Emergent Land-Use Patterns of Social-Biophysical Interactions in
Complex Systems1
Peter Deadman  *, Tom Evans , Hugh Kelly ♣, Elinor Ostrom 
♣
Department of Economics,  Department of Geography, Center for the Study of
Institutions, Population, and Environmental Change, Indiana University, Bloomington,
IN 47408, USA

Department of Geography, University of Waterloo, Waterloo, Ontario, Canada. N2L
3G1.
* corresponding author – [email protected] - Visiting Scholar at CIPEC,
Indiana University until June 11, 2004.
ABSTRACT
Landuse/landcover change is a complex process driven by interactions between human
and natural systems. Our research uses agent-based models to explore the interactions
between natural and human systems within the forested regions of the American Midwest
and the Brazilian Amazon. We use modeling as well as other methods to explore the
social and biophysical drivers of deforestation and reforestation and the interaction of
these drivers.
The natural systems in question are represented as a grid of cells which have a number of
attributes (soil quality, slope, land cover). The models include autonomous agents
(households) that make land use decisions based on their household attributes, the
attributes of their land holdings, and exogenous factors such as crop prices and climate
change. Institutional factors such as conservation plans, crop subsidies, and state and
federal forest protection programs play a role in how agents make land use decisions.
Simulated agents represent individual autonomous households making land use decisions
which generate land cover change outcomes. Land cover data derived from satellite
imagery and aerial photos is used to match modeled landscapes to observed data.
Household level surveys inform the model by identifying key household characteristics
such as occupation and household composition. A critical component to modeling
decisions is how agents learn from past experience and adapt these experiences to future
land use decisions.
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Acknowledgements: This research is supported by the National Science Foundation
(NSF# SES008351)
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Results from model and empirical analysis indicate that agent heterogeneity is key to the
complex dynamics exhibited in these study areas. In Indiana agent heterogeneities are
manifested in the preference for pecuniary vs. non-pecuniary land uses. In Brazil,
landowners from different colonist cohorts with different household characteristics
pursue different landuse strategies that lead to a spatially complex landscape.
1.
INTRODUCTION
This paper describes a series of agent based models of land use change, developed
to both explore and compare the forces driving this phenomena in two rural locations in
Southern Indiana and The Brazilian Amazon. Land use change is influenced by two
important sources of complexity. Within the natural system, attributes such as soil
quality, land cover, topography, climate, and roads can influence future land use and land
cover patterns. Those making land use decisions within the human system are influenced
by factors such as market signals, institutions, and individual preferences influence land
use decision making as well. Further, the interactions between these two systems are
frequently characterized by non-linearities and path dependencies.
Faced with the task of trying to understand these complex systems, scientists have
utilized a number of approaches for gathering empirical data regarding their behavior.
However, direct measurements on a system by themselves are seldom sufficient in
providing a clear understanding the dynamic interactions of the forces driving land use
change. Linking these observations to empirical models provides a more comprehensive
approach to understanding land-cover change (Turner et al. 1995). Of the many modeling
approaches available, a great deal of interest has been focused recently on the
development of agent based models. These models are characterized by two basic
components, a cellular model of the landscape being studied that is capable of capturing
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the biophysical and ecological aspects of a system, and an agent based model consisting
of a collection of agents, or autonomous entities, that represent the disaggregated decision
making that occurs within the human system. Early agent based models of social
phenomena, and specifically land use change were often more theoretical in nature.
However, more recent research efforts, including this work, have focused on grounding
the research in a more solid empirical foundation.
In this project, agent based models of land use change are employed that utilize
such an empirical foundation to explore theories related to the dynamics of land use
change. The dynamic implications of these models will be compared with a rich
longitudinal data set of land use patterns in the two study regions. In this paper we focus
on a discussion of the development of agent models for the two study regions, to illustrate
how these models can be used to explore theories regarding land use change. Specifically
we use a simulation developed for south-central Indiana to explore the importance of
scale and agent heterogeneity in the generation of land use patterns. A preliminary model
for the Altamira region of the Brazilian Amazon is utilized to explore a theoretical model
linking patterns of land use change to demographic trends.
In the remaining sections of this paper, we introduce the two regions being
studied in this project, outline a conceptual systems model of the land use change
problems we are studying, describe the structure of the current versions of the models that
are being developed to study these sites, outline some preliminary results of our modeling
efforts, and discuss the findings that are common to both models, as well as the future
directions for this work.
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2.
COMPARISON OF LAND USE SYSTEMS
We start with a brief introduction to the history of land use change in the two
study locations, including an outline of some of the data that has been collected in these
regions. Understanding the similarities and differences in historical patterns of land use in
both regions informs the discussion of the models that have been developed.
2.1 Indiana
One of the study areas used for this research lies in south-central Indiana. This
part of the state was primarily forested prior to the arrival of European-based settlers in
the early 1800s. These settlers cleared substantial areas of land for agricultural
production (crops and pasture) and for forest products used for construction materials. It
is estimated that in the early 1800’s more than 87% of the state was covered with forest
of some type across a wide range of topographic zones (Lindsey et al. 1965; Lindsey
1997). The process of land clearing continued until the early 1900’s at which time areas
marginal for agricultural production were gradually abandoned resulting in a pattern of
forest regrowth in areas of low agricultural suitability. The combination of agricultural
clearing and timber extraction reduced Indiana's forested land to approximately 560,000
ha (~1,390,000 acres), or about 6% of the state by the early 1920's (Nelson 1998). Since
that time, the extent of forest cover has increased to over 1.6 million ha (4 million acres)
(Nelson 1998). Today, Indiana retains only an estimated 0.06% of its old growth forest
from its estimated original forest cover at time of European-American settlement (Davis
1993; Lindsey 1997). The majority of forest cover in the state is relatively young
successional forest covering approximately 18-20% of the state. It is this afforestation
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process (abandoned agricultural land allowed to recover to a forested state) that is
explored with one of the models presented here.
