3rd Degree Polynomial Model: Is It Best Non-Linear Model for the Estimation of Growth Curve in Anatolian Black Cattle?

doi:10.21203/rs.3.rs-2310351/v1

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3rd Degree Polynomial Model: Is It Best Non-Linear Model for the Estimation of Growth Curve in Anatolian Black Cattle?

https://doi.org/10.21203/rs.3.rs-2310351/v1

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In this study, it was aimed to estimate the growth curve of live weights from birth to 24 months of age in Anatolian Black Cattle raised for conservation purposes. For this purpose, a total of 493 weight records of 113 animals at birth, 3, 6, 12, 18 and 24 months were examined in the study. Six different non-linear models were used to describe the growth curve of animals, these are 2nd degree polynomial, 3rd degreee polynomial, Logistic, Brody, Von Bertalanffy and Gompertz models. In the study, R² values of these models determined 0.997, 0.999, 0.953, 0.979, 0.924 and 0.862; corel (correlation between the observed and estimated) were 0.994, 0.998, 0.989, 0.993, 0.961 and 0.703; Residual Standart Deviation were 3.216, 1.388, 11.533, 3.561, 14.736 and 27.141 respectively. According to these values, it was determined that the 3rd degree polynomial model was best described the growth curve of Anatolian Black. As a result of the analyses, it was determined that the values predicted by this model deviated by 1–3 kg from the observed values in all periods and in all environmental factors examined (sex, dam age, parity, birth year and birth season). It was determined that these differences increased up to 4–5 kg only in the 18-month period. The results also showed that Anatolian Black continued to grow after 24 months of age. As a result, activities such as sexual maturity age, breeding age and slaughter age can be easily predicted by determining the model that best describes growth and development in herds.

Anatolian Native Black cattle

Live weight

Growth curve

Non-linear models

Cattle breeding has an important place in Turkey's animal husbandry, while there are around 18 million cattle in Turkey, approximately 8% of this is local breeds (TÜİK, 2021). Domestic cattle breeding is important in terms of rural employment and development, rural sociology and to use of poor pasture areas. Anatolian Black Cattle (ABC) is one of the breeds that is the most widely grown and spread among the domestic breeds in Turkey. This breed is grown especially in the Central Anatolian Region of Turkey and is mostly grown for meat and milk yield by breeders living in rural areas. It has adapted to these conditions since it has grown in unfavourable conditions in this region for many years. They have gained resistance to harsh winters, drought, hunger, thirst and diseases (Ünal et al. 2019; Sakar and Zülkadir, 2022). Growth and development values in Turkey's domestic cattle breeds are generally slower than that of developed cattle breeds. In studies conducted with the ABC breed in Turkey, birth weight values were found to be between 14.62 and 21.35 kg (Demirhan, 2008lıçel, 2014; Ünal et al., 2019; Sakar and Zülkadir, 2022).

Growth in cattle is a function that continues throughout the life of the animal from embryonic stages to adulthood and can be explained mathematically by growth curve models (Hartati and Pintaka Bayu Putra, 2021). The change in any of the examined features in a certain period is defined as the growth curve (Aytekin et al., 2009). The growth curve shows the statistical relationship between the weight and time or age of the living thing, which is shaped under the influence of genetic potential and environmental factors (Bilgin and Esenbuğa, 2003). Since the growth of living things does not progress at a constant rate throughout their life, it is often insufficient to express the growth rate with linear models (Efe, 1990). In the case of constant growth, linear models are used, and when the growth rate occurs at different times depending on age, nonlinear models (such as Negative Exponential, Brody, Logistics, Gompertz, Bertalanffy, Richards, Weibull) are used (Tutkun, 2019; Hartati and Pintaka Bayu Putra, 2021). The fact that living things have different growth rates in some periods necessitated the use of nonlinear models, which is a more comprehensive model (Bilgin and Esenbuğa, 2003). There is still a need for investigating whether the most commonly preferred non-linear models are sensitive to the length of the growth period prior to truncation of the data (Koncagül and Çadırcı, 2009).

