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A conceptual approach to dynamic agricultural land-use modelling
PII:
Agricultural Sysrems, Vol. 57, No. 4, pp. 505m521, 1998 cc 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain SO308-521X(98)00005-5 0308-521X198 $19.00+0.00
ELSEVIER
A Conceptual Approach to Dynamic Agricultural Land-Use Modelling P. K. Thorntona* “International ‘International
& P. G. Jonesb
Livestock Research Institute, PO Box 30709, Nairobi, Kenya Center for Tropical Agriculture, AA 6713, Cali, Colombia
(Received
30 June 1997; accepted
30 November
1997)
ARSTRACT We describe a dynamic agricultural land-use model based on a Markov process and governed by a few simple decision rules. Currently, the model is pure1.y conceptual, and was designed with one objective: to investigate the possibility of constructing top-down land-use models based on as few processes as possible that might still be useful for statistical analyses of landuse change in a region. The model appears to behave in a plausible fashion in a simulated landscape with respect to cropping patterns and the effects of initial household size on wealth distribution. If this type of simple model could be validated, the information that could be produced might be of considerable value in a wide range of applications, particularly with regard to technology adoption patterns and resultant regional production impacts. 0 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION The relationship between the complexity of a model and its usefulness is governed by many factors. How accurately a model can simulate response in a particular real-world situation is only part of the question. Whether the model can actually be applied to provide information to those who need it at a particular time in a particular place, is another matter altogether. The current status of crop models affords an illustration of this point. Models from the Dutch school of de Wit (Bouman et al., 1996), from the USA (such *To whom correspondence
should be addressed.
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as the Decision Support System for Agrotechnology Transfer, Tsuji et al., 1998), and from Australia (such as APSIM, McCown et al., 1996), have been continuously developed over a period of at least 20 years. Achievements in scientific terms have been impressive. Models tend to develop and grow through accretion (Jones et al., 1995). This is not surprising; crop models are developing along the spectrum from simulation of potential yield through attainable yield (adding in considerations of water and nutrients) to actual yield (adding yield-reducing factors such as pests and diseases). Models that develop according to this classification of de Wit (Rabbinge and van Ittersum, 1994) become successively more and more applicable to the real world, by taking into account an increasing number of the factors that affect crop growth. By the same token, the models become increasingly complex; as more and more processes are included, input data requirements increase, and the ease with which the model can be used and the outputs interpreted, tends to decline. Alternatively, the models may become unbalanced, certain processes being simulated in more detail than others (Jones et al., 1995). The study of model detail, system complexity, research cost, and utility, is a neglected area of modelling inquiry. At what stage does a model (or even a series of models) cease to be useful? When can one say with near-certainty that a model is no good and that the associated research costs should be written off and the effort abandoned? When does a model become bloated with its own detail to the point where a much simpler model might be more appropriate and more useful? These are questions that are difficult to pose and even more difficult to answer in all honesty. There are many reasons for modelling, and there will always be a need for detailed, mechanistic models of biophysical processes. Agricultural activity can have such lasting impacts on the economic, socio-cultural, and ecological environment within which it is practised that the complexity of the system poses a substantial barrier to understanding, however. Mechanistic models of many of the important interactions may well lie many years in the future, but the bottom-up approach, where complexity is added to models to take account of higher levels in the system hierarchy, is not the only route that may be followed. That relevant processes drop in and out of the analysis as the investigator changes levels in the system hierarchy is a tenet of all systems-based inquiry. There is thus considerable merit in a top-down approach to the modelling of complex systems: start with as simple a set of rules and relationships as possible, and introduce complexity only as it is necessary to model the real world for a particular purpose. One advantage of such an approach is that the resultant models can be treated as working hypotheses and developed or abandoned very quickly. In this paper, we outline such a model of a complex system: agricultural land-use in a simulated landscape (we recognise that there are many other
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ways to use the land other than agriculture, and it may be that the model will be applicable to other types of land-use in the future). This is a development of a yet simpler model outlined by Thornton and Jones (1997). We offer some evidence that the model appears to behave plausibly with respect to changing patterns of agricultural land-use over time as driven by economic factors, and this plausibility is grounds for attempting to fit the model to a real-world landscape. Finally, we discuss the potential benefits of a validated model of this type.
AGRICULTURAL
LAND-USE
MODELLING
Agricultural land-use modelling has a long history, dating back to the agricultural location theory developed in the late 1700s. The last 10 years or so has seen burgeoning activity. The driving forces for this include the ubiquity of cheaper and more powerful computer hardware and software, and the coming to the fore of natural resource management issues in many different fields of inquiry. At the same time, resources for traditional research in agriculture have been declining in real terms, and yet the scope, applicability, and impact of research results are expected to increase substantially. The need for useful models is increasingly apparent. The importance of spatial arrangements and spatial relationships in many of the processes in agriculture is receiving increasing attention. A wide variety of methods has been developed that seek to have an impact on problem solving. Examples include spatial models of herbivory (reviewed by Coughenour, 1991), human population distribution models (Deichmann, 1996) and crop distribution models (Carter and Jones, 1993), fairly simple but potentially powerful agricultural location (semi-econometric) models to explain deforestation patterns (Chomitz and Gray, 1996), explanatory Markov rule-based models of land-use dynamics in a watershed (Stoorvogel, 1995), and a whole array of statistical and simulation models that contain some spatial components to study land-use and deforestation processes, each with particular strengths and weaknesses (reviewed by Lambin, 1994). A simple Markov-based agricultural land-use model A preliminary version of the basic model was described in Thornton and Jones (1997). This has been developed and expanded. Household preferences are treated in a different way, the model has been generalised to handle more agricultural land-use options, and initial household wealth and land-holding distributions can now be specified. A summary of the model as currently implemented follows.
