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Average optimacity: An index to guide site prioritization for biodiversity conservation
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journal homepage: www.elsevier.com/locate/biocon
Average optimacity: An index to guide site prioritization for biodiversity conservation George F. Wilherea,*, Mark Goeringb, Huilin Wanga a
Washington Department of Fish and Wildlife, 600 Capitol Way North, Olympia, Washington 98501-1091, USA The Nature Conservancy, 1917 1st Avenue, Seattle, Washington 98101, USA
b
A R T I C L E I N F O
A B S T R A C T
Article history:
We developed an index, called average optimacity, that can be used to guide site prioritiza-
Received 3 April 2007
tion for biodiversity conservation. Our index has three desirable properties: (1) it indicates a
Received in revised form
site’s potential contribution to an efficient representative reserve system; (2) it equals zero
28 December 2007
if and only if a site contains no elements of biodiversity; and (3) it accommodates multiple
Accepted 31 December 2007
sets of representation targets. The index is calculated with an optimization algorithm – simulated annealing as implemented in the computer program Marxan. We ran Marxan with 10 different representation target sets and averaged the results. The 10 target sets
Keywords:
approximated equal increments in conservation value. Data from the Willamette Valley-
Conservation planning
Puget Sound-Georgia Basin Ecoregion were used to evaluate the index. To help conserva-
Conservation assessment
tion planners understand average optimacity we: (1) compared it to an average of irreplace-
Irreplaceability
ability, a well-known index of conservation value, and (2) used logistic regression models to
Marxan
relate average optimacity to other measures of conservation value (e.g., rarity, richness). Average optimacity and average irreplaceability were highly correlated (Spearman rank correlation = 0.90), suggesting substantial agreement in site prioritization. However, there were clear differences in the rankings of highly valued sites, demonstrating that these two indices provide different perspectives on priorities. Simple regression models showed that average optimacity was strongly associated with maximum representation and maximum rarity. Two of the top multiple regression models for average optimacity exhibited a moderate increase in explanatory power relative to the best simple regression model and had three independent variables, which reflects the multi-dimensional nature of this index. The average optimacity index may be preferable to an irreplaceability index when efficiency is considered an important dimension for site prioritization. Ó 2008 Elsevier Ltd. All rights reserved.
1.
Introduction
Which places are the most important for biodiversity conservation? This fundamental question encapsulates two major challenges faced by organizations trying to conserve biodiversity – one scientific and one socio-economic. First, biodiversity, for our purposes defined as all native species, plant communities, and ecological systems within a geographic re-
gion, exhibits spatial patterns that confound simple formulations of importance. Species richness, species rarity, and other variables used to map ‘‘biodiversity’’ are often poorly correlated (Lennon et al., 2004). Peaks in richness may not coincide with peaks in rarity or endemism (Prendergast et al., 1993; Orme et al., 2005). Peaks in endemism for one taxon may not coincide with peaks in endemism for other taxa (Kerr, 1997). And, regional peaks in richness are unlikely to
* Corresponding author: Tel.: +1 360 902 2369. E-mail address: [email protected] (G.F. Wilhere). 0006-3207/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.biocon.2007.12.022
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include all species in that region (Williams et al., 1996). In short, any meaningful quantification of important places will most likely be a multi-dimensional relationship. Second, the funds that society has allocated toward conservation are insufficient to protect all places with biodiversity value. Consequently, efficiency is imperative in conservation planning (Pressey et al., 1993; Ando et al., 1998), and therefore, efficiency should be another dimension influencing importance. Furthermore, because funding is typically spread over time, conservation organizations must prioritize sites for protection. The problem of site prioritization has motivated extensive research in the development and application of various analytical techniques (for reviews see Justus and Sarkar, 2002; Williams et al., 2004). The first analytical approach is attributed to Ratcliffe (1971; cited in Justus and Sarkar, 2002). Ratcliffe (1971) proposed that sites be evaluated according to well-defined criteria such as community diversity, species rarity, and naturalness and that a multi-dimensional scoring system might be developed to determine the relative importance of sites. Scoring methods for site prioritization received considerable use through the 1970s (Margules and Usher, 1981) and similar prioritization schemes are still in use (e.g., Benayas and de la Montana, 2003). The main shortcoming of scoring methods is inefficiency (Kirkpatrick, 1983). High priority sites, i.e., those assigned high scores, may contain similar assemblages of species and communities resulting in unnecessary duplication (Pressey and