Indian Creek Township, an area of approximately 10 x 10 km located in
southwest Monroe County, Indiana, comprises the spatial extent for this model. Private
landholders are the primary actors in the landscape. Indian Creek Township is
characterized by a series of rolling hills with bottomland areas suitable for agricultural
production interspersed between ridges/hills that are largely forested. Forest cover
composed 43% of the landscape in 1939 and 60% by 1998 (Figure 1). In general,
afforestation has occurred in steeply sloped areas while areas with shallow topography
remain in some type of agricultural landuse (Tables 1 and 2, Figure 2). Landowners are a
mix of households that derive a portion of their household income from extractavist
practices (agriculture, farming, haying, timber harvesting) and other households that
derive all their income from non-farm activities (Evans et al. 2001, Koontz 2001). It is
this mix of household types that we explore with the model presented here. The
development of landcover change models is often made in the context of what data are
readily available or in the context of new data acquisition efforts in which case the
modelers have an opportunity to determine at which scale data are compiled. In both
cases the modeler needs to understand the implication of running a model at multiple
scales of analysis and the potential for the model results to vary as a function of scale. To
explore the sensitivity of the model to scale effects, the ABM was run using input data at
60, 90, 120, 150, 240, 300, 480 meter spatial resolutions.
2.2
Data Collection
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A broad array of data sources are used to represent key dynamics in the land use
management system. These data sources include economic/price/wage, landcover,
demographic, and agricultural census information. Crop price, timber price, and wage
labor rate data are considered exogenous and uniform for all agents. Annual crop and
timber prices were acquired for the major types of row crops and tree species harvested
from 1940 to -present. These prices are used to determine the economic benefit from
agriculture and timber harvesting landuses in the model. Crop prices for corn and
soybeans were aggregated to a single mean price/bushel measure derived from U.S.
Agricultural Census data sources. Timber prices for a group of hardwood species were
also aggregated to a single index price per board foot for potential timber harvest income.
As described below, our landcover data does not allow us to discriminate with a sufficient
level of accuracy what crops are being grown in agricultural areas or what tree species
compose forested areas over time. Thus we simplify the model to a single agriculture
class and a single timber/forest class in our model runs.
The change in off-farm wage labor rates is represented by the minimum wage
between 1940- and the present. While household residents were employed in an array of
occupations, many of which had wages higher than the minimum wage, we have used a
coefficient term to modify the impact of the wage labor rate assuming that the trend in
minimum wage increase is indicative of changes in income levels for occupations
requiring more than a minimum wage.
Many of these data inputs are broad scale data and assumed to be homogenous
within the study area (e.g. crop prices, wage data). However, there are several datasets
that are critical to the scale issues being addressed in this research. These data include
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landcover, land ownershiop and surface topography. There to be more local level
variability in landcover and topography compared to factors such as crop prices and labor
wages and so these data have been compiled at the highest spatial resolution possible.
2.3. Landcover Data
Aerial photography was interpreted to produce a time series of landcover data for
the following dates: 1939, 1958, 1967, 1975, 1980, 1987, and 1993. The aerial
photography products ranged in scale from 1:15,000 to 1:40,000. Individual 9” x 9”index
sheets were digitally scanned and visually interpreted using a combination of heads-up
digitizing complemented by the hard copy product. A minimum mapping unit of 30 x 30
m was used to produce a series of forest/non-forest layers for each date. Sample areas
were interpreted by several individuals to assess the consistency of the visual
interpretation. An attribute code was used to tag polygons where the analyst had less
confidence in the assigned landcover code. A post-processing error assessment was
conducted to identify miscoded polygons and imprecise digitizing. From 1939 to 1992
forest cover in Indian Creek increased from approximately 43% to 60% of the landscape
(Figure X). Small patches of forest cover loss are also evident during this time period,
but the net pattern in the township is one of forest cover increase.
Figure 2 shows the pattern of land cover and topography in Indian Creek
Township. These figures, along with the data presented in Table 1, indicate that there is a
relationship showing forest cover to be predominantly located in areas of steep
topography and that afforestation in the last 60 years has occurred primarily in areas of
steep topography. This is mainly due to the fact that areas of shallow slope are more
suitable to agricultural production. However, it should be noted that many of these
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steeply sloped areas were cleared of forest at one time. It can be seen that (1) forests
account for a larger proportion of the landscape in 1992 than in 1939 and (2) the forest
regrowth has occurred in both shallow slope areas (< 4 degrees surface slope) and steep
slope areas (> 10 degrees surface slope).
2.4 Land ownership
Land ownership boundaries were derived from historical parcel maps from 1928
to 1993 and from a digital GIS dataset provided by the Monroe County Tax Assessors
office for 1997. These parcel boundaries provide the mechanism whereby agents are
assigned to the landscape over time. Vector datasets of land ownership boundaries were
constructed for each time point and converted to a raster representation for input into the
main model.
2.5 Topography
Surface topography was generated from a Digital Elevation Model (DEM) with a
spatial resolution of 10m that was constructed from contour lines extracted from 1:24,000
scale topographic base maps. A slope surface dataset was derived from the 10 meter
DEM dataset using a moving window algorithm that fits a plane to a 3x3 window around
the central output cell. Thus, the slope value for each 10 meter cell was derived from a
30 x 30 meter area but produces a slope dataset with a cell resolution of 10 m.
2.6 Brazil
Although studies on land use change in the Altamira region have focussed on the
last 35 years, some important similarities have been observed between the two study
sites. Much like the Indiana situation in the 1800’s, since the early 1970’s patterns of land
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use change in the Brazilian Amazon have been characterized by rapid deforestation. In
many parts of the Amazon, including the region west of Altamira, land use change has
been facilitated by government activities including economic incentives and the
construction of the Transamazon highway. Running from east to west through Altamira
and across the Amazon, the highway has been associated with marked changes in the
region, including deforestation, (Wood and Skole 1998; Moran and Brondizio 1998). In
the region west of Altamira, the main highway is accessed by a series of side roads at
roughly 5km intervals. Along both the main highway and these side roads rectangular
properties with an average size of 100 ha were surveyed for the purpose of creating
individual family farms.
In this region, research has begun to reveal how factors operating at different
scales influence patterns of land use. At the local scale, individual farming households
make significant contributions to the land use/cover changes. At this local scale, land-use
decisions of each household are influenced by household composition, available capital,
and soil fertility (McCracken et al., 1999). At the regional scale, land use is further
influenced by a number of socioeconomic factors, including local credit policies, market
opportunities, and inflation rates (Moran, 1981). Research has revealed the existence of
complex relationships, feedback, and interactions both between human and natural
systems, and across multiple scales.