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The landscape is made up of a set of plots of land, or facets, which may have any one of a number of agricultural land-uses. Households within the landscape operate a number of these facets. In any time period t, the household makes a choice concerning the land-use of all its facets during the season (time t). Choice is based on (1) expected gross margins of each competing land-use alternative, based on past performance, and (2) household preferences for each alternative that vary according to the household’s past experience and are sampled randomly. Expected gross margins of land-use alternatives
For each land-use, the expected gross margin for period t is calculated from:
where for the ith land-use on the jth facet, GA4 is the gross margin in arbitrary monetary units per hectare (muha-‘), Y is the quantity of product (t ha-‘), P the product price (mu t-i), I the input costs (mu ha-‘), and L the location costs (mu ha- l, described later). Yield or production, Y, for any land-use on any facet is defined to be a function of land-use in the previous time period and land quality,
Yt =f(Yt-1,
LQ)
A core feature of the model is a two-dimensional transition matrix of productivity coefficients by land quality by land-use in period t- 1. In the model, land quality is defined in terms of soil type and slope, as a one-dimensional index in the range ‘worse’ to ‘better’. The model thus needs to be initialised with production levels or yields at time t =O. The use of the first-order Markov model for production by facet is an attempt to take account of the dynamics of soil fertility through time. Thus, for a particular facet of a particular land quality index, growing low-input maize continuously will result in depletion of soil nutrient reserves and declining yields through time. Intermittent use of a legume on the same facet, however, will modify this yield decline, the appropriate matrix element for legume-followed-by-maize accounting for the residual effects of fixed nitrogen on the subsequent cereal crop. Product prices are inputs to the model, and are currently assumed to be constant through time and spatially invariant. The implementation of the model is such that it would be straightforward to relax these assumptions. Agricultural land-use options are specified by the user. Any landuse involving cropping can be associated with particular levels of inputs,
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corresponding to discrete amounts of seed, labour, and fertilizer use. A specified land-use may be associated with different sets of inputs and differing gross margins. As for product prices, the input costs are assumed to be invariant with respect to time and space. Location costs in the model primarily relate to the costs of transport of products and are made up of two components, the access cost and the distance-to-market cost. The access cost is a function of the distance, as the crow flies, from the centroid of the facet to the nearest segment of road network; each land-use can have its own access cost, at a particular cost per kilometre. The distance-to-market cost is a function of the distance from the nearest segment of road network to the nearest market in the landscape, travelling along the road network, at a certain cost per kilometre. Household preferences
The model tracks all households in the landscape through time, and cumulates and updates a proxy for wealth: Wt = Wt-1 + 2
GMjaj
(3)
j=l
where W is wealth in period t (mu), GM is the gross margin (mu ha-‘) of the jth plot of the household’s total of n plots in its chosen land-use, and a is its area (ha). The model assumes that each household has a certain predilection for utilising a facet in a particular land-use. This preference changes, depending on the household’s perception of its performance over time. An above-average gross margin will have a positive effect on the household’s preference for that particular land-use in that particular facet. In the model, these preferences are expressed as numbers between 0 and 1, and for manipulation they are transformed to normal probabilities. For choosing between land-uses in a particular time period, the current preferences are assembled into a cumulative distribution function and sampled randomly. A check is made to ensure that the household has enough wealth to purchase the necessary inputs. If not, then the choice is disallowed, and the process is repeated until the inputs for the land-use of choice can be afforded for that particular facet. (An obvious modification of the model for the future would allow the incorporation of credit and off-farm income.) In the example simulations described in this study, the agricultural land-use alternatives include fallow and pasture. For putting a facet into fallow, there are neither inputs nor direct economic returns (although there are indirect impacts, discussed later), so that the household can always afford fallow even if wealth is equal or close to zero.
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Implementing the model The model was implemented in FORTRAN, making calls to various components of IDRISI (Eastman, 1993). IDRISI is a flexible geographic information system made up of a large number of independent modules that can be called to perform various analytical procedures with spatial data. The program first generates a landscape consisting of a user-specified number of plots of land, n. It constructs facets using Dirichlet tessellations around the n randomly selected points and randomly generates a digital terrain model. The user specifies a road network and the location of the local market. The percent slope of each facet is calculated from the digital terrain model. The access distance (distance to the nearest road as the crow flies) and distance to market along the road network are calculated using the appropriate coverages and calls to various IDRISI routines. A soils coverage is also required, which again may be generated randomly or hand-digitised by the user (Table 1). To amalgamate facets into households, an empirical probability density function for the number of facets in each household is sampled randomly for each household. These facets may be contiguous (clustered around a ‘seed’ facet for each household), fragmented (located anywhere in the landscape), or a mixture of both. (These preliminary steps can be avoided by using a real-world landscape.) Each iteration of the model (Fig. 1) updates the database files that store the wealth and preference functions by household and the history file that records the evolution of land-use through time on each facet in the landscape. Initial conditions are read from a text file containing model inputs that can be changed as required (Table 2). Outputs such as land-use, production level, and realised gross margin can be mapped after each iteration. A summary file stores the total area and total production arising from each TABLE 1 Coverages Required or Derived in the Land-use Model Coverage
SOIL DEM LANDQUAL MARKETS ROADS FACETS HOUSEHOLD
Description
soils map of the landscape digital elevation model of the landscape land quality coverage, derived from the soils map and slope (from the DEM) market locations in the landscape road network; ROADS and MARKETS are used together to derive access and transport cost coverages in the landscape coverage with facet boundaries (a facet is a mapping unit with homogeneous land-use) coverage with household boundaries, made up of one or more facets
Dynamic agricultural land-use modelling
Recalculate
-1 I-
--I
Fig. 1. Outline
preference
functions
Update history file Update production file 1 of the land-use
model and file inputs and outputs.
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P. K. Thornton, P. G. Jones
512
TABLE 2
Data Stored in the Text Input Data File Section 1: land-use descriptors Six-character code for identification
and mapping
Costs and prices a Product price Fertiliser material costs Seed costs Labour costs Transport costs along road network Access costs to road network
CROP FERT SEED LABR TRANS ACCS
mu/t mu/kg mu/kg mu/h mu/km mu/km
Production levels Good land quality Medium land quality Poor land quality Ceiling production level
QUALl QUAL2 QUAW CEIL
t/ha t/ha t/ha t/ha
Inputs Fertiliser use Seed input Labour use
FERTI SEED1 LABRI
Household preferences Initial preference at t = 0 Rate of change of preference
PRED RATE
Transition coefficients Good land quality Medium land quality Poor land quality
kg/ha k/ha h/ha probability probit
QUALl 1 QUAL22 QUAL33
Section 2: household descriptors Initial facet distribution probability density function Fragmentation index (O-l) Initial wealth distribution probability density function a In arbitrary monetary units (mu).
land-use, by iteration. Alternatively, the model can be run for up to 30 iterations sequentially from a batch file. On a PC with a Pentium processor, one iteration of the model with 300 facets and 100 households takes about 1 s.