Nicholls, 1989). Hence, this method of site selection could lead to inefficient nature reserve systems. In the pursuit of efficiency, scoring methods have largely been supplanted by approaches that reformulate the siteselection problem as an optimization problem. The solution identifies a set of sites which minimize an objective function while simultaneously satisfying various constraints. Typically, the objective function is a summation of the number, land area, or ‘‘cost’’ of selected sites, and the constraints are representation targets expressed as the number of occurrences per species or area per ecological system. Sites belonging to the solution constitute the most efficient nature reserve system needed to meet the representation targets. Kirkpatrick (1983) and Margules et al. (1988) were the first to use optimization algorithms for planning a reserve system. Their algorithms and many others since then (Csuti et al., 1997; Williams et al., 2004) have been heuristic, i.e., rule-based search procedures. Heuristic algorithms do not guarantee a truly optimal solution, only nearly-optimal solutions which are considered adequate for the purposes of conservation planning (Pressey et al., 1996; Csuti et al., 1997). Every optimal site-selection method exhibits at least one of three major shortcomings. First, many optimization algorithms generate a single indivisible solution. These optimization methods assume that the entire optimal set will be protected as a reserve system, but in nearly all real-world situations this is unrealistic (Meir et al., 2004). Rarely will sufficient funds become available within a reasonable time. Second, many optimization algorithms generate binary solutions; a site is either in or out of the reserve system. Such algorithms yield no information regarding the relative importance of sites, and therefore, cannot be used for site prioriti-
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zation. Third, formulation of the optimization problem requires representation targets, which are effectively conservation goals. Conservation goals should be informed by science but are ultimately based on subjective values (Tear et al., 2005). Ideally, site prioritization would be the result of a totally objective analysis. The purpose of this study was to develop a site prioritization technique that might overcome the shortcomings of both scoring methods and optimization methods. We sought a numerical index for prioritizing site protection. The index would possess the following properties: (1) it is a good indicator of a site’s potential contribution to an efficient representative reserve system; (2) it equals zero if and only if a site contains no elements of biodiversity; and (3) it does not depend on a single set of representation targets. An index that comes close to possessing properties (1) and (2) is irreplaceability.
1.1.
Irreplaceability
Irreplaceability indicates a site’s potential to contribute to a representative reserve system. The original operational definition of irreplaceability was given by Pressey et al. (1993, 1994) – the proportion of representative reserve systems in which a site occurs. However, this definition has a problem with site redundancy. The inclusion of a site in a reserve system that is already fully representative increases that site’s irreplaceability even though the site is replaceable. Consequently, Pressey helped to develop a revised definition of irreplaceability that eliminated the redundancy problem – the proportion of representative reserve systems in which a site is necessary to be fully representative (Ferrier et al., 2000). The most direct way to calculate this revised irreplaceability is, at present, computationally impractical because the number of possible site combinations quickly approaches astronomical numbers. For this reason, Pressey et al. (1994) and Ferrier et al. (2000) developed indirect ways to estimate site irreplaceability. Irreplaceability has become a fundamental concept in conservation planning (Margules and Pressey, 2000), and the revised definition is quite elegant. Unfortunately, the appeal of ‘‘irreplaceability’’ has led to other definitions that alter its meaning (e.g., Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003). These other definitions arise from the use of optimization algorithms which offer a means to estimate a pseudo-irreplaceability. However, as it was originally defined, irreplaceability is not influenced by optimality in any way, and therefore, optimization algorithms are the wrong tool for estimating irreplaceability. This mistake is most prevalent in studies that have used the simulated annealing algorithm to create optimal reserve systems (Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003; Leslie et al., 2003). Simulated annealing performs a stochastic, heuristic search for the optimal solution (Rutenbar, 1989; Lockwood and Moore, 1993). Since it is stochastic, or random, simulated annealing can find more than one nearoptimal solution to a single optimization problem. Hence, a single site may belong to many different near-optimal solutions. The output of such algorithms enables the calculation of an index that is similar to but not the same as irreplaceabil-
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ity – the number of near-optimal solutions that include a site divided by the number of near-optimal solutions. This ratio has been misnamed ‘‘irreplaceability.’’
1.2.
