IntroductionTop

Growth is a fundamental property of biological systems and can be defined as an increase in body size with time. Understanding of the economic importance of various traits, such as body weight, weight gain, rate of maturity, and age and weight at which maximal growth occurs, has led researchers to carry out detailed studies targeting weight-age relationships (Ersoy et al., 2006). For this purpose, different mathematical growth functions have been applied and developed (Gompertz, 1825; von Bertalanffy, 1957; Richards, 1959; France et al., 1996; López et al., 2000). These functions can be used to determine the efficiency of nutrient utilization, which is the derivative of the relationship between body weight and dietary nutrient intake, and as response functions to predict daily energy, protein and amino acids requirements for maintenance and growth and efficiency of utilisation (France et al., 1996; Darmani Kuhi et al., 2009; 2011; 2012).

The general shape of the growth curve is a sigmoid form and is usually modelled using classic growth functions, which summarise the information into a few biologically interpretable parameters (Darmani Kuhi et al., 2010; Teleken et al., 2017). Modelling of animal growth has been a topic of increasing interest over the past seventy-five years. There are many studies aimed at evaluating animal growth models but, due to the nebulous nature of model development, modellers seem more willing to re-develop and adapt existing models for their own requirements than to develop new ones.

In the field of poultry science, growth modelling has been applied to the major avian species, mainly broilers and turkeys (Anthony et al., 1991; Knizetova et al., 1991; Maruyama et al., 1998; Porter et al., 2010). Much less attention has been paid to other poultry species, even though their farming is of increasing interest nowadays (Minvielle, 2004). In particular, Japanese quail (Coturnix japonica) is a species with many positive biological characteristics, such as small size, fast growth rate, early sexual maturity, short generation interval or less feed requirements and less susceptibility to some avian diseases than other poultry species (Vali, 2008). Laying performance (laying rate, incubation period) is also rather prospective, conferring some advantages to the farming of this species. Ho­wever, there is scarcity of information on the growth traits of quail and on their modelling using non-linear equations, aspects that would result in more efficient farming programmes. Therefore, it is valuable to gain insights and knowledge on the factors influencing the productive performance of this species. The objectives of the present study were to introduce a sinusoidal equation into poultry science, to apply it to temporal growth data from quail, and to compare its fitting performance with four standard growth functions, viz. the Gompertz, logistic, López and Richards.

Material and methodsTop

Data set

Twelve time course profiles representing the growth curves of six male and six female quail were obtained from two publications (Balcıoğlu et al., 2005; Sezer & Tarhan, 2005), and used in the study. The growth profiles reported by Sezer & Tarhan (2005) each contained 17 data-points, with weights given at 3-day intervals from hatching to Day 48 of age. The curves described by Balcıoğlu et al. (2005) had weekly weight recordings from hatching to Week 8, giving 9 data-points in each profile.

Sinusoidal model

The functional form of the sinusoidal equation proposed as a growth function to describe the rela­tionship between bodyweight and age in male and female quail is:

where w is body weight (g) at age t (d), wf is final weight, and the parameters b and θ are real numbers. The above equation provides a versatile function that can describe a range of predominantly sigmoidal behaviour. Initial body weight (w0) and point of inflexion (w*, t*) when describing sigmoidal patterns (Fig. 1) are given by

Figure 1. Graph of the sinusoidal equation showing its fit to the male data of the High line. BW is body weight (g), Time is age (d).

where b and θ are positive. The integer N is given by

where

denotes the greatest integer less than or equal to x.

To examine the performance of the sinusoidal equation in describing the growth patterns of quail, four conventional growth functions, viz. Gompertz, logistic, López and Richards (Thornley & France, 2007), were used as reference in this study (Table 1). These functions are the most common equations used to describe somatic growth curves in animals. The five non-linear functions were fitted to the growth curves using the non-linear regression procedure of SigmaPlot 12.0 (Systat Software, Inc., San Jose, CA, USA). The Marquardt-Levenberg algorithm was used for iterative estimation of the parameters of each model. Initial parameter values had to be provided to commence the iterative process. Final estimates were not affected by the initial values adopted.

Statistical analysis

Comparison of models was carried out by analysing model behaviour when fitting the curves using nonli­near regression and assessing statistical performance. Maximum log-likelihood estimation (MLE), mean squared error (MSE), Akaike information criterion (AIC) and Bayesian information criterion (BIC) were used to evaluate the general goodness-of-fit of each model to the different data profiles.

ResultsTop

Fitting behaviour of the five equations when applied to growth curves for quail of a high line (as an example) is illustrated in Fig. 2 (males) and Fig. 3 (females). The parameter estimates are given in Tables 2 and 3 for male and female quail, respectively.

Figure 2. Plots of body weight (g) against age (d) showing the fit of different growth functions to the data for male High line quail.

Figure 3. Plots of body weight (g) against age (d) showing the fit of different growth functions to the data for female High line quail.

As seen in s 2 and 3, final weight, and age and weight at point of inflexion were numerically different between males and females. With all quail strains and all the growth functions, point of inflexion occurred at an earlier age in males than in females, and both weight at inflexion and asymptotic or final weight were smaller in males than in females.

