Dynamical Systems Modelling of the Interactions of Animal Stocking Density and Soil Fertility in Grazed Pasture

To examine the long term e ects of fertiliser application on pasture growth under grazing a mathematical representation of the pasture ecosystem is created and analysed mathe matically From this the nutrient application level needed to maintain a given stocking rate can be determined along with its pro tability Feasible stocking levels and fertiliser application rates are investigated and the optimal combination found along with the sensitivity of this combination It is shown that pro tability is relatively insensitive to fertiliser level compared with stocking rate


Introduction
A grazed pasture ecosystem consists of pasture, climate, soil and animals interacting through many di erent processes (Snaydon 1981). It is necessary to simplify this system in order to examine the dynamics between soil nutrient and pasture mass levels under grazing.
Previous models of nutrient cycling in soils have focused on the soil-plant response to fertiliser addition (e.g., Cornforth & Sinclair, 1984Metherell et. al., 1995 and have made over-simpli ed assumptions regarding the way that the plant-animal system responds to fertiliser input. Given the feedback b e t ween pasture mass and pasture growth rate, the seasonal nature of pasture growth and animal demand, and the seasonal pattern of pasture growth response to soil fertility, a more dynamic representation of pasture and animal production response to fertiliser use would be expected to better de ne optimum production levels. Soil fertility level and pasture growth rate are controlled by a n umber of factors. Some of these are controlled by the farmer (e.g., fertiliser input, stocking rate) while others are uncontrollable (e.g., rain fall) and may v ary seasonally (e.g., temperature). In this paper we will concentrate on two factors under farmer control, namely fertiliser rate and stocking rate. We will test the qualitative b e h a viour of the model as assessed by its predictions of long term system behaviour under con- Several assumptions are made. For the purposes of this model, phosphorus (P) is used as a typical fertiliser, and it is assumed that other nutrients are freely available. The grazing animals are assumed to be sheep (ewes). These choices will principally a ect the parameter values in the model, since the biological processes are qualitatively similar for other nutrients and stock classes. The model may b e used to model other fertiliser nutrients or livestock classes by altering the appropriate parameter values.
The dynamical systems approach attempts to assemble a model by constructing and linking mechanistic sub-models of the various important i n teractions at work in the system. These sub-models are based on data from controlled eld experiments that isolate the relationships. Having constructed the model, it is analysed and veri ed using dynamical systems techniques. As necessary those relationships which need to be examined more closely are identi ed, in order to stimulate further eld experiments in these areas.

The Model
This model describes only the dynamics of pasture biomass (Y kgDM ha ;1 ) and soil fertility ( F kgP ha ;1 ) a s t h e y i n teract over several years with constant a n n ual fertiliser application and stocking levels.

Pasture Dynamics
The daily net rate of pasture accumulation is given by the amount that the plants grow less the amount that is removed by grazing and senescence. Each of the process sub-models will be documented in detail.

Net Plant Growth
A response function has been formulated for net daily pasture accumulation (growth less senescence, neglecting the time delay i n volved in the latter) which is dependent only on photosynthesising leaf area and fertiliser availability but not on seasonality. This assumes that seasonal e ects are of secondary importance and annual average e ects dominate. Thus plant growth is dependent on soil nutrient and pasture biomass independently of one another. Mitscherlich equations are often used to describe yield-soil fertility relationships (Metherell et. al. 1995, Woodward 1996, in press a, in press b).
where the variables and parameters, along with their approximate values, are described in Table 1. If seasonal patterns of growth and response are to be incorporated, a, d and K may be made time dependent with a period of 365 days. However, signi cant progress c a n b e m a d e b y taking the annual average values for each of these parameters, as our initial interest is in long-term e ects rather than intra-annual dynamics. A seasonal formulation would require a numerical analysis.

Grazing
A linear assumption gives a good approximation for intake rate where pasture mass is low (500 -1800 kgDM ha ;1 ) ( B i r c ham 1981). While a Michaelis-Menten function may be more appropriate (Woodward, in press a), fertilised pastures will tend to carry high stocking levels so the pasture mass remains low. Thus we expect nonlinear e ects in removal of pasture mass to be minimal, and this turns out to be the case for the values of m (application rate) and n (stocking rate) considered in the threshold analysis. Therefore we can write: intake rate = nrY kgDM ha ;1 day ;1 : (2) 30 I D WICKHAM, G C WA K E , S J R W OODWARD, AND B S THORROLD Similarly as in equation (1) we assume that r and n are constant o ver time. Seasonal changes in management and animal physiological state will cause r and n to vary in practice, but the consideration of seasonality is a second step in the analysis of the model.
Together with the net growth response (equation (1)) this gives a di erential equation for the rate of change of pasture mass, Y :

Fertility Dynamics
The accumulation rate of plant a vailable nutrient in the soil is the quantity that the farmer applies less the amount that is lost through net transformation to unavailable forms, erosion or leaching, less the amount removed from the system by the animals. For a typical annual fertiliser application level, the proportion of nutrient in the pasture and animal is of the order of 1-2% of the total nutrient in the system (Nguyen and Goh 1992), so the nutrient cycle through plant, animal and back t o soil is not handled explicitly, but through a net loss process as described below (Metherell et. al., 1995).