Optimacity
The number of near-optimal solutions that include a site should be a good index of a site’s conservation value. This is the assumption made by Andelman and Willig (2002) and Leslie et al. (2003). Both used a simulated annealing algorithm to calculate an index which can be expressed as Oj ¼ ð1=RÞ
R X
xjr
ð1Þ
r¼1
where Oj is the conservation value of site j, R is the number of near-optimal solutions generated by R replicate runs of the optimization algorithm, and xjr is a binary variable that equals 1 when site j is included in solution r but 0 otherwise. The input data and parameter values, including representation targets, are identical for all replicates. Oj indicates a site’s potential contribution to an efficient representative reserve system. Specifically, it is the probability that site j will be included in an optimal or near-optimal reserve system. We call this index optimacity. Optimacity is different than optimality. Optimality is a quality of the entire reserve system, but optimacity is a quality of a site. Renaming ‘‘irreplaceability’’ (sensu Leslie et al., 2003) to optimacity was partly motivated by conservation planners. We found that some planners objected to efficiency influencing a site’s ‘‘irreplaceability.’’ They believed that ‘‘irreplaceable’’ was synonymous with ‘‘priceless’’, and therefore, efficiency was irrelevant to irreplaceability. The definitions of Pressey et al. (1993, 1994) and Ferrier et al. (2000) were much closer to the planners’ conception of irreplaceability. Because Pressey et al. (1993) was the first to operationally define irreplaceability, their definitions were adopted for our study. While new technical jargon is best avoided, we believe that the word ‘‘optimacity’’ effectively conveys the computational basis and meaning of the index. Optimacity is an index based on optimization; irreplaceability (sensu Pressey et al., 1993) is an index based on representation. Optimacity is partly a function of the representation targets. Changing the representation targets changes the number of sites included in each solution and it changes the number of solutions in which each site is included. As representation targets increase, optimacity steadily increases for most sites but for some sites optimacity may fluctuate as it increases (Stewart et al., 2007). In general, low representation targets typically yield a small number of sites with high optimacity, some sites with moderate optimacity, and many sites with zero optimacity (Leslie et al., 2003). The fact that some sites go from zero optimacity to positive optimacity demonstrates that this index is somewhat misleading – at low representation targets, some sites are shown to have no conservation value when they actually do. To address this shortcoming, we created an average optimacity index defined as j ¼ ½1=ðM RÞ O
M X R X m¼1 r¼1
xjrm
ð2Þ
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where M is the number of different but closely related optimization problems. For instance, each of the M optimization problems might correspond to slight changes in a particular assumption or parameter value in the problem formulation. j , R replicates of the optimization algorithm To calculate O are run, a parameter value is changed, and R replicates are run again. This is repeated until M sets of replicate results are obtained.
2.
Study area
Our case study was associated with the Willamette ValleyPuget Trough-Georgia Basin Ecoregional Conservation Assessment (WPG ECA; Floberg et al., 2004). The WPG Ecoregion is a broad valley, approximately 810 km long, flanked by the Olympic Mountains and Coast Range to the west and the Cascade Mountains to the east. The ecoregion encompasses 3,977,000 ha of land; 82% in the United States and 22% in Canada. Its boundaries in the United States, specifically the states of Washington and Oregon, were delineated by Omernik and Gallant (1986) and the boundaries in Canada, specifically, British Columbia, followed those of Demarchi (1996). The ecoregion also encompasses Puget Sound and part of Georgia Strait, 1.5 million ha of marine waters. The effects of these marine waters, the adjacent mountain ranges, and pre-historic glaciation have created dramatic localized differences in climate, soils, and topography. From distinctive combinations of these factors spring an array of ecological systems and rare plant communities ranging from coniferous forests to herbaceous grasslands, oak savannahs, and rocky balds (Floberg et al., 2004). Historically, the predominant vegetation types were 39% conifer or conifer-deciduous forest, 25% oak or oak-conifer forest, 12% grasslands, and 23% wetlands (Johnson and O’Neil, 2001, pp. ix–xi).
3.
Methods
The methods consisted of five major stages: (1) assembling and processing land cover and species occurrence data; (2) defining the representation target levels; (3) using an optimization algorithm to generate optimacity values; (4) evaluating the average optimacity index by comparing it to other measures of conservation value; (5) building a multivariate model that explained the average optimacity and index.
3.1.
Data
A coarse filter–fine filter approach (Hunter, 1990; p. 238) was employed to represent all biodiversity in the WPG Ecoregion. This approach hypothesizes that preservation of all ecological systems and plant communities, i.e., the coarse-filter elements of biodiversity, will also preserve the majority of species that occupy them. The fine-filter elements are rare or imperiled species thought to be inadequately preserved by the coarse-filter elements. Hence, fine-filter elements of biodiversity warrant special attention in a conservation assessment.
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3.1.3.
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Sites
Two types of spatial data corresponded to coarse-filter elements – an ecological systems layer and points denoting the locations of rare plant communities. The spatial data for fine-filter elements were points denoting the locations of species. Point data are also known as occurrences. We assembled data for 84 coarse-filter elements and 325 fine-filter elements from various sources. Detailed information about the spatial data can be found in Floberg et al. (2004).
For the purposes of analysis, the locations of all biodiversity elements were referenced to a regular hexagonal grid. The grid cells were 750 ha, and were also known as ‘‘sites.’’ Site records consisted of a unique identification number and the amount of each biodiversity element at that site. The entire ecoregion had 8107 sites, but the analysis excluded sites that were in marine waters. Of the remaining 6772 sites, 6100 contained a biodiversity element of some sort.
