Lactation curves of different cattle breeds under cooperative dairying conditions in Bangladesh

Figures & data

Table 1. Mathematical models used to describe the lactation curve of dairy cows.

Model	Equation^a	Description
Gaines (Gaines 1927)	Y = ae ^−bt	A two-parameter model that takes no account of a rise to peak yield.
Wood (Wood 1967)	Y = at ^b e ^−ct	The model takes into account the rise to peak lactation, and the decay to drying off.
Polynomial	Y = a + bt + ct ^2	Polynomial can provide a very good fit to data. It is unbound; that is, as x (days in milk) increases indefinitely, the function, y increases or decreases without limit.
Sikka (Sikka 1950)		The model considers the initial rise in the milk yields.
Ali (Ali and Schaeffer 1987)	Y=a+b(t/270) + c(t/270)^2+dlog(270/t)+ klog(270/t)^2	Both random regressions and covariance functions were selected to compare the differences in model flexibility due to the number of parameters and the degree of correlation between them.
Legendre Polynomial (Brotherstone et al. 2000)	Y=aP 0 + bP 1 + cP 2	Functions of P 's of Legendre polynomials were calculated as P 0(t) = 1, P 1(t) = x and P 2(t) =1/2(3x ^2−1) where x=(2(t– (270 + 1)))/(270–1), and t is the ith day in milk.
Wilmink (Wilmink 1987)	Y=a+bt+ce ^−kt	This model combines exponential and linear functions. The parameters of this model can be reduced to a three-parameter linear model by assuming the k exponent is a suitable fixed value.
Nelder^b (Nelder 1966)	Y=t/[a+bt+ct ^2+dt ^3]	This model is a derivation of the Sikka model and an inverse exponential parabolic function.
Rook (Rook et al. 1993)	Y=a{1/[1 + b/(c+t)]}e ^−dt	The model represents the lactation curve as a multiplicative mixture of cell growth and death processes.
Dijkstra (Dijkstra et al. 1997)	Y=ae[b(1 – e ^−ct)/c – dt]	The model is based on a set of differential equations representing cell proliferation and cell death in mammary gland.
^a Y is milk yield (kg/d), t is days in milk, and a, b, c, d and k are parameters that define the shape of the lactation curve. It should be noted that the same symbol in different equations may represent a different component of lactation curves.
^bIn the original paper of Nelder (Nelder 1966), the model was presented without a cubic term (parameters a, b and c); however in this study with the addition of parameter d, a modified Nelder model was used.

Table 2. The number of cows, lactation yield calculated by BMPCUL technicians and lactation length of different breeds, estimated from test-day records.

Breed group	No. of cows	Lactation milk yield, kg (Mean±SE)	Lactation length, days (Mean±SE)
P	185	1503^y±31.15	268.63^y±3.10
A×P	53	1632^x±62.09	282.80^x±5.28
H×P	242	1619^x±28.84	280.35^xy±2.61
J×P	31	1517^y±65.39	267.48^y±2.61
S×P	207	1548^xy±24.89	271.30^y± 6.84
Note: Means with different superscripts within the same column are significantly different at p<0.05.
P, Pabna cattle; A, Australian-Friesian-Sahiwal; H, Holstein; J, Jersey; S, Sahiwal.

Table 3. Comparison of model performance using different fit statistics (±standard error).