Specifically, research over the last decade has addressed rates and patterns of
deforestation (Brondizio et al., 2002), the degree of secondary growth (Lu et al., 2002;
Moran et al., 2000; Moran et al., 1996; Mausel et al., 1993), radically differing land
patterns between neighbouring farms (McCracken et al., 2002), and the influence of soil
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quality on household success (Moran, 1995, Moran et al 2000, 2002). Multi-temporal
analysis of remotely sensed images has revealed that many once-deforested areas are now
undergoing secondary succession, indicating that the rainforest ecosystem may be more
resilient than was once thought (Moran et al. 1994). While previous research has tied
deforestation rates to population change, resulting from high rates of in-migration and
subsequent population growth and movement, these general observations do not explain
the heterogeneity of land use strategies that occur across individual farms in colonized
regions such as that near Altamira (McCracken et al., 2002). Neighbouring farms in the
Altamira region have been observed to have radically different patterns of land use. This
observation has raised research questions related to the relative importance of family and
market factors in shaping land use decisions. One approach to this problem has been to
study the individual cohorts who arrived on the Altamira frontier at different times, along
with age and period effects (see McCracken et al., 2002).
With a focus on local scale factors, an analysis of data collected in Altamira has
led to the development of a conceptual model which proposes that land use changes in the
region should be understood as a product of the age and gender characteristics of farm
households as they interact with local environments and external factors (McCracken et
al., 1999). This model maps out a trajectory for families, which relates the type of
agricultural practices pursued to the available capital resources and labour pool within
each household. Possible land use strategies include growing annual cash crops, such as
rice, maize, beans, and manioc; perennial crops, such as cacao and black pepper; fruit tree
production; agroforestry; and the transformation of land into pasture with or without
cattle.
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For the purpose of sampling households and farm lots across cohort and age differences,
five temporal stages of household composition are proposed by the conceptual model,
with each stage of development characterized by varying levels of capital and available
family and male labour (McCracken et al., 2002). Stages represented in the conceptual
model follow a trajectory of land uses that occur through the evolution of colonists
households to inform survey data collection in the region (Moran et al., 2002; McCracken
et al., 1999). According to this trajectory, young families that arrive on the frontier
typically have limited capital resources and young children. Initially, these young
families typically deforest three to five hectares of land a year to plant annual crops such
as rice, beans, or manioc. As families age, they continue to deforest their properties to
grow annual crops, while turning previously cleared land into pasture or perennial crops,
or allowing the land to become fallow or enter secondary succession. McCracken et al.
(2002) have observed that although this trajectory can be shaped by outside forces such
as credit policies, available household labour plays a significant role in the land use
strategies that are pursued by an individual family. Households with abundant available
labour gravitate towards a diversified land use with an emphasis on perennial crops such
as fruit trees, coffee, or black pepper, while those with less available labour tend to
pursue land-use strategies focussed on creating pasture and raising cattle while
maintaining small-scale annual crop production (McCracken et al., 2002). Over time, this
shift in land use strategies away from annuals, which require newly cleared land every
few years, and towards perennials or pasture results in reduced levels of deforestation and
increased secondary succession.
2.7
Conceptual Diagram of Land Use Change
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As conceived in the models developed for both study regions, the forces driving
land use change can be described as a number of separate and interacting systems, or
modules (See Figure 3). In this model, two basic types of modules are included that
represent the human and biophysical factors influencing land use decision making. These
modules are connected by a series of relationships, or flows of information, represented
here as arrows. Central to this system are multiple agents who make decisions about land
use in order to improve their utility which is composed of pecuniary and non-pecuniary
factors. The agents are represented in figure # by the stacked boxes in the center of the
diagram. The information agents use to make their land use decisions flow from the
economic, political, biophysical, and social domains. Further, agents are influenced by
externalities that are realized across agents. This influence is represented by the arrow
connecting the two agents, i.e. the center boxes. Similarly, the actions of agents may
influence the dynamic processes evolving in these outer modules. This influence is
represented by the arrows pointing outwards from the agent decision maker to the
biophysical module. In theory an agent could influence each of these outer modules;
however the current version of the model restricts the influence of agents to land cover
change. Also, the current model does not allow the outer modules to influence one
another.
Information from the biophysical module includes all data an agent may use about
landscape classification (slope, soil content) when making any possible land use decision.
From the economic module we need, farm product and land prices, as well as off farm
wages to characterize potential non-agricultural income. From the social model we need
information about the cultural preferences for farming and risk, wealth, and education
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levels. From the political module we need to know if there are set aside subsidies, taxes,
or are zoning restrictions, or particular policy objectives for the landscape.
Across different models, agents are distinguished from one another based on two
sets of characteristics: (1) The decision making strategy employed by agents; (2) Agent
specific weights for the various components of their utility function. Agents’ decision
strategy can vary, but in all cases they are concerned with improving their overall utility
which has an explicit structural representation. Within the Indiana model, three versions
of the decision making strategy currently included are: Myopic Utility Maximization;
Reinforcement learning; Hill Climbing. The agent specific weights included, describe
agents preferences for farming and tree growing, their learning rate given experience, and
how sensitive they believe profits are to land suitability/slope and across agent production
externalities. Within the Altamira model, agent decision making is represented by a
heuristic based decision tree. Agent preferences for particular land use strategies and
their knowledge of local soils can be altered by adjusting specific variables a points
within the decision tree.
As shown in Figure 3, the Indiana and Altamira models differ in the approach
their decision strategies take to exploring the land use solution space that exists in each
round of the simulation. In the Indiana model, agents are able to simultaneously compare
the expected utility of all the decisions that are available to them. The decision tree
employed in the Brazil model reduces the number of possible land use decisions with
each step through the tree; thereby reducing the amount of the solution space available to
the agent in that round.
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The structure of the land use decision making system, as outlined in this
conceptual model, has guided the design of the agent based models for Indiana and
Altamira. All the models focus on the individual household as the land use decision
making agent, which effects land use changes within the biophysical module, and is in
turn influenced by information coming from all the other modules. Specific
implementations of these agent based models are discussed in the next section.
3.
METHODS
Both the Indiana and Altamira share a basic architecture in which a collection of
autonomous agents, representing land use decision making households, interact with a
biophysical environment that is represented by a series of grids showing the spatial
distribution of land cover, soil quality, or topography. The specific implementations of
this basic architecture are outlined below.
3.1.
General Model Design: Indiana
3.1.1 Agent characteristics
In the agent based model designed for the Indiana project individual agents adapt
their labor Li and land Mi input allocations across several productive activities. Their
general goal is to maximize their weighted utility which is composed of pecuniary and
non -pecuniary items. Importantly, there are across time and across agent land-use
production externalities which endogenously influence the productivity of their own and
nearby neighbors’ lands.