3.1.1.
3.2.
Coarse-filter elements
Nineteen ecological systems (sensu Comer et al., 2003) were identified in the WPG Ecoregion. To develop an ecological systems layer for the entire ecoregion we combined the best available land cover and vegetation layers from the three geo-political jurisdictions. To combine layers within a jurisdiction and to create consistent land cover and vegetation categories across the entire ecoregion we developed a rulebased reclassification scheme. Places with no ecological systems (e.g., urban areas, farmland, etc.) were classified as null data in the final ecological systems layer. Four ecological systems could be accurately modeled and comprehensively mapped across the entire ecoregion. High quality occurrences of these four ecological systems were also mapped as points. The other 15 ecological systems, which typically occur in smaller, discrete patches, were more accurately represented as point occurrences. In total, 3840 points denoted the locations of ecological systems. Plant communities are described by the dominant plants within the community, e.g., tufted hairgrass-Henderson’s checker mallow or Douglas fir-common snowberryoceanspray. Rare plant community locations were obtained from the British Columbia Conservation Data Centre (CDC), Oregon Natural Heritage Information Center (ONHIC), and Washington Natural Heritage Program (WNHP). These locations were mapped as single points and had to meet minimum standards for ecological integrity to be included in the analysis. We identified 65 rare plant communities and their locations were represented by 439 occurrence records.
3.1.2.
Fine-filter elements
Most fine filter data were obtained from the CDC, ONHIC, WHNP, and the Washington Department of Fish and Wildlife. Occurrence data records gave spatial coordinates of actual observations of plants or animals. Occurrence records were eliminated if they were: (1) animal data collected before 1980; (2) plant data collected before 1975, (3) outside the species’ current geographic range, (4) not in the species’ habitat type or in a location where the species’ habitat was known to have been destroyed, (5) too imprecise spatially, or (6) spatially redundant. The analysis included 186 vascular plant species and 47 non-vascular plant species. The plant species had 2,055 occurrence records. We identified 127 animal species as potential fine-filter elements, but could only assemble adequate data for 92 elements: 10 mammals, 29 birds, 5 reptiles, 14 amphibians, 7 fish, 23 insects, 3 molluscs, and 1 annelid. In total, 1428 animal occurrence records were included in the analysis.
Representation targets
The average optimacity index was calculated by running the optimization algorithm with M different sets of representation targets that spanned a wide range. For this study M equaled 10. The set of largest representation targets were fixed at 100% of species and rare community occurrences and 100% of ecological systems area. Increments between successive sets of representation targets were designed to yield roughly equal increments of biodiversity conservation. These equal conservation interval (ECI) target sets produced an index called ECI average optimacity, henceforth known as average optimacity. Preserving fine-filter elements is accomplished by reducing extinction risk. Therefore, we wanted our ECI targets sets to provide roughly equal increments in risk reduction. Natural Heritage Programs (NHPs) in the United States use various factors to determine the conservation status of imperiled species (Regan et al., 2004). One such factor relates the number of occurrences to qualitative levels of extinction risk (Table 1). We assumed equal intervals of risk for this NHP factor, and equated codes A, B, C, D with levels 1, 2, 3, 4. Hence, the relationship was modeled as the function: upper threshold = 100.59d+0.11, where d is level of extinction risk (r2 = 0.99, p < 0.0001). We used this function to interpolate between the thresholds to yield 10 ECI targets for population-like occurrences: populations, subpopulations, or population segments (Table 2). For each species, if the target exceeded the total amount available in the ecoregion, then the target was set to the amount available. Some species occurrence data corresponded to the nest or territory of breeding adults. Occurrences of this type were dealt with somewhat differently. We followed the same approach as above but used a different NHP factor that relates the number of individuals in a population to extinction risk (Table 1). We converted the number of individuals to number of nests simply by dividing by 2. We assumed equal intervals of risk in this NHP factor, and equated codes A, B, C, with levels 1, 2, 3. This relationship was modeled as the function: upper threshold = 100.65d+0.76 (r2 = 0.99, p < 0.0001). Because the coarse-filter elements were intended to preserve the vast majority of species, ECI target sets for ecological systems and plant communities were based on the species–area curve. The curve follows the relationship species / areaz, where ‘‘species’’ refers to the number of species, ‘‘area’’ refers to the area of ecological systems, and z is a constant (Rosenzweig, 1995). An oft cited rule-of-thumb for z’s value is called Darlington’s Rule (MacArthur and Wilson, 1967, pp. 8–9). The rule states that a doubling of species occurs for every 10 fold increase in area, hence z = log10(2). We
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Table 1 – Thresholds for two factors used by Natural Heritage Programs (NHPs) to determine species status (Regan et al., 2004) Threshold code
Risk level
Number of occurrences Lower threshold
A B C D E
1 2 3 4 5
Number of mature individuals
Upper threshold
1 6 21 81 301
Lower threshold
Upper threshold
1 51 251 1001 2501
50 250 1,000 2,500 10,000
5 20 80 300 None
Threshold codes, defined by NHPs, were translated to risk levels. ‘‘Occurrences’’ refer to known locations of species. We derived the nest and den ECI representation targets by dividing the number of individuals by two.
