The relationship between feed intake and liveweight in domestic animals

Abstract

Feed intake changes as animals age and grow. A constraint of most functional forms used to describe this relationship is that intake is maximum only once an animal reaches its mature weight. Often such is not the case and maximum intake is achieved earlier. Our aim was to describe a form unburdened by such a constraint and to determine its utility to describe the relationship between feed intake and liveweight across multiple species. Twelve data sets representing seven domestic animal species (cattle, chicken, dog, pig, rat, sheep, and turkey) with a wide range of mature weights were used. Average daily ad libitum feed intakes and liveweights were available on either a weekly or fortnightly basis. Rates of intake were scaled to mature intake. Within each set, the quadratic regression of scaled intake on the degree of maturity in weight was fitted. This form provided a very good description of the relationship between these variables (R² > 0.86) and, for all but one case, a realistic prediction of mature intake. With one exception, intake reached its maximum value at a liveweight below its mature value. Furthermore, by appropriately scaling the relationship between intake and liveweight, the data could be described by a function with a single parameter with general relevance across species. By expressing the rate of intake as a function of its value at maturity, a quadratic form provides a robust and general description of the relationship between feed intake scaled to mature intake and degree of maturity in weight.

Introduction

Various forms have been suggested to describe how the rate of feed intake changes with either time (Parks, 1970, 1982) or weight (Taylor, 2009) for animals fed ad libitum. The functions of Parks and Taylor both have the consequence that intake has no maximum for a weight less than mature weight. This is also a property of most of the other forms that have been proposed (Pittroff and Kothmann, 2001; Whittemore et al., 2001). An exception is the equation of CSIRO (2007), which predicts a maximum intake at a fixed 0.85 of mature size.

Lewis and Emmans (2010) used data from different kinds of sheep over a wide range of degree of maturity, and on feeds of different composition, to explore the relationship between the rate of intake and weight. They found that the data were well described by a quadratic form of equation through the origin. The same form applied when the intake was scaled to mature weight raised to the 0.73 power, and when weight was expressed as a proportion of mature weight, u. The quadratic form allows a maximum scaled intake when u < 1, as was found to be the case in almost all data sets that they used.

In this study, we had two objectives. The first was to make the scaling rule of Lewis and Emmans (2010) more general. The second was to examine whether that rule was applicable to species of domestic animals other than, but including, sheep.

Materials and Methods

Animal Care and Use Committee approval was not obtained for this study since the data were obtained from the published scientific literature.

Model

Between kinds of animals of different mature weights (A, kg), intake at maturity (C, kg/d) is expected to be scaled so that C is proportional to A^0.73. A simpler, and stronger, scaling rule than that used by Lewis and Emmans (2010), where the intake was scaled by A^0.73, is to express an immature intake, dF/dt (kg /d), as a proportion of C so that the new scaled intake (S) is defined as S = (dF/dt)/C. For a given animal, weight (W, kg) can be expressed as a degree of maturity, u = W/A. The scaled quadratic equation reflecting the relationship between S and u is:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{S}=\alpha .\mathrm{u}+\beta .{\mathrm{u}}^2}\end{array}}$$ (1)

Given the scaling rules used to derive u and S when u = 1, that is the animal is mature, then S = 1 as it is now eating at its mature rate, C. From equation 1, it then follows that 1 = (α + β) so that β = (1 – α). The scaled model thus has a single parameter, α, and equation 1, therefore, can be rewritten as:

$${\displaystyle \begin{array}{l}{\displaystyle \mathrm{S}=\alpha .\mathrm{u}+\left(1-\alpha \right).{\mathrm{u}}^2}\end{array}}$$ (2)