In this study, it was aimed to obtain using non-linear models of the growth curve in the ABC cattle taken individual weights from birth to 24 months and to determine the model that best explains growth. For this purpose, 2nd degree polynomial, 3rd degree polynomial, Logistic, Brody, Von Bertalanffy and Gompertz models were used.

Animal material:

The animal material of this study consisted of Anatolian Black Cattle (ABC) grown in the “International Center for Livestock Research and Training” (39°97′ N, 33°10′ E; elevation 826 m) located in Ankara. This breed has been conserved within the scope of the project “Conservation of Domestic Genetic Resources and Sustainable Use” conducted by the General Directorate of Agriculture Research and Policies (TAGEM). The study was carried out on a total of 113 heads ABC born between 2015–2020.

ABC calves are raised with their dams from birth and they are allowed to suckle their dams freely, and the cows are not milked in the farm. Feeding of ABC breeds cows is two meals a day, morning and evening, ad libitum in the form of total mixed feed. ABC cows are given 80% barley bales and 20% dry meadow grass as roughage.

Data Set

In this study, live weights (LW) of ABC were taken for 6 periods (at birth and 3, 6, 12, 18 and 24 months). These values were determined by weighing them with precision scales up to 200 g. Characteristics of data are presented in Table 1. In this study, birth weight, 3, 6, 12, 18 and 24 month weights of 113 calves born between 2015 and 2020 were used. In addition, sex, dam age, parity, birth year and month, information were also recorded regularly.

Table 1

Descriptive statistics of body weight at different ages in ABC
Statistics	BW	3MW	6MW	12MW	18MW	24MW
N	113	93	96	98	35	58
Minimum (kg)	13.00	37.00	52.50	89.00	144.00	178.00
Maximum (kg)	30.00	99.00	153.00	283.00	332.00	444.00
Female (kg)	17.25	61.70	90.42	144.30	188.65	225.34
Male (kg)	19.55	67.66	103.08	162.75	245.50	303.90
Mean (kg)	18.57	65.10	98.33	155.60	217.89	264.64
Standard Error	0.312	1.340	2.120	3.830	7.970	7.910
Coefficient of Variation (CV%)	17.86	19.88	21.17	24.40	21.64	22.78

Notes

BW = birth weight, 3MW = 3 month weight, 6MW = 6 month weight, 12MW = 12 month weight, 18MW = 18 month weight, 24MW = 24 month weight.

Predicting The Growth Curve

In the study, six different non-linear models were used in the estimation of growth curves, and these models are presented in Table 2.

Table 2

Non-linear models used to describe the growth of ABC
Model	Equation	Reference
2nd Degree Pol.	y_t = β₀ + β₁t + β₂t²	Heinrichs and Hargrove (1987)
3rd Degree Pol.	y_t = β₀ + β₁t + β₂t² + β₃t³	Heinrichs and Hargrove (1987)
Logistic	y_t = A (1 + be ^− kt)⁻¹	(Nelder, 1961)
Brody	y_t = A (1 - be ^− kt)	(Brody, 1945)
Von Bertalanffy	y_t = A (1 - be ^− kt)³	(von Bertalanffy, 1957)
Gompertz	y_t = A exp(-be ^− kt)	(Laird, 1965)

In all equations presented;

β₀, β₁, β₂, β₃: regression coefficients of 2nd and 3rd degree polynomial models,

y_t = observed BW at age t (kg),

A: the asymptotic limit of the BW when age t approaches to infinity (kg),

b: Integration constant, related to the initial weights of the animal and without a well-defined biological interpretation,

k: ratio of the relative intensity of growth (maturation rate),

t: time (month).

Statistical analysis

Statistical analyses were carried out using the Proc Nlin procedure in SAS (2017). Growth curve models were fitted for each animal separately and then the best-fitted model parameters and the other phenotypic data were analyzed using the Proc Glm procedure. For group comparison, the Tukey test was used. In Glm analyses, year and season of birth, parity, age of dam and sex of calves were put into the model as fixed effects.