Each year agents observe the new set of exogenous and endogenous features of
their environment and re-approximate their own utility maximizing factor allocations. In
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this setting, agents’ adapt their factor allocations as the environment changes. However,
the deviation of the realized outcome from their expectations does not directly influence
their decisions, as might be the case with a deductive reinforcement adaptation strategy.
Based on research suggesting land owners may display risk aversion regarding
their income I, we posit the additively separable constant absolute risk aversion utility
function U = -e-2·RA·I, see Parks (1995). Also, we assume each agent has a time or labor
constraint regarding the labor they can supply. I represents a scaled sum of expected
pecuniary income and non-pecuniary ‘output’ derived from reforestation activities. RA
represents the degree of risk aversion, and σ2I represents the variance of income, i.e. risk.
The intuition describing this functional form is that agents derive utility from a scaled
composite of pecuniary and non-pecuniary components I. However, all else being equal,
they dislike variance in this measure.
Define λ as the Lagrange multiplier on an agent’s labor constraint and L as the
total available labor hours. The agent must then choose the Lis below in order to solve
their unique constrained expected utility maximization problem:
E(U) = E(I) – RA σ2I + λ (L – Lfarm – Ltree – Laes – Loff farm)
(3)
Agents can supply labor inputs to i = 4 activities respectively: growing crops (farm),
harvesting trees (tree), growing trees (aesthetic), or working off the farm (off farm) in a
manufacturing sector. Land can be used for three activities including: growing crops,
growing or harvesting trees, or fallowing the land.
In each year the sequence of possible actions for an individual agent are as
follows: At the beginning of the year, given biophysical conditions and their expectations
about future prices, an owner agent makes a decision about the utility maximizing labor-
15
allocation; Conditional upon this allocation an agent then selects the best cells for each
activity on which to apply their labor. They compare the payoffs possible for each cell for
each potential use, and rank all cells for each possible activity. They then sequentially
apply equal amounts of labor to the cells generating the highest potential utility until all
labor has been applied. Over the year crops grow, dynamic biophysical properties of the
landscape evolve and interact, and/or the agent earns an off farm labor supply wage; At
the end of the year, all unknown prices, wages, tax rates are observed, all output is
liquidated at the historical market price and/or agents are paid the historical nonagricultural wage; finally, net revenue and overall utility are computed.
The key aspects of the market setting are that: (1) there are multiple potentially
heterogeneous agents who can differ based on the quality and quantity of land owned,
and in their preferences and in perceptions about the productivity effects of land
suitability and land-use externalities; (2) agents make a labor then a land allocation
decision each year; (3) agents are small producers relative to the overall market
supply;(4) agents interact with one another via spatial production cost, or aesthetic
pleasure land-use externalities; and finally, (5) agents are aware of the production, cost,
and profit functions that describe the output, costs, and profits they can expect from the
various productive activities.
3.1.2 Agent Decision-Making
Integrating the functional relationships between the inputs land and labor, and the
output goods yields the constrained expected utility expression (9). Each agents’ decision
then involves choosing the labor allocations across i, in order to maximize (9) subject to
the land available for each activity and given the average parcel characteristics, i.e.
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average slope, production externalities, etc. This is similar to the solution procedure
described in Puu (1997).
Et(U) = αf ( (1 – τfarm) profitfarm – Y2farm RA σ2farm)
+ αt ( (1 – τtree) profittree – Y2tree RA σ2fpr,tree)
+ αg ( (1 – τgraze) profitgraze – Y2graze RA σ2graze)
(9)
+ αof ( (1 – τof) profitoff farm – Y2off farm RA σ2wof)
+ αaes (Yaes - Y2aes RA σ2fpr,aes)
+ λ (L – Lfarm – Ltree – Laes – Loffarm)
An agent’s maximal labor allocation is that which simultaneously satisfies the
following first order and Kuhn-Tucker conditions:
dU/dLf = 0, dU/dLt = 0, dU/dLa = 0, dU/dLof = 0, dU/dλ = 0
(10)
λ ≥ 0, dU/dλ ≥ 0, λ·dU/dλ = 0
(11)
This system of equations reduces to one equation with one unknown, the Lagrangian λ,
which is numerically estimated to solve (10) and (11) for each agent and period. This
yields an agents’ maximizing labor allocation quantities {L*farm, L*tree, L*aes, L*off farm}
which must be applied to available acres of land Mi. The labor applied to an individual
cell is then determined by the equal allocation rule to be licell = max(L*i/Mi,1) hours/cell.
3.2
General Model Design: Brazil
As with the LUCIM model, LUCITA is comprised of two sub-models designed to
represent the interactions of the ecological and human systems characteristic of the
Altamira study region. The ecological sub-model operates on a set of raster grids in
which each cell in the grid represents one hectare within the Transamazon highway
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corridor west of Altamira. In the version of the model described here, two georeferenced
grids represent land-cover and soil quality. Within the soil grid, each cell adjusts nutrient
values in response to the land use activity occurring on the corresponding cell in the landcover grid. For example, when a cell in the land-cover grid is cleared and burned, nutrient
values in the corresponding soil grid cell are altered to represent nutrient deposition.
Similarly, when a crop is planted and harvested on a particular land cover grid cell,
nutrient uptake by the crop depletes the soil nutrient values in the corresponding soil grid
cell. Due to the limited availability of soils data for this region, initial values for some
soil parameters, soil changes through land-cover clearing and burning practices, and soil
depletion and crop yield prediction are determined by regression equations developed by
Fearnside (1986, 1984, and 1998).
The landcover grid represents the current landcover state in each cell, which can
be; forest, secondary succession, annual crops, perennial crops, pasture, or road. The
landcover grid is subdivided into 100 ha farm properties in a pattern that is representative
of the farm plot configuration designed by the Brazilian Government, prior to
colonization of the Transamazon corridor west of Altamira. This pattern consists of a
rectangular network of plots that run along each side of the main highway, and along the
access roads that are set perpendicular to the main highway every five kilometres
(Fearnside 1986). Plot sizes along the highway are relatively uniform (Balmann, 1997;
Fearnside, 1986; McGrath et al., 2001; Moran, 1981) and abstracted in the model to the
designed size of 100ha.The narrow side of each plot, which is either 400 or 500 meters in
length, is situated adjacent to the road.
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During a simulation run, household agents make land use decisions within their
property, which result in changes to the landcover and soils grids. If a particular cell is
abandoned, it reverts to secondary succession and eventually mature forest.
