A simple model for optimizing rotational grazing

Agricultural Systems 41 (1993) 123-155

A Simple Model for Optimizing Rotational Grazing Simon J. R. Woodward, Graeme C. Wake Department of Mathematics, Massey University, Palmerston North, New Zealand

Anthony B. Pleasants & David G. McCall Ministry of Agriculture and Fisheries (MAFTech), Whatawhata Research Centre, Hamilton, New Zealand (Received 18 February 1992; accepted 18 March 1992)

A BSTRA CT A simple mathematical model has been formulated to model a simple rotational grazing system of two fields, where a mob of animals is grazed first in one field, then the other. By using linear functions for the rate of herbage mass accumulation and for the rate of animal herbage intake, this system may be solved explicitly to yield expressions for (1) the total intake per animal over the time period, and (2) the total remaining herbage after grazing. Regarding the swap over time as an optimal control variable, these expressions can be optimized The results are analysed for various initial conditions, including stocking rate, initial biomass in each field, and field area. Comparisons are made with continuous grazing. Extensions to the model are discussed

1 INTRODUCTION This paper deals with the allocation o f pasture to a group of animals to optimize some aspect o f production. Ecologists have usually modelled the interaction between plant and herbivore populations using the p r e d a t o r prey theory (e.g. Caughley & Lawton, 1981), an approach which relies 123 Agricultural Systems 0308-521X/92/$05.00 © 1992 Elsevier SciencePublishers Ltd, England. Printed in Great Britain

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Simon J. R. Woodward et al.

on mathematical dynamical systems. It is usual for ecologists to concentrate on the stability of such systems in order to predict population fluctuations and to identify critical sensitivities which may be exploited or managed to achieve certain outcomes (e.g. Goh et al., 1974; Stocker & Waiters, 1984; Barlow, 1987). A well known application of this theory to domestic grazing systems is that of Noy-Meir (1975, 1978a), who suggested that some continuous grazing systems might have two possible steady states. Many agriculturalists (e.g. Johnson & Parsons, 1985) support this idea that grazing systems may be discontinuously stable, a feature which it may be possible to exploit to achieve increased productivity. However, there are important differences between ecological systems and agricultural grazing systems. The wild ecosystem is by and large uncontrolled; both herbivore and herbage populations are allowed to fluctuate naturally, and this implicitly requires long time scales. The kinds of controls that are used usually entail some form of harvesting, and often it is movement towards a steady state (sustainable yield) which is desired. By contrast, in agricultural grazing systems we are often dealing with populations of animals that are intensively managed on discrete fields, most commonly by controlling the animals' access to pasture. The time scales are short, and so stability analysis is not usually relevant. While agricultural systems are of great interest due to their economic importance, they are also, unfortunately, notoriously difficult to perform experiments on. This is due to both the tremendous number of alternatives of possible grazing schedules, and the dependence of productivity on environmental conditions with significant random variability. In practice, direct measurement is difficult and costly, and the use of frequent cutting to simulate grazing is questionable. Therefore, a theoretical analysis is relevant (Chen & Wang, 1988). Computers have helped alleviate these problems by allowing large numbers of carefully controlled trials to be simulated at low cost (e.g. Christian, 1981; White et al., 1983). Computer models may be very sophisticated and take into account a huge range of extrinsic factors. However, the fundamental cause and effect mechanisms of the system can be obscured by this very sophistication. An analytical approach is required to achieve a thorough mechanistic understanding. Parsons and Johnson (1986) have provided a good summary of the theoretical understanding of continuous grazing management. Rotational grazing, on the other hand, is still not yet well understood. Experimental results seem contradictory regarding when and whether rotational grazing offers increased production over continuous grazing (e.g. Morley, 1981; Sharrow, 1983; Florez et al., 1986).

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125

Morley (1968) and Parsons et al. (1988) approached optimizing pasture production in rotational grazing by considering the optimal rest period for fields between being grazed. While an innovative and tractable approach, it avoids the issues of (1) how much herbage is produced during grazing, and (2) how these conclusions may be put together into a practical grazing programme. Noy-Meir (1976) applied an ecological stability theory to a grazing system under rotational management. He made the a priori assumption in common with most models of rotational grazing that the rest and grazing intervals should be of fixed length, and compared results using different interval lengths. Noy-Meir principally examined the long term stability of such a system. Achieving optimum performance from any modelled system must involve some form of mathematical optimization procedure (for instance, as used by Morley (1968) and by Parsons et al. (1988)). Because agricultural grazing involves intensive control of the herbivore-plant system in order to maximize productivity, this points to the use of the Applied Optimal Control theory, which was developed by engineers who wished to find optimum schedules for machines (e.g. Huffaker et al., 1989). Chen (1986) and Chen and Wang (1988) have used the applied optimal control theory to a grazing model suggested by Johnson and Parsons (1985). The model is climate-driven. Chen (1986) found the theoretical optimal sequence of cuts for a pasture harvested by mowing over a growing season of 273 days. Then Chen and Wang (1988) found the theoretical optimal stocking density (adjusted by the farmer with time) that maximized total animal intake (sum over all animals) over the same growing season. The purpose of this paper is to apply a mathematical analysis to a simple rotational grazing situation involving the movement of animals between defined fields. The particular problem of interest involves the optimal control of grazing for animal production where there are no pasture quality constraints. This is the situation in winter and early spring in year-round grazing systems. At this time pasture production limits animal production because the intake potential of animals is greater than net pasture growth. Two common objectives exist in the control of grazing over this period. The first involves pregnant animals where the objective is to optimize conservation of herbage over winter for feeding in early spring after parturition. This strategy maximizes milk production in dairy cows (Bryant, 1982) and lamb production in ewes (Bircham, 1984). The second objective is to maximize animal intake over the entire period; a goal sought by farmers raising animals for slaughter. In the above cases the farmer has a number of fields with given initial herbage cover.

Simon J. R. Woodward et al.

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The farmer may control the number of animals in the system (the stocking rate), the order of grazing the fields, and the time at which animals are moved between fields. Whereas Chen and Wang used a relatively complex model (that of Johnson & Parsons (1985)) and solved it numerically, the aim in this paper is to use a greatly simplified model at first, and to explore the behaviour of the optimal solutions of this model analytically. The advantages of this are that the strategic outcomes are found for a large range of parameters simultaneously, and that results will be derived from a mechanistic understanding. When the structure of the simple grazing system is understood, it will be possible to develop extensions to the model dealing with even more realistic situations.

2 THE MODEL This section describes the assumptions upon which the model is based and the formulation of the model itself in mathematical terms. Taking simple pasture growth and animal intake functions the model is then solved.

2.1 Assumptions (1) Suppose a farmer has two fields of area hl,h 2 (ha), respectively. At time t = 0 (days) the mass of herbage in the fields is wl(0) and w2(0) (kg ha-~). Assume that a pasture can be adequately represented by a single state variable, the herbage mass. (2) The farmer wishes to graze a mob of n animals (constant) in these two fields for a total period of T (days). This is split between t 1 (days) in field 1 followed by T - tl (days) in field 2. The swap over time tl is the point in time at which he or she transfers the animals into field 2. In this context the two fields are only a small part of a wider farm system, and alone are not expected to sustain the animals for more than the limited time period we are interested in. (3) Assume that the rate of pasture accumulation is dependent only on the current herbage mass, and label the pasture accumulation rate function 'grow' (w) (unspecified at this point). Then, in a field i which is being rested (i.e. not grazed), dw i

dt

- grow (wl)

(kg ha 1 day 1)

where wi (kg ha 1) is the herbage mass in field i.