Fertiliser Application
We assume that fertiliser application may b e a veraged over the year, so that m is the average daily application level. If more accurate predictions were required it would be necessary to introduce a function m(t) that took account of the fact that fertiliser is applied periodically (e.g. annually), and does not instantly enter the soil.

Soil Related L osses
Losses of P fertiliser can occur within the soil due to leaching, erosion or by transformation into non-available forms of P. In moderately fertilised grassland systems the major part of this loss appears to be due to transformation within the soil (Metherell et. al., 1995). Leaching of P and P losses in soil erosion are generally small (Lambert et. al., 1985). Following Metherell et. al., (1995) we assume the loss of soil P is a constant proportion (s) of the current soil P level: soil-loss rate = sF kgP ha ;1 day ;1 : FERTILISER STRATEGIES FOR GRAZING SYSTEMS 31

Animal Uptake Through Grazing
The P in pasture mass is consumed by the ewe and either converted to meat, wool and bone or returned to the pasture in dung. Conversion to meat, woolorboneisa loss from the soil P pool. A proportion of the dung is not returned to the pasture, but is transferred to stock camps and other non-productive a r e a s . In this model we combine these two animal related loss processes and assume that a constant proportion ( ) of P intake is lost from the soil. Research has shown that this proportion varies for stock t ype and farm system (Metherell et. al., 1995), but in our qualitative assessment o f m o d e l b e h a viour we can assume a constant.
The P concentration (kgP kg ;1 DM) of pasture is approximated by a Michaelis-Menten function. We already have the amount of pasture consumed by the animals (equation 2) so that we can determine the animal related loss rate as: animal-loss rate = fraction lost P in pasture pasture intake (5) = F c + F nrY (6) where the variables and parameters, along with their approximate values are given in Table 1. Together with the soil loss and application terms this gives a di erential equation for the net accumulation rate of fertiliser nutrient in the soil: dF dt = m ; sF ; F c + F nrY: (7)

Dynamical Systems Model
The dynamical systems model consists of these two di erential equations (equations (3) and (7)). Changes in one variable a ect the rate of change of the other, giving a coupled system of two equations: d Y dt = aF d + F Y ; Y 2 K ; nrY (8) dF dt = m ; sF ; F c + F nrY (9) where the variables and parameters are described in Table 1. These coupled equations capture the essence of the pasture/fertiliser system dynamics in a manner that allows direct analysis and also shows the interaction between the state variables explicitly.

Analysis and Results
A fourth-order Runge-Kutta implementation was used to generate solutions of the system from various initial conditions. The qualitative behaviour of the system depends greatly on the value of the parameters, and so stability analysis is required to fully understand the system dynamics and to determine stocking levels which may be supported by v arious fertiliser regimes, optimal stocking and fertiliser application rate. m (fertiliser application rate) and n (stocking rate) are treated as \control" parameters and we examine the behaviour of the system when these are varied. Mathematical solutions with negative n utrient or pasture biomass are ignored. For ease of understanding and interpretation in the biological context, we have analysed the equations in dimensional form. More complex extensions of this system may require us to nondimensionalise the equations, thus reducing the number of parameters (which in the present c a s e w ould reduce the number of parameters from ten to ve).