SOME SIMULATION
RESULTS
Evolution of land-use patterns through time Results from runs with a previous version of the model were reported in Thornton and Jones (1997). One aspect of interest was that particular
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conditions gave rise to what appeared to be crop rotations. Cropping on the poorer soil types in the landscape became intermittent, for 2 out of 3 years these facets being left fallow, allowing soil fertility to recover. Approximately every third year these facets were cropped, in effect the household ‘banking’ the soil fertility that had built up over the previous years. With the revised model, simulations were carried out using the inputs and initial conditions shown in Table 3 for a randomly generated landscape of 300 facets divided up into 106 households, with no fragmentation. The possible The Set of Standard
TABLE 3 Inputs Used to Run the Land-Use Model, Pasture, and Fallow Land-Use Options
Defining
Bean,
MZ
PS
FA
120.0 1.5 2.0 4.0 8.0 3.0
80.0 1.5 1.0 4.0 15.0 10.0
0.0 0.0 0.0 0.0 0.0 0.0
1.9 1.5 0.7 5.6
3.7 2.5 1.8 6.0
0.0 0.0 0.0 0.0
25.0 5.0 5.0
20.0 5.0 4.0
5.0 2.0 40.0
0.0 0.0
Preferences PRED RATE
0.5 0.2
0.5 0.2
0.5 0.2
0.5 0.0
Transition coefficients QUALll QUAL22 QUAL33
1.oo 0.90 0.75
0.93 0.85 0.70
1.04 1.06 1.04
1.12 1.16 1.13
Number of facets Probability
1 0.10
2 0.30
3 0.30
4 0.20
5 0.10
6 0.00
Fragmentation
0.000 2 0.30
3 0.30
4 0.20
5 0.10
6 0.00
Land-use
label
BN
costs CROP FERT SEED LABR TRANS ACCS
240.0 1.5 2.0 4.0 8.0 4.0
Yields QUALl QUALZ QUAL3 CEIL
1.:3 1.0 0.4 2.8
Inputs FERTI SEED1 LABRI
Maize,
o,o
Household
index
Initial wealth *5000 mu Probability
1 0.10
For a description fallow.
see Table 2. BN, bean; MZ, maize; PS, pasture;
and units of variables,
7 0.00
FA,
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P. K. Thornton. P. G. Jones
land-uses were, nominatively, a high-value bean crop, a lower-value maize crop (but higher yielding than beans), a pasture option, and a fallow option, with no operating costs whatsoever. For the pasture option, transport costs and labour inputs were set at relatively high (and arbitrary) levels; the objective was for this option to have some of the characteristics of highly labour-intensive, small-holder dairy systems, providing quickly perishable liquid milk for market. Evolution of land-use from initial conditions to the 20th iteration is summarised in Fig. 2. At t = 1, the landscape is essentially randomised, reflecting the equality in preference indices at the start of the simulation (Table 3). During later iterations, the preference indices become modified for each household, resulting in changes in the proportion of the landscape in different land-uses. The results in Fig. 2 suggest that a reasonably steady state is reached by Iteration 8 or so, at least in terms of an absence of underlying trends (although the between-iteration variability is still substantial, owing in part to highly dynamic soil fertility status). The area planted to maize decreased, probably owing to the nutrient-extractive nature of this land-use option as set up in Table 3, while the bean area increased (the crop has a higher value and is not so extractive as maize). The spatial arrangement of these agricultural land-uses is shown in Fig. 3 after 1 iteration and 20 iterations. For the pasture-dairy option, as might be expected, the heavy transport costs are penalising this land-use option for facets located away from the road network and the market (the market is
Pasture (Dairy)
Iteration,
t
Fig. 2. Changes in proportion of the landscape in various agricultural land-uses over 20 iterations of the land-use model, using the input data in Table 3.
Dynamic agricultural land-use modelling
515
located at the intersection of the roads, Fig. 3). To quantify the changes in landscape structure that occurred, we calculated four landscape metrics using the package FRAGSTATS (McGarigal and Marks, 1994, from which the verbal definitions given here and in the caption to Fig. 3 are taken): 1. Landscape patch density, defined as the total number of patches in the landscape (adjacent facets in the same agricultural land-use) per 100 ha. 2. A contagion or interspersion index, to measure dispersion or clumpiness of patches, the value of which is 100 when all patch types are equally adjacent to all other patch types (i.e. there is maximum interspersion of patches) and approaches 0 when the distribution of adjacencies among unique patch types becomes increasingly uneven. 3. Patch richness, which measures the number of patch types present in the landscape, and is unaffected by their relative abundance. 4. Simpson’s evenness index, which varies between 0, when the landscape contains only one patch (there is no diversity), and 1, when the distribution of area among patch types is perfectly even (the proportional abundances are the same). Values of these metrics are shown in Fig. 3 for each landscape. Patch density decreased substantially across iterations, while the contagion index increased. Thus, the landscape became less heterogeneous and patches tended to become more clumped through time. This is to be expected, as the landscape developed from the situation where each patch type (or agricultural land-use) was equally likely at time t = 1, to reflecting household preferences for different agricultural land-uses at time t = 20. Patch richness did not change (there are four patch types in both landscapes), but patch evenness decreased, indicating an increasing dominance of one patch type (fallow, as can be seen clearly from Fig. 3). In sum, 20 iterations of the model produced a landscape with fewer, more clumped patches of larger size, and the relative proportions of the various patch types changed considerably. The ability to quantify such landscape changes will be of considerable value in future validation of the model. Introducing a new land-use The effects of introducing a new land-use option into the landscape was simulated by adding in a fifth option that had the same characteristics as the bean crop (Table 3) except that the product price was increased by 20%. The rapidity with which households are simulated to try out the new crop in the model is dependent on the initial value of the preference index. Results are
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P. K. Thornton, P. G. Jones
MAIZE PASTURE FALLOW PLOT CENTROID
T=l
PD - 1.25 Cl = 29.00 PR-4 SEI = 1.00
T=ZO
PD - 0.83 Cl - 36.22 PR=I SEI = 0.93
Fig. 3. Simulated agricultural land-use in a generated landscape using the input data in Table 3 after 1 and 20 iterations of the model. Landscape metrics shown (definitions from McGarigal and Marks, 1994): PD is patch density (number of patches of the same contiguous agricultural land-use per 100 ha); CI is the contagion index, which equals minus the sum of the proportional abundance of each patch type and all other patch types, multiplied by the logarithm of the same quantity, summed over each patch type, divided by twice the logarithm of the number of patch types, multiplied by 100 to give a percentage; PR is the patch richness (number of different patch types present within the landscape); SE1 is Simpson’s evenness index (1 minus the sum, across all patch types, of the proportional abundance of each patch type squared, divided by 1 minus 1 divided by the number of patch types).
Dynamic agricultural land-use modelling
517
shown in Fig. 4 for two values, 0.02 and O-10, of the index for the fifth landuse option (units are probits or normal probabilities and relate to the step size by which the index is increased or decreased per iteration of the model). For the smaller-sized preference index, adoption, as measured by the cumulative proportion of households that tried the new land-use in any iteration period, started slowly; by Iteration 6, only 5% of total households had tried the new option at least once. The adoption rate increased in subsequent iterations, and by Iteration 30 was showing signs of abating. With a larger value of the initial preference index, however, adoption in the early iterations was much more rapid-this land-use option gives superior gross margins over the others, all other things being equal, so that the preference index is being quickly increased from one iteration to another. By Iteration 30, some 98% of households had tried the new option at least once. In aggregate terms, these types of responses appear reasonable. At the level of an individual household, however, uptake of a new option is likely to be based as much on availability of information and the influence of neighbours who have tried it out as on extension messages and field days. Further development of the model with respect to information transmission in a landscape originating from well-defined foci of adoption is one aspect that warrants serious attention. The ramifications for targeting the release of technology in particular circumstances could be profound.
