14.6 Landscape Phase Transitions and Percolation Theory
transition) and continuous (second-order phase
transition). The point of phase transition is usually
called the critical point. Strictly speaking, it is not
necessary that a transition be realized just at a point.
For example, an ecotone can be considered a phase
transitional area or zone on [a, b).
This section will introduce a simplest phase transition model; the percolation model, together with
fractal analysis, to assess ecotonal landscape dynamics (Loehle et aI., 1996). Percolation processes
were first developed by Flory (1941) and Stockmayer (1943) to describe how small branching molecules react and form very large macromolecules.
This process is very similar to the expansion of forest or shrubs in grassland (Li et aI., 1992). In mathematical literature, percolation was introduced by
Broadbent and Hammersley (1957). They originally dealt with the concept of the spread of hypothetical fluid particles through a random medium.
The terms fluid and medium were viewed as totally
general: a fluid can be liquid, vapor, heat flux, infection, fire or forest spreading, and so on. The
medium where the fluid is carried can be the pore
space of a subsurface, an array of trees, grassland,
or the universe. Percolation provides an intuitively
appealing and transparent model for dealing with
the unruly geometries that occur in many random
media. The percolation threshold is a prototypical
phase transition that occurs as we vary the richness
of interconnections present (a generalized density
or composition). For a detailed introduction to percolation models and theory, see Stauffer (1985) and
Sahimi (1994). In ecology, percolation models have
been applied to the spread of epidemics (Mollison,
1977, 1986; Grassberger, 1983, 1985; Cox and
Durrett, 1988), forest fire (MacKay and Jan, 1984),
microbial transport (Li et aI., 1996), and ecotone
and landscape ecology (Gardner et aI., 1987; Milne
et al., 1996).
Milne et al. (1996) used critical densities or percolation thresholds of percolation models to identify
woodland ecotonal phase transitions. In percolation
theory, the numerical value of every percolation
quantity for any percolation probability P depends
on the microscopic details of the system, such as its
configuration or neighborhood. But near the bond
or site percolation threshold Pc> most percolation
quantities obey scaling laws that are largely insensitive to the network structure and its microscopic
details. However, the percolation threshold Pc still
varies with configuration or neighborhood. Percolation theory tells us that the topological exponents
including fractal dimension near Pc are completely
universal (Stauffer, 1985; Sahimi, 1994). They are
independent of the microscopic details of the sys207
tem and depend only on the dimensionality of the
system. This discovery allows us to directly use
fractal dimensions to detect spatial phase transitions of an ecotone, regardless of the spatial configuration of the object of concern.
Based on percolation theory and computer experiment simulation of general random percolation
transitions (Stauffer, 1985; Stinchcombe, 1990;
Sahimi, 1994), the relation between critical exponents and a dimension of a fractal occurring in percolation is FD = d - {3lv, where d is the Euclidean dimension, {3 and v are the critical exponents,
and (-{3lv) can be regarded as the length-scaling
exponent. Based on currently accepted values of
critical exponents (Stauffer, 1985; Sahimi, 1994),
we have fractal dimensions, FD(P = Pc) = 1.8958,
FD(P > Pc) = 2, and FD(P < Pc) = 1.56. Here let
us consider the forest spread as a dynamic percolation process. Forest percent cover is percolation
probability. When forest cover is below the percolation threshold Pc, the landscape is dominated by
tallgrass prairie. When forest cover is above Pc> the
forest invasion can spread through the entire landscape, which landscape becomes forest dominant.
On the basis of this intuitive argument, we suggested that fractal dimensions of forest spatial distributions could be used to characterize ecotones in
landscapes and should range between 1.56 and
1.8958. This result can also be used to monitor ecotonal movement and responses to climate change.
Spatial phase transitions of an ecotone between
tall grass prairie and forest are complex. In general,
if environmental or management conditions change
in ways that are beneficial for one of the adjacent
ecosystems, patch size is likely to increase in this
system, and it is likely to invade habitats previously
unsuitable in the adjacent system (Risser, 1995).
Let us assume that 70% of the driving forces (or
controlling factors and constraints) support forest
spread. We could simply calculate the new fractal
dimension of this critical transition in the twodimensional lattice as
FD70% = 1.8958 - (1.8958 - 1.56)
X (1 - 0.7) = 1.7951.
And if there are 80% driving forces, we have
FDSO% = 1.8286. We could use the critical fractal
dimension to detect a phase transition at a forest-tallgrass prairie ecotone. Now we can state our
hypothesis in this study; that is, if the fractal dimension of forest cover is above 1.7951 or 1.8286,
the forest invasion at the forest-prairie ecotone may
spread to the entire system. We tested this hypothesis based on actual and simulated data of forest
transition) and continuous (second-order phase
transition). The point of phase transition is usually
called the critical point. Strictly speaking, it is not
necessary that a transition be realized just at a point.
