Introduction to Models

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Introduction to Models

• Lecture 8
• February 22, 2005

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Why use any models?

• Models help us to generate or test hypotheses.
• To formally organize ideas or data.
• To provide a framework for making comparisons.

• Identify areas of understanding
• Identify range of variability
• Identify sensitive parameters

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Why use any models?

• To interpolate or extrapolate understanding,
  often across scales.
• Management applications - make predictions or
  test different management scenarios.

• To explore scenarios where experiments are not
  easily conducted.

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Why use landscape models?
Input
Output
Model
???

• Spatial and temporal constraints on landscape
  studies
• Experiments on large areas are difficult.
• Even more difficult to replicate experiments or
  even "sample" and analyze replicates.
• Many large-scale processes operate slowly, so
  landscapes also change slowly.

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Three Model Types

• Operationally, useful to think of three general
  types of landscape models
• Neutral Models
• Landscape change models
• Land cover classes, ecosystem types, or habitats
• Influenced by natural or anthropogenic processes
• Includes landscape process models
• Individual-based models

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Neutral Landscape Models
Neutral landscape models generate raster maps in
which complex habitat structures are generated
with analytical algorithms. Thus, they are
neutral to the biological and physical processes
that shape real landscape patterns.
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Neutral Landscape Models
What is the value of neutral models?
Statistical How do structural properties of
landscapes deviate from theoretical spatial
distributions? Modeling How are ecological
processes affected by landscape pattern? Neutral
models DO NOT represent actual landscapes!!!!!
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Neutral Landscape Models
Neutral landscape models may be generated by
random, hierarchical, or fractal algorithms.
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Neutral Landscape Models
Simple random maps
p 0.4
p 0.6
p 0.8
One class
Multiple classes
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Percolation Theory and Neutral Models

• As an increasingly large proportion of the
  landscape is occupied, the occupied cells
  coalesce into larger patches.
• Once p 0.5928 (0.41 for the 8-neighbor rule),
  the largest
• cluster will span the map
• edge-to-edge.
• Important since all landscape
• metrics covary with p.

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Example Neighborhood Rules examined using
Neutral Landscape Models
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Neutral Landscape Models

• General insights
• Threshold effects occur as nonlinear
  relationships between patterns or processes and
  p.
• Neutral landscape models are very important for
  calibrating and understanding different measures
  of landscape pattern - what is the expected
  range?
• Concepts from Neutral Models can be applied to
  Landscape Change Models - What happens if I turn
  on/off process X?
• Specific results of neutral models do not
  necessarily apply to any actual landscapes, but
  the insights of the models do apply.

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Landscape Change ModelsIntroduction

1. Landscape change models simulate pattern change
   or state change in a landscape .
2. Most landscape models are different ways of
   conceptualizing the interactions between three
   general areas abiotic template, biotic
   interactions, disturbances.
3. Depending on needs, a model may need to include
   processes operating within any of these three
   areas.
4. All landscape change models include some
   processes.
5. Questions and scales determine which processes to
   include.

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Landscape Change Models Simple Markov Models
Markov models To predict the state of the
system at time t1, you only need to know the
state of the system at time t and the probability
of transition. (first-order)
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mid- seral closed
mid- seral open
early seral
late- seral closed
late- seral open
succession
low intensity fire
high intensity fire
Ponderosa pine forest
thinning
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Landscape Change Models Simple Markov Models

• Markov Models
• Requires a Transition Probability Matrix (TPM)
• TPM may be derived from landscape data collected
  at two time points.
• TPM may be derived from expert opinion.

Harvard Forest Dioramas
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Landscape Change Models Projecting Markov Models
The transition matrix is invoked on a
cell-by-cell basis.
The resulting projected landscape is a stochastic
outcome of the transition probabilities.
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Landscape Change Models Problems with Simple
Models
Historical Influences If the transition
probabilities depend on more than the immediately
prior state, then the system retains a memory
of antecedent conditions If so, the dynamics are
not first-order.
pBG f(time-1, time-2)
pBG
Cellxy t
Cellxy time-1
Cellxy time-2
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Landscape Change Models Problems with Simple
Models
History (time lags) The transition
probabilities become conditional based on the age
of the site (transitions occur only after the
site has been in a certain state for some
time). Time lags are particularly important for
disturbance events.
Old-field succession
Old Field
Pine Forest
Fire
Time
p 0
p 0.2
p 0.8
Transition probabilities
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Landscape Change Models Problems with Simple
Models
History (antecedent events) The transition
probabilities become conditional based on whether
a particular antecedent event occurred.
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Landscape Change Models Problems with Simple
Models
Spatial dependencies - Covariates The
transition probabilities become conditional
probabilities based on some ancillary information
about the covariates.
Land Cover Classes
Soil type
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Landscape Change Models Problems with Simple
Models
Spatial dependencies - Neighborhood Effects The
transition probabilities become conditional
probabilities based on the state of neighboring
cells surrounding the focal cell.
Cellular automata models are best suited to model
neighborhood effects (vs. Markov models).
Probability of green occupying cell 3/8
0.375 Probability of blue occupying cell
0.125 Probability of magenta occupying cell
0.125 Probability of red occupying cell
0.25 Probability of navy occupying cell 0.125
Focal cell
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Landscape Change Models Problems with Simple
Models
Nonstationarity The transition matrix varies
over time (i.e., the probabilities are not
constant) which implies that the rules
governing landscape change are changing over time.
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Landscape Change Models Problems with Simple
Models
Nonstationarity Solution Calculate new
transition matrices for each time period of
interest, or calculate transitions as functions
of time.
Harvard Forest Dioramas
Transition Matrix 1740-1850
Transition Matrix 1850-1910
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Landscape Change Models Problems with Simple
Models
Disturbance Disturbances are a special case in
modeling, because they are an integration of all
the special cases affecting transition
probabilities. Disturbances (e.g., fires) may be
physically constrained (spatial covariates), may
spread contagiously (neighborhood effects), may
be lagged in time (time lags), and may change
through time (nonstationarity), and may be
stochastic.
Insects
Fire
Wind
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Cellular Models

• Cellular automata models "systems of cells
  interacting in a simple way but displaying
  complex overall behavior" (Phipps 1992)
• System of cell networks or grids
• Cells interact with neighborhood
• Each cell adopts one of m (m may be infinite)
  possible states
• Transition rules for each state can be simple,
  deterministic, or stochastic.
• Transition rules f(abiotic constraints, biotic
  interactions, disturbances)