case of repeated measurements design.
The most common approach in lactation curve modelling is to fit suitable functions of time, y=f(t), to test day records. Such an empirical approach is essentially aimed at describing the regular and continuous component of the lactation pattern. Some of the functions used to model goat lactation curves are reported in Table 2.1.
Model Equation Reference
Cappio-Borlino et al. (1995)
y(t ) = atbe(−ct) Todaroet al. (1999), Fernándezet al. (2002), Guimarãeset al. (2006) Cobby and
Le Du (1978)
y(t ) = a−bt−ae−ct Fernándezet al. (2002), Guimarãeset al. (2006) Dhanoa (1981) y(t ) = atbce−ct Fernándezet al. (2002),
Guimarãeset al. (2006) Grossman and
Koops (1988)
y(t ) = {a bi i[ tan ( (b ti ci))]}
i
1 2
1 2
− −
∑
= Gipson and Grossman(1989), Macedoet al.
(2001) Morant and
Gnanasakthy (1989)
y(t ) = ae[b(1+t′/2)t′]+c(t′)2– (1.01/t ) Williams (1993), Macedoet al. (2001), Gonçalveset al. (2001), Guimarãeset al. (2006) Nelder (1966) y(t ) = (t /a) + bt + ct2 Guimarãeset al. (2006) Wilmink (1987) y(t ) = a + be−kt+ct Macciottaet al. (2004a) Wood (1967) y(t ) = atbe−ct Gipson and Grossman
(1990), Rotaet al. (1993), Giacconeet al. (1995), Ruvunaet al. (1995), Montaldoet al. (1997), Andonovet al. (1999), Akpaet al. (2001), Fernándezet al. (2002), Macciottaet al. (2003), McManuset al. (2003), Silvaet al. (2005), Zambomet al. (2005), Guimarãeset al. (2006) y= daily milk yield measured at time;t = days in milk;t¢= (t−150)/100;a,b,c,k= function parameters.
Table 2.1. Mathematical models used to fit goat lactation curves.
It is worth noticing that most of the mathematical functions used to describe the goat lactation curve have three parameters. This is a direct consequence of the limited average number of records per lactation available for goats, not more than six for an average lactation length of 221 days for goat breeds raised in Italy during 2005 (based on data published by AIA, 2005).
Models with a large number of parameters, such as the five-parameter multiple regression of Ali and Schaeffer (1987), have been used to fit average curves in test day repeatability models (Schaeffer and Sullivan, 1994) or individual random curves in random regression models (Bredaet al., 2006).
The alternative and more complex approach is based on the use of mecha- nistic models, aimed at translating in mathematical terms a hypothesis about the physiological and biochemical processes that regulate the phenomenon of interest (Neal and Thornley, 1983). However, the application of mechanistic models in goat lactation curve modelling has not been successful, as happened in other dairy species such as cattle and sheep. This was due to their high theoretical complexity, the large number of input variables involved and the high computation requirements.
A peculiarity of goat lactation curve modelling, which can also be found in dairy sheep, is with regard to the first phase of the lactation pattern. In intensive high-producing farming systems (typical of France, north European countries and northern Italy), kids are artificially reared and milking starts immediately after kidding. By contrast, in extensively farmed flocks (typical of southern Medi- terranean countries, Africa and Latin America), milk of the first month of lactation is usually suckled by the kid. In the second situation, early lactation yields can be obtained from partial milking (i.e. after the kid has suckled) (Wahomeet al., 1994;
Ruvunaet al., 1995) or estimated from the average daily gain of the kid (Giaccone et al., 1995). In any case, the scarce availability of data points during the first month of lactation often prevents an accurate modelling of this phase and may lead to unexpected outcomes, such as the estimation of curves without the lactation peak or a great variability in the time at which the lactation peak occurs.
Besides the prediction of total lactation yield from few test days, the estimation of lactation curve parameters may allow one to calculate its main characteristics.