(1)

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127

Pasture G r o w t h F u n c t i o n s

,v

e~

H e r b a g e Mas s , w [ k g / h a l

Fig. 1.

P a s t u r e g r o w t h rate f u n c t i o n s : , linear; . . . . (after B a r l o w , 1987).

, logistic; . . . . .

, skewed

In general, the net pasture growth function grow (w) will be convex with a single maximum. Figure 1 shows some explicit functions that have been used in previous models. (4) Assume that the intake rate of an animal~ dc/dt, depends only on the instantaneous available herbage biomass in the field f being grazed and that the pasture removal rate by the mob is the sum of the intake rates of all of the animals. This implies that animals (1) are non-selective and (2) do not influence one another's grazing except through a greater or lesser rate of pasture removal. Assume that the effects of treading and~ fouling can be neglected. We label our animal intake rate function graze (w) (also unspecified) so that the rate of intake of a single animal is dc

dt

- graze (wf)

(kg animal -1 day-L).

(2)

The parameter c can be interpreted as the cumulative intake of a single animal over some time period, expressed in (kg animal-l). The general form of an animal intake function graze (w) is that of a saturation function. Figure 2 shows some examples of intake functions from the literature.

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Simon J. R. Woodward et al.

Animal

Inlake FunctioJ~s .

.

.

.

.

.

.

/

7

./ /

g,' /

/

/ ......./~'. ................................. .- / ....'../

/

,.,./

4

,

e~

,J/ c

IZ, Ilerbage

Mass,

~ {kg/hal

Fig. 2. Animal intake rate functions: - - - , linear; . . . . . , ramped, including an ungrazable residual (e.g. Noy-Meir, 1978b); . . . . , sigmoid (e.g. Noy-Meir, 1978a); . . . . . , Michealis-Menton (e.g. Barlow, 1987).

The rate of increase or decrease of biomass in the grazed field is the difference between pasture accumulation and pasture removal by grazing, i.e. dwf n dt - grow (wf) - hr graze (w 0

(3)

where n/hf is the stocking density. Only two major assumptions have been made, namely that the animal intake rate and the pasture growth rate can be described as the single parameter functions graze (w) and grow (w) respectively. These assumptions have often been made when modelling grazing systems at this level of detail (e.g. Noy-Meir, 1975) and are considered appropriate for times of the year when pasture is in deficit to potential animal demand and is of uniform high quality. Even so, this type of system can conceptually be extended to multi-compartment grass growth models such as Johnson and Parsons' (1985), if desired. Equations (1)-(3) form a dynamical system. Given explicit functions for grow (w) and graze (w), it is theoretically possible to solve the system and predict the changes in c and w with time.

Optimizing

rotational

129

grazing

In this paper, the two quantities of interest are c(T) (kg animal-~), the total intake per animal over the time 0 to T, and W(T), where

W(T) = hlwl(T ) + h2w2(T )

(kg)

(4)

the total remaining herbage mass at the end of the grazing period. 2.2 Explicit functions

In this analysis the authors have chosen linear functions for grow (w) and graze (w) for simplicity, that is to say, the rate of herbage increase and the rate of animal intake are both proportional to the biomass, i.e.

grow (wi) = a wi

(5)

graze (wf) = k wf

(6)

and This simplification will be most appropriate where potential animal demands are greater than net pasture growth and hence herbage mass tends to a low level. The system (1)-(3) now becomes dwi _ __

dt

_

aw i

dwf n - a w f - - - k wf = a(1 dt hr

-

m f ) wf

and dc

dt

- k wf

(7)

where mf -

n k , the 'grazing intensity' ratio, has been chosen to represent ahf the ratio of maximum grazing rate to maximum pasture growth rate. The value mr is a positive constant for each field f with the following interpretation: if mf is large this implies a grazing situation with a stocking density n/hf high enough for the field to be grazed to extinction unless the animals are moved on. If mr = 1, we have an equilibrium situation where the animals are removing mass at the same rate that it is being produced. If mr < 1 we have lax grazing, where herbage accumulation in field f outstrips removal by grazing. We do not expect this last case (mr < 1) to be well represented by the linear growth eqn (5) since real pastures are limited to a ceiling yield of biomass due to the burdens of leaf ageing and senescence. Thus any

Simon J. R. Woodward et al.

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TABLE 1 Nomenclature, Units and Typical Parameter Values When the Model is Applied to a Mob of 300 kg Bulls During the Winter on a Perennial Ryegrass Pasture

Parameter

Typical value

a c

1.4%

dc dt C'mi n

f hf H i k mf n tI t~ T wi

Units

Description

day -~ kg animals -l

Pasture 'intrinsic' growth rate Cumulative per animal intake

7.5

kg animals i day-l

Animal intake rate

4

kg animals-l day ~

1-0 2

ha ha

animals days days days kg ha 1

Minimum intake rate Number of the field being grazed Area of the field being grazed Total area of all fields Number of a field being rested Proportional animal intake rate 'Grazing intensity' ratio Number of animals Swapover time Optimal swapover time Length of rotation Herbage biomass in field i at time t

kg h a ~ day 1

Pasture growth rate

kg

Total biomass at time t in all fields

0.5% 0-100 0-100 5 ? 10 1 500

dwi dt

20

W

3 000

ha animals ~ day ~

results for mf < 1 are presented here for completeness rather than realism. A lax grazing situation with high pasture mass would require one of the other curves on Fig. 1, such as the logistic growth equation, to model the real world adequately, as well as a saturation intake function such as the 'ramped' on Fig. 2. The actual forms of the functions required are difficult to investigate, and have not yet been satisfactorily established. 2.3 Parameter values

Although the mathematical analysis here is theoretical and the qualitative results we derive are not dependent on the actual values of the various parameters and constants, it is desirable to use realistic parameter values when making comparisons between grazing models or with experimental outcomes from the field. The values used in plotting the figures in this paper are listed in Table 1 with the list of symbols. In the analysis that follows, the only restriction on parameter values is that they be greater than or equal to zero.

Optimizing rotational grazing

13l

2.4 Solving the model At time t = 0 we have initial conditions c(0) = 0, wl(0 ) and W2(0). Solving eqns (7) with f - - 1 and i = 2, and applying these initial conditions, gives, at s o m e later time fi, wl(tl) = Wl(0) exp (a(1 - ml)tl) w2(t0 = w2(0) exp (atO

c(tO -

kwl(O)

(exp (a(1 - m o t 0 - 1)

a(1 - m 0

(8)

At time t = t~ the farmer moves his stock to field 2. So we have f = 2, i -- 1 in eqn (7). Solving the system again, applying the initial conditions c(tl), w~(fi) and w2(/0 at time t = fi, the solution at s o m e later time T is

wl(T ) = wl(tl) exp ( a ( T - q)) w2(T) = w2(tO exp (a(1 - m z ) ( T c(T) = C(tl) +

kw2(tO a(1 - m2)

t0)

(exp (a(1 - m 2 ) ( T - tO) - 1)

(9)

Replacing c(t O, wl(tl) and w2(t 0 with the values calculated in eqn (8) gives us the final solution:

wl(T ) = wl(0 ) exp (a(1 -- mOtO exp (a(T-- tl) ) wz(T) = w2(0) exp (atl) exp (a(1 - m 2 ) ( T -

c(T) -

kwl(O)

t0)

(exp (a(1 - mt)tl) - 1)

a(1 - m 0 +

kw2(O) exp (aq) a(1 -- rn2)

(exp (a(1 -- m2)(T-- tl) ) - 1)

(10)

and also f r o m eqn (4),

W(T) = hlwl(T ) + hzw2(T) = hlwl(O) exp (a(1 - rnl)tl) exp (a(T-- tO) + h2w2(0 ) exp (atl) exp (a(1 - r n 2 ) ( T - q))

(11)

Figure 3 shows a typical time course o f w~, w 2 and c over the d u r a t i o n o f grazing.

