Livestock production in Ukraine plays an important role in the country's agricultural economy (Stavetska et al., 2022), providing the population with significant quantities of milk (Bal-Prylypko et al., 2024; Ruban et al., 2022) and meat (Antoniv et al., 2023). According to the State Statistics Service of Ukraine, at the beginning of 2023, there were 1 352 800 cows of various breeds on all categories of farms in Ukraine (SSSU, 2024).

Intensive increase in weight and body size, commonly referred to as growth, is an important fundamental goal of animal breeding (Kozakli et al., 2022; Kramarenko et al., 2022). Scientists have described growth in three stages, stating that growth starts at a certain point and gradually increases in the first stage, which is characterized by a partially linear shape of the growth curve until it reaches the inflection point in the second stage, and, finally, in the third stage, the growth curve approaches an asymptote (Waheed et al., 2011). Ghavi Hossein-Zadeh (2017) emphasized the importance of observing the growth of animals throughout their lives, while Sakar et al. (2023) emphasized the importance of determining the optimal body weight, breeding age, slaughter age, and feeding procedures. Budimulyati et al. (2012) stated that the age of sexual maturity for first mating in cattle is determined by the optimum body weight (Sakar et al., 2023). From an economic point of view, it is essential to produce more meat in a shorter feeding period (Domínguez-Viveros et al., 2023). In addition, Ismirandy (2018) recommended that growth curves be used to select breeding strategies for high-producing animals, taking into account their growth performance. Gemuh (2020) highlighted that calves continue to grow optimally between 1–8 months. For example, the value of cattle at slaughter is highly dependent on the amount of muscle in the carcass. Body weight is strongly related to other economically important traits, including production and reproductive traits (Abdallah and McDaniel, 2000). In order to maintain the high productivity of cattle, optimization of genetic improvement programs is important (Matvieiev et al., 2023). Body weight gain after weaning in cattle represents the growth potential of the animal, which is an important trait in animal selection (Raungprim et al., 2023). A significant advantage of some breeds in this respect may lead to a decision to change the breed composition of the herd (Christensen et al., 1984). It should be noted that mating cows of Ukrainian origin with the semen of foreign sires allows one not only to increase growth energy but also to increase milk production and to reduce the percentage of stillborn calves (Bashchenko et al., 2023; Borshch et al., 2021).

Of particular interest is the experience with the Ayrshire and Norwegian Red breeds. Ayrshire crosses are known to have higher growth energy than the parent breeds (Robison et al., 1980). In addition, such crossbred animals have higher milk production. In particular, the milk yield of primiparous (F₁) cows obtained by crossbreeding Finnish Ayrshire and Danish Red was 859 and 1045 kg higher than that of purebred cows of the corresponding parent breeds (Pedersen and Christensen, 1989). In Croatia, first-generation crosses of Holstein and Norwegian Red were characterized by higher milk yield, better milk composition (fat, protein), and shorter days open than pure Holsteins (Benak et al., 2020). In addition to the increase in milk productivity, the Norwegian Red × Ayrshire also showed an improvement in milk nutrition values (Ruban et al., 2023).

In addition, reaching optimal weight by a certain age is an important goal for many breeding programs (Halvoník et al., 2023). It is also important to note that environmental factors should be taken into account when assessing the growth curve (Mokhtari et al., 2023), which can be done using individual statistical models (Weigel et al., 2017). There are various non-linear mixed mathematical models such as the Brody, von Bertalanffy, logistic, Gompertz, Richards, monomolecular, and Weibull models (Beltran et al., 1992; Waiz et al., 2019) that are used to describe growth curves.

The main objective of this study was to compare four different non-linear mathematical models used to characterize the growth pattern of Ayrshire and crossbred cattle of both sexes.

Mean body weights, standard deviations, and coefficients of variation were calculated for different ages by sex (Tables 2–5).

The birth weights of Ayrshire and crossbred calves at birth were 29.78 and 29.66 kg for males and 29.38 and 29.32 kg for females. The 330 d live weights of Ayrshire and crossbred animals were 265.00 and 289.12 kg for males and 265.98 and 289.12 kg for females, respectively. When the differences in body weight were analyzed, the males had a higher body weight than the females. Meanwhile, the crossbred genotype had a higher body weight than Ayrshire cattle.