Stability and Steady States
The long-term tendencies of the system are important to the farmer, as they show the sustainable equilibrium states of the system. These steady state solutions, or equilibrium points, are the solutions to the di erential equations that do not change with time, and are found by equating both di erential equations to zero, i.e.: aF d + F Y ; Y 2 K ; nrY = 0 (10) m ; sF ; F c + F nrY = 0 : (11) The steady state solutions are points (Y F) that satisfy these equations. One such solution occurs at: 0 m s : This point corresponds to the extinction of the pasture and an equilibrium between fertiliser input and soil P related losses. This is the theoretical long-term result when stocking rate is too great to be supported on the pasture grown.
Another equilibrium point is found by substituting (rearranging equation (10)) into equation (11) and solving for F. The relevant value of F is the solution of a quadratic given by: where: A = as B = a(m ; cs) + K n r (nr ; a) and C = K n 2 r 2 d + acm and the corresponding value of Y is: The point ( Y F ) is an equilibrium point where stocking rate, pasture, and soil nutrient status are in balance. It lies in the physical region when Y 0, which i s the only region necessary to consider as the feasible region (Y F 0) is invariant.
This is assured when: (c + F )(m ; sF ) F nr 0: That is: n a m r(ds + m) = n : Therefore when n < n there are two equilibrium points in the physical region. At n = n the two points coincide (see section 3.2), and when n > n there is a single equilibrium at (0 m s ). The next step is to determine for which parameter values the steady state points are stable (attracting) as only stable steady states are observed in physically feasible systems. Figure 1 shows that in the absence of grazing (n = 0) the pasture reaches an equilibrium point at ceiling mass (4000 kgDM ha ;1 ). At this point F is at the level F = m s , given that there are no animal related losses. As n increases (at constant m) the equilibrium Y and F decline to a point where the equilibrium F is at a minimum. The value of F for a given m is the point a t w h i c h soil losses (sF ) a r e minimised, therefore this must be the point where animal P intake (hence P loss) is maximised. As n increases further the equilibrium value of F increases while the equilibrium value of Y continues to fall. This represents falling animal intake in t h e f a c e o f l o w pasture mass, with soil related P loss again increasing. Even at the minimum equilibrium F value, soil related loss comprises about 69% of total P loss. Given maximum annual P uptake of 55 kgP ha yr ;1 (and = 0 :3) and the high soil P levels generated by long term rates of m at 40 kgP ha ;1 yr ;1 these outcomes are consistent with eld observations and other models. Jacobian because one of the eigenvalues is zero. Using a computer to trace paths numerically indicates that the point will be stable in this case.
For small n there are two equilibrium points in the physical region, one of which is stable. As n becomes larger the two equilibrium points become closer and eventually become indistinct (see Figure 4). At t h a t v alue of n, the equilibrium point is stable. As n increases further, (Y F ) moves out of the physical region. Due to the direction of evolution along the boundary to the physical region paths can neither cross the F axis at all nor cross the Y axis in the direction of decreasing F. In addition the paths are bounded because if F > m s or Y > K then dF dt or dY dt respectively will be negative. Thus any path that enters or starts in the physical region must stay there.
Thus for a given pasture system and the xed values for a, r, d, and s that accompany it, there is an impassable limit to the amount o f s t o c k that can graze on the pasture if complete depletion is to be avoided. Irrespective of the fertiliser application rate m, the largest value of n is the ratio of a, the relative pasture growth rate, to r, the relative consumption rate. Any s t o c king level less than this can be maintained given su cient fertiliser application and the result will be a stable grazed pasture system with non-zero pasture biomass and hence intake.

Thresholds
The stable point and the system behaviour are determined by the values of the parameters. Changing n, the stocking rate, or m, the fertiliser application rate, result in the position of the stable point c hanging. Threshold values of these two control parameters are values where the system behaviour changes qualitatively .
For a given (m n) pair the steady state (Y F ) yields a certain daily herbage intake level. Of particular interest are the values of (m n) for which pasture becomes extinct, and hence intake i s zero and secondly, those at which intake i s at some  shown in Figure 5 for the parameter values in Table 1. I= 0 yields the simple   relationship: n a m r(ds + m) (17) which represents the boundary where pasture biomass becomes extinct. Beneath this threshold, equilibrium pasture biomass (and hence intake per animal in kgDM ewe ;1 day ;1 ) is positive and non-zero, whereas above this threshold equilibrium pasture biomass (and hence intake) is zero. The system attains the desirable steady state of non-zero pasture biomass and plant a vailable P only when the inequality i n equation (17) is satis ed. Further it implies that stocking rates of n = a r or greater can never be sustained on any fertiliser level. The second threshold on gure 5 is a dashed line indicating where rY, the intake per animal, is equal to some minimal survival level, say I= 0 :5 k g D M e w e ;1 day ;1 . This is a survival threshold above this stocking level it is expected that the animals will not be able to survive.

