Rotational grazing in a continuously growing pasture: A simple model

R O T A T I O N A L G R A Z I N G IN A C O N T I N U O U S L Y G R O W I N G PASTURE: A SIMPLE M O D E L

I. NoY-MEm

Department of Botany, Hebrew University, Jerusalem, Israel

SUMMAR Y

The effects of rotational and continuous grazing on long-term steady-state productivity of a continuously growing pasture were examined in a simple mathematical model representing both plant growth and herbivore consumption as functions of green plant biomass. The results showed that moderate rotation, with few subdivisions and short cycles, had only minor effects on productivity, compared with continuous grazing. Intensive rotation (many subdivisions and/or long cycles) resulted in a substantial decrease in long-term productivity in the absence of an ungrazeable plant residual or when the stocking rate was moderate and the initial condition (biomass) of the pasture high. When the stocking rate was high and the initial condition of the pasture poor, but an ungrazeable residual was present, intensive rotation substantially increased the long-term productivity compared with continuous and moderate rotational grazing. INTRODUCTION

Should a pasture be grazed continuously at a uniform stock density, or should it be divided into sub-plots, to be grazed in turn by the whole herd in a rotational manner so that each sub-plot receives alternate periods of heavier grazing and of rest ? Should there be many or few sub-plots, and should the rotation cycle be long or short? These questions have for long been controversial among both pastoralists and scientists. The supporters of each method have claimed it to be advantageous for animal production and/or pasture condition, and have brought forward proofs from theory, field experience and results of grazing experiments. Both experimental evidence and theoretical considerations have been thoroughly reviewed and discussed by Heady (1961), Morley (1966b, 1967) and Spedding 87 Agricultural Systems (1) (1976)--O Applied Science Publishers Ltd, England, 1976 Printed in Great Britain

88

I. NOY-MEIR

(1970). These authors indicate that there is no general and clear-cut answer; the relative advantages of continuous and various rotational grazing schemes may depend on such factors as stocking rate, type and condition of pasture. The effect of rotational versus continuous grazing on plant and animal production is the sort of problem which is amenable to examination by mathematical models. The basic processes involved (plant growth and regrowth, consumption by animals) are the same in both cases and are fairly well known. Indeed, simulation experiments with rotational grazing were one of the first exercises to be undertaken with mathematical models of grazing systems (e.g. Freer et al., 1970; Goodall, 1971 ; Christian et al., 1972). The results from the simulation experiments in these specific cases (as with the results of field experiments) do not yet allow any general conclusions; sometimes rotational grazing was superior to continuous grazing (in terms of production or income), sometimes inferior, and sometimes there was no difference. A model of an intensive grazing system with a somewhat specialised type of rotation (with different groups of animals) was developed by Jones & Brockington (1971). Morley (1968) used a simple mathematical model to specifically answer the question of the optimal length of rest and grazing periods and the optimal number of subdivisions in rotational grazing. Plant growth was expressed as a logistic function of biomass, on the basis of growth curves measured by Brougham (1956). The optimum rest period (interval between grazings) was determined as the period over which the average growth rate was maximal; this optimum (eight to ten weeks in winter, five to six weeks in summer) was found to be much more sensitive to the relative growth rate parameter than to other parameters of the growth equation. Given the optimum rest period, the effect of the number of subdivisions (and of the length of the grazing period) on the average amount of biomass available for consumption was calculated. It was found that, insofar as the average growth rate was assumed to be lower in grazing periods than in rest periods, consumption (~ production) increased with the number of subdivisions from 2 to 5, then levelled asymptotically. All these general results were obtained not by dynamic simulation, but by some analytical short-cuts made possible by the simplicity of the model and its assumptions. The objective of this paper is to explore the general effects of rotational grazing schemes on pasture production, using a simple mathematical model which represents only the minimum essential features of the major processes involved. These are the rates of pasture growth and of animal intake and the dependence of both on total leaf area or green biomass. All other complicating factors occurring in reality have been deliberately excluded at this stage, as the aim is to test the simplest model more or less exhaustively and to see how far one could get with it. The approach to growth dynamics is similar to Morley's, but the initial conditions and consumption dynamics are treated more explicitly here; in addition, continuous

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

89

grazing is included as a reference treatment. The addition of these features precludes the use of the short-cuts used by Morley. Therefore, the response surfaces had to be explored with a dynamic simulation model. THE GENERAL GROWTH-CONSUMPTION MODEL

Assumptions

The two basic processes in a plant-herbivore system are the growth of the vegetation and the consumption by herbivores. Any model of such a system must account for the dependence of both growth and consumption rates on the quantity of vegetation available. The simplest possible model of a grazing system could be one which considers only this factor, assuming all others to be constant. Some general results on stability and productivity in such a model, under continuous grazing, can be derived by graphical analysis (Noy-Meir, 1974, 1975). The assumptions in this model were: (1) Effective vegetation density (e.g. green biomass orleafarea) can be expressed by a single variable, V. (2) Effective herbivore density can be expressed by a single variable, H. (3) Vegetation growth rate G depends only on V. (4) Consumption rate of green biomass per animal, c, depends only on V. (5) The net rate of change in V equals growth G minus consumption per unit area, C: dV dt

- G-C

= G-cH

(1)

(6) The function relating growth to biomass, G(V), is convex with a single maximum (Fig. 1). (7) The function relating consumption to biomass, c(V), is a saturation function (Fig. 1). (8) The residual plant biomass, V,, which is ungrazeable but produces growth, is constant (or zero). (9) Animal productivity is monotonically and linearly related to consumption. (10) Effective herbivore density H (stocking rate) is constant. Assumptions (5), (6), (7) and (9) are probably fair approximations of reality in most grazing systems. In assumption (9), linearity was introduced only for illustrative purposes, and is not essential to the argument; any form of monotonic increase of productivity with consumption will produce qualitatively identical results. Assumptions (1) to (4) are fully valid only for a rather special class of systems: one herbivore species grazing on one plant species, which continuously grows a single type of green organ, with constant potential growth characteristics, in a constant environment.

90

i. NOY-MEIR

i

I,,'/ I//..-ore

-

!

\

¢ "'= =-

- - ~

|

Vx V : vegetation

biomass

- -

Vm =

Fig. 1. Superimposed graphs of growth G ( - ) and consumption C ( - - ) as functions of vegetation biomass (V) and herbivore density (H). Gx, maximum growth, Vx, biomass at which

growth is maximum, Vm, maximum biomass, Cm, maximum consumption (at H = 1).