Simon J. R. Woodward et al.

132

wl~ w2~ as functions of time

c, as a function of time

_

~(T)

..

wl(0) ~ . ~ ~

"

-twl (T)

r

i

,

tl

T

i

/

i

/

c(0) tl

T i m e ldaysl

Fig. 3.

T

T i m e [daysJ

Typical c o u r s e o f wj(t), w2(t ) a n d c(t) o v e r the g r a z i n g period. T h e d o t t e d line s h o w s the s w a p o v e r time tl, at w h i c h t h e a n i m a l s are t r a n s f e r r e d i n t o field 2.

3 RESULTS The objective is to control our grazing system so as to maximize c(T) and W(T). The results derived will (in theory) be valid for any values of the parameters and so may be applied across a wide range of modelled scenarios. Nevertheless it is helpful at this point to choose some parameter values for the sake of visualization, even though our calculations will not rely on these. The values for a, k and T given in Table 1 and the sample systems in Tables 2 and 3 are used in these examples. Note that W(O) = htwl(O ) + h2w2(O)

(kg)

(12)

is the total initial herbage mass, and

M-

nk aH

is the grazing intensity over the whole area.

TABLE 2 T w o Field System U s e d in E x a m p l e s Field 1 hI wl(0) htwl(O) nk ml - ahj

= 0.8 -- 1 620 -- 1 296

- 8.93

Field 2 h2 = w2(O) -h2w2(O) = nk m2 = ~ 2 =

Total system 1.2 1 420 1 704 5.95

H = 2.0 W(0) = 3 0 0 0 nk M- aH 3-57 n = 20

Units ha kg h a l kg

animals

Optimizing rotational grazing

133

TABLE 3 Two Field System with Homogeneous Initial Pasture Mass Used in Examples Field 1

Field 2

h I = 0.8 wl(0) -- 1 500 hlwl(O) = 1 200 m 1 = 8.93

h 2 = 1.2 w2(0) = 1 500 h 2 w 2 ( 0 ) = 1 800 m2 = 5.95

Units

Total system

H -w(0) = W(0) = m= n=

2.0 1 500 3000 3.57 20

ha kg ha--I kg animals

W e n o w c o n s i d e r strategies o f m a n a g e m e n t that m a x i m i z e intake, residual pasture m a s s , or s o m e c o m b i n a t i o n o f these t w o objectives. Figure 4 s h o w s h o w c ( T ) a n d W ( T ) vary as the c o n t r o l p a r a m e t e r , i.e. the s w a p o v e r time tt, is varied b e t w e e n 0 a n d T for a typical scenario. It c a n be seen that c ( T ) is m a x i m i z e d at a local m a x i m u m m i d w a y t h r o u g h the time period, near t~ = 5 (days), a n d W ( T ) is m a x i m i z e d at o n e o f the e n d p o i n t s , that is, w h e n o n l y o n e o f the fields is g r a z e d for the total T = l0 (days).

c(T) and W(T) against tl 70

3500

60

30OO

2500

50 ...........

3o!

W(T)

........ 2000

40

W(T) IkgJ 1500

1000

2O i

5OO

I(l!

0

0

2

4

6

8

0 10

t 1. Swapovcr [days]

Fig. 4.

Typical change in c(T) ( ) and W(T) (. . . . ) achieved by varying h, the swapover time. Optimum strategies are marked (o).

Simon J. R. Woodward e t

134

al.

3.1 Maximum residual herbage Let us first suppose that the farmer wishes to manage the system to give the maximum amount of herbage remaining after grazing. Conservation of herbage is a common objective of rotational grazing management, and may be practiced either to maintain a consistent (but low) supply of feed during a period of slow pasture growth, or to accumulate pasture mass in preparation for a future need, especially parturition, when newborn lambs or calves will require sufficient feed to allow their rapid weight gain and increased survival chances. From Fig. 4 we predict that fi should be chosen to be at one of the endpoints of the feasible region. It is easy to show that 32 W(T) _>0 3tl 2

for all scenarios, so that the maximum residual herbage will indeed always be achieved by choosing the latest or the earliest feasible fi, since the function W(T) has no local maxima. The feasible region here is 0 < t~ < T

(13)

Whether to choose t~ -- 0 or t~ = T can be ascertained by evaluating eqn (11) with each of these values of t~ and observing which is greater. That is, choose t~ = 0 (i.e. graze only field 2) if W(T)[,,-o

> W(T)I,, : T

i.e.

hlwl(O) exp (aT) + h2w2(0) exp (a(1 - m2)T ) > hlwl(O) exp (a(1 - ml)T) + hzw2(0) exp (aT)

(14)

If this is not the case, choosing t~ -- T will give the maximum residual herbage, that is, we would choose only to graze field 1.

Example.

For example, using the values in Table 2, grazing only field 2

gives

W(T)[t, =0 = 2343

(kg)

and grazing only field 1, W(Z)lt,

= T =

2387

(kg)

Therefore, maximum herbage will be left if only field 1 is grazed (i.e. tl -T). This may be a little surprising since we choose to graze the field with the greater initial herbage but can be explained by the reasoning that field 2 is able to achieve greater biomass production due to its larger

Optimizing rotational grazing

135

area, which results in a greater residual bulk of herbage (though not a greater herbage density). As a comparison, if both fields were grazed equal lengths of time the model predicts a much lower residual herbage mass; W(T)[t, =~r = 2090

(kg)

F r o m this example it can be seen that it is not always possible to determine the optimum strategy intuitively, even in a simple case like this. Conditions such as eqn (14) are useful in providing quantitative predictions of the consequences of choosing various strategies. 3.1.1 Minimum intake constraint Usually when a farmer wishes to conserve herbage he or she will wish also to ensure that the animals receive some minimum level of feed. This is relevant when m~ and m2 are greater than 1, i.e. animals tend to graze to extinction. As seen in Fig. 4, choosing t~ -- 0 or T to maximize W ( T ) results in a low intake for the animals. It is therefore sensible to set a constraint to ensure that the animals achieve at least a minimal rate of intake. This can be done by requiring that

dc

--

dt

~> C'mi n

(kg ha i day l)

(15)

throughout the period. Since the animals' intake rate is proportional to the available pasture mass, this constraint will be limiting when the pasture mass becomes low, especially at the end of the animals' time in field 1 (time q) and the end of the animals' time in field 2 (time T) (see Fig. 3). In practice it is more relevant to specify a minimum average intake over some period of several days. In this case the intake on a given day could be lower than this average. While it is possible (though more complicated) to construct such a constraint mathematically, the form of constraint here is sufficient to our current purposes if it is remembered that C'min is then a conservative minimum requirement. F r o m eqn (7), de

dc

dt

- kwf. S o w e require

= kwl(q) = kwl(O) exp (a(1 - ml)tl) > C'rnin t=tl

and

dc dt t = r

= kw2(T ) = kw2(0) exp (at1)

exp (a(1 - m 2 ) ( T - tl) )

> C'mi n

(16)

Simon J. R. Woodward et al.

