A parametric measure of lactation persistency in dairy cattle

Livestock Production Science 96 (2005) 141 – 148 www.elsevier.com/locate/livprodsci

A parametric measure of lactation persistency in dairy cattle R.E. KamidiT International Livestock Research Institute, P.O. Box 30709, Nairobi, Kenya Received 20 June 2003; received in revised form 28 October 2004; accepted 19 November 2004

Abstract Current measures of lactation persistency are not uniformly applicable to the variety of lactation curves for individual cows. Moreover, they are either difficult to interpret or are functions of lactation yield and therefore inappropriate. In this study, a simple, robust parametric measure is derived from a perfect fit of cumulative milk yield data to quadratic curves. The curvilinear model is appropriate for trend analysis, a major manifestation of which is persistency. Data fitted to the model were obtained from a total of 194 lactations of 169 cows in 5 mixed-breed herds under conditions varying from a small holding herd in the tropics to research herds in Italy. A negative association between persistency and total lactation yield was observed. The proposed lactation persistency model could result in more accurate estimates and better insight into this trait. D 2005 Elsevier B.V. All rights reserved. Keywords: Lactation curve; Persistency; Dairy cattle; Linear models

1. Introduction Mathematical modelling of the lactation of dairy cows has attracted considerable interest and attention (Grossman and Koops, 1988; Morant and Gnanasakthy, 1989; Beever et al., 1991; Sherchand et al., 1992; Rook et al., 1993; Perochon et al., 1996; Gengler, 1996; Olori et al., 1999; Grossman et al., 1999; Mostert et al., 2003). Modellers seek to find parametric descriptors of the shape of the curve, factors that affect the shape and tools for predicting milk yield. Lactation curve variables significantly contribute to feeding and management decision support T Tel.: +254 20 4223000; fax: +254 20 4223001. E-mail address: [email protected]. 0301-6226/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.livprodsci.2004.11.042

systems for optimisation of dairy herd production processes. Variations in the shape of the lactation curve for dairy cows are believed to stem from both genetic and environmental factors (Wood, 1968, 1969, 1970, 1980). Olori et al. (1999) fitted data obtained from a single uniformly managed herd to five different models, none of which adequately described individual lactations. They inferred that the suitability of empirical models of the lactation curve does not depend on the mathematical form of the function alone but rather, more on the biological nature of the lactation. Use of a single one model to adequately describe dairy cattle lactations has been elusive. This empirical state is likely to remain, as long as modelled production patterns traverse combinations of climate,

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management, age, health, parity and heritability. Olori and Galesloot (1999) and later Mostert et al. (2001, 2003) observed a strong influence of calving season on the shape of standard lactation curves derived for cows in Ireland and South Africa respectively. Researchers working with animals under diverse conditions often come across atypical lactations that cannot be adequately described by standard models. Seasonal patterns in pasture availability for instance is a major environmental factor contributing to deviations from the dtypicalT curve. Of particular significance are animals in smallholder rain-fed production systems that are common in the tropical highlands where farmers are largely constrained in terms of management, lack of supplementary feeds, diseases and seasonal forage shortages. In dairy cattle, production typically rises to a peak 2 to 8 weeks postpartum and steadily declines thereafter. Apart from peak yield, persistency is a parameter of major interest in a lactation curve. It is usually described as the ability of the cow to maintain production beyond peak yield. However, production conditions vary over seasons as well as locations. In a broader definition, persistency encompasses the ability of lactating cows to rebound from effects of adverse events such as illness and climatic stress as well as feed shortages thus providing a gauge for adaptability of the dairy cow. Resilient cows will record smaller declines in yield or quickly restore production with implication for higher persistency. Grossman et al. (1999), in an effort to obtain a measure that is consistent, defined persistency as the number of days during which peak yield is maintained. To the contrary, what happens beyond the plateau certainly contributes more significantly to persistency and cannot be ignored. An appropriate persistency parameter should be holistic in describing the whole lactation and represent a general trend that shows a net decline after taking into account observed fluctuations in production including the rise to peak yield. Most of the currently used measures of lactation persistency cannot be easily interpreted or are functions of yield and therefore inappropriate (Gengler, 1996). Using the non-linear curve y t = a b e
ct to represent milk yield y t on day t, Wood (1967) defined persistency as s =
(b + 1)loge c, an expression that is difficult to interpret biologically. Cobby and Le Du

