17. Mathematical Methods for Identifying Representative Reserve Networks
299
Generally, we would like to cluster sites in a reserve system; however, there can
be compelling reasons to avoid clustering. Where catastrophes can impact large
areas and cause local extinctions, it may be less risky to conserve each species in
at least two or three separate places, rather than clustering those sites. Issues such
as boundary length, spatial constraints, and costs make the problem significantly
more complex. Mathematically, the problem becomes nonlinear, because the cost
of adding a site to the reserve system depends on which other sites are already
protected and on the spatial relationships between candidate sites and those already reserved. One formulation of a spatial reserve design problem is shown in
the following example for section 342I of the Columbia Plateau ecoregion (Bailey
1994). There are 821 sites (subwatersheds) of different sizes. The elements of
biodiversity to be conserved in this problem are 113 species, rare plant communities, and common coarse-scale vegetation types. The rare species and plant
community data are in the form of presence-absence data, whereas occurrences of
vegetation types are measured in the number of hectares of that type found in each
site. Thus, in this problem, there are two types of target representation levels, each
for different “types” of biodiversity, expressed as ones and zeros for presenceabsence, or as area.
When using greedy and rarity-based algorithms, clustering of reserve sites is
often achieved by including an adjacency constraint (Nicholls and Margules
1993). Another approach, used here, is to try to minimize the boundary length of
the reserve system. For a given area, a smaller boundary length gives a more
compact area. To minimize both boundary length and area in this problem, we
introduce a boundary length modifier (BLM). The objective to be minimized is
now area + boundary length × BLM. By varying the BLM, the relative importance
of compactness and size can be balanced. If the BLM is set to zero, the algorithm
will ignore boundary length.
A simple measure of the degree of clustering among reserves in the network is
the boundary length of the reserve system divided by the area. This is a measure of
length per unit area and is intuitive. A more advanced measure is the ratio of the
boundary length of the reserve system to the circumference of a circle with the
same area as the reserve. A circle is the most compact shape possible, so this is the
ratio of the boundary length to the theoretical minimum. As such, it is a dimensionless measure. The formula for this measure is
ratio =
boundary length
2√π × area
As shown in Figure 17.1, as the boundary length modifier is increased, both the
boundary length and boundary length/area measures decrease. This occurs at the
expense of area. Table 17.4 indicates that when the largest boundary length
modifier is used, two-thirds of the area is reserved, and the boundary length is
minimized. The best balance between total area and clustering seems to be
achieved, with a BLM between 0.5 and 1. Here, the area is increasing, but the
boundary length is decreasing at a greater rate.
299
Generally, we would like to cluster sites in a reserve system; however, there can
be compelling reasons to avoid clustering. Where catastrophes can impact large
areas and cause local extinctions, it may be less risky to conserve each species in
at least two or three separate places, rather than clustering those sites. Issues such
as boundary length, spatial constraints, and costs make the problem significantly
more complex. Mathematically, the problem becomes nonlinear, because the cost
of adding a site to the reserve system depends on which other sites are already
protected and on the spatial relationships between candidate sites and those already reserved. One formulation of a spatial reserve design problem is shown in
the following example for section 342I of the Columbia Plateau ecoregion (Bailey
1994). There are 821 sites (subwatersheds) of different sizes. The elements of
biodiversity to be conserved in this problem are 113 species, rare plant communities, and common coarse-scale vegetation types. The rare species and plant
community data are in the form of presence-absence data, whereas occurrences of
vegetation types are measured in the number of hectares of that type found in each
site. Thus, in this problem, there are two types of target representation levels, each
for different “types” of biodiversity, expressed as ones and zeros for presenceabsence, or as area.
When using greedy and rarity-based algorithms, clustering of reserve sites is
often achieved by including an adjacency constraint (Nicholls and Margules
1993). Another approach, used here, is to try to minimize the boundary length of
the reserve system. For a given area, a smaller boundary length gives a more
compact area. To minimize both boundary length and area in this problem, we
introduce a boundary length modifier (BLM). The objective to be minimized is
now area + boundary length × BLM. By varying the BLM, the relative importance
of compactness and size can be balanced. If the BLM is set to zero, the algorithm
will ignore boundary length.
A simple measure of the degree of clustering among reserves in the network is
the boundary length of the reserve system divided by the area. This is a measure of
length per unit area and is intuitive. A more advanced measure is the ratio of the
boundary length of the reserve system to the circumference of a circle with the
same area as the reserve. A circle is the most compact shape possible, so this is the
ratio of the boundary length to the theoretical minimum. As such, it is a dimensionless measure. The formula for this measure is
ratio =
boundary length
2√π × area
As shown in Figure 17.1, as the boundary length modifier is increased, both the
boundary length and boundary length/area measures decrease. This occurs at the
expense of area. Table 17.4 indicates that when the largest boundary length
modifier is used, two-thirds of the area is reserved, and the boundary length is
minimized. The best balance between total area and clustering seems to be
achieved, with a BLM between 0.5 and 1. Here, the area is increasing, but the
boundary length is decreasing at a greater rate.