Considering Adj-R2, OSE, AIC, and BIC as performance criteria, model selection is based primarily on the information criteria (AIC and BIC). On this basis, across both sexes, as shown in Table 7, the Richard model provides the best overall fit. By contrast, the Gompertz model often attains higher explanatory power (greater Adj-R2) and lower prediction error (smaller OSE). Because AIC and BIC penalize model complexity and offer a principled basis for comparative selection, the Richard model is retained as the preferred specification, while the Gompertz model's strengths in terms of Adj-R2 and OSE are acknowledged (Table 6).

It should be noted that Adj-R2 values were high for all models describing Ayrshire growth, ranging from 0.8699 to 0.9531. Based on AIC and BIC, the Richard model provided the best overall fit for Ayrshire cattle for both males and females. The body weight growth curves (observed and Richard-predicted) for Ayrshire cattle are shown in Fig. 1 (A: male; B: female).

For the models describing the growth of crossbred (F1) cattle, the adjusted R2 values ranged from 0.8125 to 0.9358. Because model selection prioritized AIC and BIC, the Richard model provided the best fit for both male and female crossbred (F1) cattle, yielding the lowest AIC and BIC values (Table 6). The body weight growth curves of crossbred (F1) cattle were plotted using the Richard model and are presented in Fig. 2.

The growth parameters for Ayrshire and crossbred F₁ cattle are presented in Table 7.

The regression coefficients are β0, β1, and β2, where β0 is the asymptotic value, interpreted as the adult weight parameter; β1 is an adjustment parameter; and β2 is the growth rate, expressing weight gain as a proportion of total weight. The standard error Se2 indicates the level of uncertainty in predictions.

The prediction of the Richard model for female Ayrshire cattle is as follows.

The data obtained were used to fit regression equations to predict weight. 10yi=966/((1-0.1669⋅exp⁡(-0.002132⋅day))∗∗(1/(-0.062))) In the Richards model equation, the double asterisk (∗∗) denotes exponentiation (raising to a power). For example, X∗∗ (1/β3) means X raised to the power of (1/β3). The asymptotic value (β0) of 966 kg is the maximum weight that purebred Ayrshire females can reach as they grow older. It represents the highest body weight they can achieve over time. β1=-0.1669 shows how the growth rate changes over time. In this case, the negative value indicates a decreasing growth rate; i.e., weight gain slows down as Ayrshire females mature. The slope of the growth rate curve (β2) is 0.002132, reflecting the rate at which the growth curve rises. A higher value indicates a steeper growth curve, implying faster weight gain during certain growth periods. The parameter (β3), with a value of -0.062, influences the shape and slope of the growth curve. A negative value of (β3) results in a concave growth curve, where the growth rate decreases over time and approaches a plateau. The error variance (Se2) of 5.5800 represents the variability or spread of actual live weights around the values predicted by the model. A lower Se2 indicates a better fit of the model to the observed data, with less variability in the predicted weights.

The prediction of the Richard model for female F1 crossbred cattle is as follows: 11yi=1371.22/((1-0.09056⋅exp⁡(-0.001855⋅day))⋅(1/(-0.03112))). The asymptotic value (β0) of 1371.22 kg is the maximum weight that female crossbred F₁ cattle can reach as they grow older. It represents the highest weight they can reach over time. β1=-0.09056 indicates how the growth rate changes over time. In this case, the negative value indicates a decreasing growth rate, which means that weight gain slows down as the crossbred F₁ cattle mature. The slope of the growth rate curve (β2) is 0.001855, reflecting the rate at which the growth curve rises. A higher value indicates a steeper growth curve, implying faster weight gain during specific periods of growth. The parameter β3, with a value of -0.03112, influences the shape and slope of the growth curve. A negative value of β3 results in a concave growth curve, where the growth rate decreases over time and approaches a plateau. The error variance (Se2) of 4.5341 represents the variability or scatter of actual live weights around the values predicted by the model. A lower Se2 indicates a better fit of the model to the observed data, with less variability in the predicted weights.