For example, an ecotone can be considered a phase
transitional area or zone on [a, b).
This section will introduce a simplest phase transition model; the percolation model, together with
fractal analysis, to assess ecotonal landscape dynamics (Loehle et aI., 1996). Percolation processes
were first developed by Flory (1941) and Stockmayer (1943) to describe how small branching molecules react and form very large macromolecules.
This process is very similar to the expansion of forest or shrubs in grassland (Li et aI., 1992). In mathematical literature, percolation was introduced by
Broadbent and Hammersley (1957). They originally dealt with the concept of the spread of hypothetical fluid particles through a random medium.
The terms fluid and medium were viewed as totally
general: a fluid can be liquid, vapor, heat flux, infection, fire or forest spreading, and so on. The
medium where the fluid is carried can be the pore
space of a subsurface, an array of trees, grassland,
or the universe. Percolation provides an intuitively
appealing and transparent model for dealing with
the unruly geometries that occur in many random
media. The percolation threshold is a prototypical
phase transition that occurs as we vary the richness
of interconnections present (a generalized density
or composition). For a detailed introduction to percolation models and theory, see Stauffer (1985) and
Sahimi (1994). In ecology, percolation models have
been applied to the spread of epidemics (Mollison,
1977, 1986; Grassberger, 1983, 1985; Cox and
Durrett, 1988), forest fire (MacKay and Jan, 1984),
microbial transport (Li et aI., 1996), and ecotone
and landscape ecology (Gardner et aI., 1987; Milne
et al., 1996).
Milne et al. (1996) used critical densities or percolation thresholds of percolation models to identify
woodland ecotonal phase transitions. In percolation
theory, the numerical value of every percolation
quantity for any percolation probability P depends
on the microscopic details of the system, such as its
configuration or neighborhood. But near the bond
or site percolation threshold Pc> most percolation
quantities obey scaling laws that are largely insensitive to the network structure and its microscopic
details. However, the percolation threshold Pc still
varies with configuration or neighborhood. Percolation theory tells us that the topological exponents
including fractal dimension near Pc are completely
universal (Stauffer, 1985; Sahimi, 1994). They are
independent of the microscopic details of the sys207
tem and depend only on the dimensionality of the
system. This discovery allows us to directly use
fractal dimensions to detect spatial phase transitions of an ecotone, regardless of the spatial configuration of the object of concern.
Based on percolation theory and computer experiment simulation of general random percolation
transitions (Stauffer, 1985; Stinchcombe, 1990;
Sahimi, 1994), the relation between critical exponents and a dimension of a fractal occurring in percolation is FD = d - {3lv, where d is the Euclidean dimension, {3 and v are the critical exponents,
and (-{3lv) can be regarded as the length-scaling
exponent. Based on currently accepted values of
critical exponents (Stauffer, 1985; Sahimi, 1994),
we have fractal dimensions, FD(P = Pc) = 1.8958,
FD(P > Pc) = 2, and FD(P < Pc) = 1.56. Here let
us consider the forest spread as a dynamic percolation process. Forest percent cover is percolation
probability. When forest cover is below the percolation threshold Pc, the landscape is dominated by
tallgrass prairie. When forest cover is above Pc> the
forest invasion can spread through the entire landscape, which landscape becomes forest dominant.
On the basis of this intuitive argument, we suggested that fractal dimensions of forest spatial distributions could be used to characterize ecotones in
landscapes and should range between 1.56 and
1.8958. This result can also be used to monitor ecotonal movement and responses to climate change.
Spatial phase transitions of an ecotone between
tall grass prairie and forest are complex. In general,
if environmental or management conditions change
in ways that are beneficial for one of the adjacent
ecosystems, patch size is likely to increase in this
system, and it is likely to invade habitats previously
unsuitable in the adjacent system (Risser, 1995).
Let us assume that 70% of the driving forces (or
controlling factors and constraints) support forest
spread. We could simply calculate the new fractal
dimension of this critical transition in the twodimensional lattice as
FD70% = 1.8958 - (1.8958 - 1.56)
X (1 - 0.7) = 1.7951.
And if there are 80% driving forces, we have
FDSO% = 1.8286. We could use the critical fractal
dimension to detect a phase transition at a forest-tallgrass prairie ecotone. Now we can state our
hypothesis in this study; that is, if the fractal dimension of forest cover is above 1.7951 or 1.8286,
the forest invasion at the forest-prairie ecotone may
spread to the entire system. We tested this hypothesis based on actual and simulated data of forest