(1978) suggested two alternatives to Wood’s curve with simpler persistency parameters. Gravert and Baptist (1976) used an approximate measure of the rate of decline of milk yield from peak yield obtained from a linear regression of data in the decline phase of lactation. Later, Rowlands et al. (1982) observed that values of the first derivative of Wood’s model evaluated at t = 25 were very similar to the linear rate of decline between 15 and 35 weeks of lactation and suggested this as an alternative way of estimating the rate of decline. This is unlikely to be satisfactory for all possible curves, as the rate of decline at the 25th week of lactation is not representative of the mean rate in general. So¨lkner and Fuchs (1987) compared different ratio and dispersion measures. They pointed out that the problem in measuring persistency is to express the shape of the lactation curve by a single parameter. This obstacle is satisfactorily overcome by considering deceleration in cumulative milk yield over the whole lactation. Deceleration in production or the decline in the rate of decline will be herein considered as persistency failure. Since lactation curves are nonlinear in time, the rate of decline is not constant but time-dependent and cannot provide a reliable measure of persistency over a whole lactation. On the other hand, deceleration is logically inherent in persistency and is constant for each complete lactation modelled, thus providing a parameter that is easy to compare across individual cows or lactations.

2. Methods 2.1. The model Cumulative milk yield generally follows a curvilinear trend that closely resembles the initial phase of a trajectory (see Fig. 1). In physics, a quadratic model is usually used to describe ballistic trajectories. Vertical distance covered by a projectile at constant acceleration and negligible friction is represented by the quadratic curve d = ut
1/2gt 2 where u is the initial velocity and g the gravitation constant (see for instance, Nelkon and Parker, 1977). An analogous relationship adequately describes the trend in cumulative lactation yield. The model represents a reduction of complex and varied lactation

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Fig. 1. Fitted quadratic curves corresponding to cumulative milk yield obtained from Anita’s five lactations.

dy ¼ b þ 2ct: dt

If the deceleration constant c =0, then in theory no decline is exhibited and the curve is reduced to a straight line. A natural decline in yield is inevitable and c will therefore be precisely negative. Persistency is inversely proportional to deceleration and minimum decline or maximum persistency will occur when c = 0. For ease of interpretation, let maximum absolute persistency be equal to the benchmark value 1 so that proportional persistency is equivalent to 1 +2c (the resultant quantity after taking into account the deceleration). This is plausible since the absolute value of deceleration |2c| cannot exceed 1 on the kilograms per day per day scale of measurement. The biological implication of which is that; a peak yield of say 100 kg cannot naturally dry out in less than 100 days. On the percentile scale therefore, percent persistency, P = 100(1 +2c) where c b 0. This parametric measure is notably independent of lactation yield levels, a property that Gengler (1996) elucidated as a desirable attribute of a good measure of persistency. It is important to note however, that units of measurement must necessarily remain the same, i.e., yield in kilograms regressed on lactation days in order to maintain a uniform scale of measurement for persistency, P.

Deceleration in yield over the whole lactation is the value of the second derivative;

2.2. Data analysis

d2 y ¼ 2c: dt 2

The first data set comprised empirical data extracted from daily milk records obtained from a

curve patterns to a simple, mathematically wellbehaved monotonic increasing function from which it is relatively easy to derive a general trend that fully defines persistency. Daily milk yield curves derived from the model are however imprecise and inappropriate as some degree of specific information loss occurs while general trend is consolidated by the yield accumulation process. Cumulative milk yield y at lactation day t is given by: y ¼ bt þ ct 2 þ e: b and c are constants and q a random error term. This simple linear model (in the parameters) is amenable to standard linear statistical analyses. The constant b is highly associated with total lactation yield. A deceleration constant c restrains milk production after the initial thrust to the peak, analogous in physics to the gravitational force attracting a trajectile towards the earth. The rate of decline in cumulative yield at lactation day t is given by the first derivative;

