The mathematical description of lactation curves in dairy cattle

Figures & data

Figure 1 Standard shape of the lactation curve for dairy cattle.

Table 1 Some of the most popular empirical models used to fit the lactation curve in dairy cattle.

Function	Parameters	Author
	2	Brody et al. (1923)
	3	Wood (1967)
	3	Cobby and Le Du (1978)
	4	Wilmink (1987a)
	5	Ali and Schaeffer (1987)
	3 per phase	Grossman and Koops (1988)
	5	Brotherstone et al. (2000)
	14	White et al. (1999)

y_t, daily milk production at time t from parturition; a,b,c, function parameters.

Figure 2 Decomposition of the Wood incomplete gamma function.

Table 2 Parameters and lactation curve traits for dairy cattle of different parities estimated with the Wood model (lactation traits have been calculated on the basis of parameter values reported in the articles).

Parity	a	b	c	ym, kg/d	yt,d	Persistency	Author
First	23.7	0.12	0.061	22.8	58	3.13	Kellogg et al. (1977)
Second	35.7	0.25	0.160	31.1	47	2.29
Third	39.1	0.32	0.186	33.8	52	2.22
First	12.6	0.17	0.002	22.6	85	7.27	Val-Arreola et al. (2004)
Second	19.1	0.15	0.004	28.3	37.5	6.35
Third or >	15.1	0.21	0.005	26.8	42	6.41
First	15.7	0.21	0.002	33.5	104	7.51	Dematawewa et al. (2007)
Third or >	23.1	0.21	0.004	43.9	57	6.82
First	13.89	0.25	0.004	31.2	69	7.02	Rekik and Ben Gara (2004)
Second	17.46	0.24	0.005	35.6	53	6.69
Third	19.56	0.23	0.005	37.9	48	6.57
First	13.01	0.27	0.003	33.5	89	7.34	Cole and Null (2009)
Second or >	22.01	0.22	0.004	42.6	54	6.74

y_m, peak production; y_t, peak occurrence.

Figure 3 Lactation curve of a dairy cow fitted with the Wood model (a), a fourth order Legendre polynomials (b) and a cubic regression spline (c) with three knots placed at 16, 69 and 252 DIM, respectively.

Table 3 Knot positions (days in milking) and Mean Squared Error of average lactation for regression splines of different orders fitted to average lactation curves of cows of three parities by using a non linear regression procedure.

	No. of knots
Spline order	1		2		3
	Position	MSE	Position, d	MSE	Position, d	MSE
			1st parity
Linear	37	0.09	35, 219	0.06	Not conv.
Quadratic	46	0.02	17, 53	0.009	20, 59, 202	0.006
Cubic	62	0.01	21, 75	0.005	26, 57, 103	0.004
			3rd parity
Linear	34	0.09	18, 41	0.03	Not conv.
Quadratic	46	0.03	16, 51	0.02	11, 38, 59	0.01
Cubic	60	0.02	26, 67	0.01	16, 69, 252	0.007

Figure 4 Average lactation curves of dairy cattle of three different parities estimated with test day model in which the DIM factor is nested within parity (a) or reconstructed by fitting the Wood model to daily yields averaged by parities (b).

Table 4 Sample variances (kg²), on the diagonal, and correlations, off diagonal, of test day records for milk yield along the lactation estimated with a mixed linear model without any assumption of (co)variance structure.

38.14
0.58	24.61
0.57	0.66	19.82
0.44	0.55	0.65	20.05
0.43	0.50	0.64	0.69	19.66
0.33	0.50	0.57	0.74	0.77	17.64
0.20	0.34	0.44	0.66	0.66	0.80	19.61
0.13	0.29	0.44	0.59	0.58	0.68	0.76	19.65
0.01	0.24	0.35	0.46	0.62	0.65	0.67	0.82	18.67
0.02	0.13	0.25	0.39	0.53	0.48	0.50	0.59	0.64	24.46

Table 5 Comparison between different (co)variance structure used in mixed models to fit milk test day yields using Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC).

Covariance structure	Covariance Parameter	N	Estimate	AIC	BIC
Unstructured		55	See Table 4	11051.9	11281.2
Compound symmetry	σ^2L	2	13.9505	11318.2	11326.5
	σ^2e		7.996
First order autoregressive	σ^2	2	21.4145	11200.0	11208.4
	ρ		0.7679
Ante -dependence		19		1124.4	11203.7

Table 6 Correlations among actual and predicted lactation yields in first parity cows with different approaches and different plans of missing data.

No. of available tests	Method
ARMA	WMTP	ATD
2	0.85	0.80	0.86
3	0.92	0.87	0.91
4	0.96	0.92	0.94
5	0.98	0.95	0.96
6	0.99	0.97	0.98
7		0.98	0.99
8		0.99	0.99
9		0.99	0.99

ARMA, Autoregressive Moving Average Model; WMTP, Multiple Trait Model based on Wood function; ATD, Autoregressive Test Day Model.

Figure 5 Plot of the eigenvalues of the first two principal components extracted from the correlation matrix of the TD records along the lactation (from Macciotta et al., 2006).

Table 7 Lactation curve traits estimated in different cattle breeds.

Breed	ym°, kg	yt^#, d	Persistency	Author
Barrosa	3.5 (5.3)	23 (56)	5.59	Silvestre et al. (2010)
Holstein x Zebu crosses	14.9	25	6.14	Cunha et al. (2010)
Guzerà	12.22	25	6.13	Cruz et al. (2009)
Sindi	11.80	22	6.17	Cruz et al. (2009)
Milking Shortorn	33.8	49	6.31	Cole and Null (2009)
Italian Simmental	27.3	29.3	6.41	Unpublished
Norwegian Red	30.2	35.3	-	Andersen et al. (2011)
US Holstein	43.9	59	6.82	Dematawewa et al. (2007)

° Yield at lactation peak;

# days at lactation peak.

Figure 6 Patterns of extended individual lactations of Italian Holstein (up) and Simmental (down) cows.

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