The prediction of the Richard model for male Ayrshire cattle is as follows: 12yi=356.5/((1-0.8762⋅exp⁡(-0.003861⋅day))⋅(1/(-0.7434))). Here, β0=356.5 kg is the asymptotic (mature) body weight that males approach with age; it represents the maximum attainable weight over time. The adjustment parameter β1=0.8762 is linked to the initial acceleration of growth; its negative sign implies that early gains are relatively fast but progressively decelerate as animals mature. The growth rate parameter β2=0.003861 governs how quickly weight increases per unit time; larger values would imply a steeper rise and an earlier approach to maturity. The shape parameter β3=-0.7434 determines the curvature of the trajectory; a negative value yields a sigmoidal curve with clear deceleration toward a plateau. The model error variance Se2=45.0315 quantifies the dispersion of observed weights around fitted values.

The Richard model prediction for crossbred F₁ male cattle is as follows: 13yi=388.34/((1-0.8970⋅exp⁡(-0.003362⋅day))⋅(1/(-0.7658))). In this equation, β0=388.34 kg denotes the mature (asymptotic) weight. The adjustment parameter β1=-0.8970 modulates the early phase of growth; its negative sign indicates rapid early gain followed by tapering. The growth rate parameter β2=0.003362 controls the speed at which body weight increases; higher values correspond to a steeper rise. The shape parameter β3=-0.7658 governs the curvature and the approach to the plateau. The error variance Se2=27.8932 reflects the residual variability around the fitted curve.

In this study, only appropriate models were selected for the description of the growth curve. The Gompertz model provided the best fit of the growth curve for male Ayrshire and crossbred cattle, and the Richard model provided the best fit for female Ayrshire and crossbred cattle due to higher Adj-R2 and lower OSS, AIC, and BIC than other models. The results of the present study are in agreement with the reports of Hartati and Putra (2021), who evaluated the growth curves in male and female Madura cattle using logistic, Gompertz, and von Bertalanffy functions due to the simplicity of their interpretation. Similarly, Inoue et al. (2020) determined the growth curve and reported that the Gompertz model gave the best fit in Japanese Black cattle. Similarly, Tutkun (2019) evaluated the Gompertz, Richard, and logistic models and selected the Richard model as that which best fitted the equation for Holstein-Friesian bulls. Júnior et al. (2022) compared six different growth models (Brody, Gompertz, logistics, Richard, Meloun, von Bertalanfy) describing the growth curve of Nelore cattle and found that the Brody function gave the best-fitting results. In addition, Adinata et al. (2022) described growth patterns in Ongole grade cattle and found that the Brody model gave the best fit to the growth curve. Meanwhile Sakar et al. (2023) compared six different non-linear models to describe the growth curve of animals (second-degree polynomial, third-degree polynomial, logistic, Brody, von Bertalanffy, Gompertz), and, according to the results obtained, the third-degree polynomial model was the best to describe the growth curve of Anatolian Black cattle.

Hartati and Putra (2021) calculated β0 in Madura cattle using the logistic, Gompertz, and von Bertalanffy models and obtained values of 218.02, 274.13, and 329.83 kg for females and 220.80, 277.72, and 333.92 kg for males. Júnior et al. (2022) determined β0 values of 357.51 and 334.00 kg for male and 306.93 and 289.00 kg for female Nelore cattle using the Gompertz and logistic models. Domínguez-Viveros et al. (2023) evaluated β0 using the Gompertz, logistic, and Brody models, obtaining values of 491.0, 408.2, and 1645.9 kg for males and 715.1, 402.9, and 352.3 kg for females in Limousin cattle. Tutkun (2019) estimated β0 in Holstein-Friesian bulls using Gompertz, Richards, and logistic models, obtaining 986.440, 1110.24, and 672.940 kg. Sakar et al. (2023) calculated β0 in Anatolian Black cattle using different models: logistic – 206.09 kg, Brody – 225.56 kg, and Gompertz – 129.05 kg. Budimulyati et al. (2012) found that 29- and 21-month-old Holstein-Friesian heifers had β0 values of 343.6 and 306.3 kg (logistic model) and 354.5 and 319.1 kg (Gompertz model). Adinata et al. (2022) calculated β0 in Ongole grade cattle using Brody, logistic, and Gompertz models and obtained 857.639, 596.868, and 640.136 kg for males and 692.346, 04.237, and 537.936 kg for females, respectively. The β0 was also calculated using logistic, Gompertz, and Brody models for Swamp buffalo (Bubalus b. carabanensis): 675.10, 698.18, and 902.00 kg (Raungprim et al., 2023). On the other hand, the β0 values obtained using different models for some breeds of Bos indicus cattle were similar to those of Domínguez-Viveros et al. (2023) for Limousin cattle but did not agree with the results of Adinata et al. (2022) for Ongole grade cattle, Tutkun (2019) for Holstein-Friesian bulls, and Sakar et al. (2023) for Anatolian Black cattle.