Some of the mathematical functions reported in Table 2.1 have parameters that have a clear meaning in terms of shape of the lactation curve. An example is the Wood (1967) equation, which is probably the most popular function of the lacta- tion curve. In the Wood model,ais a scale parameter that regulates the general level of the curve,bcontrols the type and the magnitude of the curvature of the function, andcregulates the decrease of yield after the lactation peak. Table 2.2 reports values of parametersa,bandcof the Wood function estimated for some goat breeds. Values of these parameters can be used to calculate some essential features of lactation curve shape such as the DIM at which the lactation curve is attained (tm), the peak yield (ym) and lactation persistency (p) (France and Thornley, 1984):
tm = −b m= = − +
c y a
c b e p b c
b b
;
( / ) ; ( 1) ln
Some characteristics of the lactation curve shape for different goat breeds, esti- mated with the Wood function, are reported in Table 2.3. The time at which the
lactation peak occurs is highly variable, ranging from about 2 weeks to more than 60 days. Such differences could be mainly due to breed characteristics, even if the role of sampling effects (considering the great difference in the num- ber of lactations among the considered studies) should not be neglected. The lac- tation peak occurs earlier in local and low selected breeds (at about 15–30 DIM) than in highly selected breeds such as Saanen, Alpine and their crosses (about 2 months). These results are in agreement with those observed in tropical breeds, which have lactation peaks at about 3 weeks from parturition (Akpa et al., 2001), and in their crosses with high-producing breeds, which show a delayed peak (Ruvunaet al., 1995). First-kidding goats tend to have later peaks than higher parities in Saanen and Murciano-Granadina breeds (Table 2.3), in agreement with results reported by Gipson and Grossman (1990). Peak milk yields reflect the different productive characteristics of the breeds.
Values of lactation persistency (Table 2.3) express a non-dimensional mea- sure of the time interval during which milk yield is maintained at a value similar to the peak (Cappio-Borlinoet al., 1989). It can be observed that such values tend to decrease from first to greater parities. These results are in agreement with those reported in other dairy species such as cattle, sheep and buffaloes: younger animals show lower peaks and greater persistency than older animals, because the maturation process which is still in progress during the first lactation counter- acts the normal decline in milk yield (Stantonet al., 1992).
Most studies on goat lactation curve modelling deal with average curves of homogeneous groups of animals (parity order, kidding season, number of kids at parturition). When average curves are modelled, almost all functions give good fitting performances, withR2values often higher than 0.80 (Montaldoet al., 1997;
Breed a b c Reference
Alpine, La Mancha, Nubian, Saanen, Toggenburg
2.316 0.230 −0.005 Gipson and Grossman (1990)
Derivata di Siria 1.388 0.163 −0.005 Giacconeet al. (1995) East African, Galla 0.345 0.149 −0.082 Ruvunaet al. (1995) Crosses European×
local Mexican breeds
3.756 0.641 −0.0109 Montaldoet al. (1997) Crosses Saanen×
local Brazilian breeds
1.056 0.383 −0.0123 Macedoet al. (2001) Murciano-Granadina 2.287 0.129 −0.029 Fernándezet al. (2002) Red Sokoto 0.586 0.316 −0.023 Akpaet al. (2001)
Sarda 1.007 0.182 −0.007 Macciotta (unpublished
data)
Small East African 0.333 0.227 −0.0052 Wahomeet al. (1994)
Verata 1.290 0.207 −0.0052 Rotaet al. (1993)
Table 2.2. Estimates of parametersa, b and c of the Wood function obtained in several goat breeds.
Todaroet al., 2000; Macedoet al., 2001; Fernándezet al., 2002). These perfor- mances are no longer maintained when individual patterns are fitted, despite the increasing interest in this topic, particularly for breeding purposes. Actually the genetic modification of the lactation curve shape, to obtain economical advantages, is one of the major challenges for geneticists and professionals interested in the genetic improvement of milk production traits (Rekayaet al., 2000).