Stability properties Graphical stability analysis (Rosenzweig & MacArthur, 1963) of this general growth-consumption model has yielded some general conclusions about stability of production in a continuously-grazed, continuously-growing pasture (Noy-Meir, 1974, 1975): (1) At low herbivore densities (H) there is always a stable plant-animal equilibrium (at high V) and productivity per unit area increases with H. (2) At very high H the vegetation either becomes extinct or (if V, > 0) equilibrates with the herbivores at a very low level of V and productivity (both decreasing with H). (3) The behaviour in the intermediate range of H differs between two types of plant-herbivore systems: (a) Continuously stable systems: At any level of H there is a single stable equilibrium; the equilibrium V (and beyond a certain H also the stable productivity) decreases continuously with H. (b) Discontinuously stable systems: In the intermediate range of H, there is at any level of H both a stable equilibrium (at high V), and an unstable equilibrium or critical threshold (at lower V: Vt). If V goes below lit it will decrease further, until plant extinction or (if V, > O) a stable equilibrium at low biomass and productivity is reached. Thus in a discontinuously stable system there is the possibility of a 'crash' in the system.

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

91

The discontinuity is also reflected in a sharp (rather than continuous) drop in productivity as H goes above a critical threshold. These results from the general graphical model can be made somewhat more specific by introducing explicit mathematical growth and consumption functions and examining the effect of plant and animal parameters and of function types (Noy-Meir, in prep.). This stability analysis was based on a simple combination of growth and consumption functions, which is possible only for continuous, not for rotational grazing. However, one tentative qualitative conclusion that has been drawn from it (Noy-Meir, 1975) is that rotation may have significant effects on production, particularly in discontinuously stable systems, in the range of H where a threshold separates a domain of stable high productivity from a domain of extinction or stable low productivity. Then, if the system is initially in the high domain (and would remain in it if continuously grazed), the temporary heavy grazing in a rotational scheme can only push the pasture across the threshold and down into the low domain. However, if the pasture is already in the low domain, the rest periods in rotation could push it across the threshold into the high domain, thus allowing a higher production than under continuous grazing. The aim of the present paper is to verify and specify this tentative conclusion by directly testing the effect of rotational schemes on stability and production of an explicit growth-consumption model. The same basic growth and consumption functions were used (or rather special cases of them), as those employed in the stability analysis of continuous grazing. However, the combination of the two processes into a model of leaf biomass dynamics requires, in the rotational case, separate consideration of grazing and rest periods and the biomass transfers between them. Since there is no simple analytical solution, the test has to be made by a series of simulation experiments. For simplicity, only predetermined rotation schemes (not allowing for dynamic decision-making based on condition of pasture or animals) have been considered; of these, a wide range has been tested.

AN EXPLICIT MATHEMATICAL MODEL" LOGISTIC/M1CHAELIS

Assumptions There are a number of mathematical functions ('ramp', Michaelis, negative exponential, logistic) which can be used to describe the growth function and which have the general 'optimum curve' form (assumption (6)). Similarly, there are several possible saturation functions ('ramp', Michaelis, negative exponential) to describe consumption (assumption (7)). At present there is no overwhelming experimental or theoretical evidence in favour of any particular function for general use. The effects of selecting different functions on the properties of the growth-consumption

92

I. NOY-MEIR

model (under continuous grazing) are being examined elsewhere (Noy-Meir, in prep.). For the purpose of testing the effects of rotational grazing on the growthconsumption model, one specific growth function (logistic) and one consumption function (Michaelis) were chosen. The assumptions of the explicit model are thus the same as those for the general model, except that two assumptions are now more specific. (6(a)) Growth is a logistic function of vegetation biomass:

where g = maximum relative growth rate; Vm = maximum plant biomass. (7(a)) Consumption is a 'Michaelis' function of vegetation biomass above an ungrazeable residual: c = c t t = c~ ( v -

- I(Iv, k - vr) H; v ,V) +

if V < 1 I , , C = 0

(3)

where cm = maximum (satiation) consumption rate per animal; Vt = ungrazeable residual plant biomass; Vk = 'Michaelis constant', characteristic plant biomass at which consumption is half of satiation. This function may alternatively be written as: s(V-V,)

C = cm s ( V - V,)+Cm

H;

if V < V t, C = O

(4)

where s = Cm/(Vk - V,) = the maximum slope of the consumption function (at low V), i.e. the maximum grazing (searching) rate (area per animal per unit time). The logistic equation for pasture growth is consistent with certain theoretical considerations on competition for space (light) between plants (Shinozaki & Kira, 1956; de Wit, 1960): it has been fitted to the growth curves of Brougham (1956) and used by Morley (1968) in his analysis of rotational grazing. Recent work suggests that it may exaggerate the rate of decline in net growth rate as the biomass (leaf area) exceeds the 'optimum' (de Wit et al., 1970). The 'disc equation' developed by Holling (1959) to describe the function relating the rate of prey consumption by predators to prey density is formally identical to the Michaelis equation for enzyme kinetics, or the Langmuir absorption equation. Theoretically it can be derived from several alternative assumptions about the process of predation or herbivory, e.g. by assuming a constant prey handling time (Holling, 1959); or by assuming that the effective search rate (or the 'success rate', i.e. the proportion of found prey which is actually eaten) decreases linearly with relative satiation (c/c~). Some of the consumption functions found experimentally for sheep (e.g. Allden & Whittaker, 1970) at least superficially resemble a Michaelis curve.

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

93

Stability properties under continuous grazing The stability properties of any explicit growth-consumption model can be examined by setting the equilibrium condition: dV

dt - ~ ( v ) - C(V) = G ( V ) - c(V)H = 0

(5)

and considering the values of H and V at which this equilibrium is possible (the zero net pasture growth isocline). For the logistic-Michaelis model, the equilibrium levels of H and V are related by the equation: H* = g V*(Vm- V*)(V* ~- Vk) VmC,.(V*-- Vr)

(6)

The analytical solution of this for V* (as well as the solution for maxima or minima dH*/dV* = O) involves third-order polynomials. However, for the case when V, = 0 (no ungrazeable residual) the solutions are simpler: H* -

g

(7)

( V m - V ) ( V + V~)

Vmcm

d H * _ _ _ g [-2V*+(V m-Vk) ] = 0 dV* VICm

when

V * - Vm--Vk 2

(8)

:0

(9)

The equation for V* is quadratic

V2-(Vrn--Vk)V-.[-Vrn(~H*-Vk)

V* = ½ [(Vm-- Vk)~ J(Vm-- V~)2-'4Vm ( ~ H*-- Vk)]

(10)

The following results can be derived:

(1) The plant-herbivore system is discontinuously stable if Vm > Vk. Since a grazing system in which Vk > Vm (consumption is less than half satiated at the maximum possible plant biomass) would be of little practical interest even if it did exist, this means that the logistic-Michaelis model is practically always discontinuously stable (at least when Vr = 0). (2) The system has only a single stable equilibrium if:

H* < gVk =

em

g

S

(11)

Thus, the 'safe carrying capacity'--the animal density below which the system is

94

I. NOY-MEIR

always stable (no danger of crashing to extinction) is:

(12)

n s - gVk _ g Cra S

(3) The system has no stable equilibrium (and the vegetation becomes extinct) if: g H * > 4~mV m [(Vm - Vk)2 +4VkVm]

(13)

Thus the 'maximal carrying capacity', above which a crash to extinction is certain, is: Hx

_

g

[(I'm- Vk)2+4VkV.,] = . g

4¢m V,n = Hs+

(I'm+ Vk)2

(14)

t4Cm I/m

g..._.~__( V m - Vk) 2 4Cm Vm

When animal density is between H~ and Hx, there are two equilibrium points (two solutions of eqn. (10)): the higher one (lie) is stable, and the lower (Vt) is an unstable threshold point; if V is below Vt the pasture goes to extinction.