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smallholder 12-ha farm located near Kitale, Kenya. The data are considered to be both opportune and appropriate because livestock records are rarely kept on smallholder land units in the developing world, a factor that has led to a critical information gap. Nearly all published research results are pertinent to pedigree cows under high management whereas livestock research in developing nations is appropriately being re-focused to target smallholders with limited access to production resources. As noted elsewhere in this paper, atypical lactation curves are likely to be more prevalent among smallholder herds due to genetic, feeding, health and management constraints as well as climatic variability and forage seasonality. Daily milk records taken between May 1995 and February 2003 for 11 cows covering a total of 31 lactations were used. The cows of Friesian or Ayrshire parentage were all bred at the farm by artificial insemination over 2 to 5 generations. Cows at the farm graze on approximately 5 ha of Rhodes grass + stylosanthes pasture and are milked twice daily. Supplement feed comprising approximately 1 kg of concentrates mixed with chopped Napier grass and an additional 0.1 kg of dairy mineral supplement is offered to each cow during milking. Calliandra leaves and molasses are added to the ration whenever they are available. The climate is characterised by a main rainy season from April to June with a short intermittent dry spell that lasts until August. Some rains are received between August and October followed by a dry season that usually lasts until March. The stocking rate is much higher than recommended and the pasture gets completely depleted during the dry season when supplementation with cut Napier grass, coffee pulp and purchased hay is necessary. Quadratic curves were fitted to cumulative daily milk yields for up to 305 days of each lactation to obtain the deceleration constants and subsequently derive persistency. Persistency values were compared by statistical significance of differences between corresponding deceleration constants using the general linear hypothesis. Statistical models for any two lactations may be written as: y1i ¼ b1 ti þ c1 ti2 þ ei : y2i ¼ b2 ti þ c2 ti2 þ ei :

Where numerical subscripts refer to the different lactations and model parameters are as described in the previous section. We require to test the hypothesis: H 0 : c 1 = c 2 against the alternative H 1 : c 1 p c 2, that may be equivalently written as: H0 : c1
c2 ¼ 0; H1 : c1
c2 p0: This hypothesis is a special case of the general linear hypothesis: H0 : C c ¼ 0; H1 : Ccp0: Where C is the r  p contrast matrix of rank r V p, c the transposed vector of the p parameters and 0 the null matrix. For a contrast involving two lactations C = [0 0 0 0 1
1] and c is comprised of b 1, b 2, c 1 and c 2 respectively. A statistical test for linear contrasts is provided by Graybill (1976) and may be computationally obtained using the General Linear Models (GLM) procedure in Statistical Analysis System SAS (1999). The second data set comprised data extracted from weekly milk yield records kept at a large-scale farm located a few kilometres from the smallholding. The data were derived from lactations of 96 cows in a mixed-breed herd that calved between May 1984 and September 1985. A total of 101 lactations lasting 38 weeks or longer were considered with data from longer lactations being limited to the first 44 weeks of lactation. A third data set comprised data obtained from cows of Italian brown and Italian Holstein breeds in three herds reared under Italian conditions. A total of 62 complete lactations recorded between October 2001 and September 2003 lasting 32 weeks or more were considered but limited to the first 44 weeks of lactation. A few gaps in the data were linearly interpolated without loss of general trend.

3. Results and discussion Results obtained from analysis of the first data set including model fit, deceleration constants and corresponding deceleration and persistency levels are presented in Table 1. Perfect fits of cumulative yield data to the curvilinear model were obtained for all cows and lactations with coefficients of determination averaging 0.998. Model fit was notably equally good

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Table 1 Model fit, deceleration constants and derived persistency, corresponding standard errors and lactation yields obtained from cumulative milk yield data fitted to quadratic curves for various lactations of 11 cows Cow

Adema Amina

Anita

Baraka Fatuma

Jahenda Kaliyesa

Kamonya Marion

Minayo Sadaka

Lactation Number

Days

1 3 4 5 6 1 2 3 4 5 1 2 3 4 5 6 1 5 6 7 1 1 2 3 4 5 1 2 1 2 3

305 305 266 271 224 305 305 293 305 245 305 305 305 305 301 305 305 305 305 305 305 305 305 305 305 305 305 218 305 305 305

Model R 2

Deceleration constant, c

SE c

Persistency, P (%)

SE P

Lactation yield (kg)