136

These may be rearranged to give limits on the swapover times t~ which can be chosen if the m i n i m u m allowed intake level in eqn (15) is to be achieved: 1 In

mza

c min

( kw2(O)e x p'~ ( l

< tl < (1

1

-

ml)a In

- m2) T

t

( C'min ) (kw,(O))

(17)

Example. Returning to the previous example, consider the effect of requiring at least a m i n i m u m daily intake of C'mi. = 4

(kg animal -I day -1)

Then as well as condition (13) we are limited to values of tl which satisfy 1-4 < t~ < 6.4

(days)

from eqn (17). Clearly t~ = T cannot now be chosen because the minim u m intake constraint would be violated and the animals' condition would not be maintained. The best strategy will instead be to choose one of the endpoints of condition (17). To determine which, compare 2209

(kg)

W(T)],, : 6.4 = 2122

(kg)

W ( T ) It,

:1.4

•

and

Therefore the best strategy to maximize residual herbage while providing a minimal diet is to choose tl = 1.4 (days). It is interesting that whereas for the unconstrained question we chose to graze only field 1, we now choose to graze mostly field 2. It should be noted that for too high values of C'm~n the interval in eqn (17) vanishes and so it is not possible to find a swapover time tl that can meet the constraint in eqn (15). This is due to the unrealistic expectations put u p o n the grazing system. One would need to graze fewer animals to remedy this. An expression for the m a x i m u m allowed n u m b e r of animals is given in the appendix.

3.1.2 Which field to graze first? A final question to consider in this section is whether greater herbage may be conserved if the two fields are grazed in one or the other order. Specifically, which field should be grazed first? When there is no m i n i m u m intake constraint, the optimal strategy is to graze only one field, as chosen by condition (14), so the question of

Optimizingrotationalgrazing

137

which field to graze first is not relevant. However, it may be necessary to graze both fields to satisfy a minimum intake constraint if one exists. When a minimum intake was specified there was a constraint on the allowed choices of the swapover time as described by condition (17). If field 2 is to be grazed first, we derive a similar expression limiting t~, where t~ is now the time spent in field 2; 1 In

m-~a

C'min

<

kw~(O)exp(-~l - ml)T

t~ ___

-

In (1 - mz)a

~,kwdO))

There is no simpler way to determine which will be the best strategy than to compare the predicted results from the model using each strategy in turn: (1) Graze field 1 first, for tl _

1 In (

m2a

C'm,n

(2) Graze field 1 first, for t~ _ 1 In ( C'min )

mOa

~,kwl(O))

(3) Graze field 2 first, for t~ - 1 ln(

C'min

(1

-

mid

i)

kw2(O)exp (a(1 - ma)T

(days)

(days)

))(days)

kwl(O) exp (a(1 -- mOT

(4) Graze field 2 first, for t~ _ 1 I n ( c'min ~ (1 - rn2)a ~,kw2(0))

(days)

There is no simple a priori method to determine which of these four strategies will be optimal in a given case.

Example.

As before, with C'min = 4.0 (kg animal -l day-l), compare the results for each of the 4 strategies above. Strategies (1) and (2); graze field 1 first: (1) I4I(T) 1,, = 1 . 4 = 2209 (kg) (2) IV(T)[,,

= 6.4 --

2122

(kg)

Strategies (3) and (4); graze field 2 first: (3) w ( r ) l , , = 32 = 2139 (kg) (4)

w(r)l,

, __ 8.3

= 2187

(kg)

So strategy (1) is the optimum in this scenario.

Simon J. R. Woodwardet al.

138

3.1.3 Comparison with continuous grazing Rotational grazing is sometimes advocated as a means to conserve herbage. In continuous grazing herbage conservation is not possible except by removing animals from the system. This analysis can compare theoretical results of these two strategies. Assuming Wl(0 ) = w2(0), the two fields can be combined to m a k e one field of size H and initial herbage mass W(O)/H. Then the grazing intensity ratio is M and the residual herbage mass is W~(T) = W(0) exp (a(1 - M ) T ) The m i n i m u m intake constraint will be satisfied if dc dt

t, : r

k W~(T) H

For this we require n<

all( lc

1-

k W(0) exp (a(1 - M)T) > C'min H

1 In (C'minH')"]

Example.

Take the example in Table 3 with wl(0 ) = w2(0 ) = 1500 (kg ha-I). Combining the two fields gives one field of area 2 (ha) and initial mass 1500 (kg ha 1). We require n < 30 (animals) to satisfy a m i n i m u m intake requirement of C'mi n = 4 (kg animals ~ day 1) for continuous grazing. This being met since n = 20 (animals), we calculate the residual herbage from continuous grazing to be

Wc(T) : 2093

(kg)

By comparison we require n < 29 animals to satisfy this same minim u m intake on our two field rotation system (see appendix for method). Testing the four rotational strategies gives the o p t i m u m strategy to be (1) W(T)I,, :08 = 2211 kg, i.e. to graze field 1 first and swap over into field 2 at tl -- 0.8 days. By comparison, W(T)I,, : 5 = 2104 kg. Therefore a significant improvement in W(T) is achieved in this case by the optimal rotational strategy over the continuous strategy. It is possible however that this improvement would be masked by r a n d o m variability in practice.

3.2 Optimal animal intake An alternative objective is to find the swapover time i~ which gives the m a x i m u m cumulative intake per animal, c(T). Mathematically the maxim u m of c(T) shown in Fig. 4 is located by setting the partial derivative of eqn (10) equal to zero, i.e.

Optimizing rotational grazing

ac(T) o~tl

- kwl(O) exp +

(a(1

139

ml) tl

kw2(O) exp (at)) @xp

(a(1 - m z ) ( T - tl)

1)1

m2 - kw2(0) exp -- kw2(0) exp 1 -

(at1) exp (a(1 - m2)(T- tl)) (atl) I w,(O) exp (_m,at,)_ l - m2 exp (a(1- m2)(T- t,)) ]

(18)

1 ---m 2

We(0 )

= 0 for extrema. It is reasonable to assume that the first factor, nonzero, and so we can rearrange eqn (18) to give (1 -

m2)r exp

(-mtatl) = 1

kw2(0 ) exp (atl), is

m 2 exp (a(1 - m2)(T

i,))

(19)

where r = wl(O)/w2(O), the ratio between the initial herbage masses. In Table 2, r -- 1620/1420 = 1-14. Then if W(0) is as defined in eqn (12), the original herbage masses m a y be recovered by w,(O) = w ( o )

h~r +

h2

and

w2(0) = w ( 0 ) -

h~r + h2

Equation (19) has a unique solution i~ corresponding to a maximum o f c(T). While it gives the mathematical o p t i m u m of c(T), this must be constrained by condition (13) to give a feasible swapover time. The maxim u m intake per animal is then achieved by swapping the animals over at this q. Unfortunately it is not possible to obtain an explicit expression for i~. Because of this, graphical methods are used to show how it depends on the other parameters.

