Behavioural aspects of intake at pasture in ruminants

(13)

And Equation (10) re-arranges to yield:

                  (14)

Substituting into Equation (13) gives:

             (15)

Equation (15) shows that calculating intake rate for various biomass levels at constant cover (h. h_r >h_x) will now give a constant intake rate in contrast to the results shown in Figure 1. The animal is grazing to its maximum bite depth and the only change occurring in the sward canopy is an increase in biomass by way of increasing height. Furthermore, calculating intake rate as a function of cover at constant biomass will now predict increasing intake rate with increasing cover (see Figure 2).

Figure 2. A comparison of hourly intake rate as a function of sward cover assuming constant biomass density or constant biomass height (grazing-process model A)

Equations (12) and (15) represent two extreme caricatures of sward development and intake, one based on constant height and the other on constant density. In reality, sward development involves simultaneous changes in cover, height and density. Both extremes have shown that biomass distribution or spatial heterogeneity is an important factor in determining intake, but the qualitative behaviour of these two models is fundamentally different.

A further limitation of this preliminary grazing process model is the determination of the parameter p . the proportion of perceived food items that are selected. Not only is it unreasonable to assume that p remains constant over a wide range of pasture conditions, but it is also problematic to ascribe any value at all where there is no recognition of heterogeneity within food items. Grazing process model B includes a more realistic description of spatial heterogeneity of the sward and thereby enables the optimisation of the parameter p.

Grazing-process model B

Description of the model

In developing model B, two major changes were made in the description of the sward canopy. Firstly, a distribution function of biomass per unit area at the food-item site is introduced. Secondly, a more realistic function is used to describe the vertical allocation of biomass within a food-item site (though pasture height remains uniform as in model. A).

Inclusion of a distribution function for biomass per unit area at the selection site permits a study of selective foraging and the model can therefore be used as a simple optimal foraging model. The underlying assumption is that the animal selects for biomass concentration in the horizontal plane, which is essentially the same as selecting for bite density aril thus bite weight, if the model assumptions are taken as a whole.

The normal distribution function is used to describe spatial heterogeneity of biomass per unit area. A preliminary analysis of biomass data from Migda indicates that the frequency distribution of biomass per unit area of samples based on 100 quadrats of 100 cm² approaches a normal distribution approximately mid-way through the growing season. Thus choice of this function does have relevance to the situation in the field albeit at a stage in the season when availability is unlikely to limit intake. A further reason for using this function is the relative ease with which it can be handled mathematically as opposed to skewed functional forms that would be more appropriate for the early stages of the growing season. Characterisation of this aspect of spatial heterogeneity in the model requires the definition of two parameters; mean biomass per unit area and the standard deviation of the mean.

Selectivity is defined in terms of the number of standard deviations frown the mean (Z_l) above which the animal selects all food items it encounters. The algorithm calculates intake for Z_l values of 3 to . 3 in steps of 0.1 standard deviation. A value of . 3 approximates zero selectivity and a mean selected biomass density equal to that of the field mean.

Vertical biomass distribution is defined in the form of a table giving the cumulative fraction of biomass from the top of the canopy as a function of fraction canopy depth from the top. Values for this function are based on Milne et al (1982) (see Table 1).

Handling time is defined more simply in model B. Receding (ruminating) time is no longer incorporated in the calculations since there seems to be little evidence in the literature of ruminating time limiting the time spent in active grazing. Furthermore, excluding rechewing time simplifies the calculation of the daily active-grazing time needed to meet a given intake requirement. Biting and chewing time are no longer differentiated and a linear function is used to relate total handling time to dry-matter intake per bite. The terms bite and selected food item are synonymous in model B, implying that the animal perceives bite-sized food items as opposed to there being n potential bites in a food item as in model A. (Note, however, that the parameter n was set to 1 for all numerical examples of model A.)

Searching time is a function of the density of food items in the horizontal plane (Equation (5): model B also assumes that food items are randomly distributed i.e. biomass distribution is not clumped). The density of selected food items is the proportion of cover of selected food items divided by the area of the food item. Using the normal distribution curve, the proportion of cover of selected food items is the area under the curve from Z_l to infinity. This is not soluble analytically but a fifth-order polynomial expression is used to approximate this integral.