SIMULATION EXPERIMENTS WITH ROTATIONAL GRAZING

General design The explicit model (eqns (2) and (3)) was used to compute daily growth, consumption and plant biomass in a unit area of a growing pasture which was either continuously grazed or was the first sub-plot in a rotational grazing scheme. The rotational scheme was defined by the two parameters: n = number of sub-plots (degree of subdivision); t, = length of a whole rotation period or cycle (in days). For continuous grazing (n = 1), animal density in the pasture, H, was set equal to the average density H throughout the simulation. For rotational grazing it was set to H = n H for the first tg = t,/n (grazing period) days in each cycle of tr days, and to H = 0 for the remaining ts = t,(n- l)/n days (rest period). The model was written as a CSMP-II program. Integration was rectilinear with a time-step of DELT = 0.1 day. Simulation was continued for 100-250 days. Output variables printed out at one to five day intervals were plant biomass V and animal consumption per unit area C. The average of C over a rotation cycle equals average utilised plant productivity and can also be interpreted as a rough index of average 'gross' animal productivity, to which it is almost certainly monotonically (if not linearly) related. The model thus had nine parameters which could be changed (Table 1), including

ROTATIONALGRAZING IN A CONTINUOUSLYGROWING PASTURE

95

the initial plant biomass V . In each experiment or series of simulation runs one, two or three parameters were changed from their 'standard' values. Most of the experiments were concerned with the management options, i.e. the effects of changing the grazing scheme (n, t,), the stocking rate (H) and the initial biomass at the start of the grazing period (Vo), and the interactions between these. The effects of two parameters characterising the plant-animal system were also tested to some extent: V,, the ungrazeable plant residual, and Vk, which inversely expresses the grazing or searching efficiency of the herbivore. TABLE

1

PARAMETERS OF THE MODEL

Symbol g

Vm Vr V~ C,n Vo /7 n tr

Meaning maximum relative growth rate maximum plant biomass ungrazeable residual plant biomass 'Michaelis constant' for consumption maximum consumption rate initial plant biomass average animal density number of subdivisions length of rotation period

Units

Standard value(s)

day-1 g/m2 g/m2 g/m2 kg/animal/day g/m2 animals/1000 m 2 -days

0.10 500 0(10; 5-100) 100(10-500) 3 250(50, 150, 400) 5(6, 8) (1-25) (10-80)

Three parameters were not changed throughout. Vm = 500 g/m 2 (dry matter) and g = 0.1 d a y - 1 can be regarded as characteristic of a grassland of high productivity. The satiated consumption rate cm = 3 kg/animal/day may be interpreted as a fairly high estimate for sheep, which includes 30-50 ~ wastage and damage and 5 0 - 7 0 ~ actual intake. Satiation intakes for sheep in green pasture have been reported to vary between 1 and 2 kg dry matter/animal/day (Willoughby, 1958; Allden, 1962; Arnold, 1963; Allden & Whittaker, 1970). Animal density H is expressed in animals/1000 m 2 (animals/dunam), which is consistent with the units chosen for V and c m. The levels chosen should be related to the threshold levels for continuous grazing, which can be computed from plant and animal parameters (eqns (12) and (14)). Thus the maximum carrying capacity is Hx = 0.1(500+ 100)2/(4 × 3 x 500) = 6 animals/1000 m 2, and the safe carrying capacity is: Hs - 0.1 × 100 _ 3.33 animals/1000 m 2 3 The standard stocking rate in the first set of experiments, _H = 5, is thus just below the maximum for continuous grazing, which may be a reasonable choice o f ' n o r m a l ' stocking in a commercial pasture. (The actual values of H here are significantly higher than in most real systems, as the model assumes continuous optimum growth conditions, while in most pastures growth is limited by weather conditions to part of the year.)

96

i. NOY-MEIR RESULTS OF EXPERIMENTS

1. The effect of rotation: no plant residual, normal stocking In this experiment plant and animal parameters were at their standard values, the plant residual was assumed to be V, = 0, and the stocking rate was just below the maximum for continuous grazing, H = 5. The degree of subdivision was changed from n = 1 (continuous) to n = 2, 5, 10, 25 and the rotation period was t, = 10, 25, 50, 80 days (and, in one case, 125); in addition, all combinations of n and t, were tried. Plant biomass at the beginning of the first grazing period was Vo = (Vm/2) = 250 g/m 2, which is the biomass giving maximal growth. Under continuous grazing (Fig. 2), V increased with time and asymptotically approached a stable level of 322 g/m 2. Under rotational grazing, V of course

6O0

R = 5

~

continuous

................ t r =

Vr=O

......

Vo = ;~50

ZS,

n = 5

tr = 5 0 ,

n=2

= 50,

n=5

--~--tr

,oo

,', - -

3oo

:

Z

i,

4.

t t ' . . iX "

.

~

,.,

I

"

..."

;¢",.:., ~. • ~ ~:

i#

/

/

:•

: .<#

.Q

'

I

•"

i

".. k

".,7

t~

/11

Vi

.

":

tt

:..'#

#l"...,"~

/~"

;~

:"I

',,

/

X

t/

~

X

.-

',/

_

~

x

#

"',,'

u

>

I

I00

t 0

",,,,,/i I ~i

t

/

I

\ 50

~

i

\

\~

/ I00

t i m e (days)

Fig. 2.

i

i,

\

150

•

Time courses of plant biomass under continuous and rotational grazing: output from simulations (moderate animal density, no residual, V0 = 250).

decreased (in grazing periods) and increased (in rest periods) alternately in a 'sawtooth' fashion. Over several rotation cycles, however, the trend superimposed on these fluctuations could be observed. In some cases, both the maximum level of V at the end of a rest period (Vmax) and the minimum level at the end of a grazing period (Vmin) increased steadily, approaching stable values (Fig. 2); thus rotational grazing led to stable fluctuations. In other cases, Vmax and Vmin decreased steadily from each rotation cycle to the next, approaching zero; in those cases rotation led to a 'crash' and to vegetation extinction, at a stocking rate at which stable production was still possible under continuous grazing. Thus the outcome of a rotational scheme over several cycles may be either stable production or instability and extinction. Which is the outcome depends both on the