		Equations
Breed group	Item	Wood	Legendre polynomial	Polynomial	Sikka	Ali	Nelder	Wilmink
P	AIC	21.47±0.71	43.04^xy±0.63	22.27±0.69	22.43±1.40	11.50±0.49	19.64±0.79	25.94±0.71
	R ^2	0.78^xy±0.01	0.95±0.004	0.82^x±0.01	0.71±0.02	0.89^y±0.011	0.82^x±0.01	0.78^x±0.017
	RMSPE	0.24^xy±0.08	1.30±0.06	0.63^x±0.09	1.2^y±0.52	0.55±0.03	0.20^y±0.05	0.70^y±0.22
	CCC	0.87^x±0.01	0.70^x±0.02	0.87^y±0.01	0.82±0.02	0.87^xy±0.005	0.88^xy±0.03	0.86^y±0.14
A×P	AIC	20.22±1.30	44.96^x±1.17	23.96±1.31	21.69±2.38	10.31±0.85	18.59±1.48	23.92±1.20
	R ^2	0.80^xy±0.03	0.95±0.007	0.82^x±0.02	0.67±0.03	0.93^x±0.018	0.84^x±0.02	0.73^y±0.03
	RMSPE	0.23±0.01	1.45±0.11	0.58^x±0.15	0.79^x±0.68	0.54±0.05	0.17±0.06	0.62±0.30
	CCC	0.89^x±0.02	0.65^y±0.03	0.90^x±0.02	0.77±0.03	0.89^y±0.01	0.90^xy±0.03	0.84^y±0.12
H×P	AIC	20.80±0.58	45.58^x±0.52	23.60±0.58	23.28±1.18	11.54±0.42	19.40±0.66	25.31±0.59
	R ^2	0.84^x±0.01	0.96±0.003	0.85^x±0.01	0.73±0.017	0.92^x±0.009	0.86^x±0.01	0.81^x±0.01
	RMSPE	0.21^x±0.07	1.31±0.05	0.66^y±0.08	0.80^x±0.48	0.54±0.022	0.17^x±0.04	0.62^y±0.20
	CCC	0.88^x±0.01	0.67^xy±0.01	0.86^y±0.02	0.80±0.015	0.88^y±0.003	0.92^x±0.01	0.84^y±0.12
J×P	AIC	19.65±1.74	40.67^y±1.56	21.99±1.76	20.34±3.09	11.27±1.10	20.12±1.95	24.17±1.56
	R ^2	0.73^y±0.03	0.94±0.009	0.78^y±0.03	0.74±0.04	0.85^y±0.023	0.75^y±0.03	0.78^x±0.04
	RMSPE	0.28^y±0.02	1.51±0.18	0.59^x±0.19	0.82^x±0.77	0.55±0.06	0.20^y±0.07	0.54^x±0.33
	CCC	0.85^y±0.009	0.66^y±0.03	0.86^y±0.02	0.78±0.04	0.84^x±0.02	0.87^y±0.03	0.87^x±0.31
S×P	AIC	21.08±0.54	44.62^x±0.48	23.16±0.54	21.56±1.18	10.68±0.42	18.76±0.61	25.50±0.59
	R ^2	0.80^x±0.01	0.96±0.003	0.85^x±0.01	0.72±0.017	0.91^x±0.009	0.84±0.01	0.79^x±0.014
	RMSPE	0.21^x±0.06	1.18±0.05	0.66^y±0.08	0.87^x±0.69	0.53±0.022	0.18^xy±0.04	0.65^y±0.20
	CCC	0.89^x±0.009	0.68^x±0.01	0.86^y±0.01	0.79±0.015	0.88^y±0.004	0.89^xy±0.01	0.85^y±0.12
Note: Means with different superscripts are different at p<0.05 between breeds within model.
A, Australian-Friesian-Sahiwal; H, Holstein; J, Jersey; P, Pabna cattle; S, Sahiwal; AIC, Akaike information criteria, R ^2, RSquare; RMSPE, root mean square prediction error; CCC, concordance correlation coefficient.

Figure 1.  Lactation curves for all the cows of different genotypes obtained from the test-day data and by fitting the Nelder equation.

Table 4. Parameter estimates (±standard error) and other measures when Nelder model was fitted to the different breed groups in Bangladesh.

Breed group	Parameter	Estimated value
P	a	0.66^y±0.15
	b	0.12^x±0.04
	c	0.00019^y±0.00015
	d	1.465E-6^y±5.003E-6
	Plmy	1509.07^y±58.34
A×P	a	0.82^x±0.15
	b	0.06^y±0.017
	c	0.00061^x±0.00025
	d	−3.1E-7^x±8.6E-6
	Plmy	1649.91^x±98.46
H×P	a	0.55^y±0.11
	b	0.11^x±0.007
	c	0.00004^y±0.0001
	d	1.597E-6^y±3.481E-7
	Plmy	1823.43^x±48.88
J×P	a	0.73^y±0.51
	b	0.12^x±0.03
	c	0.00016^y±0.00047
	d	1.233E-6^y±1.583E-6
	Plmy	1531.01^y±126.84
S×P	a	0.54^y±0.08
	b	0.11^x±0.006
	c	0.00011^y±0.00008
	d	1.685E-6^y±2.896E-7
	Plmy	1626.90^xy±49.01
Note: Means with different superscripts are different at p<0.05 between breeds within model.
H, Holstein; J, Jersey; P, Pabna Cattle; S, Sahiwal, a, b, c and d are parameters that define the scale and shape of the lactation curve. Plmy, predicted lactation milk yield.

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