The mean selected biomass per unit area is needed in order to calculate both bite weight and handling time. This is not analytically soluble for the normal distribution function and so a set of solutions was calculated numerically and incorporated into the model.  

To summarise, the equations used in model B to calculate intake rate are as follows. (Constants for the conversion of area and time units have been omitted for the sake of clarity.)

    Bite radius:                                         (16)

    Biting depth: h' = min (h_x, max (0,h-h_r))                     (17)

    Fraction of canopy height that is grazed: f_h = h'/h     (18)

    Fraction biomass grazed, vertical plane (Table 1) f_v = f (f_h) (19) 

    Fraction cover of selected biomass (polynomial) : c = f (Z₁) (20)

    Density of selected items: d = c/a                                         (21)

   Searching time: t_s = 1/(2rdu) + _                                            (22)

   Mean normalised selected biomass level (table): Vn = f (Z ₁)(23)

   Mean absolute selected biomass level: Vs = Vn * SD + V     (24)

where: SD = standard deviation of mean biomass per unit area, 

V = mean biomass per unit area

Mean bite weight: w = fv * Vs * a                            (25)

Handling time: t_h = 1 + m w                                    (26)

where: 1 = intercept of handling time function,

m = slope of handling time function.

Intake rate: I = w / (t_s + t_h)

Model B also calculates the mean biting rate and mean walking speed during active grazing, which are both more easily related to grazing behaviour as observed in the field than many of the basic parameters used in the model.

Mean biting rate: B = 1/(t_s + t_h)                                        (28)

Mean walking speed: = u(1-(B*t_h)) = u(1 - t_h/(t_s+t_h))   (29)

The energy cost of grazing (Eg) was included in the model to enable calculation of the net rate of energy gain (E_n). The energy cost of grazing is taken to be the energy cost of walking (E_w). No estimate for the energy most of feed harvesting and chewing could be found in the literature, though these are probably very small relative to the walking cost. An energy concentration (Ep)is ascribed to the pasture dry matter. Thus:

Energy cost of grazing: E_g = E_w *                  (30)

Energy intake rate: I_e = I * E_p                              (31)

Net rate of energy gain: E_n = I_e - E_g                    (32)

Results

The standard run uses parameter values shown in Table 1, with a pasture biomass of 50 g/m² and standard deviation of 50 g/m². A coefficient of variation of 100% (between sampling quadrats of 10 x 10 cm) is quite typical for pastures at Migda throughout the growing season. Biting depth is the maximum potential of 6 cm, which is 50% of the pasture height and represents 20% of the biomass per unit area. Figure 3 shows the form of relationship between various variables.

Figure 3. A selection of relationships between output variables of grazing-process model B under the standard parameter set. (i) Relationships with Z₁ (all symbols defined in Table 1)

As the degree of selectivity declines from the highest values, searching time decreases relatively rapidly compared with the moderate decline in handling time and bite weight. The very rapid decline in searching time more than compensates for the effect of declining bite weight, and the net effect is a sharp increase in intake rate. Searching time declines less rapidly at Z₁ values below about 1.2 whilst handling time and bite weight continue to fall. Frown the functional form of Equation (27) above, it is clear that the net effect is to reduce intake. At extremely low degrees of selectivity, searching time, handling time and bite weight all remain fairly constant and thus intake rate changes little with further decline in selectivity. The resultant relationship between intake rate and selectivity indicates that the cost to the animal of a small deviation frown the optimum selectivity level is greater if the animal is over-selective rather than under-selective.

Throughout all runs of the model it was found that the optimum selectivity level (OSL, defined in Z units) is only slightly altered when the optimum is defined in terms of maximising the net rate of energy gain (OSLg) as opposed to maximisation of intake rate (OSL_i). This is due to the fact that the ratio of enemy expenditure rate to energy intake rate rapidly becomes very low as the degree of selectivity declines. At zero selectivity, the energy cost of grazing per unit time (E_g) represents about 3% of the energy intake per unit time (I_e). At OSL_i (Z₁ = 0.5; standard run), Eg/I_e = 4.8%.

