18.2 Models of Ecosystem Structure
Markov models, in their traditional form, are
limited by the assumption of stationarity (i.e., probabilities do not change with time), as well as their
lack of mechanism for transitions, fixed time step,
and lack of spatial and historical effects. Modifications to the models have been implemented to
overcome these limitations. However, model extensions can significantly increase data requirements and processing time. Nonstationarity of transition can be incorporated by constructing more
than one stationary matrix, with a process in the
model for switching among matrices (Baker,
1989a). The influence of environmental variables
has been incorporated by making the transition
probabilities functions of these variables, including
nonlinear functions. Examples of environmental
variables that have been included in Markov models are climate attributes, fire frequency, and probability of insect attack (Henderson and Wilkins,
1975; Marsden, 1983; Woolhouse and Harmsen,
1987). Semi-Markov models can be used to simulate transition probabilities that vary with time
(Ginsberg, 1971, 1972). Functions are developed
that determine the duration of stay or sojourn time
in each state (Baker, 1989a). The effects of landscape heterogeneity on changes in state, in which
transitions may depend on location in the landscape, can be simulated by developing individual
transition matrices for homogeneous subareas of
the landscape (Baker, 1989a). The effect of previous history can be modeled by defining matrices
that include both current and preceding states. This,
however, results in an exponential increase in the
number of transition probabilities that must be estimated (Baker, 1989a).
Spatial automata offer the advantage of explicit
inclusion of spatial effects within a neighborhood
(Shugart, 1998). However, some choices of modeling structure can lead to intractable complexity.
Model complexity is determined by the number of
possible neighborhood configurations, a function
of the number of states, neighborhood size, and
method of tallying neighborhood states. Childress
et ai. (1996) present formulas for calculating the
number of neighborhood configurations. The
method of tallying neighborhood states can include
identifying each cell uniquely by its spatial position, the unique neighbor method, or tallying the
number of cells without considering their spatial
position, the voting method (Childress et aI., 1996).
Use of the voting method can reduce the number
of neighborhood configurations by several orders
of magnitude and is likely to meet the needs of
many ecological applications, with the exception of
those in which a gradient is included in the model
grid (Childress et aI., 1996).
259
Pastor et ai. (1993) developed a Markov model
to evaluate historical landscape changes caused by
beaver pond construction in Voyageurs National
Park, Minnesota. Aerial photographs taken over a
50-year period were the basis for delineating four
hydrologic regimes in maps of floodable areas.
Baker (1989b) used a set of Markov models to determine whether a stable mosaic of fire-induced
vegetation patches was present in the Boundary
Waters Canoe Area, Minnesota. Acevedo et ai.
(1996) implemented a semi-Markov model for forest dynamics at a landscape scale, estimating transition probabilities and holding times within states
for tree functional types from a series of runs of an
individual-based plant model (ZELlG, discussed
later). This enabled integration of both landscape
and local scales in the modeling effort. Hall et ai.
(1988) believed that landscape transition models
could be coupled to general atmospheric circulation models to project changes in vegetation as a
result of global climate change; they proposed substituting observed vegetation transition probabilities over a spatial climatic gradient for probabilities of transition as a result of climate change over
time. Often, insects or disease-causing organisms
operate on very different spatial and temporal
scales than their hosts (Turner and Dale, 1991).
Dale et ai. (1991) developed two separate Leslie
matrix models to couple the dynamics of a tree
species, the Fraser fir, with an insect predator, the
balsam woolly aphid, in the southern Appalachians. Trees were modeled with an annual time step,
whereas insects were modeled using a two-day time
step. Green (1989) used spatial automata to simulate changes in tree locations in response to the effects of fire, seed dispersal, and environmental gradients on a rectangular grid. Thiery et al. (1995)
developed a spatial automata model to examine
change in tiger-bush vegetation in Niger, which has
a distinctive striped spatial pattern. The model was
able to reproduce the vegetation pattern when initialized with a random pattern (Shugart, 1998).