3.2.1 The relation oft1 to T (Fig. 5) By rearranging eqn (19) to give T=il

+ a ( 1 - -1 m 2 )

In

(-m2 1 -- -

1 --m 2

r - -

m2

exp

(--mlatl^ ,)]'~

(20)

we m a y get a feel for h o w i 1 depends on T. Figure 5 shows two graphs of T(i~); one for r > 1 and one for r < 1. The dotted lines on these graphs show the location o f the physical constraints and of two simple bounds to

140

Simon J. R. Woodward et al. r>l

5O

r7Z 40i! 30 i

r<1

60-5O

"7

."

..t"

40203"010/

D

/

{} '-' 20

4()

60

0

20

40

61)

T.FolalTimeIdaysl

"1. Total Time da~sl

Fig. 5. Optimal t I against T for the two cases r < 1 and r > 1, from eqn (20). - - , optimum constrained by constraint (13); . . . . , solution to eqn (19) where this violates constraiIit (13); . . . . . , constraint (13) and bounds as described in the Appendix.

T ( i 0 . Since it is not possible to find the exact il explicitly, it m a y be useful to have these linear b o u n d s which allow a g o o d a p p r o x i m a t i o n . The b o u n d s s h o w n are derived and explained in the appendix. The physical constraints are given in eqn (13) and are the lines il = 0 and il -- T respectively on Fig. 5. The conditions necessary to satisfy these constraints are also derived in the appendix. The heavy curve in Fig. 5 is defined by eqn (20) and gives the optimal i~ as a function o f T for an arbitrary set o f parameters. The shape o f this curve depends non-linearly on r, m~ and m 2. 3.2.2

Two identical fields

In general, all of the parameters could take any non-negative value. Because o f difficulties involved in finding the optimal solutions and analysing their dependence on the various parameters, we start by assuming that the two fields are identical, that is hi = h 2 =

H

and w~(0) = w2(0) -

Then r=l and m l = m 2 = 2M

where M-

nk aH

w(o) H

Optimizing rotationalgrazing c(T)

141

against tl

100 90

n=l

80 ...................... 70

~ ............................ o

.............. ...........

r~

60

.-~ .

.

.

.

.

.

.

.

.

o ...... .

.

50

.

.

.

"

n=5 ......... n=10

. .... .

.

.

.

.

.

. n=20

~

40 30

..........

.e- . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

n=100

20 o 10

n = 2013"" n=500

0

tl, Swapover Time [daysl

Fig. 6. c(T) against t~ for identical fields, i.e. r = 1 and h1 = h 2 (equal size fields with uniform initial herbage mass). Each curve corresponds to a number of animals n as shown. Optimal strategies in each case are marked (o).

Using H = 2 ha and W(0) -- 3000 kg, Fig. 6 shows how the optimum shifts as different numbers o f animals are grazed in a typical system, from n - 1 to 500 bulls. At stocking densities less than 100 animals ha -~, the optimum is near to 5 days, which would be the usual strategy adopted by a farmer grazing animals on two identical fields. In practice, a farmer would use a stocking density this high only over a very short time span. Considerably lower intake would be achieved by choosing t~ = 0 or t~ = T. This is to be expected, since only half o f the total area is then available to be grazed. Clearly this would be a poorer strategy than continuous grazing. However, is even the optimum strategy poorer than continuous grazing? Since w~(0) = w2(0) it is possible to test this hypothesis.

3.2.3 Comparison with continuous grazing If instead of dividing the H (ha) of land into two fields, the farmer left it as a single field and grazed the animals on it for T days, their intake would be, from eqn (8)

Simon J. R. Woodward et ai.

142

Comparison of Managements 80 70 60

50

401 30

206

20

40

60

8'0

100

n, Number of Animals [an] Fig. 7. C o m p a r i s o n o f g r a z i n g m a n a g e m e n t s in identical fields, i.e. r -- 1 a n d h I = h 2 (equal size fields w i t h u n i f o r m initial h e r b a g e mass): , c o n t i n u o u s grazing; . . . . , o p t i m a l r o t a t i o n a l strategy; . . . . . , e q u a l time r o t a t i o n (partially o b s c u r e d b y o p t i m a l

rotation curve).

cc(T ) _

kw(O)

(exp (a(1 -

a(1 - M)

M)T)-

1)

where w(0) - W(0) H

(kg ha ~)

In this case the farmer has exercised no control over the grazing. Figure 7 shows how intake per animal varies as a function of the number of animals for three strategies: (1) continuous, (2) equal time rotation, and (3) optimal swapover time rotation. This graph shows that for identical fields the optimal strategy is no better than the usual strategy of tl = 5, but neither is it significantly better than the continuous strategy. The differences increase as stocking rate gets large, but even if n could feasibly be as large as 600 animals, the optimal strategy would provide only 5% more intake per animal than the continuous strategy. It should be noted that to completely optimize the two field rotational treatment we would also need to calculate the optimal field sizes ht, h2, which may not turn out to be ideally equal. We have assumed the division hi = h2.

143

Optimizing rotational grazing

3.2.4 Different sized fields, w I (0) = we(O) N o w consider the effect o f changing the size o f the fields. If the total a m o u n t o f land, H, is fixed then

hi + h2 = H Then nk

nk

nk

am~

am 2

aM

i.e. 1

1

1

ml

m2

M

Since we are considering the case where r = 1, the o p t i m u m tl described by eqn (19) will always lie in the feasible region (eqn (13)) (see Appendix). Figure 8 shows how the optimal choice o f tl depends on m~ and m 2, and Fig. 9 shows the corresponding optimal c(T). The dotted lines on Fig. 9 are lines o f constant stocking rate, i.e. constant M. F r o m Fig. 8 notice that there are two qualitatively different situations. The first is where M is less than about 60. This would be the case in most practical scenarios since it corresponds to

Optimal tl against ml,m2 ..-"

103

~

~e'" i,//"

..J

/..

,'/

/."..................................

,/

//"

//'

,

~F

--~ 101

10o

10-1 1(1-1

l0 0

101

102

103

104

ml, Field 1 Grazing Intensity,

Fig. 8. Optimal t~ when r = 1 (uniform initial herbage masses). - - , contours of optimal t~ -- 1, ,,~ ..., 8, 9 days; . . . . . , contours of optimal t~ = 0.001, 0.01, 0.1, 9-9, 9.99, 9.999 days.

144

Simon J. R. Woodward et al. Optimal c(T) against ml,m2 10 4

i i ::ii!i i!

ii

10 ~

,.

",,,i

i: :iil ':. '. " ".,.................................... 10 2

L~ eq

i ,

I(P

",,, "-?"?:_:.:....................LiSLLL2.~.~

~7 r-

%..

-. . . . . . . . . . . . .