3.2.1 Agents
Human decision making regarding land use on this agricultural frontier is
represented by a collection of agents, where one agent represents a single household
owning a single farm plot. The decision making mechanism takes the form of a decision
tree, where individual decision points on the tree represent heuristics that guide the
decisions of a typical farmer within the trajectory. The agents make choices regarding
general land use strategies to pursue, including the growing of annuals, perennials, or
pasture, on a cell-by-cell basis. Once these basic land use strategies are selected, more
detailed decision trees are employed to select specific annual or perennial crops. The
heuristics were designed to represent the decisions made by the typical farmer depicted in
the conceptual trajectory. The decisions are governed by the subsistence requirements of
the household, the capital and labour resources of the family, and the quality of the local
soils.
Household subsistence requirements are calculated in terms of required capital
inputs and land cultivated in annuals for family consumption. Cash requirements for
commodities and services such as seed, clothing, medicine, transportation, and other
necessities are represented by a fixed price of $1615.30 Cruzeiros per adult (half that
value for each child) per year (Fearnside, 1986). Each household must produce one
hectare of annual crop for each adult and ½ hectare of annual crop for each child.
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Households are able to incur debt in their annual production provided that it is met by
profit from pasture or perennial production.
LUCITA assumes that all farmers have the ability to detect good or poor soil
quality. Moran (1981, 1995) documents the ability of farmers to classify or categorize
soil to determine and map areas of soil quality based on existing vegetation, soil color
and texture.
Knowledge of crop price influences household selection of annual crops for
planting. In this version of LUCITA, households have perfect knowledge of static crop
prices (per kg). While calculated profit returns seem to be the ideal determinant for crop
selection, this is typically not done in the Altamira region. Instead farmers use crop prices
as a proxy for return revenues such that a crop with a higher price per kg will yield
greater returns than those with a lower price per kg (Moran 1981). Strictly using this
price gradient would create a deterministic selection of crops to farm. To improve
variability and avoid unrealistic path dependencies, LUCITA uses crop prices to create a
weighted probabilistic form of crop selection.
Initial farming households colonizing the study area were selected by the
Brazilian government using a process that favoured families that were both young and
large (Moran, 1981). Within the simulation, households amongst the first wave of settlers
are composed of a husband (aged 20-30), wife, and 4 to 10 children; while successive
families arrive with 0-4 children. Based on observations by Moran and Brondizio,
children may begin to assist with household tasks as early as age 7. However, most
children begin to contribute noticeable amounts of farm labour after the age of 10
(Siqueira et al., 2002). The range of possible labour provided by children is increased in
20
LUCITA to accommodate the small contributions by the very young (7-10 years of age)
and the increasing contributions by adolescents (11-19 years of age).
Endowments exist in the form of labour and capital. Households utilize both
endowments collectively. Each household member produces a specified amount of
labour, for the sole purpose of increasing the household’s level of farming. Similarly the
household head is provided with a constrained range that is defaulted to a maximum of
2250 hrs/yr (45hrs/wk * 50wks/yr). The random allocation of labour provides for the
incorporation of heterogeneous household factors such as health issues (injuries,
sickness), varying cultural norms related to religious activities in some cases, and gender
related norms defining who participates in which farm related activities (Siqueira et al
2002, Moran 1981).
Agent interaction occurs through a local labour pool composed of farmers that
have failed or have been removed from their plot due to the incurrence of excessive debt.
Only labour produced by the household head is available each year. A household is able
to seek out labourers when it has sufficient capital to meet its labour demand. The price
per hour for wage labour is homogeneous for all labourers.
3.3.
Model Validation: Indiana
The primary goal of the analyses of the Indiana model is to investigate the
empirical descriptiveness of the agent-based-model. We conduct a by-agent parameter
fitting exercise exploring the extent to which hypothetical utility maximizing land owner
agents can be parameterized make land-use decisions that are representative of the actual
owners’ historical decisions. This allows us to quantify the degree of heterogeneity in the
21
agents fitted parameters, thereby addressing whether a homogenous agent assumption is
accurate.
3.3.1 Model Fitting
For parameter fitting we estimate the by-agent preference, suitability, and
externality parameters in order to minimize the deviation between the historical and
simulated landscapes. The across time characteristics of the landscapes are summarized
by the spatial metrics percent forest and forest patch edge. Estimated parameters include
the agricultural and timber harvesting production preferences αfarm and αtree; the land-use
spatial externality weights for farming, tree harvesting, and reforestation, respectively
γe,farm, γe,tree, and γe,aes; and the land suitability weight describing the extent to which the
slope of the landscape is perceived to influence output, i.e. γslope. For simplicity we
impose the restriction that the farming and tree harvesting cost externality effects are
equal, i.e. γe,farm = γe,tree. We focus on these five parameters because they summarize each
agent’s motivation for pursuing farming or timber activities, the primary land-uses in this
area. We use the Matlab fminsearch iterative parameter-search algorithm to estimate all
free parameters. There were 7 actual cover years for each agent’s 5 parameter estimation
procedure providing 2 degrees of freedom.
The parameters are chosen to maximize a goodness of fit measure, for each agent,
denoted Null_R2. This measure compares the across-time sum of squared error between
simulated landscape metric values and the actual landscape metric values relative to the
variation in the actual historical landscape from 1940. This measure is based on Granger
and Newbold (1976) and is calculated as:
Null_R2 = 1 – SSESim / SSENull
(12)
22
SSESim = ∑6t = 1(ActMetrict - SimMetrict)2
(13)
SSENull = ∑6t = 1(ActMetrict – ActMetrict=1)2
(14)
For this measure SSESim represents the sum of the squared deviations of a particular
metric value (percent forest, forest edge, or a weighted average of these two) for the
actual landscape ActMetric, from the metric value for the simulated cover, SimMetric,
across time. SSENull represents the overall variation in the actual landscape across time
relative to the cover at t=1, i.e. the null model. The second term in (14), ActMetrict=1,
represents the actual metric value for the cover in the first year for which we have data.
For the R2 described in Granger and Newbold (1976) the second term in (14) would be
the mean value for the metric over the time period. However, in the LUCC literature, one
common means to compare the goodness of a land-use/cover model is relative to what no
model, or a null model would provide. This null model uses the landscape metric value at
the first time period (t = 1) to predict all future dates, and then calculates the resulting
null SSE. We chose to remain consistent with this literature, and the Null_R2 we report is
in fact a comparison of our model predictions relative to the sum of squared error that
would obtain in the absence of any model. Null_R2 > 0 indicates our model does better
than nothing, Null_R2 < 0 indicates that our model is worse than doing nothing, i.e.
relative to using the 1940 cover to predict all future dates.