18.2.2 Individual-based Plant Models
The group of stochastic individual-based plant
models derived from the JAB OW A model (Botkin
et aI., 1972) has been used to model vegetation distributions throughout the world (Hunsaker et aI.,
1993). Models have been developed for a wide
range of forests (Urban and Shugart, 1992), in addition to a grassland (Coffin and Lauenroth, 1990,
1996) and an alpine meadow (Humphries et aI.,
1996). These models track the establishment,
growth, and death of individual plants on a small
plot using an annual time step. As a consequence
Markov models, in their traditional form, are
limited by the assumption of stationarity (i.e., probabilities do not change with time), as well as their
lack of mechanism for transitions, fixed time step,
and lack of spatial and historical effects. Modifications to the models have been implemented to
overcome these limitations. However, model extensions can significantly increase data requirements and processing time. Nonstationarity of transition can be incorporated by constructing more
than one stationary matrix, with a process in the
model for switching among matrices (Baker,
1989a). The influence of environmental variables
has been incorporated by making the transition
probabilities functions of these variables, including
nonlinear functions. Examples of environmental
variables that have been included in Markov models are climate attributes, fire frequency, and probability of insect attack (Henderson and Wilkins,
1975; Marsden, 1983; Woolhouse and Harmsen,
1987). Semi-Markov models can be used to simulate transition probabilities that vary with time
(Ginsberg, 1971, 1972). Functions are developed
that determine the duration of stay or sojourn time
in each state (Baker, 1989a). The effects of landscape heterogeneity on changes in state, in which
transitions may depend on location in the landscape, can be simulated by developing individual
transition matrices for homogeneous subareas of
the landscape (Baker, 1989a). The effect of previous history can be modeled by defining matrices
that include both current and preceding states. This,
however, results in an exponential increase in the
number of transition probabilities that must be estimated (Baker, 1989a).
Spatial automata offer the advantage of explicit
inclusion of spatial effects within a neighborhood
(Shugart, 1998). However, some choices of modeling structure can lead to intractable complexity.
Model complexity is determined by the number of
possible neighborhood configurations, a function
of the number of states, neighborhood size, and
method of tallying neighborhood states. Childress
et ai. (1996) present formulas for calculating the
number of neighborhood configurations. The
method of tallying neighborhood states can include
identifying each cell uniquely by its spatial position, the unique neighbor method, or tallying the
number of cells without considering their spatial
position, the voting method (Childress et aI., 1996).
Use of the voting method can reduce the number
of neighborhood configurations by several orders
of magnitude and is likely to meet the needs of
many ecological applications, with the exception of
those in which a gradient is included in the model
grid (Childress et aI., 1996).
259
Pastor et ai. (1993) developed a Markov model
to evaluate historical landscape changes caused by
beaver pond construction in Voyageurs National
Park, Minnesota. Aerial photographs taken over a
50-year period were the basis for delineating four
hydrologic regimes in maps of floodable areas.
Baker (1989b) used a set of Markov models to determine whether a stable mosaic of fire-induced
vegetation patches was present in the Boundary
Waters Canoe Area, Minnesota. Acevedo et ai.
(1996) implemented a semi-Markov model for forest dynamics at a landscape scale, estimating transition probabilities and holding times within states
for tree functional types from a series of runs of an
individual-based plant model (ZELlG, discussed
later). This enabled integration of both landscape
and local scales in the modeling effort. Hall et ai.
(1988) believed that landscape transition models
could be coupled to general atmospheric circulation models to project changes in vegetation as a
result of global climate change; they proposed substituting observed vegetation transition probabilities over a spatial climatic gradient for probabilities of transition as a result of climate change over
time. Often, insects or disease-causing organisms
operate on very different spatial and temporal
scales than their hosts (Turner and Dale, 1991).
Dale et ai. (1991) developed two separate Leslie
matrix models to couple the dynamics of a tree
species, the Fraser fir, with an insect predator, the
balsam woolly aphid, in the southern Appalachians. Trees were modeled with an annual time step,
whereas insects were modeled using a two-day time
step. Green (1989) used spatial automata to simulate changes in tree locations in response to the effects of fire, seed dispersal, and environmental gradients on a rectangular grid. Thiery et al. (1995)
developed a spatial automata model to examine
change in tiger-bush vegetation in Niger, which has
a distinctive striped spatial pattern. The model was
able to reproduce the vegetation pattern when initialized with a random pattern (Shugart, 1998).
18.2.2 Individual-based Plant Models
The group of stochastic individual-based plant
models derived from the JAB OW A model (Botkin
et aI., 1972) has been used to model vegetation distributions throughout the world (Hunsaker et aI.,
1993). Models have been developed for a wide
range of forests (Urban and Shugart, 1992), in addition to a grassland (Coffin and Lauenroth, 1990,
1996) and an alpine meadow (Humphries et aI.,
1996). These models track the establishment,
growth, and death of individual plants on a small
plot using an annual time step. As a consequence