Iteration,
t
Fig. 4. Household adoption curves in the landscape of Fig. 3 over 30 iterations, for two values of the initial preference index for a new land-use option consisting of the same data as for ‘Beans’ in Table 3 except with a product price increased by 20%.
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518
Changing the distribution of household size The simulation runs were carried out using a particular initial distribution of household size (in this context, the number of facets that belong to each household, Table 3). The distribution of wealth, calculated using eqn (3) changes markedly after a number of iterations of the model, tending towards a highly-skewed distribution where a minority of households holds a majority of the wealth. We investigated the possibility that this is related to the initial distribution of wealth in the community. We carried out four sets of runs with different initial wealth distributions (standard, uniform, identical, and right-hand skewed). Results are shown in Fig. 5. There seems to be little impact of initial distribution on final wealth distribution; the end result is a
(A)
I
I
Distribution
Facets per Househdd 1
2
3
4
5 0.1
StOltdWd
0.1
0.3
0.3
0.2
Ulififam
0.2
0.2
0.2
0.2
0.2
Singl*valu*
0.0
0.0
1.0
0.0
0.0
Ri~htM-handrkewd
0.0
0.1
0.1
0.3
0.6 ,
I
(B) ??Standard Uniform
cl
? ?Single-value ?? Right-hand skewed
2-4
4-6
6-8
WEALTH, 1 O5 mu Fig. 5. The impact
of (A) initial household size distribution on (B) simulated distribution after 30 iterations of the model.
wealth
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small proportion of well-endowed households owning most of the wealth in the landscape. The results of the standard run in Fig. 5 show that less than 22% of households own more than 50% of total household wealth, and this result is echoed for the other initial distributions. The reasons why a threeequation Markov land-use model run over 21 iterations should illustrate so graphically the development of familiar inequalities in wealth distribution are worth elucidating.
VALIDATION
AND APPLICATION
We are starting to apply this land-use model to real situations, to assess what kind of explanatory power it might have. Validation of top-down models is no less difficult than for mechanistic models, simply because most if not all of the processes are highly integrative and thus subject to considerable variation. In general, we cannot run controlled experiments with which to test the model, so we have to make do with historical data. Most of the processes should be comparatively straightforward to test in a real landscape. In terms of generating input data for the model, crop simulation models could be used to derive the transition matrix for particular land quality conditions. The household preference functions pose major problems, because they cannot easily be derived directly but have to be induced from observed behaviour. Even fairly rudimentary testing would allow many of the revealed characteristics of the model to be assessed. Until this testing is carried out, we are continuing to develop the model. The next stages are to include livestock more explicitly in terms of land-use options and their impacts on household wealth, and to include the options of transferring ownership of facets, either through inheritance (which would allow experimentation concerning land-use and wealth distributions as a function of different inheritance and landtenurial factors) or through trading in facets, land prices being a function of land quality and distance to markets, for example. A certain amount of further model development would open the way to a number of interesting and potentially useful applications of the model with regard to real landscapes. One of these would be to interpret possible ecological consequences of changes in input conditions on the landscape. This could lead to the derivation of some simple indices or measures of potential ecological impact of technological and economic change on agricultural land-use, which could be of real value in a range of impact assessment studies. Another application would be to use the model to derive supply curves. For a region with a wide array of land-use options, this would involve
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steadily dropping the product price and observing changes in aggregate production. There is scope for a contribution to the debate on the form of the supply curve, especially at relatively low prices (Pachico et al., 1987). As Elbasha (1997) shows clearly, despite much conventional wisdom on the subject, the shape of the supply curve can have an enormous impact on any calculation of consumer and producer benefit arising from adoption of new technology. Parallel shifts in a linear supply curve as a result of technological change may make for algebraic simplicity, but they may not be very realistic. As already noted, another application of potential value would be the testing of hypotheses concerning the spatial spread of information in a community concerning new technology or management options. The study of the dynamics of diffusion in a landscape could have substantial impacts on the targeting and delivery of technology.
CONCLUSIONS The merits of taking a simple (even a simplistic) approach to the modelling of highly complex systems consist of the following: (1) the approach forces the analyst to define what are really the overriding factors that bring about change in the systems under study and (2) the approach minimises the modelling overhead in such a way that the effort can be abandoned very quickly if need be, with minimal waste of resources. The potential strength of these merits will be unrealised in the absence of real-world testing of the model. This is also required if it is to be demonstrated that useful information (in the sense of its being of value to a decision maker) can come from such simple models. The performance of this land-use model suggests that it is worthwhile to pursue its validation with respect to some real landscapes where sufficient data exist to enable an attempt to explain variation in land-use over a number of years. It may be that the approach is too simplistic to produce any useful information or insights. On the other hand, the construction and validation of simple models that explain useful proportions of the variation observed would be a contribution of some value towards an understanding of the functioning of complex systems.
ACKNOWLEDGEMENTS Without implicating her in any way, we would like to thank Robin Reid for very useful comments on an earlier draft of this paper.
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REFERENCES
Bouman, B. A. M., van Keulen, H., van Laar, H. H. and Rabbinge, R. (1996) The ‘school of de Wit’ crop growth simulation models: a pedigree and historical overview. Agricultural Systems 52, 171-198. Carter, S. E. and Jones, P. G. (1993) A model of the distribution of cassava in Africa. Applied Geography 13, 353-371. Chomitz, K. M. and Gray, D. A. (1996) Roads, land use, and deforestation: a spatial model applied to Belize. World Bank Economic Review 10, 487-512. Coughenour, M. B. (1991) Spatial components of plant-herbivore interactions in pastoral, ranching, and native ungulate ecosystems. Journal of Range Management 44, 530-542.
