Assessing the logistic growth function in connection to growth metrics

This study set out to establish which of the different non-linear growth curve model - Von-Bertalanffy, Gompertz, Richards, Weibull, exponential function, and logistic - was most suitable for characterising the growth curve. A total of 715 female and 340 male kids body weight records, taken every three months from birth to the twelveth month of life, were included. Males had a higher asymptotic live weight (

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

Non-linear Growth Curve ComparisonClose ×

AI-generated conceptual illustration (goat kids and lambs). It explains the idea and does not show study data.

Open full-size image ↗

Introduction

A variety of non-linear mathematical functions have been used to define the growth curve for evaluating response to a particular treatment at different times, interaction between and within populations for identifying heavier animals at an early age, and Gompertz (Laird, 1965) [8] , Bertalanffy (Bertalanffy, 1957) [3] , Logistic (Nelder, 1961) [11] , and negative exponential (Bathaei and Leroy, 1998) [1] (Magotra et al., 2021) [9] . Numerous studies have been undertaken in animal research to define the growth patterns of the animals. In addition to fitting standard models like simple linear or exponential growth to size-at-age data, these research also took into account models with sigmoidal (S-shaped) growth curves, the Brody model, and negative exponential growth models. The latter is modelled by Richards, von Bertalanffy, Gompertz, and Verhulst (logistic growth). A Google Scholar search yielded about 22,500 and 15,500 results, respectively, for articles on the application of the Brody model for sheep and goats. The Verhulst model gave approximately 5500 and 3500 hits, the Gompertz model produced roughly 4000 and 2000 hits, and the von Bertalanffy model produced roughly 2500 and 1500 hits (Brunner and Kühleitner, 2020) [5] .

Materials and Methods

The sheep and goats were maintained on a vast field grazing method. The goats grazed on the pasture for six to eight hours every day. The trees, bushes, and grasses that are available for the goat are classified as follows: Monsoon (Kair, Dhaman, Dudh, Patharchatta, Motha, Akra, and Thur), winter (Neem, Motha, Akra, Keekar, and Beri), and summer (post-harvest leftover residue of Gramme pea (Chickpea), Babul, Kair, and Khejri).

Statistical Analysis

Body weights were standardized for 30, 60, 120,150, 210, 240, 300 and 330 days using the following methodology (Warwick and Legates, 1979) [21] .

Pi = Pneari + ADG (i -age Pneari)

where Pi is the standardized weight at standard age (i), Pnear is the weight nearest to standard age (i), ADG is average daily gain considered among the weights after standard age (i) and before standard age (i), (i) is age to which weight is standardized, and age Pneari, age to weight nearest to standard age (i) considering. Average daily gain in the body weight of individual animal was calculated by using the following formula (Brody, 1964) [4] .

Where: W2= Final body weight (kg); W1= Initial body weight (kg); t2= Age of the animal at the end of the period (days); t1= Age of the animal at the beginning of period (days) Different mathematical models were used to estimate growth curve parameters using Sistastics 10 software. www.extensionjournal.com Non-linear growth curve models Equations Logistic (Nelder, 1961) [11] Where: Wt = the expected body weight (Kg) at 't' time; A= is the asymptotic weight; B= the folding point of growth; K = the rate of growth; m = Shape parameter; Ɛ= random error; e = the base of natural logarithm; t = time (birth to 12 th month of age). Residuals were plotted graphically which gave an accuracy of the model to fit the growth curves.

Where: = Observed body weight at time "t"; = Predicted body weight by regression model at time "t". The Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) was calculated as below (Topal and Balukbasi, 2008) [17] .

Where: = Observed body weight at time "t"; = Predicted body weight by regression model at time "t"; n = Number of observations (data points). In the case the sample size is smaller than the number of model parameter (N⁄K<40), the AIC might not be accurate then after use of Akaike's information Criteria (AICC) was appropriate and calculated as using the equation Motulsky and Christopoulus (2004) [10] .

Where: AIC = Akaike's Information Criteria. Therefore, AICC is a good static for comparison of models of different complexity because it adjust the residual sum of squares (RSS) for number of parameters in the model. A smaller numerical value of AICC indicates a better fit when comparing models. The Chi-square (x 2 ) values is used to designate the relationship between actual and predicted body weights whether there is a significant difference between the predicted and the observed body weights.

Where: Yi = Observed body weight at time "t"; = Predicted body weight by regression model at time "t".