ROTATIONALGRAZING IN A CONTINUOUSLYGROWING PASTURE

97

length o f the cycle t,, and the degree o f subdivision n (Table 2). In general, stable production was attained with short cycles (10 or 25 days), but not with long cycles TABLE 2 THE EFFECT OF ROTATION PARAMETERS n AND tr ON STABILITY OF PRODUCTI ON, STARTING FROM Vo = 250g/m ~ ( g = 5, Vr = 0 ) , COMBINATIONS W H I C H RESULT IN STABLE PRODUCTION + ; W H I C H RESULT IN DETERIORATION TO EXTINCTION - -

Number of subdivisions n

Rotation period t (days)

10

25

50

80

1

+

+

+

+

2 5 10 25

+ + + +

+ + + +

+ ---

----

(50 or 80 days). With a cycle o f 50 days, stable production was attained when subdivision was moderate (n = 2) but when it was more intensive (n = 5, 10, 25), extinction occurred. The explanation o f this result relates to the unequal effect o f rotation on biomass depletion and recovery. As the intensity o f subdivision is increased, and as the rotation period is lengthened, the amplitude o f the fluctuations in V, (Vm~x- Vm~n), increases (Fig. 3). However, the decrease in Vm~n (end of grazing period) is larger 600[500

T

4OO

R --=5

L

----

Vo=250

V

min

t~ = 5 0 _

tr= I0

x •

V I

Vr=0 V max

\

--

~z...........

,,,

•

•

. . . .

~ a

•

~__

tr = 10

D

200

"\

""It

"o.

100

" " ---

""'

"-"--t'2.25

-~,,,~O. I

2

5 10 25 (continuous) n = number of sub-plots .1Io9 stole) Fig. 3. The effects of rotation parameters n and tr on the extremes of plant biomass, Vmaxand

Vmin (moderate density, no residual, V0 = 250).

98

~. NOY-ME1R

than the increase of Vmax (end of recovery period), and is more strongly affected by t, and n (Fig. 3). Thus the longer recovery period allowed by longer cycles or more subdivisions does not fully compensate for the stronger depletion of biomass and growth potential caused in the longer and more intensive grazing periods. If this depletion is too strong (4 and/or n too high), plant biomass will not be able to maintain itself above its critical 'turning point', even during rest periods, and will fluctuate towards extinction. Table 2 suggests that this will occur when the rest period ts = t,(n- 1)In is longer than about 30 days. The long-term effect of the rotation scheme (n and t,) on animal production can be assessed by considering the average rate of plant consumption by animals (per unit area) when the fluctuations in V have stabilised, averaged over the rotation period. It appears (Fig. 4), that insofar as stable consumption and production is Cm=15

: 5

%--250

%:0

l

,

- ,-

""-" -~t~......

It

vr-- ~o

tr= I0

'l,

rr = ,':3

II

-- •

" " ~-,,~r= 50

10

E

u

25

2 n

Fig. 4.

~

The effectsof n and tr on the consumption rate per unit area at steady state, C, (moderate density, with and without residual), 1Io = 250.

possible at all, the average consumption rate Ce is only slightly affected by the grazing scheme. It is highest for continuous grazing (11.4 g/m 2 day) and decreases with degree of subdivision and length of rotation cycle, but the decrease is of the order of less than 1 0 ~ over a wide range of n and 4-

ROTATIONAL GRAZING

IN A CONTINUOUSLY

GROWING

PASTURE

99

2. The effect of initial biomass Since the possibility of stability under rotational grazing seemed to be related to a critical threshold of plant biomass below which deterioration occurred, it was decided to test the effect of the initial V on the results. Thus Vo was changed to either a high value (400 g/m 2) or a low value (150 g/m 2) over a range of values of n and tr and their combinations (Table 3). Other parameters were as in Experiment 1. TABLE 3 THE EFFECT OF ROTATION PARAMETERS ON STABILITY OF PRODUCTIONs STARTING FROM VO = 1 5 0 AND 4 0 0 g / m s ( H = 5, Vr = O)

V0 150

Number of subdivisions n

10

25

1 2 5 10

+ + + +

+ + -

+ -

tr (days) 50

80

?

-

-

1

+

+

+

+ +

2 5 10 25

+ + + +

+ + + +

+ + -

+ -

25 400

Rotation period

Under continuous grazing, V decreased from 400 g/m 2 towards the same stable value of 332 g/m 2 as before. From 150 g/m 2 it increased asymptotically towards it (the turning point below which there is extinction under continuous grazing is 78 g/m 2, from eqn. (10)). Under rotational grazing with very short or very long cycles, the starting point did not affect the eventual result, but in some intermediate cases it did (Fig. 5). In some cases (n = 5, t, = 25; n = 2, t, = 50), stable fluctuations could be attained from Vo = 250 but not from Vo = 150, extinction occurring instead. In other cases (n = 5, t, = 50), deterioration occurred from both Vo = 150 and V0 = 250, but when the initial biomass was 400 g/m 2 stable fluctuations could be maintained. Thus the initial conditions determined the qualitative outcome in a range of 'marginal' rotation schemes. These initial conditions do not, however, overrule the effect o f n and tr in the more clear-cut cases (Fig. 6).

3. The effect of Vk To test the effect on stability and production of the 'Michaelis constant' of the consumption function, VR (or the grazing rate, s = Cm/Vk), the model was run with V~ = 10, 50, 100, 200, 500 g/m 2 under continuous grazing and under two rotational schemes. F o r comparison, values of Vk that can be roughly estimated from experimental

100

I. N O Y - M E I R

6oo lr

~.5

!

5oo

,.:50

v,--o

n--2

v , : 150, 250, 4oo

1,oo V ~00 2OO I00

0

50

I00

150

200

time

Fig. 5. The effect of initial biomass (V0) on biomass dynamics under rotational grazing with tr = 50, n = 2.

H:5

400

----

t~ v

n

Vo

50

25

400

25

2

150

/

2Oo 10o

Vr:O

tr

^/ /iX/ \ V / /// " , / ;I /

/

I I

"7~ / II // /

/

I I

..,s

/

/

/

,4I

I I

I L _ _ _ _ ~

50

I00 time

150

~

2(~)0

=

Fig. 6. An example of the effect of rotational scheme overriding initial conditions. consumption curves range from 30 g/m 2 (Arnold, 1963) to 100 g/m 2 (Allden & Whittaker, 1970). The results (Table 4) show that the more efficient the animals are in grazing (higher s or lower Vk) the lower are the stable levels of plant biomass (Ve, or Vmax and Vmin). Under rotational grazing, Vmln is more affected than Vrnax and, if the herbivore is an efficient grazer, instability and extinction will occur. On the other hand, provided that Vk is not as low as that, lower Vk results in higher Ce, and thus higher animal production.