Data

The data sets used came from species having a wide range of mature weights. Average ad libitum daily feed intakes and liveweights were available on a weekly or fortnightly basis. The feed intakes used were those reported by their source; they were not converted to a dry matter basis or to energy contents. For rats, the first data set came from Table A-11, and the second from Table A-12, in Appendix A of Parks (1982). Data for two kinds of male chickens were taken from Tables A-36 (layer) and A-24 (broiler) in Appendix A of the same source. The data for male and female turkeys of one strain came from R. M. Gous (University of Kwazulu-Natal, Pietermaritzburg, South Africa, personal communication) and were from birds described in Gous et al. (2019). Data for a breed of dog, Great Dane, were taken from Table A-6 in Appendix A of Parks (1982). The data for two kinds of sheep, male and female Suffolk sheep from a selected line fed a high-quality feed ad libitum, were from among those used by Lewis and Emmans (2010). They were chosen because they had the longest run of data. The data for cattle were taken from two sources for growth after weaning. The first was Table A-3 in Appendix A of Parks (1982) and the second was Table 1 from Taylor et al. (1986). Two sets of data on swine were also investigated. The data from Table 20 of Shull (2013) for gilts were consistent with the Gompertz form for growth and with the quadratic form for intake. Whittemore et al. (2003) described their data by a quadratic relationship between intake and weight. However, in neither case was the heaviest weight a large enough degree of estimated maturity to allow an accurate estimate of mature intake. These data on swine were, therefore, excluded from the analysis of scaled intake. The data of Shull (2013) were, however, used to estimate the values of the Gompertz parameters and to fit the quadratic form to the intake data. No data set was excluded because it failed to be described by a quadratic function.

Methods

The W by time (t, d) data from each of the 12 sets were first used to estimate A using the assumption that a Gompertz function applied (Gompertz, 1825). The range in the degrees of maturity for the 12 sets is in Table 1. The data were then used to calculate the Gompertz variable G where G = −ln(−ln(W/A)). The linear regression of G on t was fitted, with both the slope (B) and intercept estimated. An example is in Figure 1.

Table 1.

Summary statistics for 12 data sets obtained from the fit of Gompertz and scaled intake regressions

			Gompertz regression3				Scaled intake regression4
Species1	Data set	Range in u2	A, kg	B, /d	R 2	B*	C, kg/d	C*	R 2	α
Rat	One	0.17:0.95	0.44	0.0245	0.996	0.0196	0.0230	41.9	0.898	2.88
	Two	0.16:0.96	0.41	0.0365	0.997	0.0287	0.0197	37.8	0.931	3.35
Chicken	Layer	0.03:0.96	2.2	0.0239	0.998	0.0296	0.136	76.5	0.955	1.95
	Broiler	0.02:0.98	5.6	0.0267	0.990	0.0425	0.208	59.1	0.880	2.81
Turkey	Female	0.008:0.80	15	0.0256	0.997	0.0532	0.408	56.5	0.963	2.95
	Male	0.004:0.75	28	0.0212	0.997	0.0521	0.571	50.1	0.954	3.54
Dog		0.09:0.90	55	0.0167	0.968	0.0493	1.08	57.9	0.951	2.88
Sheep	Female	0.34:0.82	100	0.0108	0.998	0.0374	1.72	59.6	0.856	4.30
	Male	0.25:0.79	130	0.00915	0.999	0.0341	2.00	57.3	0.938	4.66
Pig		0.03:0.55	230	0.0126	0.999	0.0547
Cattle	One	0.18:0.72	520	0.00457	1.000	0.0247	7.29	75.9	0.987	3.43
	Two	0.13:0.75	750	0.00341	0.999	0.0204	6.04	48.1	0.981	5.65
Average						0.0372		56.4
CV, %						34.8		21.5

¹Data sources: on rats from Tables A-11 (set one) and A-12 (set two) in Appendix A of Parks (1982); on male chickens from Table A-36 (layer) and A-24 (broiler) in Appendix A of Parks (1982); on female and male turkeys from R. M. Gous (University of Kwazulu-Natal, Pietermaritzburg, South Africa, personal communication) as described by Gous et al. (2019); on the Great Dane dog breed from Table A-6 in Appendix A of Parks (1982); on female and male Suffolk sheep from a selected line as described by Lewis and Emmans (2010); on gilts from Table 20 of Shull (2013); on cattle after weaning from Table A-3 (set one) in Appendix A of Parks (1982) and Table 1 (set two) from Taylor et al. (1986).