Malhado et al. (2009) pointed out that β1 is an integration constant with no specific biological meaning. In this study, the estimated β1 values in the Gompertz and Richard models were 2.4064 and -0.0134 for Ayrshire cattle and 2.3610 and -0.01163 for crossbred cattle for males and females, respectively, and were lower than those obtained by Hartati and Putra (2021), who evaluated the growth curve in Madura cattle using the logistic model, obtaining values of 7.10 and 7.31 for males, and the Gompertz model, obtaining values of 2.48 and 2.51 for females. Júnior et al. (2022) determined the value of β1 for female Nelore cattle using the Brody, Gompertz, and logistic models (0.9216, 2.0002, and 4.6810 in males and 0.9067, 1.9472, and 4.5291 in females, respectively). Domínguez-Viveros et al. (2023) found that the best-fitting model for estimating growth in Limousin cattle was the von Bertalanffy model, with a parameter β1 of 0.5949 for males and 0.5666 for females. Sakar et al. (2023) calculated β1 in Anatolian Black cattle using the Brody model (0.83) and the Gompertz model (0.40). Tutkun (2019) estimated the β1 value in Holstein-Friesian bulls using the Gompertz model (3.354), the Richard model (0.299), and logistic model (13.760).

Using the Gompertz model, β2 values were calculated for Ayrshire cattle: 0.006707 and 0.004153. The Richard model shows that the β2 values of crossbred cattle were 0.006316 and 0.004546 for males and females, respectively. The estimated value for β2 in the present study was in line with the findings of Tutkun (2019), who estimated the β2 value in Holstein-Friesian bulls using different models: Gompertz – 0.004, Richard – 0.003, and logistic – 0.008. Júnior et al. (2022) determined the β2 value in Nelore cattle using the Gompertz and Brody models, obtaining 0.0051 and 0.0019 for males and 0.0059 and 0.0025 for females. Domínguez-Viveros et al. (2023) estimated the β2 value using the Brody, von Bertalanffy, Gompertz, and logistic models to be 0.000618, 0.00583, 0.0117, and 0.00400 for male Limousin cattle and 0.00151, 0.00656, 0.0124, and 0.0477 for female Limousin cattle. However, the β2 values obtained in this study were significantly lower (0.18 and 0.09 for females and 0.19 and 0.09 for males) than those calculated by Hartati and Putra (2021) for Madura cattle using the logistic and Gompertz models.

This study evaluated the growth performance of Ayrshire and crossbred (F1) cattle using four non-linear growth models. While the Gompertz model sometimes provided higher explanatory power (Adj-R2) and lower prediction error (OSE), the Richard model consistently yielded the lowest AIC and BIC values, making it the most reliable and parsimonious choice. Across both breeds and sexes, growth curves showed that females attained substantially higher mature body weights than males, with crossbred cattle outperforming purebred Ayrshire in terms of asymptotic weight and goodness-of-fit precision. The lower error variance observed in crossbred cattle further emphasizes the predictive accuracy of the Richard model. Overall, the Richard model provides a robust framework for describing and predicting cattle growth, offering valuable insights for genetic selection, herd management, and breeding strategies aimed at improving productivity.