The growth curve parameters as derived from 6 different models using 493 weight records for ABC are presented in Table 3. In the study, R² (coefficient of determination) and RSD (Residual Standart Deviation) and corel (correlation) between the observed and estimated growth curves were used to compare the models. The 3rd degree polynomial model showed the highest R² and corel and lowest RSD indicating the best goodness of fit. On the other hand, the Gompertz model was the least fitted to estimate the ABC weight based on its lowest value of R². That is, the 3rd degree polynomial model with four parameters (β₀, β₁, β₂, β₃), which has the highest R² and smallest RSD value, best explains the change in live weight according to age in ABC. In Fig. 1, the growth curve of the observed weights by gender and estimated according to the models is presented. As can be seen in Fig. 1, the model most compatible with the values observed in ABC from birth to 24 months is the 3rd degree polynomial model.

Table 3

Model comparison for growth of Anatolian Black Cattle
Model	β₀	β₁	β₂	β₃	R²	RSD	corel
2nd degree polynomial	21.80 ± 0.619	0.44 ± 0.0132	0.01 ± 0.001	-	0.997	3.216	0.994
3rd degree polynomial	18.72 ± 0.361	0.33 ± 0.099	0.01 ± 0.002	0.01 ± 0.001	0.999	1.388	0.998
	Wa	B	K	t_i
Logistic	206.09 ± 8.417	-	-1.79 ± 1.788	8.84 ± 8.839	0.953	11.533	0.989
Brody	225.56 ± 15.625	0.83 ± 0.027	0.01 ± 0.001	-	0.979	3.561	0.993
Von Bertalanffy	169.96 ± 15.650	-0.19 ± 0.124	0.01 ± 0.005	-	0.924	14.736	0.961
Gompertz	129.05 ± 4.530	2.751 ± 0.079	0.40 ± 0.000	-	0.862	27.141	0.703

In similar studies, the 3rd degree polynomial model was determined as the most suitable model in Holstein breed by Heinrichs and Hargrove (1987); Ayrshire, Brown Swiss and Shorthorn breeds by Heinrichs and Hargrove (1994); Holstein and Brown Swiss breeds by Akbulut (1999). On the other hand, the most suitable models were Gompertz and Von Bertalanffy model (R² = 0.70) in Madura breed by Hartati and Putra (2021); Richards model (R² = 0.999) in Holstein by Tutkun (2019); Logistic model in pre-weaning period; Gompertz and Richards models in post-weaning period in the Holstein breed by Koşkan and Özkaya (2014); Richards model (R² = 0.968, 0.960) in Brown-Swiss and Holsteins by Bayram and Akbulut (2009); the Richards model (R² = 0.976) in Anatolian Buffaloes by Şahin et al. (2014). The most suitable models differ in studies conducted with different breeds and environmental conditions. In practice, determining the weight-age relationship of cattle requires a lot of expense and time (Bayram and Akbulut, 2009). In order to make reliable estimations in different regions and different breeds and to use the obtained parameters for selection purposes, first of all, the selection of the appropriate model is necessary.

In the rest of this paper, the results of the 3rd degree polynomial model were presented due to reason that it was determined as the best model for the evaluation of the environmental effects on the data obtained from animals. Table 4 reflects the least square means and their corresponding standard errors of β0, β1, β2, β3 parameters by environmental factors. As a result of the analysis, R² values were found between 0.999-1.0 in all environmental factors, while corel values ranged between 0.997-1.0.