4. The effect of Vr The ungrazeable residual of plant biomass, Vr was changed from 0 to 5, 10, 50

10l

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

and 100 g/m 2, under continuous grazing and two rotational schemes. Other parameters were at their standard values. Measured values of Vr are scarce, but from some field observations one might estimate 0--10 g/m 2 for annuals (higher for prostrate than for erect plants), and up to 50 g/m 2 for perennials. TABLE 4 THE EFFECTS OF PARAMETER Vt(g/m s) oN STABLEPLANT BIOMASSLEVELS, Ve, graax, Vmln (g/m s) AND ON STABLE TOTAL CONSUMPTION Ce (g/ms/DAY) Vk

10

50

100

200

500

240 14.0

295 12"8

332 11 "4

370 9"8

418 6.8

0 0 0

348 133 12"4

408 191 11"1

435 254 9"4

468 338 6"6

0 0 0

0 0 0

0 0 0

477 173 8-8

495 290 6.4

Continuous: V~ G* Rotation: tr = 25, n = 5

Vmax Vmtn G Rotation: tr = 50, n = 5 Vmx Vmin G * Cra = c m H = 15"0 kg/1000

mS/day

With increasing 1I, (Table 5), the steady-state levels of plant biomass (Ve, Vmx and particularly Vmin) increase both under continuous and rotational grazing, as might be expected. Total consumption and production at steady-state usually decreases somewhat, as the increase in stable plant biomass is less than the unavailable V,. THE

TABLE 5 EFFECTS OF RESIDUAL PLANT BIOMASS Vr (g/m s) (SEE TABLE 4)

Vr

0

5

10

50

100

332 11 "4

334 11 "4

339 11"3

362 10' 1

386 8"8

408 191 11"1

410 197 10"9

413 203 10"8

430 244 9"7

446 287 8"4

0 0 0

? ? ?

446 97 9"4

478 165 8'8

486 224 7"6

Continuous: Ve

Ce Rotation: tr = 25, n = 5 Vmx Vraln G Rotation: tr = 50, n = 5 Vmx //max Ce

In a rotational scheme which caused extinction with II, = 0 (t, = 50, n --- 5), the presence of a residual allowed the system to settle into stable fluctuations and to maintain a stable level of production (which was, however, lower than in the two other schemes). To examine further this qualitatively new result, a simulation experiment was set up to test the effect of management parameters and initial biomass when a small residual is present.

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I. NOY-MEIR

5. The effect of rotation and initial biomass: with residual, normal stocking This experiment differed from Experiments 1 and 2 in that 1I, was set to 10 g/m z. Stocking rate remained 5 animals/1000 m 2. Continuous grazing was compared with long-period (t, = 50) rotational grazing at various degrees of subdivision (n = 2, 5, 10, 25), from two initial states (Vo = 50 and 400 g/m2). Under continuous grazing, biomass converged from both starting points to a stable level of 330 g/m 2 (Fig. 7). Convergence from Vo = 50 was very slow (about 600

H 5 Vr : I0 Vo= 50, 400

F 500L

continuous ----- tr=50 , n =25

/ 4

v

/

4

oo ~

300-

/

oo, I r

~i 0

iI /I

/

I

/ I

t

,IA

I I

,,/q i? t

.~II

.I

//

I I

1~

4

/~

~

I

l

I/

I

~ i

I'

~

i1

.....~/ II ii I

I

.<.

i

50

100

/

/i / I ~ I

~

!

I

/

#

,,"

"

/

I

I

I

V

I

I

I

I

150

I

200

time Fig. 7.

Time courses of biomass under continuous and intensive rotational grazing, when an

ungrazeable residual is present (moderate density).

200 days). It should be remembered that without a residual the initial biomass of 50 g/m 2 was below the turning point and led to extinction even under continuous grazing; however, the presence of the residual reduced V t below 50 g/m 2. Also with rotational grazing at all degrees of subdivision the fluctuations in plant biomass converged from both 50 and 400 g/m 2 to the same stable limits Vmin and Vma~. This also happened in the rotation schemes with high subdivision (t, = 50, n = 5, 10, 25), which, in the absence of the residual, had caused extinction. Indeed, convergence was faster with high subdivision than with low subdivision or continuous grazing (Fig. 7). Thus, if V, is not zero, stable production is possible also under rotation with a long period and intensive subdivision. The steady-state consumption (,,,animal productivity) still decreases with n and t, (Fig. 4), but the decrease is gradual and not precipitous. It was next decided to try the effect of increasing animal density on the effects of rotation in the presence of a plant residual.

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

103

6. The effect of rotation and Vo: maximal stocking, with residual present In this experiment again various combinations of t,, n and Iio were tested, but with an average stocking rate of H -- 6 animals/dunam, i.e. about the maximum carrying capacity for continuous grazing (Hx is 6 exactly for 1I, = 0, but should be only slightly higher with V, = I0). V, was 10 g/m 2 as in Experiment 5. Time courses of plant biomass from Vo = 50 and 400 under continuous grazing, moderate subdivision and intensive subdivision are shown in Fig. 8. Under continuous grazing V stabilises either at a near optimum (262 g/m 2) or at a very low

[

A = 6 Mr = I0 Vo: 50, 400

.... continuous ----- tr:50 , n =2 . . . . It= 50, n=25

I li I!

\

z.'/ i \ /// ! \

\

//

tII ,, ,::, ),, ,, \,/

I0011-

IL 0

,.;#

//

/IN

/'

I"-,..""

50

,7. i , i x

/

/

/1

./

:L

I\

,~ t

,

/ ,' } )/ /

i \'./

100

/: 15o

time Fig. 8. Time courses of biomass under continuous and rotational grazing at maximum animal density, with residual. (32 g/m 2) level, depending on the initial V. Under rotational grazing with intensive subdivision (t, = 50, n = 25) the system converged, already after one cycle, from the two extreme starting conditions to the same pattern of fluctuations with a wide amplitude (Vmin = 13, Vmax = 360). In this scheme the unit was very heavily stocked ( H = net = 150 animals/1000m2!) for two days and was stripped practically down to the ungrazeable residual in any case; therefore recovery was determined solely by the residual, and the starting Vo had no effect. Under moderate subdivision (n = 2), biomass continued to fluctuate in the low rangewhen startingfrom 1Io = 50, and in the high range when starting from Vo = 400 (although a very slow converging trend was noticed). In general, biomass levels under rotation were higher than under continuous grazing when the initial level was low, and lower than under continuous grazing when the initial level was high. The effects of degree of subdivision and initial biomass on the extreme levels of biomass when fluctuations had stabilised are summarised in Fig. 9. When the

104

]. NOY-MEIR

initial biomass is high, the main effect of increased subdivision is to reduce Vmi,; when l/o is low, the main effect is to increase l/max. Already at n = 5 there is rapid convergence to the same Vmln and Vmax from any starting point and further subdivision has little effect. Vmln is then limited by the residual l/r, and I/max by the

500-

1

H=6 Vr=lO tr = 50

----

'" Vo= 50 Vo = 4 0 0

4OO

..---- .-...- .-..- . . . . | ~ . ~ . ~ . . . , . . . , . + m . . ~ . ,

V

V max

,ool :i t~ I

? 2

~ ""

1

f

5

10

n

Fig. 9.