There were also relevant differences among models in the growth parameters that are common to all the models (w0, wf, t*, w*). For female quail (Table 3), the overall estimates of final weight from the sinusoidal and logistic equations were lower than those from the other equations. Fitting the growth equations to both male and female data profiles mostly led to the lowest MSE, AIC, BIC and highest MLE values with the sinusoidal and Richards equations, indicating that the data are better described by these equations than by the others (Table 4). Comparing the sinusoidal with the Richards, the sinusoidal equation was superior to the Richards for 66.7% of the male data profiles, whereas for the females it was superior to the Richards in all cases. The López, Gompertz and logistic equations led to an inferior fit when compared to the sinusoidal. Inferiority of the three aforementioned equations occurred in less cases with the male data profiles than with the female ones, which is in agreement with the results obtained for the Richards equation. The Richards showed a statistically better fit to male than to female growth curves when compared to the sinusoidal. In general, the sinusoidal was the best model compared to Richards, logistic, Gompertz and López (Table 4).

DiscussionTop

Quail farming is generating much interest in many parts of the world as it provides low energy/high protein meat of high biological value. It also offers an opportunity for poultry farmers to undertake a new and profitable enterprise (Minvielle, 2004). In addition to its lean meat, the quail’s egg is lower in cholesterol compared to that of the chicken (Musa et al., 2008).

Growth curves are useful tools to represent the evolution of body weight during the growing period and to provide useful and practical information for breeding and feeding purposes (Maruyama et al., 1998; Aggrey, 2004). Growth curves may be used for pre-selection of animals as they can be applied to prediction of future growth at any age (Anthony et al., 1991, 1996; Tekel et al., 2005). Brody (1945) suggested that final or mature weight, rate of attainment of mature weight, and standardised age at which an animal attains the point of inflexion (maximum growth rate) of the curve are parameters that can be suitable and relevant objectives for genetic improvement (Raji et al., 2014). Different mathematical functions have been used to describe growth curves. Furthermore, accurate estimation of daily weight gain using growth functions will allow for better calculation of the bird’s energy and nutrient requirements, and thus the formulation of more fit-for-purpose diets. The most commonly used animal growth functions are Brody (Brody, 1945), von Bertalanffy (von Bertalanffy, 1957), Richards (Richards, 1959), logistic (Nelder, 1961) and Gompertz (Laird, 1965).

To the best our knowledge there is little information available on the use of trigonometric (such as the proposed sinusoidal) functions (Darmani Kuhi et al., 2018) to describe the evolution of growth in poultry. In this study, a sinusoidal equation was evaluated with regard to its ability to describe the relationship between body weight and age in quail. Fitting performance observed with the sinusoidal was compared with four standard growth functions, viz. the Gompertz, logistic, López and Richards. The sinusoidal equation has the ability to describe a range of different curve shapes.

A comparison among models involves a contrast of the parameter estimates obtained with each model. The greatest discrepancy between models was observed in initial weight, as the smallest values of w0 were obtained with the Gompertz and sinusoidal equations and the largest with the logistic and López. Reported hatching chick weights (Daikwo et al., 2011; Farghly et al., 2015) are closer to the values estimated with the Gompertz and sinusoidal equations. Relative differences among models were narrower for final (asymptotic) weight, with López and Gompertz giving the greatest and sinusoidal the smallest values. Average reported mature weights for male and female Japanese quail (Balcıoğlu et al., 2005; Raji et al., 2014) are closer to the estimates obtained with the sinusoidal equation. As for age and weight at point of inflexion, López and Gompertz estimated earlier ages and lighter weights at inflexion, in contrast to the logistic and sinusoidal for which values of both parameters were higher. With all five functions the estimates of age and weight for females at point of inflexion, in theory related to the onset of sexual maturity, were substantially lower than the values reported for hens at the onset of laying or first egg age (Zelenka et al., 1984; Steigner et al., 1992). Furthermore, all the functions represented the differences between males and females, and the different comparisons among quail lines/strains. Males reached the point of inflexion at an earlier age than females, with lighter weights and lower growth rates at that age, and showed a smaller final weight and consequently a shorter growth period than female quail (Steigner et al., 1992; Du Preez & Sales, 1997). The comparison between heavy and light lines of quail within the same sex was similar with all the models.

Comparison of the models based on their behaviour and statistical performance showed that all functions gave a suitable fit to the data profiles. In general, assessment of the growth models based on the statistical criteria indicated some relevant differences between the functions for describing future growth of quail. In spite of the fact that the appropriateness of the models was dependent on the data and sex, the estimated statistical criteria demonstrate the suitability and superiority of the sinusoidal equation over the others. The comparison of growth functions in terms of the goodness-of-fit criteria revealed that the sinusoidal was the most appropriate for describing the age-related changes of bodyweight in quail. Nevertheless, selection of the best function requires special attention to characterise the growth patterns of animals under different environmental conditions (Dogan et al., 2010). This is especially important when a particular data set is obtained under certain specific conditions (for a given bird strain, for particular feeding regimes, or when quail are farmed in certain climatic or environmental conditions) that were not defined precisely (Darmani-Kuhi et al., 2003; Beiki et al., 2013). Therefore, it is advisable to compare different functions to fit prior to model selection.

In conclusion, the sinusoidal equation is a suita­ble alternative to conventional growth functions to

represent the growth curves for a range of strains/lines of male and female Japanese quail.