10 o

--"~'.... 10 I ,"~'~ra . . . . 10-1 10 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

10 2

10 3

10 4

m l , Field 1 Grazing Intensity

Fig. 9. Optimal c(T) when r = 1 (uniform initial herbage masses), corresponding to the optima in Fig. 8: . . . . , contours of optimal c(T) = 10, 20..... 80 kg animal-I; . . . . . , lines of constant M, proportional to constant stocking rate.

n

a -

H

k

M < 110 (animals ha -l)

In this case the optimal swapover time is determined by the relative sizes of the fields, hi~h2, and changes in stocking rate affect only the animals' intake, not the optimal strategy. The other situation is where M is greater than 60. In this case the lines of constant i I are horizontal. Here we have a situation of extremely high stocking rate where the pasture would be grazed to extinction in a short time. In this case there is an advantage to be gained by detaining the animals in field 1, allowing some extra production in field 2 prior to swapover. The optimal swapover time in this case is determined solely by m2 (since field 1 is considered a holding area or 'sacrifice field') and is near '

il ~ T (which is Earlier attention sequence

a(1

-- m2)

one o f the bounds on the optimum derived in the appendix). we noted that to properly optimize the rotational strategy, must be paid to both swapover time and field sizes. The cono f Fig. 9 is that the division o f field sizes is not significant pro-

Optimizing rotationalgrazing

145

Optimal tl against r 10 i 9 --

8!

~

7

V.

6

~.

5

4

-5

i lO-I

10 o

,

i

b , i ii 101

r, R a t i o o f I n i t i a l M a s s e s

Fig. 10.

Optimal t I when h I -- h2 (equal sized fields): - - , optimal tl against r for n -- 1, 5, 10, 20, 50, 100, 200, 500, respectively.

vided the optimal swapover time is used. Because the lines of optimal c(T) are parallel to the lines of constant M, optimal intake is shown to be completely determined by stocking rate. F r o m Fig. 9 it can be established that for r = 1, it is not significant which field is grazed first, and that no further advantage m a y be gained in terms of animal intake by adjusting the field sizes than when the fields were o f equal size.

3.2.5 Equal sized fields with wl(O) ~ w:(O) Apart from size, the other significant difference between the two fields which m a y influence the choice of tl is their initial herbage cover. This can be reduced to the consideration of the parameter r as noted in eqn (19). So far we have only examined cases where r -- 1. Figure 10 shows how fl varies with r when m~ = m 2. Figure 11 shows the corresponding optimal c(T), and c(T)lt ' =~T As r increases from 1 we see that a greater intake c(T) will be achieved by increasing the length of time the animals spend in field 1 as opposed to field 2. At a high stocking rate this effect is moderated since the animals will consume most of the herbage mass in a short time anyway. However, some advantage is gained at high stocking rate by holding the animals in field 1 even after the herbage level has been reduced to near

Simon J. R. Woodward et al.

146

Optimal 160r

....

T

ctT)

. . . . . .

against

-,-

~

•

~

r ~ T ~

I i

il

/ I 00 (

"

"

""

121.1___,.

6o.~(~L

/

_ .

"

.-""

"

.. . . . . . . . . . . . . . . . . . .

"5

4o

i tg_~t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 LJ200

LSOO oL I0 I

Fig. 11. Fig. 10.

. . . . . . . . . . . .

. . . . . . . . . . . IO n

101

Optimal c(T) when h I = h 2 (equal sized fields), corresponding to the optima in c(T) against r for n = 1, 5, 10, 2 0 , 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , respectively; ..... , corresponding c(T) when the non-optimal t~ : 7"/2 strategy is used. - - - -, o p t i m a l

zero as mentioned before, in order that field 2 may accumulate a little more leaf before the animals are grazed there.

3.2.6 Graze which field first? From Fig. 11 we can again explore the question of which field to graze first, now in terms of whether to graze that with greater or lesser initial herbage cover first. As the graph is closely symmetrical, we conclude that it is not important which field is grazed first provided the optimal t~ is used. The dotted lines on Fig. 11 correspond to the intake when the nonoptimal strategy t l -- IT is used. This strategy gives a much lower intake than the optimal strategy when the fields start with very different quantities of herbage. 3.3 Optimal compromise So far we have looked at the scenarios where the farmer wishes to maximize either the residual herbage W(T), or the per animal intake c(T). In many cases it would be desirable to achieve some sort of compromise between these objectives. A possible way of approaching this is to note that the total intake o f the mob in kg is, from eqn (10)

147

Optimizing rotational grazing

nc(T) -

nk wl(O) - (exp ((1 - ml)atO - 1) a l-ml

nk w2(O) exp (atO + -(exp ((1 - m 2 ) a ( T - tO) -- 1) a 1 - m2 = hlwl(O) 1

ml (exp ((1 -- m O a t O - - 1) --ml

+ h2w2(O) m 2 exp (atO (exp ((1 - m 2 ) a ( T - tO) - 1) 1 - m2 Similarly the net gain of herbage mass (also in kg) over the time period is, from eqn (11) W(T) - W(O) = (hlwl(O) exp (a(1 -- m r ) t 0 exp ( a ( T - tO) + h2w2(0) exp (atl) exp ((1 - m 2 ) a ( T - tO)) - (hlwl(O) + h2w2(O)) These two expressions represent the output o f the pasture system in terms of animal intake and herbage production respectively. We desire to maximize some weighted sum of these, with the weighting factor, 0 < c~ < 1, being chosen to reflect the relative importance o f feeding the animals or conserving herbage. We therefore construct a composite output function J = a n c ( T ) + (1 - ot)(W(T) - W(O))

(kg)

which is optimized with respect to t~ as before:

aJ

ac(T) -

an

Ot~

. . . .

OW(T) +

(1

-

a)

-

0t]

cgtl -- 0

-

for optimum.

Differentiating and simplifying, using the fact that this is equivalent to

him 1 -

h2m 2 -

nk a

Wl(O) exp ((1 - ml)at 0 [o~ - (1 - a) exp ( a ( T - tO)] +wz(0 ) exp (atl)[1----am,. (m2 exp~ ( ( 1 - m z ) a ( T + (1 -- or) exp ((1 -- m z ) a ( T - - tl))J = 0 Solving this formula for tl gives both To find the absolute m a x i m u m value necessary to evaluate J at tl = 0, at which satisfy eqn (21) to find the one

tl)) - - 1 ) (21)

maximal and minimal choices of tl. of J over the range 0 ___tl - T it is tl = T, and at all the values o f tl which gives the m a x i m u m output.

148

Simon J. R. Woodwardet at.

The question of how to choose a is not easy to answer, ot -- 0 is equivalent to maximizing W(T) and o~ -- 1 is equivalent to maximizing c(T), but apart from these it is hard to know how to best choose this weighting parameter. A marginal (e.g. monetary) value attached to c(T) and W(T) would be helpful for this.

4 DISCUSSION This study has approached the question of optimizing grazing management from two points of view. These are the maximum conservation of herbage for feeding at a later time, and maximizing the cumulative intake of grazing animals over a defined time period. 4.1 Conservation of herbage Analysis of this model showed that conservation of herbage is achieved almost at the direct expense of animal intake. Herbage production is of a much smaller magnitude than herbage removal by grazing over a short time period such as that considered here. As expected, rotational grazing was shown to be superior to continuous grazing for conserving herbage in the examples considered. This is mainly because it allows restriction to animal intake, rather than because it promotes greater herbage growth. For the case when fields are of different initial pasture mass or of different size, the order in which fields are grazed affects optimal herbage conservation (Section 3.1). However, the optimal strategy for grazing two particular fields for herbage conservation depends in a non-trivial way on a range of factors including initial pasture mass, field size, minimum intake requirements, stocking rate, etc. Because of this, no simple guidelines can be given. The consideration of maintaining a minimum intake level for the animals has been dealt with in two ways. The first is by setting a minimum daily intake for each animal, which if violated forces transfer of the animals into the next field. The second method considered was optimization of a composite measure of herbage conservation and animal intake. 4.2 Maximizing animal intake A simple rotation gives a small advantage over set stocking in maximizing animal intake (Fig. 7). The results support claims by Bryant (1990a,b) based on field observations that different systems of rotational grazing operating at moderate stocking rate have little effect on the intake of cows over a period of time.