For the standard parameter set, OSL_i = 0.5 (standard deviations above the mean). This corresponds to a field cover of selected items (c) of 31% and a mean selected biomass level (V_s) of 105 g/m² (2.1 times the mean biomass of the field). At OSL_i the following results were obtained. Searching time, handling time and bite weight are 0.31 s, 1.1 s, and 63 mg, respectively. Intake rate, mean biting rate and mean walking speed are 161 g/h, 43 bites/min, and 11.2 cm/s, respectively.

The effect of biomass heterogeneity (at constant mean biomass) is shown in Table 2. OSL_i and intake rate increase with heterogeneity. At constant heterogeneity, OSL_i is independent of mean biomass level (Table 3). t_s remains constant since it is a function of selected item density which is in turn a function of Z₁. t_h does not increase proportionately with biomass and thus the net effect is for intake rate to increase with biomass. Figure 4 shows various iso-intake contours (at OSL_i) for biomass and heterogeneity.

Table 2. Results of grazing-process model B. Sensitivity analysis to coefficient of variation of biomass. Mean biomass = 50 g/m². Other parameter values and symbols defined in Table 1.

Table 3. Results of grazing process model B. Sensitivity analysis to biomass per unit area. Coefficient of variation = 100%. Other parameter values and symbols defined in Table 1.

V  (g/m2)	OSLj   (. )	Vs   (g/m2)	w  (mg)	Th  (s)	Ts  (s)	I  (g/h)	B  (/min)	U  (cm/s)
20	0.5	42	25	0.88	0.31	76	50	13.2
30	0.5	63	38	0.95	0.31	108	47	12.4
40	0.5	84	50	1.02	0.31	136	45	11.8
50	0.5	105	63	1.09	0.31	161	43	11.2
60	0.5	126	76	1.16	0.31	184	41	10.6
70	0.5	147	88	1.23	0.31	205	39	10.2
80	0.5	168	101	1.31	0.31.	224	37	9.7
90	0.5	189	113	1.38	0.31	241	35	9.3
100	0.5	210	126	1.45	0.31	257	34	8.9

Figure 4. The response surface of hourly intake rate (g/hour) to mean biomass (V, g/m²) and biomass heterogeneity (SD,%) at the optimum selectivity level (grazing-process model B )

The functional response of intake to biomasss is that of a saturation function. Increasing sward heterogeneity steepens the initial ascending section of the function and raises the satiation intake level. On an absolute basis, sensitivity of intake to heterogeneity increases with biomass, but on a relative basis, it decreases. For example, at a biomass of 25 and 250 g/m², an increase in the coefficient of variation of biomass frown 75% to 100% increases intake rate by 10 and 21 g/h, respectively. On a relative basis, the increases are 12% and 6%, respectively. The result may partly explain the quantitatively very different biomass-intake relationships reported in the literature, even for a given experimental technique carried out using the same animals and plots but in different years.

As noted earlier, the normal distribution function is not an appropriate functional form for describing early-season spatial heterogeneity of natural pasture biomass in environments typified by Migda. Grazing process model C aims primarily to improve the description of heterogeneity.

Grazing-process model C

Introduction

Grazing-process model C attempts to improve the description of spatial heterogeneity in the horizontal plane. Figure 5 shows a typical progression of biomass distribution curves for the early-season growth phase of natural pastures as observed at Migda. A conceptually and mathematically simple method was developed to describe this wide range of distribution skewness.

Figure 5.Correlation between the alpha heterogeneity parameter and time from pasture emergence (t).(Based on field data from Migda, various fields ungrazed)

The normalised mean percentile method

Biomass distribution is expressed as a function relating the mean normalised biomass level (V') to the area percentile (P). Assume that a data set consists of 100 biomass estimates for a given field sampling. If these 100 estimates are sorted in descending order, then the first number represents an estimate of the biomass per unit area of the top 1% (area basis) of the field. The mean of the first two numbers represents the mean biomass up to the second area percentile, and so on. The mean of the 100 numbers is the mean biomass level up to the 100th percentile and is therefore the field mean. Since the variability of biomass per unit area is greater than zero, the function of V' against P will always yield a downward-sloping curve. In order to facilitate comparisons of heterogeneity between samplings and fields, the original 100 estimates are normalized by dividing through by the mean.  Thus V' = the mean normalised biomass level in multiples of the field mean and V' = 1 at p = 100. Since the distribution functions of biomass are generally skewed to the right, the function of V' against P will be concave with respect to the origin.