Deichmann, U. (1996) Population data for Asia: Asia population database documentation, a review of spatial database modelling and design. NCGIA, University of California, Santa Barbara, California, USA. Eastman, J. R. (1993) IDRISI version 4.1 technical reference manual. Clark University, Worcester, MA. Elbasha, E. H. (1997) The calculation of research benefits with flexible functional forms for supply and demand. Selected paper, American Agricultural Economics Association Meetings, Toronto, Canada, July 1997. Jones, P. G., Thornton, P. K. and Hill, P. (1995) Agro-meteorological models: crop growth and stress indices. In Proceedings, EU/FAO Expert Consultation on Crop Yield Forecasting Methods, Villefranche-sur-Mer, 2427 October 1994, pp. 5366. FAO, Rome. Lambin, E.F. (1994) Modelling deforestation processes: a review. TREES Series B: Research Report No. 1, Ispra, Italy. McCown, R. L., Hammer, G. L., Hargreaves, J. N. G., Holzworth, D. P. and Freebairn, D. M. (1996) APSIM: a novel software system for model development, model testing and simulation in agricultural systems research. Agricultural Systems 50, 255-27 1. McGarigal, K. and Marks, B. J. (1994) FRAGSTATS: spatial analysis program for quantifying landscape structure. Version 2.0, March 1994. Forest Science Department, Oregon State University, Corvallis, Oregon, USA. Pachico, D., Lynam, J. K. and Jones, P. G. (1987) The distribution of benefits from technical change among classes of consumers and producers: An ex ante analysis of beans in Brazil. Research Policy 16, 279-285. Rabbinge, R. and van Ittersum, M. K. (1994) Tension between aggregation levels. In The Future of the Land: Mobilising and Integrating Knowledge for Land Use Options, eds L. 0. Fresco, L. Stroosnijder, J. Bouma and H. van Keulen. pp. 31-40. Wiley, Chichester, UK. Stoorvogel, J. J. (1995) Geographical information systems as a tool to explore land characteristics and land use, with reference to Costa Rica. PhD thesis, Wageningen Agricultural University, Wageningen, The Netherlands. Thornton, P. K. and Jones, P. G. (1997) Towards a conceptual dynamic land use model. In Applications of Systems Approaches at the Farm and Regional Levels, eds P. S. Teng, M. J. Kropff, H. F. M. ten Berge, J. B. Dent, F. P. Lansigan and H. H. van Laar, Vol. 1, pp. 341-356. Kluwer Academic Publishers, Dordrecht, The Netherlands. Tsuji, G. Y., Hoogenboom, G. and Thornton, P. K. (eds) (1998) Understanding Options for Agricultural Production, pp. 399. Kluwer Academic Publishers, Dordrecht, The Netherlands.
Agricultural Sysrems, Vol. 57, No. 4, pp. 505m521, 1998 cc 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain SO308-521X(98)00005-5 0308-521X198 $19.00+0.00
ELSEVIER
A Conceptual Approach to Dynamic Agricultural Land-Use Modelling P. K. Thorntona* “International ‘International
& P. G. Jonesb
Livestock Research Institute, PO Box 30709, Nairobi, Kenya Center for Tropical Agriculture, AA 6713, Cali, Colombia
(Received
30 June 1997; accepted
30 November
1997)
ARSTRACT We describe a dynamic agricultural land-use model based on a Markov process and governed by a few simple decision rules. Currently, the model is pure1.y conceptual, and was designed with one objective: to investigate the possibility of constructing top-down land-use models based on as few processes as possible that might still be useful for statistical analyses of landuse change in a region. The model appears to behave in a plausible fashion in a simulated landscape with respect to cropping patterns and the effects of initial household size on wealth distribution. If this type of simple model could be validated, the information that could be produced might be of considerable value in a wide range of applications, particularly with regard to technology adoption patterns and resultant regional production impacts. 0 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION The relationship between the complexity of a model and its usefulness is governed by many factors. How accurately a model can simulate response in a particular real-world situation is only part of the question. Whether the model can actually be applied to provide information to those who need it at a particular time in a particular place, is another matter altogether. The current status of crop models affords an illustration of this point. Models from the Dutch school of de Wit (Bouman et al., 1996), from the USA (such *To whom correspondence
should be addressed.
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as the Decision Support System for Agrotechnology Transfer, Tsuji et al., 1998), and from Australia (such as APSIM, McCown et al., 1996), have been continuously developed over a period of at least 20 years. Achievements in scientific terms have been impressive. Models tend to develop and grow through accretion (Jones et al., 1995). This is not surprising; crop models are developing along the spectrum from simulation of potential yield through attainable yield (adding in considerations of water and nutrients) to actual yield (adding yield-reducing factors such as pests and diseases). Models that develop according to this classification of de Wit (Rabbinge and van Ittersum, 1994) become successively more and more applicable to the real world, by taking into account an increasing number of the factors that affect crop growth. By the same token, the models become increasingly complex; as more and more processes are included, input data requirements increase, and the ease with which the model can be used and the outputs interpreted, tends to decline. Alternatively, the models may become unbalanced, certain processes being simulated in more detail than others (Jones et al., 1995). The study of model detail, system complexity, research cost, and utility, is a neglected area of modelling inquiry. At what stage does a model (or even a series of models) cease to be useful? When can one say with near-certainty that a model is no good and that the associated research costs should be written off and the effort abandoned? When does a model become bloated with its own detail to the point where a much simpler model might be more appropriate and more useful? These are questions that are difficult to pose and even more difficult to answer in all honesty. There are many reasons for modelling, and there will always be a need for detailed, mechanistic models of biophysical processes. Agricultural activity can have such lasting impacts on the economic, socio-cultural, and ecological environment within which it is practised that the complexity of the system poses a substantial barrier to understanding, however. Mechanistic models of many of the important interactions may well lie many years in the future, but the bottom-up approach, where complexity is added to models to take account of higher levels in the system hierarchy, is not the only route that may be followed. That relevant processes drop in and out of the analysis as the investigator changes levels in the system hierarchy is a tenet of all systems-based inquiry. There is thus considerable merit in a top-down approach to the modelling of complex systems: start with as simple a set of rules and relationships as possible, and introduce complexity only as it is necessary to model the real world for a particular purpose. One advantage of such an approach is that the resultant models can be treated as working hypotheses and developed or abandoned very quickly. In this paper, we outline such a model of a complex system: agricultural land-use in a simulated landscape (we recognise that there are many other
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ways to use the land other than agriculture, and it may be that the model will be applicable to other types of land-use in the future). This is a development of a yet simpler model outlined by Thornton and Jones (1997). We offer some evidence that the model appears to behave plausibly with respect to changing patterns of agricultural land-use over time as driven by economic factors, and this plausibility is grounds for attempting to fit the model to a real-world landscape. Finally, we discuss the potential benefits of a validated model of this type.
AGRICULTURAL
LAND-USE
MODELLING
Agricultural land-use modelling has a long history, dating back to the agricultural location theory developed in the late 1700s. The last 10 years or so has seen burgeoning activity. The driving forces for this include the ubiquity of cheaper and more powerful computer hardware and software, and the coming to the fore of natural resource management issues in many different fields of inquiry. At the same time, resources for traditional research in agriculture have been declining in real terms, and yet the scope, applicability, and impact of research results are expected to increase substantially. The need for useful models is increasingly apparent. The importance of spatial arrangements and spatial relationships in many of the processes in agriculture is receiving increasing attention. A wide variety of methods has been developed that seek to have an impact on problem solving. Examples include spatial models of herbivory (reviewed by Coughenour, 1991), human population distribution models (Deichmann, 1996) and crop distribution models (Carter and Jones, 1993), fairly simple but potentially powerful agricultural location (semi-econometric) models to explain deforestation patterns (Chomitz and Gray, 1996), explanatory Markov rule-based models of land-use dynamics in a watershed (Stoorvogel, 1995), and a whole array of statistical and simulation models that contain some spatial components to study land-use and deforestation processes, each with particular strengths and weaknesses (reviewed by Lambin, 1994). A simple Markov-based agricultural land-use model A preliminary version of the basic model was described in Thornton and Jones (1997). This has been developed and expanded. Household preferences are treated in a different way, the model has been generalised to handle more agricultural land-use options, and initial household wealth and land-holding distributions can now be specified. A summary of the model as currently implemented follows.