A study carried out on individual lactation curves of Frisa goats using the Wilmink function (Macciotta et al., 2004a) showed a great range ofR2values Breed Observations Parity tm(days) ym(kg) p Reference Alpine, La Mancha,
Nubian, Saanen, Toggenburg
– 1 53 3.0 6.68 Gipson and
Grossman (1990)
– ≥2 46 4.4 6.51
Derivata di Siria 6 1 45 1.65 6.63 Giaccone
et al. (1995)
22 2 63 1.89 6.82
38 ≥3 47 2.05 6.40
Crosses Saanen, Alpine,
49 1 58 1.78 7.53 Montaldo
et al. (1997)b
Toggenburg×local 221 2 61 2.38 7.57
Mexican goats 355 3 54 2.54 7.17
Crosses Saanen× local Brazilian breeds
31 1–4 34 2.24 6.27 Macedo
et al. (2001)b
Florida 968 1–7 13 2.68 4.32 Pena Blanco
et al. (1999)
Murciano-Granadina 190 1 4.5a 1.66 3.75 Fernández
et al. (2002)b
167 2 2.4 2.35 3.81
376 ≥3 2 1.78
Red Sokoto 22 1 20 0.76 5.45 Akpaet al.
(2001)b
17 2 20 0.95 5.44
13 3 19 1.64 5.39
Saanen 150 1 64.6 3.22 7.04 Groenewald and
Viljoen (2003)
211 2 54.4 4.21 6.84
253 3 58.8 4.53 7.02
Sarda 161 1 32 1.35 6.07 Macciotta
(unpublished data)
222 2 31 1.57 6.04
152 3 38 1.53 6.01
aWeeks.
bValues ofym,tmandpin the works by Akpaet al. (2001) and Macedoet al. (2001), andpin the works by Montaldoet al. (1997) and Fernándezet al. (2002), have been calculated by using Wood parameter values reported by the authors.
Table 2.3. Days in milk at which the lactation peak occurs (tm), peak yield (ym) and lactation persistency (p) calculated from parameters estimated by fitting the Wood function in different goat breeds.
(Table 2.4). This result is in agreement with findings in dairy cattle, where the great variability of individual lactation curve shapes has been ascribed to both biological differences among cows and the interaction between the structure of the data analysed and the mathematical properties of the models used (Olori et al., 1999; Landete-Castillejos and Gallego, 2000; Macciottaet al., 2005b). In the same study, some atypical shapes (Fig. 2.3), i.e. without the lactation peak, were also observed. Detection of atypical shapes, also reported in cattle (Shanks et al., 1981) and sheep (Cappio-Borlinoet al., 1997), is based on the signs of the estimated function parameters. In the case of the Wood function, atypical shapes are characterized by negative values of the parameterb(abeing positive andc negative), whereas for the Wilmink model atypical shapes have positive values ofb(abeing positive andcnegative).
The frequency of atypical shapes (about 30%) observed by Macciottaet al.
(2004a) for goats is similar to that reported for dairy cattle and sheep. Fernández et al. (2002) found an atypical average lactation curve for goats at third or
R2class Absolute frequency Relative frequency
<0.20 65 0.14
0.20−0.40 35 0.08
0.40−0.60 52 0.12
0.60−0.80 106 0.24
>0.80 190 0.42
Table 2.4. Distribution of individual lactation curves of Frisa goats among different R2 classes. (From Macciottaet al., 2004a.)
1.0 1.5 2.0 2.5 3.0 3.5
0 50 100 150 200
Days in milk
Milk yield (kg/day)
Atypical Standard
Fig. 2.3. Example of standard and atypical lactation curves for milk yield of Frisa breed goats estimated by the Wilmink function. (Adapted from Macciottaet al., 2004a.)
greater kidding and attributed it to the occurrence of the lactation peak very close to parturition, which therefore could not be recognized by the mathematical func- tion used. Reasons for the atypical lactation curve are: (i) biological differences between animals; (ii) mathematical properties of the model used; and (iii) the already mentioned structure of the data analysed (mainly the distance from par- turition of the first record available and the distribution of records throughout lactation). The main consequence of an atypical shape is the change in sign and, therefore, in meaning of lactation curve parameters. This makes the interpretation of outcomes very difficult when individual values of lactation curve parameters are analysed with linear models to estimate the effect of fixed and random factors on lactation curve shape traits (Shankset al., 1981), or when mathematical func- tions are used to fit individual curves in random regression test day models (Jamrozik and Schaeffer, 1997).
Finally, although most studies on goat lactation curve modelling have dealt with milk yield, some work on modelling of the lactation pattern of milk components has also been done. For example, Rotaet al. (1993) fitted the Wood function to curves for fat and protein content and for SCC in Verata goats, obtaining good fitting performances also for SCC (R2= 0.97).