The normalised mean percentile functions used in model C were derived as follows:

1. Biomass data for the 1979/80 season at Migda were used. The data consist of 111 biomass sampling sets for a number of fields (grazed and ungrazed) and dates through the season. Each sampling data set contains 100 biomass estimates. Estimates were sorted in descending order and normalised.
2. A series of 20 V'. p data pairs at 5 percentile intervals was calculated for each sampling set. A third-order polynomial function was fitted to each set of 20 V'. p data pairs, constrained to ensure that V'=1 at p=100.
3. The coefficients of the third-order polynomial functions showed a high degree of linear correlation. Each function can therefore be characterised by one coefficient which would be used in correlation analyses with grazing history. The remaining three coefficients of the polynomial function are predicted from this one. The intercept coefficient  was selected as the predictor since it is biologically the most meaningful.

The relationship between grazing history and alpha

In order to establish initial conditions at the commencement of grazing, a linear regression analysis was carried out between   and the number of days from emergence (t, taken as a fixed date for all fields) for ungrazed data sets only (Figure 5). This function simply expresses in terms of   the progression of distribution functions shown in Figure 6.

Figure 6. A typical progression of biomass distribution curves for the early-season growth phase of natural pastures under Migda conditions (schematic)

A regression analysis was carried out between and the total number of grazing days (see Figure 7). The correlation is very weak, yielding a decrease in of approximately 0.11 per 100 grazing days. It is unlikely that stocking density and grazing period can be meaningfully reduced to a single factor of grazing days where the stocking density ranges from 3.3 to 15 sheep/ha, as is the case in the data set used. However, this analysis indicates that a larger and well dominated data base would facilitate a useful study of the relationship between grazing history and heterogeneity.

Figure 7. Correlation between the alpha heterogeneity parameter and the total number of grazing days (G). (Based on field data from Migda, 1979/80,various fields)

Description of the model

To summarise, the following changes were made in proceeding from model B to model C:

1. The mean selected biomass level (normalised) (V_n) is defined by a third-order polynomial expression in which the intercept is defined as a parameter in the program and the remaining three constants are predicted from this parameter by a set of linear regression equations.
2. The mean selected biomass level (absolute) (V_s) is defined as the product of V_n and the mean biomass per unit area.
3. The calculations are made for 100 percentile values that directly provide the selectivity level in terms of percentage area (c as defined in model B).

Results

The standard run of model C uses parameter values shown in Table 1. Mean biomass and   were set at 50 g/m² and 3, respectively. A number of relationships generated in the standard run of models B and C are compared in Figure 8. On the whole, the shape of response is very similar for the two models. A skewed biomass distribution seems to make the I . w response more sharply peaked, though the optimum selectivity level is not altered materially. A maximum intake rate of 142 g/h is achieved at a selectivity level equivalent to the top 23% of the area, with a mean selected biomass level of 98 g/m². Bite weight and mean biting and walking rates during active grazing are 50 mg, 40 bites/min and 14 cm/s, respectively.

Figure 8. A selection of relationships between output variables of grazing process models B (---) and C ( . )under the respective standard parameter sets

The functional response of intake to biomass for various degrees of distribution skewness is shown in Figure 9. Maximum intake rate increases from 127 to 200 g/h as   increases from 2 to 6, at a mean biomass of 50 g/m². It should be noted that under early-season grazing conditions,   would generally decrease through time, and hence with increasing biomass. Furthermore, the animal can extend grazing time to compensate for a low hourly intake rate. Both these factors will tend to shift the shape of the daily intake-biomass response towards that of a ramp-type function.