________

]Vmin 25

=

The effects of n, tr and Vo on extremes of biomass: maximal density, with residual.

amount of recovery growth that can take place, starting from II,, during the rest period. The effects of t, and n on the stable consumption rate, Ce, are inverted by starting from low 110 (Fig. 10). When Vo = 400 (in general, above 250) the conclusions of Experiments 1 and 5 remained valid: Ce (and hence animal productivity) was highest under continuous grazing, and decreased as the degree of subdivision was increased and/or the rotation period lengthened. However, starting from Vo = 50 the opposite was true: Ce was highest with fairly intensive subdivision and long rotation periods. In fact, with t, = 50 days and n = 5 or more the same stable consumption level was rapidly converged on from both high and low Vo. This level was about 40 % of satiation, i.e. much lower than steady consumption levels under continuous and moderate rotational grazing schemes starting from high Vo (70 % of satiation) but higher than the levels achieved by the latter schemes starting from low Vo (20-25 %).

7. The effects of rotation and Vo: overstocking, with residual In this experiment the average stocking rate was raised to B = 8 animals/1000 m 2, i.e. well above the maximum carrying capacity under continuous grazing. At this

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

105

density extinction would, of course, have occurred under both continuous and rotational grazing, in the absence of an ungrazeable plant residual. Combinations of t,, n and Vo were tested with a residual biomass of V, = 10 g/m z.

/

,,,r vo=4oo

t=lO

i:;;_._........ Ce

, ......_~

tr= 5 0

tr_.=5 0

5

tr=25

o

.........

1

2

-.--_

tr = 10

I

I

1

5

10

25

n

Fig. 10.

The effects of n, tr and V0 on steady consumption per unit area: maximal density with residual.

Under continuous grazing, biomass decreased rapidly from any initial level to a very low level just above V, (Fig. 11). In rotational schemes with short periods or few subplots fluctuations stabilised near a similar very low level. However, with long t, (50 days) and more intensive subdivision (n > 5), the rest period was long enough for the pasture to 'break away' from V, to a reasonably high Vmax (250350 g/m2). In no case in this experiment did initial conditions affect the eventual result. These results are reflected also in the steady consumption rates (Fig. 12). Under continuous or short-period rotational grazing, Ce is extremely low (2 g/m2/day = 8 % of satiation) representing the available growth which is constantly produced by V,, and almost immediately consumed. However, with long-period rotation (50 days) Ce increases with n to a level of about 7 g/m2/day, which is nearly attained already at n = 10 (rest period of 45 days). Thus, when the stocking rate is above the maximum carrying capacity but plant extinction is prevented by a residual, the only grazing scheme to maintain any

106

I. NOY-MEIR

5OO

= 8

V r = I0

continuous

Vo=50,400

1,oo

------

tr= 50, n=5

v

,' 200F

I

0

z

~

50

~

/

I00 n ~

150

200

Fig. 11. Time courses of biomass under continuous and intensive rotational grazing; excessive density, with residual.

I0 L-

R = 8

V r = I0

T Ca

tr: 50

5

+

i

I

I

I

2

5

I0

25

n

,-

Fig. 12. The effects of n and tr on steady consumption per unit area: excessive density with residual. reasonable level of animal consumption is long-period rotation with intensive subdivision. However, even then consumption ( = utilised pasture production) per unit area is only 60 ~ of what can be achieved at lower densities under continuous grazing. Consumption per animal is only about 25 ~ of maximum consumption, i.e. probably at a level below animal maintenance, which might cause animal mortality (two factors not considered in the model).

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

107

CONCLUSIONS AND DISCUSSION

Summary of results The results of the simulation experiments can be summarised as follows: (1) Rotational grazing with a small number of subdivisions (2-3) and a short rotation period (less than 10--20 days) does not differ much from continuous grazing in terms of average production. (2) When the average stocking rate is low or moderate, and/or when the initial condition of the pasture is good (biomass above critical threshold l/t), maximum stable production is attained by continuous grazing. With increasing subdivision and rotation period there is a decrease in average production, and an increase in the probability of a crash to extinction where no ungrazeable residual is present. (3) When the average stocking rate is high (near or above the maximum continuous carrying capacity) and/or when the initial condition of the pasture is poor, rotational grazing can increase average production in a discontinuously stable system, provided there is an ungrazeable plant residual. Then the average production increases with increasing subdivision and rotation period, tending asymptotically to a maximum (which is identical to that obtained from an initially good pasture under intensive rotation). These simulation experiments always considered only the pasture in one of the subplots in a rotational scheme, namely the first subplot to be grazed. The effect of rotational schemes on the average production over all subplots was not directly examined. However, the qualitative extension of the results to this effect is easy, since the successively grazed subplots in a rotational scheme differ only in the initial biomass at the beginning of the first grazing period. Therefore, the above results will be modified only in those cases when the final equilibrium in a subplot depends on the initial biomass. Consider, for example, a situation in which the stocking rate is low or moderate, the initial biomass is fairly low (below the optimum but above Vt) and there is no residual (e.g. Fig. 5). Then under rotational grazing of medium intensity (n and t, both not either very low or very high) the first grazed subplot(s) may deteriorate to extinction over a few cycles, while later grazed plots may settle to stable production. In any case the average production will be less than under continuous grazing. Several other cases where the steady-state production depends on the initial biomass can be seen in Fig. 10 (H = 6, V, = 10). There under rotational schemes of medium intensity (e.g. n = 2, t, = 50; n = 10, t, = 25) it is possible that the first grazed subplots in an initially poor pasture remain indefinitely on the low production curve, while later ones converge to the high curve. In these cases the average stable production of the whole pasture may be in the intermediate range, i.e. significantly lower than under continuous grazing starting

108

L NOY-MEIR

from high biomass, but significantly higher than under continuous grazing starting from low biomass.

