Application of growth models to describe the lactation curves for test-day milk production in Holstein cows

Figures & data

Table 1. Growth equations used to describe the lactation curve of Holstein cows.

Equation	Functional form
Brody
Logistic
Gompertz
Schumacher
Von Bertalanffy
Morgan

Notes: y represents milk yield traits; PY, the maximum value for milk yield traits; PT , the PT; a, b, c and k are parameters that define the scale and shape of the lactation curve; t , the time from parturition.

Table 2. Parameter estimates for the different lactation equations of the first parity dairy cows (standard errors are in parentheses).

Trait	Parameter	Model
		Brody	Logistic	Gompertz	Schumacher	Von Bertalanffy	Morgan
MY	a	31.48 (0.01)	0.00500 (0.00002)	4301.00 (20.66)	1647.50 (9.41)	31.480 (0.008)	38,057.5 (120.1)
	b	0.50 (0.01)	8327.40 (36.22)	0.00700 (0.00003)	188.00 (0.63)	−1.68 (0.02)	793.60 (3.22)
	c	–	24,264.90 (61.88)	0.00400 (0.00001)	0.01600 (0.00008)	–	1.1600 (0.0007)
	k	0.12 (0.002)	–	–	–	0.130 (0.002)	–
FCM	a	28.06 (0.01)	0.00400 (0.00002)	5962.80 (45.33)	2727.10 (23.02)	28.060 (0.008)	49,667.2 (280.4)
	b	0.29 (0.01)	10,595.60 (74.97)	0.00500 (0.00003)	279.20 (1.56)	−2.28 (0.05)	1290.90 (9.13)
	c	–	27,394.6 (127.1)	0.00300 (0.00001)	0.01000 (0.00007)	–	1.1000 (0.0008)
	k	0.150 (0.005)	–	–	–	0.150 (0.005)	–
ECM	a	27.78 (0.01)	0.00400 (0.00002)	5718.10 (41.13)	2710.80 (22.09)	27.78 (0.01)	53005.8 (315.2)
	b	0.23 (0.01)	10085.10 (66.77)	0.00500 (0.00003)	286.50 (1.56)	−2.50 (0.04)	1404.10 (10.31)
	c	–	26,830.0 (116.9)	0.00300 (0.00001)	0.01000 (0.00007)	–	1.1000 (0.0008)
	k	0.120 (0.004)	–	–	–	0.120 (0.004)	–

Notes: MY, Unadjusted milk yield; FCM, 4% Fat-corrected milk yield; ECM, energy-corrected milk yield; a, b, c and k are parameters that define the scale and shape of the lactation curve.

Table 3. Comparing goodness of fit for average standard curves of milk yield traits for Brody, Logistic, Gompertz, Schumacher, Von Bertalanffy and Morgan models.

Trait	Statistics	Model
		Brody	Logistic	Gompertz	Schumacher	Von Bertalanffy	Morgan
MY		0.95	0.95	0.95	0.95	0.95	0.95
	DW	0.95	0.91	0.90	0.89	0.95	0.88
	RMSE	7.34	7.19	7.17	7.14	7.34	7.11
	AIC	16,136,045	16,097,425	16,093,776	16,086,287	1,613,5969	16,079,399
	BIC	3,632,947	3,594,327	3,590,678	3,583,188	3,632,870	3,576,300
FCM		0.94	0.94	0.94	0.94	0.94	0.94
	DW	1.08	1.05	1.05	1.05	1.08	1.05
	RMSE	7.15	7.07	7.07	7.06	7.15	7.05
	AIC	16,088,019	16,067,598	16,066,849	16,065,307	16,088,013	16,063,194
	BIC	3,584,921	3,564,499	3,563,750	3,562,209	3,584,915	3,560,096
ECM		0.94	0.94	0.94	0.94	0.94	0.94
	DW	1.06	1.05	1.04	1.04	1.06	1.04
	RMSE	6.80	6.73	6.73	6.73	6.80	6.72
	AIC	15,995,572	15,978,511	15,977,761	15,976,391	15,995,565	15,975,020
	BIC	3,492,474	3,475,413	3,474,662	3,473,293	3,492,466	3,471,921

Notes: MY, unadjusted milk yield; FCM, 4% fat-corrected milk yield; ECM, energy-corrected milk yield; , adjusted coefficient of determination; RMSE, root means square error; DW, Durbin–Watson; AIC, Akaike information criteria; BIC, Bayesian Information Criteria

Figure 1. Predicted MY (a), FCM (b) and ECM (c) lactation curves by Brody, logistic, Gompertz, Schumacher, Von Bertalanffy and Morgan equations.

Table 4. PT and PY for average standard lactations of MY, FCM and ECM predicted by Brody, Logistic, Gompertz, Schumacher, Von Bertalanffy and Morgan models.

Trait	Statistics	Model
		Brody	Logistic	Gompertz	Schumacher	Von Bertalanffy	Morgan
MY	PT	68	130	140	95	63	84
	PY	31.48	30.33	36.42	31.93	31.48	33.68
FCM	PT	50	115	170	111	50	81
	PY	28.06	27.39	34.84	30.90	28.06	29.24
ECM	PT	60	127	170	124	61	88
	PY	27.78	26.83	33.41	31.37	27.78	28.69

Note: MY, unadjusted milk yield; FCM, 4% fat-corrected milk yield; ECM, energy-corrected milk yield; PY, peak yield; PT, peak time.

Table 5. Different measures of persistency and 305-d yields for average standard lactations of dairy cows predicted by Brody, Logistic, Gompertz, Schumacher, Von Bertalanffy and Morgan models.

Trait	Variable	Model
		Brody	Logistic	Gompertz	Schumacher	Von Bertalanffy	Morgan
MY	305TY (kg)	9399	8692	10,342	8838	9404	9358
	P2:1	1.02	1.03	1.07	1.00	1.02	1.00
	P3:1	1.04	0.95	0.99	0.89	1.03	0.91
	PWeller (%)	100	90	91	82	100	84
FCM	305TY (kg)	8418	7959	10,113	8878	8418	8348
	P2:1	1.01	1.01	1.07	1.02	1.01	1.00
	P3:1	1.02	0.95	1.05	0.96	1.02	0.94
	PWeller (%)	100	91	99	90	100	89
ECM	305TY (kg)	8331	7818	9698	9057	8329	8248
	P2:1	1.01	1.02	1.07	1.04	1.01	1.00
	P3:1	1.02	0.97	1.05	0.98	1.02	0.96
	PWeller (%)	100	93	99	93	100	91

Notes: MY, unadjusted milk yield; FCM, 4% fat-corrected milk yield; ECM, energy-corrected milk yield; 305TY, predicted 305-d yield for MY, FCM and ECM; P_2:1, Ratio between the milk yields of the second 100 days of lactation and those of the first 100 days; P_3:1,  ratio between the milk yields of the third 100 days of lactation and those of the first 100 days; P_Weller , milk yield persistency measure proposed by Weller et al. (2006).

Weller JI, Ezra E, Leitner G. 2006. Genetic analysis of persistency in the Israeli Holstein population by the multitrait animal model. J Dairy Sci. 89:2738–2746. doi:10.3168/jds.S0022-0302(06)72350-5

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