Table 2 – Ten sets of representation targets for two types of species occurrences Occurrence type
‘‘Populations’’ Nests/dens
Target set 1
2
3
4
5
6
7
8
9
10
1 6
2 10
3 16
5 25
8 43
13 70
20 125
31 191
49 315
All All
The difference between each set represents an equal conservation interval. Values are number of occurrences, which are points in the spatial data. Population occurrences were actually populations, subpopulations, or population segments. For each element, if the target exceeded amount available, then the target was set to the amount available.
used this value to derive targets that would represent equal increments in the proportion of all existing species. That is 0.05%, 0.5%, 1.8%, 4.8%, 10%, 18%, 31%, 48%, 70%, and 100% of each ecological system should represent 10%, 20%, 30%, . . . , 100% of currently existing species. Representation targets for plant communities, which were point data, followed the species–area curve (i.e., 0.05, 0.5, 1.8, . . . percent of points) or the top row in Table 2, whichever was larger.
3.3.
The optimization algorithm
We generated optimacity indices using Marxan, a computer program developed to identify optimal reserve systems (Ball and Possingham, 2000; McDonnell et al., 2002). Marxan, or its closely related predecessor SPEXAN, have been applied to reserve design problems throughout the world (e.g., Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003; Cook and Auster, 2005). Appendix A explains the optimization problem and our use of Marxan. To generate the optimacity values, we did 25 replicates for each set of representation targets. Hence, the product MR equaled 250. Because the largest target set was 100% for all biodiversity elements, we did not use Marxan to calculate the optimacity values for that target set. The value for all sites containing a biological element must equal 1 according to the optimization constraints.
3.4.
Other measures of conservation value
We evaluated the average optimacity index by comparing it to other measures of conservation value. The measures used were richness, rarity, representation, normalized representation, and a novel measure called irredundancy (see Appendix
B). Maximum and average values were calculated for all measures except richness.
3.5.
Average irreplaceability
Because irreplaceability has been confused with optimacity, we also compared average optimacity to an average of irreplaceability values. Irreplaceability values are partly a function of the representation targets. For this reason, we calculated an average irreplaceability using the same ten sets of ECI representation targets: Ij ¼ ð1=MÞ
M X
Ijm
ð3Þ
m¼1
where Ijm is the irreplaceability value of site j for one set of targets. Irreplaceability was calculated using CPlan (Pressey et al., 2005). Because the largest target set was 100% for all biodiversity elements, we did not use CPlan to calculate the irreplaceability values for that target set. By definition, irreplaceability for all sites containing a biodiversity element equals 1 when the representation target is 100%.
3.6.
Statistical analysis
The usefulness of the average optimacity index for conservation planning can be judged, in part, by its relationship to other measures of conservation value. To compare average optimacity to other measures of conservation value we constructed simple regression models (i.e., models with one independent variable) for average optimacity versus each measure. We did the same for average irreplaceability. Optimacity and irreplaceability are proportions. Hence, we did a logistic regression based on a generalized linear model assuming a binomial error distribution and with the link
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function set to logit. The resulting regression equation is of the form: j ¼ egðX1 ;X2 ;X3 Þ =ð1 þ egðX1 ;X2 ;X3 Þ Þ O
ð4Þ
where g(X1, X2, X3) = b0 + b1X1 + b2X2 + b3X3. b2 and b3 equal zero in the simple regression model. The well-known coefficient of determination, r2, cannot be calculated for generalized linear models. Hence, to quantify the strength of the each regression relationship we calculated an analog to r2, r2L , based on the ratio of log-likelihoods (Menard, 2000). An analysis of deviance assuming a chi-squared distribution was done to determine whether each simple regression relationship was significantly different than the corresponding null model (Crawley, 2002, pp. 525–528). For all statistical tests we chose a = 0.05. We further explored the relationship between average optimacity and other measures of conservation value by building logistic multiple regression models. We did the same for average irreplaceability. All combinations of the nine conservation value measures resulted in 511 possible models for average optimacity and 511 possible models for average irreplaceability. We constructed and evaluated all 1022 models. Many candidate models were eliminated from further consideration because their r2L was less than the best simple regression model. More candidate models were eliminated because of high multi-collinearity amongst the independent variables. Multi-collinearity was dealt with in two ways. First, all of the simple regression models had positive slope coefficients, and therefore, a negative slope coefficient in a multiple regression model was a sign of high multi-collinearity. Second, we calculated the variance inflation factor (VIF; Kleinbaum et al., 1988, pp. 210–211) for each model and eliminated those that had a maximum VIF greater than 4. The ‘‘best’’ models were based on Akaike’s information criterion (AIC), which indicates the best compromise between goodness-of-fit and model parsimony (Burnham and Anderson, 2002). Models with delta AIC less than 7 were considered to be among the best possible models (Burnham and Anderson, 2002; p. 70). Using AIC for model selection can result in models with less than optimal parsimony (Crawley, 2002; p. 452). To determine whether terms in each of the best models made a significant contribution to the model, we dropped each term and compared the smaller model to the original model using an analysis of deviance assuming a chi-squared distribution. All statistical analyses were done with R statistical software (RDCT, 2005; Fox, 2006)
4.