²Degree of maturity.

³Linear regression of the Gompertz variable, G = −ln(−ln(W/A)), on time t (d), where W (kg) is liveweight at t, A is mature weight, B is a rate parameter, and B* = B.A^0.27.

⁴Quadratic regression of scaled feed intake on the degree of maturity (equation 1), where C is the maximum feed intake, C* = C/A^0.73, and α is the scaled parameter value form the fit of the regression (equation 2).

Figure 1.

Gompertz variable, G = −ln(−ln(W/A)), and degree of maturity, u = W/A, plotted against time, t (d), in male layer chickens. W (kg) was liveweight at time t, and A (kg) was mature weight. Data were from Table A-36 in Appendix A of Parks (1982). The slope, which estimates the rate parameter, B, was 0.0239 ± 0.000246/d (R² = 0.998).

Actual intake in a time period, dF/dt, usually weekly intervals, was plotted against the mean weight in that time period, taken as W. The quadratic relationship of dF/dt to W, with zero intercept, was fitted. The regression coefficients were then used to estimate dF/dt when W = A, our value for C. Actual intakes were then scaled to C and S plotted against the degree of maturity in weight, u. The values of the linear coefficients of the quadratic regression of S on u through the origin were used to estimate the value of α.

Results and Discussion

The results of the Gompertz regressions are in Table 1. The quadratic relationship between intake and weight was used to estimate the values of C, that is intake when W = A. The quadratic relationships between scaled intake, S, and scaled weight, u, were then used to estimate the values of α, which are provided in Table 1 with the R² values of the regressions. The quadratic relationship captured the great majority of the variation as seen in Figure 2a–f and allowed an estimate of α. In no case was α significantly less than 2; the layer chicken had a value of 1.95. The value of α = 2 is a critical one. When α < 2, the predicted maximum intake is not achieved before reaching mature weight.

Figure 2.

Scaled intake, S, plotted against scaled weight, u. S is (dF/dt)/C, where dF/dt (kg/d) was immature intake and C (kg/d) was intake at maturity; u is W/A, where W (kg) was weight at time t and A (kg) was weight. The linear coefficient from the quadratic regressions of S on u estimates α. Data in two sets of rats (a) were from Tables A-11 (set one; ▲) and A-12 (set two; ◼) in Appendix A of Parks (1982). Estimates of α were 2.88 ± 0.0920 (R² = 0.898) and 3.35 ± 0.0868 (R² = 0.931) for sets one and two, respectively. Data on male chickens (b) were from Table A-36 (layer; ▲) and A-24 (broiler; ◼) in Appendix A of Parks (1982). Estimates of α were 1.95 ± 0.0999 (R² = 0.955) and 2.81 ± 0.167 (R² = 0.880) for layer and broiler chickens, respectively. Data on female (▲) and male (◼) turkeys (c) were from R. M. Gous (University of Kwazulu-Natal, Pietermaritzburg, South Africa, personal communication). Estimates of α were 2.95 ± 0.184 (R² = 0.963) and 3.54 ± 0.208 (R² = 0.954) for female and male turkeys, respectively. Data on the Great Dane dog breed (d) were from Table A-6 in Appendix A of Parks (1982). The estimate of α was 2.88 ± 0.0803 (R² = 0.951). Data on female (▲) and male (◼) Suffolk sheep (e) were from a selected line as described by Lewis and Emmans (2010). Estimates of α were 4.30 ± 0.0555 (R² = 0.856) and 4.66 ± 0.0823 (R² = 0.938) for female and male sheep, respectively. Data on cattle after weaning (f) were from Table A-3 (set one; ▲) in Appendix A of Parks (1982) and Table 1 (set two; ◼) from Taylor et al. (1986). Estimates of α were 3.43 ± 0.0623 (R² = 0.987) and 5.65 ± 0.0585 (R² = 0.981) for sets one and two, respectively.

A value for α could not be estimated in pigs. Although the data from Shull (2013) were well described by a quadratic form (Figure 3), the relatively small range in degree of maturity in weight did not allow C to be well estimated.