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The landscape is made up of a set of plots of land, or facets, which may have any one of a number of agricultural land-uses. Households within the landscape operate a number of these facets. In any time period t, the household makes a choice concerning the land-use of all its facets during the season (time t). Choice is based on (1) expected gross margins of each competing land-use alternative, based on past performance, and (2) household preferences for each alternative that vary according to the household’s past experience and are sampled randomly. Expected gross margins of land-use alternatives
For each land-use, the expected gross margin for period t is calculated from:
where for the ith land-use on the jth facet, GA4 is the gross margin in arbitrary monetary units per hectare (muha-‘), Y is the quantity of product (t ha-‘), P the product price (mu t-i), I the input costs (mu ha-‘), and L the location costs (mu ha- l, described later). Yield or production, Y, for any land-use on any facet is defined to be a function of land-use in the previous time period and land quality,
Yt =f(Yt-1,
LQ)
A core feature of the model is a two-dimensional transition matrix of productivity coefficients by land quality by land-use in period t- 1. In the model, land quality is defined in terms of soil type and slope, as a one-dimensional index in the range ‘worse’ to ‘better’. The model thus needs to be initialised with production levels or yields at time t =O. The use of the first-order Markov model for production by facet is an attempt to take account of the dynamics of soil fertility through time. Thus, for a particular facet of a particular land quality index, growing low-input maize continuously will result in depletion of soil nutrient reserves and declining yields through time. Intermittent use of a legume on the same facet, however, will modify this yield decline, the appropriate matrix element for legume-followed-by-maize accounting for the residual effects of fixed nitrogen on the subsequent cereal crop. Product prices are inputs to the model, and are currently assumed to be constant through time and spatially invariant. The implementation of the model is such that it would be straightforward to relax these assumptions. Agricultural land-use options are specified by the user. Any landuse involving cropping can be associated with particular levels of inputs,
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corresponding to discrete amounts of seed, labour, and fertilizer use. A specified land-use may be associated with different sets of inputs and differing gross margins. As for product prices, the input costs are assumed to be invariant with respect to time and space. Location costs in the model primarily relate to the costs of transport of products and are made up of two components, the access cost and the distance-to-market cost. The access cost is a function of the distance, as the crow flies, from the centroid of the facet to the nearest segment of road network; each land-use can have its own access cost, at a particular cost per kilometre. The distance-to-market cost is a function of the distance from the nearest segment of road network to the nearest market in the landscape, travelling along the road network, at a certain cost per kilometre. Household preferences
The model tracks all households in the landscape through time, and cumulates and updates a proxy for wealth: Wt = Wt-1 + 2
GMjaj
(3)
j=l
where W is wealth in period t (mu), GM is the gross margin (mu ha-‘) of the jth plot of the household’s total of n plots in its chosen land-use, and a is its area (ha). The model assumes that each household has a certain predilection for utilising a facet in a particular land-use. This preference changes, depending on the household’s perception of its performance over time. An above-average gross margin will have a positive effect on the household’s preference for that particular land-use in that particular facet. In the model, these preferences are expressed as numbers between 0 and 1, and for manipulation they are transformed to normal probabilities. For choosing between land-uses in a particular time period, the current preferences are assembled into a cumulative distribution function and sampled randomly. A check is made to ensure that the household has enough wealth to purchase the necessary inputs. If not, then the choice is disallowed, and the process is repeated until the inputs for the land-use of choice can be afforded for that particular facet. (An obvious modification of the model for the future would allow the incorporation of credit and off-farm income.) In the example simulations described in this study, the agricultural land-use alternatives include fallow and pasture. For putting a facet into fallow, there are neither inputs nor direct economic returns (although there are indirect impacts, discussed later), so that the household can always afford fallow even if wealth is equal or close to zero.
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Implementing the model The model was implemented in FORTRAN, making calls to various components of IDRISI (Eastman, 1993). IDRISI is a flexible geographic information system made up of a large number of independent modules that can be called to perform various analytical procedures with spatial data. The program first generates a landscape consisting of a user-specified number of plots of land, n. It constructs facets using Dirichlet tessellations around the n randomly selected points and randomly generates a digital terrain model. The user specifies a road network and the location of the local market. The percent slope of each facet is calculated from the digital terrain model. The access distance (distance to the nearest road as the crow flies) and distance to market along the road network are calculated using the appropriate coverages and calls to various IDRISI routines. A soils coverage is also required, which again may be generated randomly or hand-digitised by the user (Table 1). To amalgamate facets into households, an empirical probability density function for the number of facets in each household is sampled randomly for each household. These facets may be contiguous (clustered around a ‘seed’ facet for each household), fragmented (located anywhere in the landscape), or a mixture of both. (These preliminary steps can be avoided by using a real-world landscape.) Each iteration of the model (Fig. 1) updates the database files that store the wealth and preference functions by household and the history file that records the evolution of land-use through time on each facet in the landscape. Initial conditions are read from a text file containing model inputs that can be changed as required (Table 2). Outputs such as land-use, production level, and realised gross margin can be mapped after each iteration. A summary file stores the total area and total production arising from each TABLE 1 Coverages Required or Derived in the Land-use Model Coverage
SOIL DEM LANDQUAL MARKETS ROADS FACETS HOUSEHOLD
Description
soils map of the landscape digital elevation model of the landscape land quality coverage, derived from the soils map and slope (from the DEM) market locations in the landscape road network; ROADS and MARKETS are used together to derive access and transport cost coverages in the landscape coverage with facet boundaries (a facet is a mapping unit with homogeneous land-use) coverage with household boundaries, made up of one or more facets
Dynamic agricultural land-use modelling
Recalculate
-1 I-
--I
Fig. 1. Outline
preference
functions
Update history file Update production file 1 of the land-use
model and file inputs and outputs.