0.9995 0.9998 0.9997 0.9999 0.9998 0.9998 0.9988 0.9999 0.9896 0.9999 0.9999 0.9999 0.9983 0.9996 0.9951 0.9997 0.9999 0.9999 0.9979 0.9999 0.9977 0.9998 0.9994 0.9996 0.9933 0.9998 0.9986 0.9999 0.9998 0.9876 0.9876


0.00734
0.01052
0.01953
0.02102
0.01843
0.01122
0.00263
0.02031
0.01346
0.01086
0.01471
0.00986
0.01412
0.01279
0.02157
0.01351
0.00559
0.01503
0.00953
0.01185
0.00726
0.00823
0.01031
0.01453
0.01544
0.01113
0.01127
0.01275
0.00945
0.0128
0.0101

0.00019 0.00013 0.00020 0.00013 0.00022 0.00014 0.00028 0.00010 0.00058 0.00013 0.00006 0.00009 0.00031 0.00016 0.00038 0.00012 0.00008 0.00013 0.00042 0.00010 0.00035 0.00011 0.00016 0.00018 0.00043 0.00011 0.00018 0.00015 0.00009 0.00052 0.00015

98.53 97.90 96.09 95.80 96.31 97.76 99.47 95.94 97.31 97.83 97.06 98.03 97.18 97.44 95.69 97.30 98.88 96.99 98.09 97.63 98.55 98.35 97.94 97.09 96.91 97.77 97.75 97.45 98.11 97.44 97.98

0.04 0.03 0.04 0.03 0.04 0.03 0.06 0.02 0.12 0.03 0.01 0.02 0.06 0.03 0.08 0.02 0.02 0.03 0.08 0.02 0.07 0.02 0.03 0.04 0.09 0.02 0.04 0.03 0.02 0.10 0.03

2577.5 2792.5 2275.5 2520.0 1892.5 2520.0 2330.5 2567.5 1540.5 3162.0 2396.0 2455.0 2177.5 2396.0 1235.0 2000.5 2132.0 3293.5 2696.0 2396.0 2165.0 2087.5 1804.0 2475.5 1433.0 2424.0 1313.4 2361.5 1820.5 1290.0 2531.5

for lactations when cows were treated for a major illness. Anita was treated for theileriosis during the 9th month of her 4th lactation while Sadaka received treatment for a similar illness during the 3rd month of her 2nd lactation and Fatuma during the 7th month of her 5th lactation. Marion was treated for another tickborne ailment, cowdriosis during the 5th month of her 4th lactation. Total yields were drastically depressed for these lactations when cows were under extreme stress. Quadratic curves fitted to data from Anita’s five lactations are depicted in Fig. 1. Persistency significantly differed for lactations among cows as well as within cows. Anita produced less milk during 305 days of her 2nd lactation compared to her first lactation, although production

was significantly more persistent during the 2nd lactation during which feed resources were limited close to parturition. Kaliyesa’s 6th calving that also resulted in a relatively high lactational persistency occurred only a month earlier. Tekerli et al. (2000) also observed higher persistency for Holstein cows in Turkey that calved during summer and fall when temperature increases and fodder declines are experienced. Summary statistics of the results obtained from analyses of daily records in data set 1 and separate analyses of weekly test day data derived from all data sets are provided in Table 2. No significant change in model parameter estimates or fit was detected when weekly test day yields from the first data set were used

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Table 2 Summary statistics for model fit, deceleration constants and derived persistency and lactation yields obtained from fits of cumulative milk yield to quadratic curves for various data sets Model R 2

Deceleration constant, c

SE c

Parameter b

SE b

Persistency, P (%)