Results
Average optimacity is different than optimacity calculated with any of the 10 representation target sets (Table 3). For low targets (the first 5 sets), 75% of sites had optimacity equal to zero, but the third quartile for average optimacity was at 0.31. For moderate targets (sets 6 and 7), half of all sites had optimacity equal to zero, but the median value for average optimacity was 0.16. These comparisons show that optimacity can be a misleading index – for a wide range of representation targets, many sites are shown to have no conservation value when they actually do. For moderate and high targets (sets 6 through 9), 770–1621 sites had optimacity equal to 1,
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but only 90 sites had average optimacity equal to 1. This suggests that average optimacity is better for identifying the most important sites. Average optimacity and average irreplaceability showed some similarities in their distributions. There was little difference between their first quartiles, medians, and means (Table 3). The shapes of the two distributions began to diverge between the second and third quartiles – the third quartile equaled 0.31 for average optimacity but equaled 0.14 for average irreplaceability. Average irreplaceability had more high value sites than average optimacity. Two-hundred fifteen sites had average optimacity greater than 0.90 and 408 sites had average irreplaceability greater than 0.90. Only one site had average optimacity greater than 0.90 but average irreplaceability less than 0.90. Site prioritization implies ranking. The Spearman rank correlation between average optimacity and average irreplaceability was 0.90, suggesting little difference in how sites would be prioritized using either of these two indices. However, the rank correlation for high value sites is much less – 0.70 for sites with average optimacity greater than 0.90. In other words, prioritization of the most valuable sites was different. Potential differences in site prioritization become more apparent when plotting average optimacity versus average irreplaceability (Fig. 1). Simple logistic regression showed a strong association (r2L ¼ 0:797) and a fairly linear relationship between the two indices. However, when average irreplaceability is approximately 1, average optimacity is seen to range from 1 down to 0.7. For average irreplaceability in the range of 0.1–0.2, average optimacity ranges from 0.1 up to 0.4. Maps of average optimacity and average irreplaceability show different patterns of conservation value (Fig. 2). The maps generally agree on the location of high value sites, but keep in mind that the rankings within the high value categories are different. Many sites with intermediate value (0.2–0.4) on the average optimacity map have lower value (<0.2) on the average irreplaceability map. The analysis of deviance showed that all simple regression models for both average optimacity and average irreplaceability were significant. Average optimacity was most strongly associated with maximum representation and maximum rarity, r2L ¼ 0:744 and 0.738, respectively (Table 4, Fig. 3). In comparison, average irreplaceability had much stronger associations with these two measures, r2L ¼ 0:939 and 0.938, respectively. This makes sense because the definition of irreplaceability is based on representation and maximum rarity was highly correlated with maximum representation (Pearson’s correlation coefficient = 0.98). But for one exception, the associations of the all other conservation value measures with average irreplaceability were stronger than their associations with average optimacity. The one exception was maximum normalized representation. This makes sense because maximum normalized representation indicates which sites contain the largest amount of an element and one would expect such sites to be selected by an optimization algorithm. Based on AIC and r2L , multiple regression models were superior to simple models for both average optimacity and average irreplaceability (Table 5). The best simple regression model for average optimacity was based on maximum repre-
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Table 3 – Summary statistics for optimacity calculated with each representation target set, average optimacity, and average irreplaceability Statistic
Representation target set 1
2
3
4
5
6
7
Average 8
9
10
Optimacity
Irreplaceability
Mean optimacity
0.03
0.04
0.06
0.08
0.11
0.15
0.23
0.33
0.488
0.90
0.24
0.21
Minimum 1st quartile Median 3rd quartile 90th percentile Maximum
0 0 0 0 0 1
0 0 0 0 0 1
0 0 0 0 0 1
0 0 0 0 0.08 1
0 0 0 0 0.52 1
0 0 0 0.12 1 1
0 0 0 0.32 1 1
0 0 0.08 0.68 1 1
0 0 0.48 0.96 1 1
0 1 1 1 1 1
0 0.10 0.16 0.31 0.57 1
0 0.11 0.12 0.14 0.64 1
Sites equal to 0 Sites equal to 1
6283 90
6249 192
6149 298
6038 419
5344 566
4441 770
3625 924
3003 1049
2384 1621
672 6100
672 90
672 102
N equals 6774, the number of sites.