Optimizing rotational grazing

149

Under heavy stocking, however, pasture considerations become limiting and appropriate management may give improved production as suggested by McMeekan (1960). The present analysis supports qualitatively different management at high stocking rates (Figs 8 and 10). The trend towards spending a greater proportion of the time in the first field as stocking rate increases adds weight to the practice adopted by some farmers of sacrificing one field. This allows other fields to accumulate greater pasture mass in order to benefit the system as a whole. There has been some debate as to whether rotational grazing can achieve higher production than continuous grazing management in practice (e.g. Morley, 1981; Florez et al., 1986). Experiments comparing continuous with rotational managements have given ambiguous results. However, there is a fundamental fallacy in the design of many experiments comparing these strategies. In general the rotation length and number of fields are chosen arbitrarily (albeit based on experience). Usually animals spend equal lengths of time in each field. However, the philosophy of controlled management does not claim that any control is superior to no control (e.g. that in general any rotational treatment will increase performance relative to a corresponding continuous one) but rather that the optimal control policy will perform better than no control, or at least no worse, in the case where no control turns out to be the optimal policy. The simple system modelled here is ideal for such comparisons. At moderate stocking rates, and equal field area and initial herbage mass, animal intake is maximized when the fields are grazed for equal lengths of time (Fig. 6). This is intuitively reasonable, and is the strategy generally adopted by farmers. However, pasture consumption is relatively insensitive to timing of shift about the optimum in a two field system (Fig. 6). This indicates some latitude for farmers which may be important in practical situations where flexibility is required. At moderate stocking rates, when two fields are different in size but have equal initial herbage mass, the optimal strategy is to graze the larger one for a longer period of time (Fig. 8). Reversing the order of grazing the fields does not give a significant increase in animal intake. Provided the optimal strategy is used, it is not important how the total area is divided into two fields. It follows from this that we cannot achieve significant increases in animal intake from a two field division over and above set stocking (Fig. 9). This suggests that little can be gained in terms of production by altering the sizes of existing fenced fields. When the two fields are of the same size but of unequal initial mass and are grazed at moderate stocking rate, the optimum strategy to maximize intake is to graze the fields for time lengths in proportion to their masses (Fig. 10). Much greater animal intake is achieved by following

150

Simon J. R. Woodward et al.

this rather than an equal time policy. This is because at low pasture mass such as considered here, non-nutritional pasture availability factors limit intake. This is relevant to practical systems which normally graze fields for equal time intervals but rotate stock through a series of unequal mass fields. 4.3 Limitations of the model

By describing the grazing of farm animals in terms of linear functions this analysis has been able to deal with the local effects of grazing systems only. However, empirical evidence suggests that the linear hypothesis holds well locally for animal intake (McCall et al., 1986). While linearity does not hold for net pasture growth, especially at high levels of biomass, linearity can be regarded as approximately true over a range of herbage mass 500-1300 kg DM ha -~ (Bircham & Hodgson, 1983). This is within the range of pasture mass at which many practical systems operate. It was found that production from different managements did not differ greatly, as shown in Fig. 7. This may be due to this assumption of exponential pasture growth. When more realistic growth models are used, production from the different managements may diverge. Also, as the system is extended to encompass multiple fields and rotations it may be that these small differences will compound, giving a more significant advantage when the optimal treatment is used. 4.4 Future directions

This paper sets out a basic methodology that may be used to analyse real world problems. While differing in style from other analyses on this subject, this study demonstrates the interaction of biological factors underlying this simple rotational grazing system and how optimal performance may be achieved. The obvious extension is to increase the number of fields in a system, to formulate the system to include multiple rotations where each field is visited more than once, and to consider how more realistic growth and grazing functions may be included, or to assess if indeed the choice of these functions is critical. Another refinement would be to allow for the time variation in pasture growth rate. This is the ongoing aim of research the authors are undertaking.

ACKNOWLEDGEMENTS This work is supported by the Ministry of Agriculture and Fisheries, Whatawhata Research Centre, Hamilton, New Zealand.

Optimizing rotationalgrazing

151

REFERENCES Barlow, N. D. (1987). Pastures, pests and productivity: Simple grazing models with two herbivores, N.Z.J. Ecol., 10, 43-55. Bircham, J. S. (1984). Pattern of herbage growth during lactation and level of herbage mass at lambing: Their significance to animal production. Proceedings of the New Zealand Grasslands Association, 45, 177-83. Bircham, J. S. & Hodgson, J. (1983). The influence of sward condition on rates of herbage growth and senescence in mixed swards under continuous stocking management. Grass and Forage Science, 38, 323-31. Bryant, A. M. (1982). Developments in dairy cow feeding and pasture management. Proceedings Ruakura Farmers Conference, 34, 75-81. Bryant, A. M. (1990a). Present and future grazing systems. Proceedings of the New Zealand Society of Animal Production, 50, 35-42. Bryant, A. M. (1990b). Optimum stocking and feed management practices. Proceedings Ruakura Farmers Conference, 42, 55-9. Caughley, G. & Lawton, J. H. (1981). Plant herbivore systems. In Theoretical Ecology. Principles and Applications, 2nd edn, ed. R. M. May, Blackwell Scientific, Oxford, pp. 132-66. Chen, J. L. (1986). Optimal cutting frequency and intervals derived from Johnson and Thornley's model of grass growth. Agric. Syst., 22, 305-14. Chen, J. L. & Wang, Q. 0988). A theoretical analysis of the potential productivity of ryegrass under grazing. J. Theor. Biol., 133, 371-83. Christian, K. R. (1981). Simulation of grazing systems. In Grazing Animals-World Animal Science B1, ed. F. H. W. Morley. Elsevier, Amsterdam, pp. 36177. Florez, A., Bryant, F. C. & Schlundt, A. F. (1986). Grazing strategies and sheep production in the Andes of Peru. In Rangelands: A Resource Under Siege. Proceedings of the 2nd International Rangelands Congress, Adelaide 1984, eds P. J. Joss, P. W. Lynch & O. B. Williams. Cambridge Press, Cambridge, pp. 236-7. Goh, B. S., Leitmann, G. & Vincent, T. L. (1974). Optimal control of a predatorprey system. Math. Biosci., 19, 263-86. Huffaker, R. G., Wilen, J. E. & Gardner, B. D. (1989). Multiple use benefits on public rangelands: an incentive-based fee system. Am. J. Agr. Econ., 71, 670-8. Johnson, I. R. & Parsons, A. J. (1985). A theoretical analysis of grass growth under grazing. J. Theor. Biol., 112, 345-67. McCall, D. G,, Townsley, R. J., Bircham, J. S. & Sheath, G. W. (1986). The interdependence of animal intake, pre- and post-grazing pasture mass and stocking density. Proceedings of the New Zealand Grassland Association, 47, 255-61. McMeekan, C. P. (1960). Grazing management. Proceedings of the 8th International Grasslands Congress, University of Reading, 11-21 July 1960, ed. C. L. Skidmore. The British Grasslands Society, The Grassland Research Institute, Hurley, Berkshire, pp. 21-6. Morley, F. H. W. (1968). Pasture growth curves and grazing management. Aust. J. Exp. Agric. & Anita. Husb., 8, 40-5. Morley, F. H. W. (1981). Management of grazing systems. In Grazing Animals