Table 4

Least square means and standard errors of the 3rd degree polynomial model parameters in ABC by different environmental factors
Factor	Group	n	β₀ ± SE	β₁ ± SE	β₂ ± SE	β₃ ± SE
Sex	Female	48	17.61 ± 0.608	0.01 ± 0.182	0.011 ± 0.0028	0.0009 ± 0.0004
Sex	Male	65	20.25 ± 0.573	0.35 ± 0.171	0.005 ± 0.0027	0.0005 ± 0.0003
Dam Age	2–3	31	21.18 ± 1.773	-0.52 ± 0.530	0.006 ± 0.0083	-0.0006 ± 0.0011
	4–7	46	17.45 ± 0.797	0.80 ± 0.238	0.002 ± 0.0037	0.0006 ± 0.0005
	8–10	17	18.12 ± 1.037	0.17 ± 0.310	0.006 ± 0.0048	0.0007 ± 0.0006
	11+	19	18.99 ± 0.979	0.26 ± 0.293	0.019 ± 0.0046	0.0020 ± 0.0006
Parity	1	29	14.40 ± 1.704	0.92 ± 0.510	0.010 ± 0.0080	0.0023 ± 0.0010
	2	24	18.00 ± 0.985	-0.45 ± 0.295	0.016 ± 0.0046	0.0012 ± 0.0006
	3	19	19.95 ± 0.995	-0.34 ± 0.297	0.018 ± 0.0046	0.0008 ± 0.0006
	4	15	19.48 ± 1.084	-0.39 ± 0.324	0.012 ± 0.0050	-0.0001 ± 0.0007
	5	12	20.90 ± 1.189	0.13 ± 0.355	0.003 ± 0.0055	-0.0002 ± 0.0007
	6	8	20.33 ± 1.511	0.63 ± 0.452	-0.002 ± 0.0071	0.0001 ± 0.0009
	7	6	19.47 ± 1.683	0.73 ± 0.503	-0.002 ± 0.0079	0.0005 ± 0.0010
Year	2015	18	17.81 ± 1.026	0.49 ± 0.307	0.003 ± 0.0047	0.0005 ± 0.0006
	2016	23	19.03 ± 0.876	0.15 ± 0.262	0.007 ± 0.0041	0.0001 ± 0.0005
	2017	17	18.39 ± 0.967	0.17 ± 0.289	0.003 ± 0.0045	-0.0001 ± 0.0006
	2018	24	19.70 ± 0.784	0.21 ± 0.234	0.008 ± 0.0036	0.0011 ± 0.0005
	2019	10	19.27 ± 1.147	-0.10 ± 0.343	0.012 ± 0.0054	0.0016 ± 0.0007
	2020	21	19.39 ± 1.041	0.14 ± 0.311	0.013 ± 0.0047	0.0009 ± 0.0006
Season	Winter	17	17.49 ± 0.960^a	-0.15 ± 0.287	0.179 ± 0.0044^a	0.0019 ± 0.001
	Spring	55	19.78 ± 0.696^ab	0.37 ± 0.208	0.002 ± 0.0032^b	0.0002 ± 0.000
	Summer	26	20.58 ± 0.777^b	0.41 ± 0.232	0.004 ± 0.0036^b	0.0001 ± 0.000
	Autumn	15	17.88 ± 1.038^ab	0.08 ± 0.310	0.008 ± 0.0048^ab	0.0006 ± 0.001

^a,b,c The means with the different superscripts within the factor in the same column are different (P < 0.05).

In Figs. 2, 3, 4, 5 and 6 the growth curves of animals according to sex, dam age, parity, birth year and birth season are presented by using the 3rd degree polynomial model. As Fig. 2 is examined, it has been determined that males have a higher weight than females in both observed and predicted values in all periods. Sahin et al. (2014) found that adult live weight was higher in males in all models (Lojistik, Gompertz, Richards, Brody) examined in Anatolian buffaloes. Hartati and Putra (2021) reported that the animals had similar growth characteristics in all models (Logistik, Gompertz, Von Bertalanffy) they examined in both sexes in Madura cattle. Growth in both males and females in the study continued until the age of 24 months, which can be clearly as seen in the linearly plotted growth curve in Fig. 2. Akbulut (1999), using the 3rd degree polynomial model, determined that the growth in Holstein and Brown Swiss breeds continued linearly up to 18 months.