Figure 9. The functional response of intake to biomass for various levels of biomass heterogeneity (grazing-process model C)

The iso-intake contour map for biomass and heterogeneity is very similar to that obtained with model B, and the sensitivity of intake rate to heterogeneity, dI/d, increases with increasing V. It may, however, be more useful to look at the sensitivity of I to in terms of the relative change in I. Model C was used to numerically estimate (dI/d/I (at the optimum selectivity level), for combinations of and V. The contour map of this relationship is shown in Figure 10. The percentage increase in hourly intake rate per unit change in decreases with increasing biomass. At a given biomass, the relative change in I per unit change in is greatest in the region = 3 to 4. In the context of early-season pasture dynamics, V and are generally located in the region of greatest sensitivity to heterogeneity.

Figure 10. The response surface of the relative change (%) in hourly intake rate per unit change in alpha to mean biomass (V, g/m²) and biomass heterogeneity , at the optimum selectivity level (grazing-process model C)

Conclusions from the grazing process models

The simple models described above have provided a rudimentary description of the plant-animal interface and have produced results consistent with those of empirical field studies of intake. Qualitatively, hourly intake shows a saturation functional response to herbage availability, which can be explained in terms of the relationship between bite weight, searching time and handling time and the way in which these components change with availability. Quantitatively, the response of hourly intake rate to availability is less steep than that of daily intake rate generally reported in the literature. This indicates the importance of grazing time as a buffer against declining availability, as shown by Allden and Whittaker (1970) and others.

This study confirms the consensus in the literature that biomass alone is not a sufficient predictor of intake rate. However, this is demonstrated by considering spatial heterogeneity of biomass distribution in the horizontal plane rather than in the vertical plane via height. Over a wide range of conditions, the mean selected biomass level (for the optimal foraging strategy) is approximately twice the field mean. This is consistent with results of van Keulen and Benjamin (unpublished data) in a study of the simulation model ARID CROP under grazing. It was found that reasonable simulation of primary production could only be achieved when defoliation was restricted to the top end of the biomass distribution curve.

Further development of the grazing-process model will need to incorporate heterogeneity of canopy height. This requires an algorithm of considerably greater complexity since a non-linear biomass. height relationship (at the selection site level) results in the uncoupling of bite weight and biomass density in the horizontal plane. Furthermore, even if a reasonable prediction of intake for a given canopy structure can be achieved, dynamic simulation of a growing canopy under defoliation would, at present, be highly speculative.

References

Allden W G and Whittaker I A Mad. 1970. The determinants of herbage intake by grazing sheep: the interrelationship of factors influencing herbage intake and availability. Aust. J. Agric. Res. 21:755. 766.

Milne J A, Hodgson J, Thompson R, Souter W G and Barthram G T. 1982. The diet ingested by sheep grazing swards differing in white clover and perennial ryegrass content.Grass and Forage  Sci. 37: 209. 218.

Discussion

Question . Has this model been validated and if not, would that be the aim?

Answer . No, it hasn't been validated. The approach was that of a theoretical ecologist looking at an agricultural problem, and you would find that it is virtually impossible to observe the relevant parameters in the field. So in terms of coming to a model that can be used in simulation models of grazing systems I am not very optimistic, but it certainly gives insight into what is happening in the field.

Question . If you put a number of reasonable estimates of the parameters in the model, do you get figures of between 1 and 2 kg of intake per sheep per day for an 8 hour grazing day?

Answer . No, not from those tables, but from a simple graph of functional response from grazing. When you look at Figure 9, you will see that the values are quite reasonable.

Question . Is the model suitable to simulate intake of an animal in a given situation or is it rather a means of analysing the interaction between the behaviour of the animal and pasture characteristics?

Answer . This model can help us to understand what is happening in the field, but only on a qualitative level. It will be very difficult to introduce this type of model into an overall model of grazing systems because you also have to simulate the response of the sward to do that.

Question . Is it right that the higher the skewness of dry matter in your pasture the earlier you can put in the animals?

Answer . If the goal is a certain level of intake yes, but from the point of view of stability, the reverse holds, so that you may have to wait a little longer.