Economic Optimisation
A simple economic optimisation may be overlaid onto the system dynamics, and this illustrates the use of analytic models in economic analysis. By making some simplifying assumptions about animal production and given estimates of product value and fertiliser cost (C), we can calculate the values of n and m which produce an optimal pro t. We assume that income from sheep is derived only when their intake is greater than the intake ( I) needed for survival. We further assume that intake a b o ve m a i n tenance is linearly related to production of saleable product 40 I D WICKHAM, G C WA K E , S J R W OODWARD, AND B S THORROLD (meat, wool) and from this derive the revenue (R) per kgDM consumed above I. From these we can calculate the e ect of fertiliser rate and stocking rate on farm income net of fertiliser costs. This assumes no variable costs associated with stocking rate changes. It is assumed that an animal on a survival intake earns nothing, and an animal eating 550 kgDM yr ;1 earns $36:75. The gure of one stock unit eating 550 kgDM yr ;1 is taken from Coop (1965) and the value of $36:75 is typical of gross income for stock units on sheep and beef units in the northern North Island of New Zealand (MAF 1996). We i n terpolate linearly to give the total annual revenue per hectare as 365n(rY ; I )R where I = 0:5 kgDM ewe ;1 day ;1 is a typical survival intake and R = 0:10 $ kgDM ;1 is the ultimate value of the additional food eaten. Subtracting fertiliser costs, annual pro t is then AP = 365 ; n(rY ; I )R ; mC $ha ;1 yr ;1 : The parameters and variables are given in Table 2. With this we can compute pro t over the range of n and m. It transpires that there is a maximum value for AP over the (n m) plane. This point is marked in gure 5 and represents an optimal pro t of $877 ha ;1 yr ;1 , achieved when there is a stocking rate of n = 23 ewes ha ;1 and annual fertiliser input of m = 6 8 k g P h a ;1 yr ;1 . At this point, steady pasture mass is Y = 2348 kgDM ha ;1 and annual intake per ewe i s 642 kgDM.
These gures are somewhat high, re ecting the constant growth rate parameter, a = 0 :05 day ;1 , which implies a maximum rate of pasture grow t h o f 5 0 k g D M h a ;1 day ;1 when soil P is not limiting. The monetary values given for this calculation are typical for New Zealand conditions (with stock units equating to a 50 kg ewe) in recent y ears. These are just estimates to illustrate the procedure.
The contours of pro t in gure 5 suggest that AP is less sensitive to changes in fertiliser rate m than to changes in stocking rate n (over much of the parameter range), in line with conventional wisdom (McMeekan 1956). However, the shape of the contours has the potential to be in uenced by the relative prices of fertiliser and animal product, so that changes in the prices could change the importance of fertiliser rate compared to stocking rate. We note the even spacing of the stocking contours.

Discussion
The model presented in this paper has been constructed from component k n o wledge of biological and ecological interactions. This mathematical representation allows easy analysis of the coupled dynamics of the two state variables with a view to conducting steady state analysis and examining the thresholds.
Steady state analysis showed that the system exhibits a single stable steady state where a constant annual fertiliser application may sustain stocking rate, pasture yield and soil nutrient status inde nitely. However, when stocking level exceeds a critical value pasture tends to extinction, so that animal intake falls below biologically reasonable levels. In practice this equilibrium (0 m s ) is of little interest for two reasons. Firstly, in managed systems animals are either dead or removed before pasture mass collapses to zero. Secondly, unfertilised natural grasslands and longterm eld trials show t h a t e v en with no fertiliser input, a certain level of pasture yield is maintained for very long periods of time (> 50 years). This occurs through the slow release of P from naturally occurring soil minerals, which is not considered in this model.
Analysis of the parameter thresholds showed that increased fertiliser input is needed to maintain higher stock levels at the same level of feeding, and that there is a stocking level which c a n n o t b e m a i n tained regardless of the rate of fertiliser input. Mathematical analysis of this simple model has therefore provided insight i n to the sustainability of a grazing system in extreme conditions and at the same time gives a theoretical basis for stocking rate and fertiliser policies currently used by farmers.
At the optimum stocking rate, varying P inputs by 1 0 k g P h a ;1 has little impact on net pro t, and the loss from over-application is less than the loss from underapplication. This has two practical implications. A risk-averse farmer may tend to over apply fertiliser rather than risk under-application. However, if there is a further cost to over-application (e.g., pollution) then this may suggest lower application rates. Coupling this simple model with stochastic or seasonal production models and land use water quality models may produce some useful insights into farmer behaviour and the impacts of management c hoices.
This model illustrates the use of dynamical systems in exploring agricultural management. Considerable re nements of the model formulation are of course possible, but the simple model here captures the most important features of the interacting dynamics of pasture and fertility. More detailed models of soil-plant-animal systems would be expected to share the same qualitative behaviour as this simple model. This re ects the fact that this simple model reproduces the qualitative behavior observed in the eld. More detailed models may h o wever be useful in determining the e ect of seasonality, fertiliser timing and release rate, and animal management or optimum management strategies. It is our intention to further develop and apply this model to these issues.