SE P

Data set 1 (daily) Mean 294 Minimum 218 Maximum 305

0.9981 0.9876 0.9999


0.01249
0.02157
0.00263

0.00020 0.00006 0.00058

11.14 7.50 15.20

0.42 0.02 0.14

97.50 95.69 99.47

0.04099 0.01267 0.11507

2227.8 1235.0 3293.5

Data set 1 (weekly) Mean 43 Minimum 31 Maximum 44

0.9983 0.9865 1.0000


0.01272
0.02179
0.00380

0.00057 0.00019 0.00148

11.18 7.21 15.27

0.13 0.04 0.36

97.46 95.64 99.24

0.11329 0.03871 0.29600

2227.8 1235.0 3293.5

Data set 2 (weekly) Mean 43 Minimum 38 Maximum 44

0.9995 0.9971 1.0000


0.01915
0.04702
0.00034

0.00086 0.00027 0.00287

20.13 11.04 33.45

0.38 0.06 0.68

96.17 90.60 99.93

0.17235 0.05332 0.57400

4341.3 2704.5 6480.5

Data set 3 (weekly) Mean 30 Minimum 24 Maximum 38

0.99986 0.99890 1.00000


0.02420
0.05748
0.00359

0.00101 0.00026 0.00289

33.43 21.46 43.24

0.22 0.06 0.64

95.16 88.50 99.28

0.20159 0.05153 0.57800

7343.9 7641.7 11221.4

Observations N

in place of daily records to derive cumulative yields. Figs. 2 and 3 show fitted curves corresponding to the most persistent and least persistent lactations observed in data sets 2 and 3. Lactation persistency levels were on average lower than those observed for data set 1. Herd management and stocking rates were however

Lactation yield (kg)

more favourable at the large-scale farm that provided data set 2 and for the temperate climate herds of data set 3 compared to the smallholder-farm data set 1. Milk yield levels were also much higher for the relatively more dtypicalT lactations of data sets 2 and 3. Differences in performance among herds are clearly

Fig. 2. Fitted quadratic curves corresponding to the most persistent and least persistent lactations of data set 2.

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Fig. 3. Fitted quadratic curves corresponding to the most persistent and least persistent lactations of data set 3.

demonstrated by typically corresponding values of parameter b. The Pearson correlation coefficient matrix for lactation persistency, lactation yield and model parameter b for all data as well as lactations of optimal and sub-optimal lactation lengths is provided in Table 3. Model parameter b positively correlates with lactation yield and is negatively associated with persistency. A negative correlation was observed between persistency and lactation yield. So¨lkner and Fuchs (1987) found a significant linear regression of various measures of persistency on milk yield indicating a positive correlation. Their conclusion, like that based on most currently used persistency

Table 3 Pearson correlation coefficient matrix for persistency, lactation yield and model parameter b for data of sub-optimal (218–301 days) and optimal (305–308 days) lactation length Persistency Lactation yield Sub-optimal lactations n =87 Optimal lactations n =107 All lactations n =194 Parameter b Sub-optimal lactations n =87 Optimal lactations n =107 All lactations n =194

Lactation yield


0.23
0.44
0.39
0.53
0.69
0.65

0.93 0.95 0.94

measures, is not surprising given that the measures used are directly proportional to production levels. Mean persistency and production values attributed to different herds (Table 2) are in general agreement with the observed negative association between these variables. A major implication of this discordant association is that the production potential in high lactation persistency can only be realised through longer lactation periods for lower-yield cows. Dekkers et al. (1998) observed that longer lactations generated higher profits when persistency was high, because of greater milk yield beyond 305 days. Both production and persistency depend on genetics and prevailing conditions, but the relationship is confounded by a genotype by environment interaction. The fact that the highest and the third lowest persistency levels observed in the herd of the first data set were attributed to the same cow is remarkable. It is possible that this is a pointer to a stronger environmental influence on lactation persistency. Nevertheless, plausible economics of lactation cannot ignore persistency. So¨lkner and Fuchs (1987) observed that cows with low persistency require more quantities of concentrates to produce an equal quantity of milk compared to those with high persistency under similar conditions. The calving-to-conception period is another factor likely to have significant effect on the economics of lactation. Tekerli et al. (2000) observed

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that cows that conceived shortly after calving had lower lactation yield and persistency. On the other hand, it may be profitable to milk highly persistent cows well beyond 305 days. It seems appropriate therefore that both persistency and service period be factored into decisions regarding the most profitable lactation length.

4. Conclusion The fitting of quadratic curves to cumulative milk yield data resulted in a simple, precise and robust parametric measure of lactation persistency in dairy cows. The proposed lactation persistency model may subsequently lead to more accurate estimates and understanding of the quantitative expression of heritable and environmental influences in the trait.

Acknowledgements The author is grateful to Dr. John Kariuki for his comments on the initial draft of the manuscript and the reviewers for the suggestions that lead to a quality revised article. He would also like to thank Mrs. Gladys Wabuge for providing access to her private farm data records and Dr. Mauro Povinelli of Padova University for kindly providing lactation records of the Italian research herds.

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