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512
TABLE 2
Data Stored in the Text Input Data File Section 1: land-use descriptors Six-character code for identification
and mapping
Costs and prices a Product price Fertiliser material costs Seed costs Labour costs Transport costs along road network Access costs to road network
CROP FERT SEED LABR TRANS ACCS
mu/t mu/kg mu/kg mu/h mu/km mu/km
Production levels Good land quality Medium land quality Poor land quality Ceiling production level
QUALl QUAL2 QUAW CEIL
t/ha t/ha t/ha t/ha
Inputs Fertiliser use Seed input Labour use
FERTI SEED1 LABRI
Household preferences Initial preference at t = 0 Rate of change of preference
PRED RATE
Transition coefficients Good land quality Medium land quality Poor land quality
kg/ha k/ha h/ha probability probit
QUALl 1 QUAL22 QUAL33
Section 2: household descriptors Initial facet distribution probability density function Fragmentation index (O-l) Initial wealth distribution probability density function a In arbitrary monetary units (mu).
land-use, by iteration. Alternatively, the model can be run for up to 30 iterations sequentially from a batch file. On a PC with a Pentium processor, one iteration of the model with 300 facets and 100 households takes about 1 s.
SOME SIMULATION
RESULTS
Evolution of land-use patterns through time Results from runs with a previous version of the model were reported in Thornton and Jones (1997). One aspect of interest was that particular
513
Dynamic agricultural land-use modelling
conditions gave rise to what appeared to be crop rotations. Cropping on the poorer soil types in the landscape became intermittent, for 2 out of 3 years these facets being left fallow, allowing soil fertility to recover. Approximately every third year these facets were cropped, in effect the household ‘banking’ the soil fertility that had built up over the previous years. With the revised model, simulations were carried out using the inputs and initial conditions shown in Table 3 for a randomly generated landscape of 300 facets divided up into 106 households, with no fragmentation. The possible The Set of Standard
TABLE 3 Inputs Used to Run the Land-Use Model, Pasture, and Fallow Land-Use Options
Defining
Bean,
MZ
PS
FA
120.0 1.5 2.0 4.0 8.0 3.0
80.0 1.5 1.0 4.0 15.0 10.0
0.0 0.0 0.0 0.0 0.0 0.0
1.9 1.5 0.7 5.6
3.7 2.5 1.8 6.0
0.0 0.0 0.0 0.0
25.0 5.0 5.0
20.0 5.0 4.0
5.0 2.0 40.0
0.0 0.0
Preferences PRED RATE
0.5 0.2
0.5 0.2
0.5 0.2
0.5 0.0
Transition coefficients QUALll QUAL22 QUAL33
1.oo 0.90 0.75
0.93 0.85 0.70
1.04 1.06 1.04
1.12 1.16 1.13
Number of facets Probability
1 0.10
2 0.30
3 0.30
4 0.20
5 0.10
6 0.00
Fragmentation
0.000 2 0.30
3 0.30
4 0.20
5 0.10
6 0.00
Land-use
label
BN
costs CROP FERT SEED LABR TRANS ACCS
240.0 1.5 2.0 4.0 8.0 4.0
Yields QUALl QUALZ QUAL3 CEIL
1.:3 1.0 0.4 2.8
Inputs FERTI SEED1 LABRI
Maize,
o,o
Household
index
Initial wealth *5000 mu Probability
1 0.10
For a description fallow.
see Table 2. BN, bean; MZ, maize; PS, pasture;
and units of variables,
7 0.00
FA,
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P. K. Thornton. P. G. Jones
land-uses were, nominatively, a high-value bean crop, a lower-value maize crop (but higher yielding than beans), a pasture option, and a fallow option, with no operating costs whatsoever. For the pasture option, transport costs and labour inputs were set at relatively high (and arbitrary) levels; the objective was for this option to have some of the characteristics of highly labour-intensive, small-holder dairy systems, providing quickly perishable liquid milk for market. Evolution of land-use from initial conditions to the 20th iteration is summarised in Fig. 2. At t = 1, the landscape is essentially randomised, reflecting the equality in preference indices at the start of the simulation (Table 3). During later iterations, the preference indices become modified for each household, resulting in changes in the proportion of the landscape in different land-uses. The results in Fig. 2 suggest that a reasonably steady state is reached by Iteration 8 or so, at least in terms of an absence of underlying trends (although the between-iteration variability is still substantial, owing in part to highly dynamic soil fertility status). The area planted to maize decreased, probably owing to the nutrient-extractive nature of this land-use option as set up in Table 3, while the bean area increased (the crop has a higher value and is not so extractive as maize). The spatial arrangement of these agricultural land-uses is shown in Fig. 3 after 1 iteration and 20 iterations. For the pasture-dairy option, as might be expected, the heavy transport costs are penalising this land-use option for facets located away from the road network and the market (the market is
Pasture (Dairy)
Iteration,
t
Fig. 2. Changes in proportion of the landscape in various agricultural land-uses over 20 iterations of the land-use model, using the input data in Table 3.
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515
located at the intersection of the roads, Fig. 3). To quantify the changes in landscape structure that occurred, we calculated four landscape metrics using the package FRAGSTATS (McGarigal and Marks, 1994, from which the verbal definitions given here and in the caption to Fig. 3 are taken): 1. Landscape patch density, defined as the total number of patches in the landscape (adjacent facets in the same agricultural land-use) per 100 ha. 2. A contagion or interspersion index, to measure dispersion or clumpiness of patches, the value of which is 100 when all patch types are equally adjacent to all other patch types (i.e. there is maximum interspersion of patches) and approaches 0 when the distribution of adjacencies among unique patch types becomes increasingly uneven. 3. Patch richness, which measures the number of patch types present in the landscape, and is unaffected by their relative abundance. 4. Simpson’s evenness index, which varies between 0, when the landscape contains only one patch (there is no diversity), and 1, when the distribution of area among patch types is perfectly even (the proportional abundances are the same). Values of these metrics are shown in Fig. 3 for each landscape. Patch density decreased substantially across iterations, while the contagion index increased. Thus, the landscape became less heterogeneous and patches tended to become more clumped through time. This is to be expected, as the landscape developed from the situation where each patch type (or agricultural land-use) was equally likely at time t = 1, to reflecting household preferences for different agricultural land-uses at time t = 20. Patch richness did not change (there are four patch types in both landscapes), but patch evenness decreased, indicating an increasing dominance of one patch type (fallow, as can be seen clearly from Fig. 3). In sum, 20 iterations of the model produced a landscape with fewer, more clumped patches of larger size, and the relative proportions of the various patch types changed considerably. The ability to quantify such landscape changes will be of considerable value in future validation of the model. Introducing a new land-use The effects of introducing a new land-use option into the landscape was simulated by adding in a fifth option that had the same characteristics as the bean crop (Table 3) except that the product price was increased by 20%. The rapidity with which households are simulated to try out the new crop in the model is dependent on the initial value of the preference index. Results are
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P. K. Thornton, P. G. Jones
MAIZE PASTURE FALLOW PLOT CENTROID
T=l
PD - 1.25 Cl = 29.00 PR-4 SEI = 1.00
T=ZO
PD - 0.83 Cl - 36.22 PR=I SEI = 0.93
Fig. 3. Simulated agricultural land-use in a generated landscape using the input data in Table 3 after 1 and 20 iterations of the model. Landscape metrics shown (definitions from McGarigal and Marks, 1994): PD is patch density (number of patches of the same contiguous agricultural land-use per 100 ha); CI is the contagion index, which equals minus the sum of the proportional abundance of each patch type and all other patch types, multiplied by the logarithm of the same quantity, summed over each patch type, divided by twice the logarithm of the number of patch types, multiplied by 100 to give a percentage; PR is the patch richness (number of different patch types present within the landscape); SE1 is Simpson’s evenness index (1 minus the sum, across all patch types, of the proportional abundance of each patch type squared, divided by 1 minus 1 divided by the number of patch types).