Comparison with Morley's model The present results appear at first to conflict with those of Morley's (1968) simplified model of growth under rotational grazing. Morley found that production always increased with the length of the rest period ts = t,(n- 1)In up to an optimal length of 40-60 days; and that it also increased asymptotically with increasing degree of subdivision (at a given rest period), particularly from n = 2 to n = 5. There are two assumptions or simplifications in Morley's model which may explain this result. (1) The biomass at the beginning of each growth period, i.e. the end of the grazing period, Wo ( = Vmin here) is assumed to be a constant (in Brougham's curves Wo = A/(1 +B) = 30-70 lb/acre = 3-7 g/m 2) which is independent of the rotation scheme. In the present model (and probably in reality) this is not generally true (see Fig. 3), but is a good approximation when some conditions are fulfilled: (a) there is an ungrazeable residual, and (b) the stocking rate is high, and/or (c) the initial biomass is low. Then in each grazing period the pasture will be grazed down to very near the residual V,, under all grazing schemes (see Fig. 9). Indeed, if we restrict our attention to the results of those simulation experiments where these conditions are fulfilled (Experiment 6 with Vo = 50, Fig. 10; Experiment 7, Fig. 12) we find that production does increase with n between 2 and 5 and with tr up to 50 days (higher values not having been examined), just as in Morley's simplified model. The experiments in which the opposite result was obtained (production decreasing with n and tr) are those in which either the grazing pressure was only moderate, or an ungrazeable residual was absent, so that Morley's assumption of constant Wo (= Vmin) was inapplicable. (2) The process of consumption during the grazing period was not explicitly modelled by Morley, but its possible effect on biomass--and hence on growth rate-during that period was introduced by a reduction factor k. This k was allowed to vary between 0 (no growth during grazing period) and 1 (average growth rate during grazing period same as during rest period). The latter will occur if the relative distribution of biomass values over time is the same in the decreasing curve during grazing as in the increasing curve during rest. Morley found that the increase in production with increasing number of subdivisions n (at the optimal t~, i.e. with decreasing grazing period) was substantial only for low k(0; ~), at k = 1 production was independent ofn at given t~. In the present model, with its particular assumption on the consumption dynamics, the shapes of the increasing and decreasing curves did not turn out to be very dissimilar, so that k would not have been much lower

ROTATIONAL GRAZING IN A CONTINUOUSLY GROWING PASTURE

109

than 1. Trampling damage (mentioned by Morley as a factor which may lower k) was not considered as a separate process in the present model, although quantitatively it was included in consumption (Cm). Therefore, in those situations where an increase in production with increasing n (at constant t,) does appear here (Figs. 10 and 12), it seems to be related to the longer rest period ts = tr(1- I/n) rather than to the shorter grazing period. Thus the results of the explicit simulation model confirm the results of Morley's simple model for those situations where both models are equally applicable: high grazing pressure, ungrazeable residual present, not too severe reduction of growth rates during grazing period.

Validity of the model To what extent can the results of both models be accepted as a realistic and generally applicable answer to the question of rotational grazing? Obviously, direct a priori applicability is limited by the simplifying assumptions. There are several factors in real pasture systems which were not included in the models, and which might qualitatively affect the results of the comparison between continuous and rotational grazing. For instance: 1. Changes in plant species composition induced by the grazing scheme and affecting productivity (e.g. Morley et aL, 1969). 2. Changes in plant morphology induced by grazing and affecting availability (e.g. Smith et al., 1972). 3. Changes in leaf age distribution affecting growth rates or nutritive value. 4. Effects of dead plant material on current growth or availability. 5. Effects of grazing on growth not through leaf area, but through damage to growing points (McMeekan, 1960) or through reserve dynamics (the latter may be the reason for the consistent advantage of rotational grazing in lucerne pastures; Morley, 1967; Smith, 1970). 6. Factors related to animal health (Morley, 1967; Spedding, 1970). 7. Damage to pasture (other than consumption, e.g. trampling) which is nonlinearly related to animal density. 8. Seasonal changes in plant growth or animal requirements which may interact with the grazing scheme (e.g. Morley et al., 1969). The effect of each of these factors (and their interactions) on the continuous/ rotational question certainly deserves further study, including analysis of mathematical models (for a simple analytical approach to the last point, see also Morley, 1968). However, it seems reasonable to state that these are usually secondary effects in the plant-herbivore interaction, while the primary and dominant effect is through the leaf area/growth relation. Therefore models which realistically represent this primary interaction should at least be the best possible first approximation to real grazing systems and to the solution of practical problems such as the choice

110

i. NOY-MEIR

of the grazing scheme. Once this interaction, and its effect on the solution, are well understood, the other factors could gradually be brought into the model and their specific modifying effects on the results could be tested. The qualitative results of this first approximation model are consistent with much of the experimental evidence on rotational grazing. Heady (1961), in his fairly comprehensive review of experimental results in North America, noted that a favourable response of pasture condition and production to rotational grazing was usually observed only in initially poor, depleted pastures. In pastures in initially good condition rotational grazing either had no effect on production, or reduced it significantly compared with continuous grazing. Further field evidence from Australia (Morley, 1966a, 1967; Morley et al., 1969) also indicates that benefits of rotation tend to appear, if at all, only in pastures in a low-production stage (e.g. in winter). Then the rest-periods may, in some cases, allow the pasture in each sub-plot to 'extricate' itself from the low-production (low leaf area) state into a high-production state earlier than under continuous grazing. On the other hand, damages to production from rotation, if they occur, are most noticeable when the number of subdivisions and/or the length of the rotation period are high (Willoughby, 1959; Morley, 1967). As shown above, both results are explainable and predictable from a model considering only the effects of grazing on biomass (leaf area) and growth. Practical conclusions The simple leaf area-growth-consumption model used here is clearly deficient as a quantitative practical management tool at least as long as the effects on its predictions of several neglected factors (which certainly operate in real grazing systems) have not been examined. But its qualitative predictions seem already to be consistent enough with many observations on real systems to allow some tentative and rough practical guidelines. They should be accompanied by a cautionary note that they may be quite inapplicable in pastures where problems of species composition or of reserve dynamics are predominant. (1) Rotational grazing with few subdivisions (2-3) and short cycles (10-20 days) is unlikely to cause serious damage to production in any case, but it will never improve it much, either, over the production from continuous grazing. (2) Rotation with many subdivisions (more than 5) and/or long cycles (more than 30 days) will, in most pasture situations, reduce plant and animal production, in some situations seriously so. In this respect the mathematical growth-consumption model essentially confirms the verbal arguments by Willoughby (1958, 1959) and by Morley (1966b, 1967). (3) However, rotation with many subdivisions and/or long cycles may significantly increase production in pastures of poor condition (but resistant to being totally grazed out) or under high grazing pressure (McMeekan (1960) already arrived at a similar conclusion, from partly different considerations). This it does

ROTATIONALGRAZING IN A CONTINUOUSLYGROWING PASTURE

1 11

by enabling the pasture, which is trapped by continuous grazing in a low-biomass, low-production state, to j u m p (temporarily or permanently) to its potential higher productivity, during the rest periods. The higher grazing pressure during the grazing period cannot, in an already poor (but resistant) pasture, cause proportionally-higher damage any more. However, this means that at least during the first few grazing periods the animals will be grazing most of the time in extremely poor pasture, and their gain from this activity may be very low. It is possible that the same, or better, results for both pasture and animals, could be obtained by removing all or some of the animals from the whole pasture for a short period (usually a few weeks) until the critical biomass is reached, and then to graze continuously (Bishop & Kentish, 1966; Brown, 1970; Smith & Williams, 1973; Arnold & Bennett, 1974). Only where such deferment or spelling is not feasible, for economical or technical reasons, would the next best solution be to use rotational grazing until all subplots have passed the threshold biomass, Intensive rotational grazing (in a pasture with ungrazeable residual) allows some level of average production to be maintained also at supra-optimal animal densities which cause a 'crash' under continuous grazing. But then the production is much lower than that which can be obtained at near-optimal density by continuous grazing.

