14
Applications of Fractal Geometry
and Percolation Theory to
Landscape Analysis and Assessments
Bai-Lian Li
14.1 Introduction
Landscapes are ecologically distinct because the
flows of resources, organisms, pollutants, and sediment through space are controlled to some extent
by the connections between points on the landscape. Landscape composition and pattern influence the nature and magnitude of ecological
processes at a variety of spatiotemporal scales
(O'Neill et aI., 1988; Wiens, 1992; Li and Archer,
1997). This situation results from a set of biophysical constraints and processes, including geologic history, soils, topography, and climate.
Changes in the distribution and pattern of ecological resources (e.g., woodlands, rangeland, streams
and wetlands) and human activities can alter fundamental ecological processes, including the flows
and balances of water, nutrients, energy, and biota.
These changes in ecological processes can, in tum,
influence many aspects of the environment valued
by society. Characterizing and assessing landscapes at multiple spatiotemporal scales enables
managers, ecologists, and designers to identify the
causes of ecological change according to the scale
at which the relevant ecological forces operate.
Management decisions can also be made once the
ecological implications of the location, juxtaposition, and the flow of resources, pollutants, and
species are known.
Many thanks to Steve Archer, Craig LoeWe and Hsin-i
Wu for collaboration on these research subjects during
the last eight years. Mike Fuller and How Passell provided editorial assistance. This work was partially supported by the U.S. National Science Foundation (BSR91-09240, DEB-93-06679, and DEB-94-11976), DOE!
Sandia National Laboratories (BE-0229), the University
of New Mexico, and USDA Forest Service-Northern Region. This is Sevilleta L TER publication no. 136.
200
For two decades or more, spatial (landscape)
analysis has been dominated by a style of model
building that has sought high predictive understanding in numerical terms, but has paid little attention to the geometry of spatial form. Mandelbrot's (1983) concept of a fractal, one of the
fast-moving research fronts coupled with concepts
of complexity, criticality, and self-organization, extends our usual ideas of classical geometry beyond
those of point, line, circle, and so on, into the realm
of the irregular, disjoint, and singular. Fractals represent many kinds of patterns, including density,
diversity, dendritic stream networks, geometrical
shapes, mountainous terrain, and size distributions
of islands (Mandelbrot, 1983). It has the potential
to provide us with a new way to understand and
analyze such natural spatial phenomena, which are
not smooth, but rough and fragmented to selfsimilarity or self-affinity at all scales. Percolation
represents the simplest model of a disordered system. Percolation transition is a simple example of
a geometrical phase transition phenomenon (Stauffer, 1985; Sahimi, 1994). The percolation model,
together with fractal analysis, can provide useful
insight to assess landscape dynamics (Loehle et aI.,
1996).
Landscapes are spatially heterogeneous, and the
structure, function, and change of landscapes are
themselves scale dependent. Heterogeneity of environmental resources, succession, and disturbance
result in landscape patches of diverse size, shape,
type, and ecotone (or boundary) characteristics.
The patch characteristics may be important factors
in ecological diversity, stability, and function. The
geometric features of heterogeneity, multiple
scales, and self-similarity-affinity are characteristic of a variety of patch spatial patterns in landscapes. More generally, there are pansymmetries of
Applications of Fractal Geometry
and Percolation Theory to
Landscape Analysis and Assessments
Bai-Lian Li
14.1 Introduction
Landscapes are ecologically distinct because the
flows of resources, organisms, pollutants, and sediment through space are controlled to some extent
by the connections between points on the landscape. Landscape composition and pattern influence the nature and magnitude of ecological
processes at a variety of spatiotemporal scales
(O'Neill et aI., 1988; Wiens, 1992; Li and Archer,
1997). This situation results from a set of biophysical constraints and processes, including geologic history, soils, topography, and climate.
Changes in the distribution and pattern of ecological resources (e.g., woodlands, rangeland, streams
and wetlands) and human activities can alter fundamental ecological processes, including the flows
and balances of water, nutrients, energy, and biota.
These changes in ecological processes can, in tum,
influence many aspects of the environment valued
by society. Characterizing and assessing landscapes at multiple spatiotemporal scales enables
managers, ecologists, and designers to identify the
causes of ecological change according to the scale
at which the relevant ecological forces operate.
Management decisions can also be made once the
ecological implications of the location, juxtaposition, and the flow of resources, pollutants, and
species are known.
Many thanks to Steve Archer, Craig LoeWe and Hsin-i
Wu for collaboration on these research subjects during
the last eight years. Mike Fuller and How Passell provided editorial assistance. This work was partially supported by the U.S. National Science Foundation (BSR91-09240, DEB-93-06679, and DEB-94-11976), DOE!
Sandia National Laboratories (BE-0229), the University
of New Mexico, and USDA Forest Service-Northern Region. This is Sevilleta L TER publication no. 136.
200
For two decades or more, spatial (landscape)
analysis has been dominated by a style of model
building that has sought high predictive understanding in numerical terms, but has paid little attention to the geometry of spatial form. Mandelbrot's (1983) concept of a fractal, one of the
fast-moving research fronts coupled with concepts
of complexity, criticality, and self-organization, extends our usual ideas of classical geometry beyond
those of point, line, circle, and so on, into the realm
of the irregular, disjoint, and singular. Fractals represent many kinds of patterns, including density,
diversity, dendritic stream networks, geometrical
shapes, mountainous terrain, and size distributions
of islands (Mandelbrot, 1983). It has the potential
to provide us with a new way to understand and
analyze such natural spatial phenomena, which are
not smooth, but rough and fragmented to selfsimilarity or self-affinity at all scales. Percolation
represents the simplest model of a disordered system. Percolation transition is a simple example of
a geometrical phase transition phenomenon (Stauffer, 1985; Sahimi, 1994). The percolation model,
together with fractal analysis, can provide useful
insight to assess landscape dynamics (Loehle et aI.,
1996).
Landscapes are spatially heterogeneous, and the
structure, function, and change of landscapes are
themselves scale dependent. Heterogeneity of environmental resources, succession, and disturbance
result in landscape patches of diverse size, shape,
type, and ecotone (or boundary) characteristics.
The patch characteristics may be important factors
in ecological diversity, stability, and function. The
geometric features of heterogeneity, multiple
scales, and self-similarity-affinity are characteristic of a variety of patch spatial patterns in landscapes. More generally, there are pansymmetries of