1.0
but the best multiple regression model for average irreplaceability increased r2L by only 0.015.
average optimacity
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
average irreplaceability Fig. 1 – Average optimacity versus average irreplaceability, both calculated with ECI target sets. Points represent 6100 sites that had non-zero average optimacity and the solid line is the fitted logistic regression curve. Dashed line shows average optimacity equal to average irreplaceability and is shown for comparison only.
sentation, however, the dominant term in the best multiple regression model was maximum rarity. The best multiple regression model explained an appreciable additional amount of the variation in average optimacity relative to the best simple regression model; r2L increased from 0.744 to 0.884. The model for average optimacity with the highest r2L and nearly the lowest AIC value had three terms: maximum rarity, maximum normalized representation, and richness. The best models for average optimacity all included either maximum or average normalized representation. The best multiple regression model for average irreplaceability had maximum representation as the dominant term. Maximum rarity was the dominant term in the next best model. Irredundancy was a term in one of the best models for average irreplaceability but it had a small coefficient. The best average irreplaceability model had a greater r2L value than the best average optimacity model (0.954 versus 0.884),
5.
Discussion
5.1.
Average optimacity
We developed an index for prioritizing site protection. The index possesses the following properties: (1) it indicates a site’s potential contribution to a near-optimal representative reserve system; (2) it equals zero if and only if a site contains no biodiversity elements; and (3) it does not depend on a single set of representation targets. Optimacity possesses the first property because it was generated with an optimization algorithm. Irreplaceability does not possess this property. The index possesses the second and third properties because: (a) the optimization algorithm was run with a wide range of representation targets; (b) the largest representation targets were set to 100% of all biodiversity elements; and (c) the resulting optimacity values were averaged thus eliminating dependence on a single set of targets. When we introduced average optimacity to conservation planners they were curious about its interpretation. The planners wanted this index explained in terms they already understood, e.g., representation, rarity, and richness. To help planners understand average optimacity we (1) determined the statistical relationships between it and other well-known measures of conservation value, and (2) compared it with average irreplaceability. The regression models were not intended to be predictive. They were simply a means to gain insights regarding the new index. Using data from the WPG ECA and the ECI representation target sets, we found that average optimacity was strongly associated with maximum representation and maximum rarity and moderately associated with other measures of conservation value. The statistical modeling suggested that average optimacity is a somewhat more ‘‘multi-dimensional’’ index than average irreplaceability. The two and three variable models explained appreciably more variation in average optimacity than the best one variable model. The same could not be said for average irreplaceability.
B I O L O G I C A L C O N S E RVAT I O N
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
777
Fig. 2 – Maps of average irreplaceability (A) and average optimacity (B) for central portion of WPG Ecoregion. Six shades of gray represent six ranges of index values. Thick black line is ecoregion boundary and thin line is border between Washington and Oregon States. Each site is a 750 ha hexagon.
Table 4 – Values of r2L from simple logistic regression of average optimacity and average irreplaceability versus various measures of conservation value Average optimacity Max. representation Max. rarity Avg. representation Avg. rarity Max. norm. representation Max. irredundancy Avg. norm. representation Avg. irredundancy Richness Average irreplaceability
0.744 0.738 0.682 0.652 0.624 0.576 0.556 0.543 0.324 0.797
Average irreplaceability 0.939 0.938 0.936 0.853 0.415 0.695 0.601 0.727 0.370 –
(1) (2) (3) (4) (8) (6) (7) (5) (9)
Regressions included only sites that had non-zero average optimacity and nonzero average irreplaceability (N = 6100). First column is ordered by rank of r2L values. The numbers in parenthesis equal rank of r2L within that column.
5.2.
Site prioritization and conservation policy
Why did we need a new index for prioritizing site protection? Site prioritization techniques have been the subject of extensive research (e.g., Margules and Usher, 1981; Kershaw et al., 1995; Freitag et al., 1997; Pressey and Taffs, 2001; Benayas and de la Montana, 2003), and consequently, many different techniques were available to us. We found that none of the currently available techniques met our needs. A new technique was needed because a government agency played a major role in the WPG ECA, and government agencies must make clear distinctions between science and policy. The agency desired a conservation assessment free from any subjective decisions that might be misconstrued as an expression of government policy.