As Figs. 3 and 4 are examined, the differences according to dam age and parity became more pronounced after 18 months of age. According to the chosen model, it was estimated at higher live weight calves born from 11 + age group in periods BW, 3M and 24M; 8–10 age group in periods 6M and 12M; 2–3 age group in 18M. When the graph of differences according to the parity is examined, it was determined to have higher weights animals born from cows 5th lactation in period BW; 7th lactation in periods 3M, 6M and 12M; 2nd lactation in periods 18M and 24M.

When Figs. 5 and 6 are examined, the differences according to the year of birth and season of birth began to appear mostly from 6M. Differences between years were found to have higher weights calves born in 2020 in periods birth, 6M and 12M; 2018 in periods 3M; 2019 in periods 18M and 24M. Differences between seasons were estimated to be heavier calves born in summer in periods birth and 6M; spring in periods 6M and 12M; autumn in periods 18M and 24M.

According to this model, the estimated values showed a deviation of around 1–2 kg in female and male animals at all periods compared to the observed values, while in males they showed a deviation of 3–5 kg only in the 12M and 18M periods. In other graphs, the differences between the generally estimated values and the observed values are between 1–3 kg, and the differences were found to be around 4–5 kg only in 18M periods. This indicates that the 3rd degree polynomial model is the most appropriate model for the growth values of ABC.

Monitoring the growth and development of animals during some periods in the growth process will be of great benefit to the farms in terms of herd management, care and feeding regulation (Şahin et al., 2014). In order to obtain reliable estimates of the growth curve parameters, it may be necessary to collect growth data until the point when the growth curve starts to flatten or the growth rate slows down (Koncagül and Çadırcı, 2009). Changes in body weight in animals reflect the influence of environmental factors and management systems, particularly nutrition (Entwistle et al., 2012). The fact that there is a difference in live weights according to the environmental conditions generally examined during the weighing periods in the ABC shows that the animals can change in the desired direction by selection. In addition, by monitoring the growth of the animals, early intervention can be made for animal that has a problem in their development.

According to the results of the study, the 3rd degree polynomial model was determined as the most suitable model in ABC according to the R², RSD and corel criteria to create the growth model. By using the 3rd degree polynomial model in the farm, the general growth and development of the animals can be followed, and conditions such as sexual maturity age, breeding age and appropriate slaughter age can be easily predicted. Examining the growth curve is important for breeders to decide on the optimum body weight of the animals, the appropriate age and the ideal weight for maximum profit. Growth curve parameters can be successfully applied in animals and may benefit the development and design of selection strategies.

Author contributions: The project managers are ÇMS and İÜ. Conceptualization and methodology were made by ÇMS, SK. Applications and data collection processes were made by ÇMS, İÜ. Statistics analysis was made by SK. Original draft of the article was written by ÇMS and final check also was performed by SK.

Acknowledgements: The population of the study consisted of Anatolian Black breed practiced connected to the “Conservation of Domestic Genetic Resources and Sustainable Use”. Therefore, the authors kindly acknowledges the contribution of the General Directorate of Agricultural Research and Policies (Ministry of Agriculture and Forestry) of the Republic of Turkey who have given the necessary permission to provide the animal material that used in the study.

Funding: The material of the study consisted of animals within the scope of protection and no extra cost was provided.

Disclaimers: The author of the study kindly declares no competing interest. In addition, necessary permissions were obtained from the relevant institution for the use of data.

Consent for publication: Required publication permission was obtained from the Institute.

Statement of animal rights: The measurements of the study were conducted under the supervision of the Animal Experiments Local Ethics Committee of the International Center for Livestock Research and Training.

Availability of data and material: The data is transparent and accessible.

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3rd Degree Polynomial Model: Is It Best Non-Linear Model for the Estimation of Growth Curve in Anatolian Black Cattle?

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Abstract

Figures

Introduction

Material And Methods

Animal material:

Data Set

Predicting The Growth Curve

Statistical analysis

Results And Discussion

Conclusions

Declarations

References

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