ACKNOWLEDGEMENTS This work was supported by Ford Foundation Grant 7/E-3 through the Israel Foundation Trustees. I am very grateful to Dr N o a m Seligman for his encouragement and his comments on the manuscript.

REFERENCES ALLDEN,W. G. (1962). Rate of herbage intake and grazing time in relation to herbage availability, Proc. Aust. Soc. Anim. Prod., 4, 163-6. ALLDEN, W. G. & WmTrAKER, I. A. Mc. D. (1970). The determinants of herbage intake by grazing sheep : The interrelationships of factors influencing herbage intake and availability, Aust. J. Agric. Res., 21, 755-66. ARNOLD,G. W. (1963). Factors within plant associations affecting the behaviour and performance of grazing animals. In Grazing in terrestrial and marine environments. (Ed. D. J. Crisp) (Br. Ecol. Soc. Symp. 4), 133-54. ARNOLD,G. W. & BENNETt,D. (1974). The problems of finding an optimum solution. Symposium on The Study of Agricultural Systems, University of Reading. BISHOP,A. H. & KENTISH,T. D. (1966). The effects of liveweight and wool production of autumn deferment with hay feeding, Proc. Aust. Soc. Anim. Prod., 6, 164-8. BROUGHAM,R. W. (1956). The rate of growth of short-rotation ryegrass pastures in the late autumn, winter and early spring, N . Z . J . Sci. & Technol., A38, 78-87. BROWN,T. H. (1970). The effect of autumn saving of pasture on wool production in a Mediterranean environment in southern Australia. Proc. X I Int. Grassland Congress, 869-73. CHRISTIAN,K. R., ARMSTRONG,J. S., DONNELLY,J. R., DAVIDSON,J. L. & FREER, M. (1972). Optimization of a grazing management system, Proc. Aust. Soc. Anim. Prod., 9, 124-9. FREER, i . , DAVIDSON,J. L., ARMSTRONG,J. S. & DONNELLY,J. R. (1970). Simulation of summer grazing, Proc. X I Int. Grassland Congress, 913-7.

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GOODALL,D. W. (1971). Extensive grazing systems. In Systems analysis in agricultural management (Eds J. B. Dent & J. R. Anderson), Sydney, Wiley. 173-87. HEADY, H. F. (1961). Continuous versus specialized grazing systems. A review and application to the California annual type, J. Range Manage. 14, 182-93. HOLLING,C. S. (1959). Some characteristics of simple types of predation and parasitism, Can. Entomol., 91, 385-98. JONES, J. G. W. & BROCrONGTON,N. R. (1971). Intensive grazing systems. In Systems analysis in agriculture management (Eds J. B. Dent & J. R. Anderson) Sydney, Wiley, 188-211. McMEEKAN, C. P. (1960). Grazing management, Proc. VIII Int. Grassland Congr., 21, Reading, 1960. MORLEY, F. H. W. (1966a). Stability and productivity of pastures, Proc. N.Z. Soc. Anita. Prod., 26, 8-21. MORLEY, F. H. W. (1966b). The biology of grazing management, Proc. Aust. Soc. Anim. Prod., 6, 127-36. MORLEY, F. H. W. (1967). Pasture and grazing management. In Pasture improvement in Australia (Ed. B. Wilson) CSIRO, 81-95. MORLEY, F. H. W. (1968). Pasture growth curves and grazing management, Austr. J. Exp. Agric. and Anita. Husbandry, 8, 40-5. MORLEY,F. I'I. W., BENNETt, D. and McKINNEY, G. T. (1969). The effect of intensity of rotational grazing with breeding ewes on phalaris-subterranean clover pastures, Austr. J. Exp. Agric. and Anita. Husbandry, 9, 74-84. NoY-MEIR, I. (1974). The use of models in the study of grazing systems, Proc. Israel-France Symposium on the study of arid mediterranean ecosystems, Bet Dagan, 65-77. NoY-MEIR, I. (1975). Stability of grazing systems: An application of predator-prey graphs, J. Ecol., 63 (in press). NoY-MEIR, I., Stability conditions in simple plant-herbivore models: The effect of explicit functions (in prep.). ROSENZWEIG,M. L. & MAcARTHUR, R. H. (1963). Graphical representation and stability conditions of predator-prey interactions, Amer. Natur., 97, 209-23. SHINOZAKI,K. & KIRA, T. (1956). Intraspecific competition among higher plants. VII. Logistic theory of the C-D effect, J. Inst. Polytechn. Osaka, Set. D., 7, 35-72. SMITH, M. V. (1970). Effects of stocking rate and grazing management on the persistence and production of dryland lucerne on deep sands, Proc. XI Int. Grassland Congress, 624-8. SMITH, R. C. G., Bn)DISCOt~mE,E. F. and STERN,W. R. (1972). Evaluation of five mediterranean annual pasture species during early growth, Aast. J. agric. Res., 23, 703-16. SMITH, R. C, G. • WILLIAMS,W. A. (1973). Model development for a deferred-grazing system, J. Range Manage., 26, 454-60. SPEDDtNO, C. R. W. (1970). Sheep production andgrazing management, 4th end. London, Bailli6re, Tindall & Cassell, 435 pp. WILLOUGHBY,W. M. (1958). A relationship between pasture availability and animal production, Proc. Aust. Soc. Anim. Prod., 2, 42-5. WmLOU~HBY, W. M. (1959). Limitations to animal production imposed by seasonal fluctuations in pasture and by management procedures, Aust. J. Agric. Res., 10, 248-68. WIT DE, C. T. (1960). On competition, Versl. Landb. Onderz., The Hague, 66, 8. WIT DE, C. T., BROUWER,R. & PEYr,aN~V~r.S DE, F. W. T. (1970). The simulation of photosynthetic systems. In Prediction and measurement of photosynthetic productivity. Proe. IBP/PP Techn. Meeting, Trebon. Pudoc, Wageningen, 47-70.