3.4
Model Validation: Brazil
Simulation runs in LUCITA are set up to run for 30 iterations, such that the first iteration
represents 1970 and the beginning of colonization in the area, iteration 15 corresponds to
1985, and so on. Households are allocated to plots at 0-50 per iteration to a site
comprising 234 one hundred hectare plots on a grid representing an area of 15km by
23
20km. The assignment of 50 households per iteration allows all plots to be potentially
filled by incoming farmers within 5 years; however, it is most likely that all plots are
allocated within the 6-8 year range.
A set of thirty simulations were run in which the parameters were left at a
constant settings. Differences between individual simulation runs can be attributed to the
stochastic elements of the model including; the timing and location of the introduction of
individual households to the simulation run, the individual attributes (family size and
capital) of each household agent, and the individual household decision making regarding
crop selection that occurs throughout the simulation.
Within the field of MAS/LUCC modelling, a great deal of interest is currently
focussed on the validation of these models (Manson 2002). In early MAS/LUCC
modelling efforts, validation was not often extensively addressed. In some cases, these
early models were highly theoretical in nature, being based on artificial landscapes.
However, as the field has expanded, greater attention has been paid to the need quantify
or explain the validity of model output. Model validation should be approached with a
view to the overall purpose of the model. Bradbury (2002) argues that since complex
adaptive systems are not, by definition, predictable, then models of these systems should
not be validated against real world outcomes. If we take this view, then models become
explanatory tools, designed to explore the relationships between different elements in the
study system. The models serve to illustrate and explore particular phenomena, while
acting as a tool for generating and testing new theories. However, other researchers view
model as predictive tools, and argue in favour of a role for quantitative validation in
which simulation output is rigorously compared, by both the quantity and location of
24
change, to known real world conditions (Pontius 2002, Pontius and Schneider, 2001).
These approaches to validation typically employ spatial statistical measures to compare
binary maps.
LUCITA is viewed as a model whose role is more explanatory than predictive.
The model is seen ultimately as a tool for exploring and explaining land use change
phenomena in the Altamira region. Specifically, it is intended that the model will
eventually serve as a tool for understanding how the decisions of farmers influence land
use change in the study area. In the face of these alternate validation techniques, the
approach taken here has been to compare overall trends in the preliminary output of
LUCITA to quantitatively measured trends in land use, and to theoretical models of
individual household decision making.
4.
RESULTS
The histograms at Figures 4, 5 and 6 report the estimates of the by-parcel
parameters which provide the best fit of the utility maximizing land owner agent to the
historical owners’ land-use decisions. Also provided in each Figure is the histogram of
the by-parcel Null_R2s comparing the model’s land-use history to null model. Figure 4
reports the fits using the percent forest metric to quantify the landscape. Figure 5 reports
fits using the edge of forest to quantify the landscape, and Figure 6 reports fits
maximizing a equal weight average of the previous two R2s, i.e. Null_R2WtAve =
0.5*Null_R2PerForest + 0.5*Null_R2Edge.
25
4.1
Heterogeneity of Indiana land owners
The first thing we observe in all figures is that there is a wide distribution of byparcel fitted parameters indicating substantial across parcel, i.e. agent, heterogeneity. In
general these results indicate that in order for our model to accurately reproduce each
parcels sequence of land-use change, agents must have parcel specific parameters.
Further, the magnitudes of the estimated parameters, and the goodness of fits, also vary
depending upon which metric value we use.
More specifically, considering the fits using percent forest that are reported Figure
4, we observe that for the majority of these parcels the Null_R2PerForest lies within the
range 0.50 to 0.80, representing substantial improvements of our model relative to the
null model. However, for 7 of the 51 agents simulated, our model performs poorly. The
majorities of the externality parameter estimates are positive indicating that there are cost
reducing externalities associated with proximate agricultural and timber production. And
similarly, the estimates for γe,aes are positive indicating positive externalities may exist for
the non-pecuniary utility derived from the presence of contiguous forests. Finally, 48 of
the 52 estimated slope parameters are greater than 0 indicating that higher slope is
associated with reduced agricultural and timber harvesting productivity and therefore
land usage, as predicted by earlier empirical results, see Mokma and Sietz (1992).
Figure 5 reports the model fits using the forest edge metric to compare
landscapes. Here we observe that for the majority of these parcels the Null_R2Edge lies
within the range 0.30 to 0.80, representing improvements of our model relative to the null
model. However, for this metric there are substantially more Null_R2s in the lower
region, and for 9 agents, our model does worse than the null model. Overall, these results
26
indicate that the fits to forest edge are more variable. Importantly, the median Null_R2 for
edge is 0.73 and is actually larger than the median for percent forest which is 0.63. This
suggests that we obtain an overall better fit when using the edge metric to quantify
goodness of fit. The estimates for the various parameters using the edge measure are for
the most part are very similar to those for percent forest and therefore tell a similar story.
Figure 6 reports the model fits using an equal weight, weighted average of the
percent forest and the forest edge Null_R2s, in order to compare the landscapes. Overall,
these two metrics are capturing different spatial aspects of the landscape, and therefore
simultaneously minimizing the Null_R2s of these two metrics represents a more difficult
task. Accordingly, we observe that many of the agents’ Null_R2WtAve lie within the range
0.30 to 0.80, representing improvements of our model relative to the null model.
However, now for 12 of the agents our model does not provide an improvement in
predictability relative to the null model. Further, the median Null_R2 is now 0.39,
substantially below the earlier two median values. Again, the signs and relative
magnitudes of the various estimated parameters are quite similar to those estimated for
either metric in isolation. There are perhaps slightly larger tails in the distributions of
these parameters indicating increased variation in parameter estimates.
4.2
Scale dependence of Indiana results
Scale issues have significant implications for the analysis of social and
biophysical processes in complex systems. These same scale implications are likewise
considerations for the design and application of models of landcover change. Scale
27
issues have wide-ranging effects from the representativeness of data used to validate
models to aggregation errors introduced in the model structure (Evans et al. 2002).