Figure 3.

Daily feed intake (g/d) plotted against weight (kg) in pigs. The data were from Shull (2013). The fit of the quadratic regression (zero intercept) had R² = 0.997.

The regressions of the natural logs of B and C on the natural log of A had slopes of −0.250 ± 0.0443 (R² = 0.737) and 0.767 ± 0.0251 (R² = 0.989), respectively. Neither was significantly different from the scaling values used in the current study of −0.27 and 0.73 nor from the −0.25 and 0.75 widely used.

Brody and Procter (1932), Kleiber (1932), and Brody (1945) pointed out that much of the variation between mammals, and more widely, could be accounted for if due allowance was made for differences in mature size, A. In particular, mature metabolism was related to A^0.73. The mean value of the scaled Gompertz rate parameter B* = B.A^0.27 estimated from the 12 sets of data used here was 0.0372. That this was numerically close to the mean value of 0.0353 estimated by Emmans (1997), using pre-weaning data for eight species of domestic mammals, must be seen as coincidental. The higher values in animals selected for high growth rates in Table 1 were offset by lower values in others, probably due to suboptimal conditions of feeding or environment. Nevertheless, the mean value of this key variable is well estimated when both data sets, one pre-weaning and one post-weaning, are considered together.

Lewis and Emmans (2010), using data from different kinds of sheep, found that a quadratic equation with zero intercept gave a very good description of the relationship between the degree of maturity in weight, u, and the rate of feed intake scaled to A^0.73. With intake now scaled to mature intake, this was confirmed for all 11 sets of data as shown in Table 1. The functional form agrees with the data (Figure 2a–f) that intake can, and usually does, have a maximum when u < 1. This is in distinction to the other forms of function that have been widely used (Parks, 1970, 1982; Pittroff and Kothmann, 2001; Whittemore et al., 2001, Taylor, 2009). The equation of CSIRO (2007), only for cattle and sheep, does predict a maximum intake at u < 1 but predicts an invariant value of u = 0.85 at which the maximum occurs. As seen in Table 2, the variable value of α leads to variable values of u*, at which intake is at a maximum.

Table 2.

The degree of maturity (u*) when feed intake reaches its maximum value (S_max) and the degree of maturity (u**) when scaled feed intake (S) equals one, with respect to the value of the parameter α

α 1	u* 2	Smax	u**3
2	1.000	1.000	1.000
3	0.750	1.125	0.500
4	0.667	1.333	0.333
5	0.625	1.563	0.250
6	0.600	1.800	0.200

¹Parameter obtained from the fit of the quadratic regression of scaled rate of feed intake on the degree of maturity (equation 2).

²Obtained as u* = −α/(2.(1 – α)).

³Obtained as u** = 1/(α – 1), based on solving equation 2 for u when S = 1.

The scaled relationship is extremely powerful in that it has the single parameter α. When feed intake is maximum, dS/du = 0. Thus, from equation 2, the degree of maturity (u*) at that maximum value for intake (S_max) is then −α/(2.(1 – α)). The critical value of α is thus 2. For α < 2, there is no maximum for u < 1; for α > 2, there is. The effects of values of α of 2, 3, 4, and 5 on the values of u* and S_max are provided in Table 2. The particular degree of maturity (u^**) at which intake first reaches its mature value (S = 1) is also shown. The quantitative relationship between intake and weight for an animal with given values of A and C will thus depend only on the value of α.

The value of C was closely related to that of A with an exponent not different from the expected value of 0.73 (Figure 4). The causes of the appreciable residual variation in C*, shown in Table 1, and of the variation in α, are of interest and the focus of a further study (Emmans and Lewis, in preparation).

Figure 4.

Natural log of intake at maturity, C (kg/d), plotted against the natural log of mature weight, A (kg), for 11 data sets. The slope of the linear regression of log C on log A was 0.767 ± 0.0251 (R² = 0.989).

Conflict of interest statement

The authors declare that there is no conflict of interest regarding the publication of this article.