Results and Discussion

Male, female, and both sexes had estimated asymptotic mature live body weight parameters (A) of 22.49±0.88, 20.99±0.88, and 21.46±0.87 kg, respectively. The results were in close agreement with the reports of Yadav et al., (2009) [22] as 20.60 Kg in both sexes of Marwari sheep whereas, In contrary the present result that lower parameter (A) was observed by Thirunavukkarasu et al., (2017) [16] as 19.34, 16.76 and 17.50 Kg in male, female and both sexes of Mechari sheep, respectively. The study by Ganesan et al. (2015) [7] reported higher estimates of parameter (A) as 36.44 and 25.99 kg in male and female Madras red sheep. On the other hand, Tsukahara et al. (2008) [18] reported higher parameter (A) for both sexes, with values of 27.00 kg in Kambing Katjang goat, Tatar et al. (2009) [15] reporting 32.03 kg in young hair goat, Paul et al. (2016) [23] reporting 32.53 kg in non-descript goat, and Waheed et al. (2016) [19] reporting 27.10 kg in Thalli sheep. For the male, female, and both sexes, the Folding Point of Growth Curve (B) for the Logistic Growth Model was found to be 3.84±0.58, 3.87±0.63, and 3.89±0.62, respectively. The results of Paul et al. (2016) [23] , which measured 3.75 in non-descript goats, and Thirunavukkarasu et al. (2017) [16] , which measured 3.98 in Mechari sheep for both sexes, supported the current investigation. The higher parameter (B) was found to be 4.38 in Kambing Katjang goat by Tsukahara et al. (2008) [18] and 4.04 in both sexes of Young hair goat by Tatar et al. (2009) [15] ; on the other hand, Behzadi et al. (2014) [2] observed as 5.09 and 5.09 in Baluchi sheep and 4.05 and 4.06 in Mechari sheep for male and female by Thirunavukkarasu et al. (2017) [6] . For the male, female, and both sexes, the rates of maturing and growth rate (K) for the Logistic Growth Model were found to be 0.41±0.05, 0.41±0.05, and 0.42±0.05, respectively. The current result was consistent with results from Nimase et al. (2018) [13] , showing 0.47 and 0.50 in Madgyal sheep, respectively, in the male and female. Behzadi et al. (2014) [2] found lower estimates of parameter (K) for male and female Baluchi sheep, namely 0.02 and 0.02; Thirunavukkarasu et al. (2017) [16] found lower estimates, namely 0.01 and 0.01 for male, female, and both sexes of Mechari sheep, respectively. Contrary to the current findings, Tatar et al. (2009) [15] reported a higher parameter (K) for both sexes in young hair goats, with a value of 0.66, while Waheed et al. (2016) [19] reported a value of 0.60 in Thalli sheep.

Conclusion

The study discovered that a variety of factors contribute to variation in growth curve model parameters, including but not limited to flock size, species, breed, selection techniques, environmental and managerial conditions, farmer socioeconomic position, and nutritional traits of breeds and species. www.extensionjournal.com

1. Bathaei SS, Leroy PL. Genetic and phenotypic aspects of growth curve characteristics in Mehraban Iranian fattailed sheep. Small Ruminant Research. 1998;29:261269.
2. Behzadi MR, Aslaminejad AA, Sharifi AR, Simianer H. Comparison of mathematical model for describing the growth of Baluchi sheep. Journal of Agricultural Sciences Technology. 2014;14:57-68.
3. Bertalanffy LV. Quantitative laws in metabolism and Growth. Quarterly Review Biology. 1957;32:217-230.
4. Brunner N, Kühleitner M. The growth of domestic goats and sheep: A Meta study with Bertalanffy-Pütter models. Veterinary and Animal Science. 2020;10:100135. https://doi.org/10.1016/j.vas.2020.100135
5. Burnham KP, Anderson DR. Multi-model inference: understanding AIC and BIC in model selection. Social Methods Res. 2004;33:261-304.
6. Ganesan R, Dhanavanthan P, Balasubramanyam D, Kumarasamy P, Kiruthika. Growth Modeling and Factors Affecting Growth Traits in Madras Red Sheep. Indian Journal Animal Research. 2015;49(1):20-25.
7. Laird AK. Dynamics of relative growth. Growth. 1965;29:249-263.
8. Magotra A, Bangar YC, Yadav AS. Growth curve modeling and genetic analysis of growth curve traits in Beetal goat. Small Ruminant Research. 2021;195:106300.
9. Nelder JA. The fitting of a generalization of the logistic curve. Biometrics. 1961;17:89-110.
10. Pires LC, Machado TMM, Carneiro PLS, Silva JBL, Barbosa ADHB, Torres RA, et al. Growth curve of Repartida goat reared in the Caatinga region, Brazil. Semina Ciencias Agrarias, Londrina. 2017;38(2):10411050.
11. Nimase RG, Kaandalkar YB, Bangar YC. Non-linear modeling for estimation of growth curve parameters in Madgyal sheep. Journal of Entomology and Zoology Studies. 2018;6(2):463-465.
12. Tariq MM, Iqbal F, Eyduran E, Bajwa MA, Huma ZE, Waheed A, et al. Comparison of non-linear functions to describe the growth in Mengali sheep breed of Balochistan. Pakistan Journal of Zoology. 2013;45(3):661-665.
13. Tatar MA, Tekel N, Ozkan M, Bartci I, Dellal G. The determination of growth function in young hair goat. Journal of Animal and Veterinary Advances. 2009;8(2):213-216.
14. Thirunavukkarasu M, Balan C, Kathiravan G, Jeichitra V. Non-linear growth modeling in Mechari breed of sheep. Journal of Entomology and Zoology Studies. 2017;5(5):2005-2008.
15. Topal M, Balukbasi SC. Comparison of nonlinear growth curve models in Broiler Chickens. Journal Applied Animal Research. 2008;8:149-152.
16. Tsukahara Y, Chomei Y, Oishi K, Panandam JM, Mukherjee TK, Hirooka H, et al. Analysis of growth patterns in purebred Kambing Katjang goat and its crosses with the German Fawn. Small Ruminant Research. 2008;80:8-15.
17. Waheed A, Eyduran E, Tariq MM, Ahmad S, Hameed T, Bukhari FA, et al. Comparison of the non-linear models defining the growth of Thalli sheep under desert conditions. Pakistan Journal of Zoology. 2016;48(2):423-426.
18. Waheed A, Khan MS, Ali S, Sarwar M. Estimation of Growth Curve Parameters in Beetal Goats. Leibniz Institute of Farm Animal Biology, Dummerstorf, Germany. Archiv Tierzucht. 2011;54(3):287-296.
19. Yadav DK, Singh G, Jain A, Singh S, Paul AK. Fitting of growth models and evaluation of Marwari sheep under field conditions. Indian Journal of Animal Sciences. 2009;79(12):1242-1244.
20. Paul J, Modi A, Patel J. Predicting green product consumption using theory of planned behavior and reasoned action. Journal of retailing and consumer services. 2016 Mar 1;29:123-34.