Dynamic agricultural land-use modelling
517
shown in Fig. 4 for two values, 0.02 and O-10, of the index for the fifth landuse option (units are probits or normal probabilities and relate to the step size by which the index is increased or decreased per iteration of the model). For the smaller-sized preference index, adoption, as measured by the cumulative proportion of households that tried the new land-use in any iteration period, started slowly; by Iteration 6, only 5% of total households had tried the new option at least once. The adoption rate increased in subsequent iterations, and by Iteration 30 was showing signs of abating. With a larger value of the initial preference index, however, adoption in the early iterations was much more rapid-this land-use option gives superior gross margins over the others, all other things being equal, so that the preference index is being quickly increased from one iteration to another. By Iteration 30, some 98% of households had tried the new option at least once. In aggregate terms, these types of responses appear reasonable. At the level of an individual household, however, uptake of a new option is likely to be based as much on availability of information and the influence of neighbours who have tried it out as on extension messages and field days. Further development of the model with respect to information transmission in a landscape originating from well-defined foci of adoption is one aspect that warrants serious attention. The ramifications for targeting the release of technology in particular circumstances could be profound.
Iteration,
t
Fig. 4. Household adoption curves in the landscape of Fig. 3 over 30 iterations, for two values of the initial preference index for a new land-use option consisting of the same data as for ‘Beans’ in Table 3 except with a product price increased by 20%.
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518
Changing the distribution of household size The simulation runs were carried out using a particular initial distribution of household size (in this context, the number of facets that belong to each household, Table 3). The distribution of wealth, calculated using eqn (3) changes markedly after a number of iterations of the model, tending towards a highly-skewed distribution where a minority of households holds a majority of the wealth. We investigated the possibility that this is related to the initial distribution of wealth in the community. We carried out four sets of runs with different initial wealth distributions (standard, uniform, identical, and right-hand skewed). Results are shown in Fig. 5. There seems to be little impact of initial distribution on final wealth distribution; the end result is a
(A)
I
I
Distribution
Facets per Househdd 1
2
3
4
5 0.1
StOltdWd
0.1
0.3
0.3
0.2
Ulififam
0.2
0.2
0.2
0.2
0.2
Singl*valu*
0.0
0.0
1.0
0.0
0.0
Ri~htM-handrkewd
0.0
0.1
0.1
0.3
0.6 ,
I
(B) ??Standard Uniform
cl
? ?Single-value ?? Right-hand skewed
2-4
4-6
6-8
WEALTH, 1 O5 mu Fig. 5. The impact
of (A) initial household size distribution on (B) simulated distribution after 30 iterations of the model.
wealth
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small proportion of well-endowed households owning most of the wealth in the landscape. The results of the standard run in Fig. 5 show that less than 22% of households own more than 50% of total household wealth, and this result is echoed for the other initial distributions. The reasons why a threeequation Markov land-use model run over 21 iterations should illustrate so graphically the development of familiar inequalities in wealth distribution are worth elucidating.
VALIDATION
AND APPLICATION
We are starting to apply this land-use model to real situations, to assess what kind of explanatory power it might have. Validation of top-down models is no less difficult than for mechanistic models, simply because most if not all of the processes are highly integrative and thus subject to considerable variation. In general, we cannot run controlled experiments with which to test the model, so we have to make do with historical data. Most of the processes should be comparatively straightforward to test in a real landscape. In terms of generating input data for the model, crop simulation models could be used to derive the transition matrix for particular land quality conditions. The household preference functions pose major problems, because they cannot easily be derived directly but have to be induced from observed behaviour. Even fairly rudimentary testing would allow many of the revealed characteristics of the model to be assessed. Until this testing is carried out, we are continuing to develop the model. The next stages are to include livestock more explicitly in terms of land-use options and their impacts on household wealth, and to include the options of transferring ownership of facets, either through inheritance (which would allow experimentation concerning land-use and wealth distributions as a function of different inheritance and landtenurial factors) or through trading in facets, land prices being a function of land quality and distance to markets, for example. A certain amount of further model development would open the way to a number of interesting and potentially useful applications of the model with regard to real landscapes. One of these would be to interpret possible ecological consequences of changes in input conditions on the landscape. This could lead to the derivation of some simple indices or measures of potential ecological impact of technological and economic change on agricultural land-use, which could be of real value in a range of impact assessment studies. Another application would be to use the model to derive supply curves. For a region with a wide array of land-use options, this would involve
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steadily dropping the product price and observing changes in aggregate production. There is scope for a contribution to the debate on the form of the supply curve, especially at relatively low prices (Pachico et al., 1987). As Elbasha (1997) shows clearly, despite much conventional wisdom on the subject, the shape of the supply curve can have an enormous impact on any calculation of consumer and producer benefit arising from adoption of new technology. Parallel shifts in a linear supply curve as a result of technological change may make for algebraic simplicity, but they may not be very realistic. As already noted, another application of potential value would be the testing of hypotheses concerning the spatial spread of information in a community concerning new technology or management options. The study of the dynamics of diffusion in a landscape could have substantial impacts on the targeting and delivery of technology.
CONCLUSIONS The merits of taking a simple (even a simplistic) approach to the modelling of highly complex systems consist of the following: (1) the approach forces the analyst to define what are really the overriding factors that bring about change in the systems under study and (2) the approach minimises the modelling overhead in such a way that the effort can be abandoned very quickly if need be, with minimal waste of resources. The potential strength of these merits will be unrealised in the absence of real-world testing of the model. This is also required if it is to be demonstrated that useful information (in the sense of its being of value to a decision maker) can come from such simple models. The performance of this land-use model suggests that it is worthwhile to pursue its validation with respect to some real landscapes where sufficient data exist to enable an attempt to explain variation in land-use over a number of years. It may be that the approach is too simplistic to produce any useful information or insights. On the other hand, the construction and validation of simple models that explain useful proportions of the variation observed would be a contribution of some value towards an understanding of the functioning of complex systems.
ACKNOWLEDGEMENTS Without implicating her in any way, we would like to thank Robin Reid for very useful comments on an earlier draft of this paper.
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