The scale dependence of the Indiana model is explored by varying the resolution
of the input data used to calibrate the model (observed landcover), ancillary datasets that
affect land suitability (topography), and the resolution of the model landscape on which
agents make decisions. To explore the impact of these scale relationships the model is
run with input datasets constructed at the following spatial resolutions: 60 m, 90 m, 120
m, 150 m, 240 m, 300 m and 480 m. The results show that the distribution of landusepreference weights differs as a function of scale (Evans and Kelley in review). The finest
spatial resolution model runs generated a more diverse set of agent types than the coarser
resolutions. Agent-type variance was lost at coarse spatial scales because (1) some
agents dropped out of the landscape due to aggregation effects and (2) overall, agents
have fewer cells in their parcels, simplifying the heterogeneity of landcover in their
landscapes. These findings have implications for agent-based modeling development and
how sensitive ABMs are to land cover changes observed at varying scales of analysis.
4.3
Results from Brazil
In the simulations documented here, all farm properties are typically allocated to
farmers within the first 6 to 8 iterations (years) of a run. This pattern is somewhat
characteristic of what has been observed. In 1971 in Altamira there were less than 500
families, by June of 1972 there were 1834 families and by June of 1974 there were 3036
(Moran, 1981). Dale and Pearson (1999) also discuss the rapid arrival of colonists in their
28
study (similar to the one presented here) where initial designs for settlement of 500
families in 1971 boomed to 4000 by 1974.
Land cover trends (represented by the average of 30 runs) are shown in Figures 7
and 8. At the commencement of colonization, a significant portion of the land is
deforested and put into annual production. Over the first few years colonist agents
establish themselves in the frontier accumulating wealth and labour (through aging
children) that allow them to increase perennial and pasture production. In a typical
simulation run, the total amount of land in perennial and pasture uses surpasses annual
production in year 7. After this point the model continues in the expected trajectory in
which the next period of cultivation is dominated by perennial crops. In the later
iterations of the simulation, decreases in available household labour occur as children
begin to leave the household in search of their own economic livelihoods. This constraint
on available labour results in a levelling off of perennial production, and a decrease in
pasture production. The observed decrease in pasture production in the simulation does
not reflect observed trends. At this point in the simulation, successful farmers are those
that have accumulated enough capital and are able to hire wage labourers to continue
work in cash crops such as Black Pepper, Cacao, or Pasture.
<Insert Figure 7 approximately here>
<Insert Figure 8 approximately here>
29
Figure 8 displays the trend of mature forest and secondary succession and fallow.
These two land cover types show an inverse relationship such that when mature forest is
decreasing, secondary succession and fallow are increasing, and visa versa. Deforestation
continues, although at a decreasing rate, until year 20 in which the conversion of mature
secondary succession becomes mature forest. It is at this time of conversion (delineated
by secondary succession growth time of 17 years) that the level of mature forest in the
study area begins to rise while levels of secondary succession and fallow fall.
In Figure 9, the percentages of land cover type found by Mausel et al. (1993) are
compared to the corresponding percentages, and standard deviations, produced by the
simulation. The patterns of deforestation in LUCITA reflect land use patterns found by
Mausel et al. (1993) and Brondizio et al. (2002). While an exact numerical match is not
found, trends in percentages of land cover type recorded over time are similar (See Figure
9). Deforestation results of 43% (shown in Figure 7) at iteration 15 approach 1985
observed values by Mausel et al. (1993) of approximately 55% in a subset of their study.
The higher level of deforestation in the Mausel et al study may be due to the proximity of
their study site to Altamira, and the generally better soils found in the area. Mausel et al.
(1993) also note that from 85-88 and 88-91 an additional 4% and 1.5% are deforested,
respectively. Under iterations 15-18 and 18-21, LUCITA shows a lower decrease in
deforestation with corresponding values of 2% and 1.5%, respectively.
<insert Figure 9 approximately here>
30
Mausel et al. (1993) also measure percentage of additional land cover types:
secondary succession, fallow, crop, water, and wetland. Aggregating fallow and
secondary succession recorded by Mausel et al. (1993) to match categorization within
LUCITA (fallow, secondary succession, and perennials) we find a close similarity in
1985 of 33% and 37.7%. Difference is minute in 1988 (42% vs. 41.1%) yet increases
over time to show similar long term trends with greater variance (1991 – 53% vs. 43%).
Despite differences between observations and LUCITA the percentage of secondary
succession and fallow show similar increasing trends over periods from 1971-1991.
The third category of land-cover type is defined by the ratio of area in annual
production over the total study area is crop type. Crop percentages recorded by Mausel et
al. (1993) and LUCITA demonstrate a higher level of agreement than comparisons made
between deforestation or secondary succession and fallow levels. From 1985 to 1991
LUCITA produces a slow decrease in the amount of area in crop production (1.7-1.4%).
Throughout this period of decline in annuals, perennial production begins to stabilize.
Despite a similar decline in cropping observed by Mausel et al. (1993) from 2.4 to 1%
over the same period, measurements in 1988 spike to almost 5% of their sub-site in crop.
It is not unlikely that such spikes occur as they may be attributed to a number of external
or stochastic variables, such as droughts, that are not represented within LUCITA.
Lastly, pasture trends decrease in time from 1985 to 1991. While values observed by
Mausel et al. (1993) are markedly higher, with a dramatic decline (19-6%) that resulted
from a downturn in the Brazilian economy. Such exogenous economic fluctuations are
not represented in LUCITA. In the simulations runs, the amount of area in pasture
decreases slightly from 3.1 – 1.8%.
31
5.
Discussion: What have we learned from comparing these models?
Our work with these models outlines the explanatory power of agent based
models in which land use decision making can be modeled as a localized phenomena and
then aggregated across a landscape for the purposes of analysis. Empirical results from
the Indiana model indicate that the appropriately parameterized agents, sensitive to
economic and biophysical incentives, can be accurately fit to the 50 year sequence of land
use in our sample area. Indeed, many of the other advantages attributed agent based
modeling (Parker et al. 2002) are also illustrated in the models outlined here. Foremost,
their flexibility permits the representation of heterogeneity in both in the human and
natural systems being modeled. In the models outlined here, this means that households
with varying attributes can be captured in a single model. In the simulations described
here only a single type of agent, representing individual households, are modeled. This
makes sense in land use case studies, such as the ones in both Indiana and Altamira,
where the landscape is divided into a series of parcels owned by individual households.
However, this architecture could be modified to address cases where a variety of different
decision making entities, including individuals and private or public organizations, and
land parcels of various sizes.