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Worm Animal Science B1, ed. F. H. W. Morley. Elsevier, Amsterdam, pp. 379400. Noy-Meir, I. (1975). Stability of grazing systems: an application of predatorprey graphs. J. Ecol., 63, 459-81. Noy-Meir, I. (1976). Rotational grazing in a continuously growing pasture: a simple model. Agric. Syst., 1, 87-112. Noy-Meir, I. (1978a). Stability in simple grazing models: effects of explicit functions. J. Theor. Biol., 71, 347-80. Noy-Meir, I. (1978b). Grazing and production in seasonal pastures: analysis of a simple model. J. Appl. Ecol., 15, 809-35. Parsons, A. J. & Johnson, I. R. (1986). The physiology of grass growth under grazing. In Grazing." The Proceedings of the British Grassland Society's Occasional Symposium No. 19, Malvern, 5-7 November 1985, ed. J. Frame. British Grasslands Society, The Grasslands Research Institute, Hurley, Berkshire, pp. 3-13. Parsons, A. J., Johnson, I. R. & Harvey, A. (1988). Use of a model to optimize the interaction between frequency and severity of intermittent defoliation and to provide a fundamental comparison of the continuous and intermittent defoliation of grass. Grass Forage Sci., 43, 49-59. Sharrow, S. H. (1983). Rotational vs continuous grazing affects animal performance on annual grass-subclover pasture. J. Range Man., 36, 593-5. Stocker, M. & Walters, C. J. (1984). Dynamics of a vegetation-ungulate system and its optimal exploitation. Ecol. Model., 25, 151-65. White, D. H., Bowman, P. J., Morley, F. H. W., McManus, W. R. & Filan, S. J. (1983). A simulation model of a breeding ewe flock. Agric. Syst., 10, 149-89.

APPENDIX Maximum number of animals allowed to achieve minimal intake

If a minimum intake C'minis desired, the swapover time t I must be within the interval given by eqn (17). For this to be possible the interval in eqn (17) must exist, that is 1 in /k C'min )/< 1 I n ( C'min "] m2~ w2(0 ) exp (a(1 -- m2)T (1 - ml)a k~(2(O)J

(ml)

If the n u m b e r of animals is too great this will be impossible. Using n --

mlhl (a/k) = m2h2 (a/k) we can rearrange eqn (A1) to give the m a x i m u m n u m b e r of animals. Using B and C as d u m m y variables, let

B= ah2 ( 1 - 1 1aT n( and

~ kw~(O))J + 2-kahl ( 1 1- aT

Ink~)J( ¢'min ~)

Optimizing rotational grazing

C = k-2zhlh2

1-

aT

In

153

\ kw2(O) JJ

Then eqn (A1) will be satisfied if and only if the number of animals, n< B +.~B2-C This result assumes a grazing pressure sufficient that m~, m 2 are both greater than 1. Example Consider again the example in Table 3. If the farmer requires £'min ----4 kg animal -~ day -~ to be the minimum intake rate, we then derive from the above formula that (B = 15.4, C = 41.3) n < 29 (animals) to satisfy condition (15)

Bounds on T(tt) Since the optimal t~ to maximize c(T) cannot be found explicitly from eqn (19), it is useful to find limits inside which tl is known to lie. These give us a good approximation to where the optimum is. The bound shown in Fig. 5 as the steep dotted line to the right of the optimal solid curve is found by letting m~at I ~ oo in eqn (20). Notice that the second term increases monotonically with tl to the asymptote T=ll+

a(1 - m2)

in(l)

This gives the bound ~I>T

a(1 - m2)

(A2)

which is close to the exact solution at large T and m 2. A second bound is obtained by linearization of T(tl) near il = 0. This bound will be close to the exact solution for low values of T. As preliminaries, from eqn (20) we differentiate to give 3T 3i 1

and

1 exp (mlail) - 1 + ml + m2 r 1 r

- exp ( m l a i l ) - 1 + m 2

154

Simon J. R. Woodward et al. ml2a

- --r

o~2T

exp (mlail)

exp ( m l a i l ) -

3i12

1+m

F r o m these we get, assuming m2 > 1

3T

oi,

>0,

--3 T ,I

-~ 1

oil J i,_,.

;,=o

From these and eqn (20) we now have

TI;, = o

--

1 a(1

ln(

-- m2)

1

r

m 2

-

c92T

-< 0 O3il2

, m2) -

-

-

m 2

(1+ m 1)2 +-m -, r

aT 3il

and

i, =0

(1__1)

+m2

cgT il

---) ~

and

02T 0i1~ < 0

for all il

(A3)

Because o f the latter, any tangent to T(i~) is a bound of the function, including the tangent at t~ -- 0; T<

T[;,=o + i,. O T

-

-0~1 [ i, : o

Inverting gives the bound

T i~ >

T[;, =o (A4)

aT Oit

i,=0

where the relevant values can be calculated from results (A3). In Fig. 5 this bound is represented by the less steep dotted line which touches the solid/dashed curve at il --- 0.

Example As in Table 2, r = 1.14, m I = 8.93, m 2 = 5-95, a = 0.014 day 1, T = 10 days. Then the bounds from eqns (A2) and (A4) are il > -15.7 and i~ > 4.6

Optimizing rotational grazing

155

respectively. The former is not very helpful in this case, but the latter is quite close to the o p t i m u m , since T is fairly small. Numerically calculated, the true o p t i m u m in this case is i~ -~ 4-7. C o n s t r a i n t violations on TOO

When optimizing c(T), the mathematical m a x i m u m is found by solving eqn (19). This solution does not always satisfy the constraint (13), as observed in Fig. 5 when the solid curve leaves the feasible region 0 < t~ < T. In this case we have shown a dashed curve on the figure. The conditions for this constraint to be violated can be derived as follows. Firstly, we find that t~ < 0 only if r < 1, and when T< a(1-

1 m2) l n ( l

2

--.,l-m2 -rm~-~ /

(by putting t~ = 0 into eqn (20)). Secondly, t~ > T only if r > 1 and when a(1

exp ( - m l a T

- r - -

m2)

<0

m 2

(by putting tl = T into eqn (20)). This simplifies to T < In (r) mla By corollary, if r 5 1 then the solution to eqn (19) will always satisfy ij < T, and if r > 1 then it will always satisfy i I ___0. If r = 1, the solution to eqn (19) is always feasible.

Example In the previous example we had r = 1.14. In this case tl > T if T < 1.0

(days)

T is m u c h larger than this so spurious solutions to eqn (19) do not occur. If T was 0.5 (days), however, we should ignore the solution tl and graze only field 1 (i 1 = T).