5.1 The Importance of Heterogeneity
An important finding from the Indiana model is that the best model fits are
produced from a set of heterogeneous agents with varying preferences for diverse
landuses. This indicates that an agent-based approach allowing household heterogeneity
may provide more meaningful insights for developing polices. Modeling approaches that
32
aggregate agent dynamics are not able to represent these local level reactions and
individual responses to policies (Kelley and Evans under review).
The Altamira model also explicitly incorporates agent heterogeneity. Preliminary
analysis of an early version of the Altamira model which consisted of homogenous agents
produced model results that did not appear to accurately represent the actual land cover
outcomes from observed data. This led to the development of a subsequent version of the
model that allows for agent heterogeneity. In particular, agents now have diverse
household compositions (structure, age distribution, size) and land use decisions are in
part a function of these characteristics. For example, a particular threshold of household
labor supply can lead households to pursue particular decision paths that lead to different
land cover change outcomes (e.g. forest clearing). A more formal statistical assessment
of this approach is currently being conducted.
5.2Comparing Decision Strategies
The alternate decision making strategies outlined in the examples shown here
differ in a fundamental aspect. Despite the fact that all of these decision making
approaches have a common goal of maximizing the utility of the agent, they take
different approaches to exploring the solution space that exists in each round a
simulation. In the Altamira model, as the agent moves down through the decision tree the
number of final landuse decisions available to the agent diminish, thus reducing the area
of the solution space that is accessible to the agent in that particular round. In the Indiana
model, all possible landuse decisions are evaluated simultaneously, thus allowing an
agent to explore a larger portion of the solution space in any particular round of the
33
simulation. These alternate approaches can produce fundamental differences in the
behavior of the simulations.
As this research moves forward, we will be continuing to explore the effects of
alternate decision making strategies on the effectiveness of the simulations, while
continuing to explore techniques for grounding this research in empirical data. We have
been developing different decision strategies for agent behavior and are in the process of
a analytical comparison of these results (Kelley et al. in progress). These decision
strategies include a hill-climbing model, reinforcement learning and neural network
approaches. Initial findings from this line of research indicate that different decision
strategies can produce different landscape dynamics and outcomes dependent on various
factors. These differences are dependent on responses to exogenous prices, land
suitability and parcel context (externalities and household characteristics). These
alternative decision strategies are also being explored in a spatial experimental context
drawing on theories from experimental economics, geography and psychology. In this
analysis, model results and simulated agent decisions are compared to decisions by
subjects in computational experiments, allowing us to explore the diversity of decision
making strategies in the lab and in the model. Finally, we have tentative plans to take
these experiments to the field to test with landowners in our respective study areas.
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39
Figure 1: Location of Indian Creek Township, Indiana, USA
40
Figure 2: Location of Altamira, Brazil.
41
Political Module
Institutions/
Land use Policy
(Tax/Subsidy)
Social Module
Preferences, Risk
Attitude, Demographic
Cultural/Survey data
Agent i’s Decision
Goal: Improve U
Action: Alter Land Use
Agent 1
Biophysical Module
Land cover change
Slope/suitability,
Tree Growth, Soil
Agent 2
Economic Module
Prices
Land Market
Non-Ag Wages
Agent n
Externalities
Across Agents
Agent Decision
Making Strategies
Indiana
Expected Utility Maximization
Altamira
Decision tree based on heuristics
Simultaneous Comparison of
Alternate Available Decisions
Sequential Comparison of
Alternate Available Decisions
Figure 3: Conceptual systems model of land use decision making.
42
Alpha_farm
median = 16.4
Externality_farm
median = 2.1
Frequency
Frequency
10
8
6
4
2
0
-1
5
9
8
7
6
5
4
3
2
1
0
11 17 23 29 35 41 47 53 59 65
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Externality_aes
median = 1.6
10
10
8
8
Frequency
Frequency
Alpha_aes
median = 1.6
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Null RSQR
median = 0.65
10
10
8
8
Frequency
Frequency
Slope Dependence
median = 1.4
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Bin
Figure 4
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
RSQR Catagory
Histograms of by-parcel fitted parameters and Null_R 2s using Percent Forest spatial metric
to summarize actual and simulated landscapes.
43
Alpha_farm
median = 11.6
Externality_farm
median = 2.1
10
8
Frequency
Frequency
10
8
6
4
6
4
2
2
0
0
-1
5
11 17 23 29 35 41 47 53 59 65
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Externality_aes
median = 1.7
10
10
8
8
Frequency
Frequency
Alpha_aes
median = 1.8
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Null RSQR
median = 0.73
10
10
8
8
Frequency
Frequency
Slope Dependence
median = 1.8
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Bin
Figure 5
RSQR Catagory
2
Histograms of by-parcel fitted parameters and Null_R s using Forest Edge spatial metric
to summarize actual and simulated landscapes.
44
1
Alpha_farm
median = 15.5
Externality_farm
median = 2.1
10
8
Frequency
Frequency
10
8
6
4
6
4
2
2
0
0
-1
5
11 17 23 29 35 41 47 53 59 65
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Externality_aes
median = 2.4
10
10
8
8
Frequency
Frequency
Alpha_aes
median = 0.9
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Parameter magnitude catagory
Parameter magnitude catagory
Null RSQR
median = 0.39
10
10
8
8
Frequency
Frequency
Slope Dependence
median = 2.2
6
4
2
0
6
4
2
0
-1
0.75 1.75 2.75 3.75 4.75 5.75 6.75 7.75
Bin
Figure 6
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RSQR Catagory
Histograms of by-parcel fitted parameters and Null_R 2s using an equal weight,
weighted average of the Percent Forest and Forest Edge spatial metrics to summarize
actual and simulated landscapes.
45
1
Land-Cover Trends
% Annuals
% Perennials
4.00%
% Pasture
3.50%
Percent of Study Area
3.00%
2.50%
2.00%
1.50%
1.00%
0.50%
0.00%
1
2
3
4
5
6
7
8
9
10
11 12
13
14 15
16
17 18
19
20 21
22
23 24
25
26 27
28
29 30
Iteration
Figure 7: Crop & Pasture Land-Cover Trends in Percent of the Study Area over Time
46
Land-Cover Trends
% Mature Forest
100.00%
% Fallow & Secondary
Succession
90.00%
Percent of Study Area
80.00%
70.00%
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Iteration
Figure 8: Mature Forest & Secondary Succession and Fallow Land-Cover Trends in
Percent of the Study Area over Time
47
Figure 9: Comparison of Percentage of Land-Cover Types between Results of LUCITA
and Findings by Mausel et al. (1993).
48