The degree to which efficiency should influence site prioritization is a decision for policy makers, not scientists. Nevertheless, we believed the preferred policy was rather obvious, and consequently, we adopted an optimal site-selection algorithm for our conservation assessment. Such algorithms have three potential shortcomings that were unacceptable to the government agency. First, optimization techniques can be used to generate single indivisible solutions, which implies that the entire solution should be protected. Using optimization techniques, some conservation assessments have identified potential reserve systems covering 30–40% of an ecoregion (e.g., Klahr et al., 2000; Rumsey et al., 2003). A commitment of that magnitude would require a demanding public process, but we were not vested with the authority nor given the resources to engage in such a process. Second, most optimization algorithms generate binary solutions – a site is either in or out of a reserve system. This type of solution is too rigid. Most conservation organizations need flexibility to respond to changing circumstances and unforeseen opportunities. Furthermore, an ‘‘in or out’’ solution might be misinterpreted by the public as an authoritarian decree – all sites in the solution must be protected and all sites out of the solution need no protection. We wished to avoid the damaging loss of public confidence that such misunderstandings often engender. Third, formulation of the optimization problem requires representation targets, which are effectively conservation goals. Conservation goals are a policy decision (Tear et al., 2005), and therefore, should be established in collaboration with policy makers. We did not have access to the policy makers who possessed the requisite authority. Marxan generates an output, which we called optimacity, that partly circumvents the first two shortcomings, but optimacity did not meet our needs in two respects. First, even though optimacity is not binary, a single set of reasonable representation targets will result in zero optimacity for many sites. The government agency was concerned that zero
778
A
B I O L O G I C A L C O N S E RVAT I O N
0.8
optimacity would be misinterpreted to mean zero value for biodiversity conservation. In other words, optimacity could mislead policy and management decisions related to land use regulation and land acquisition. Second, optimacity is generated using one set of representation targets. Again, the government agency was concerned that one set of targets might be misconstrued as the government-sanctioned conservation goals. Average optimacity overcame the technical shortcomings of optimacity and effectively avoided major entanglements with conservation policy. We attempted to avoid the policy decisions implicit in a single set of representation targets. However, developing the representation target sets still entailed subjective judgments but of a different kind. The value for M, deciding to use the species–area curve, the value for z, and deciding to use the NHP risk levels were all subjective judgments. These judgments largely determined the representation targets, and therefore, strongly influenced average optimacity values. However, none of these subjective judgments could be described as policy decisions. These were technical decisions intended to yield roughly equal increments of biodiversity conservation.
0.6
5.3.
0.4
The purpose of our study was not to pit optimacity against irreplaceability (for an in-depth comparison see Carwardine et al., 2007). Clearly, the two indices are defined differently, convey different information, and will result in different site rankings. However, their applications in conserving planning can be quite similar. Both indicate conservation value, but according to Margules and Pressey (2000) and Pressey and Taffs (2001) conservation value alone is insufficient for prioritizing site protection. They suggest that site vulnerability is the other critical dimension to prioritization and that a graph of irreplaceability versus vulnerability should be used to prioritize sites. We have taken a similar approach in ecoregional conservation assessments by graphing average optimacity versus vulnerability (e.g., Floberg et al., 2004; Pryce et al., 2006). Conservation actions were guided by the relative positions of sites in the two-dimensional average optimacity–
1.0
average optimacity
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1. 0
maximum representation
average irreplaceability
B
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
1.0
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1. 0
maximum representation Fig. 3 – Results of simple logistic regression with maximum representation as independent variable: (A) dependent variable is average optimacity, (B) dependent variable is average irreplaceability. Points represent 6100 sites that had non-zero average optimacity and solid line is the fitted logistic regression curve. Equations for curves given in Table 5.
Applying average optimacity
Table 5 – Best multiple logistic regression models Dependent variable
Regression equation g(X1, X2, X3) = b0 + b1X1 + b2X2 + b3X3
r2L
Max. VIF
AIC
Average optimacity
2.44 + 9.04 max_RARE + 1.95 max_NREP 2.54 + 8.29 max_RARE + 1.89 max_NREP + 1.72 RICH 2.50 + 7.60 max_REP + 1.85 max_NREP + 1.84 RICH 1.76 + 8.98 max_REP + 9.37 avg_NREP 1.46 + 12.61 max_REP
0.884 0.890 0.888 0.796 0.744
1.21 1.39 1.40 1.42 1
3033 3034 3038 3040 3067
Average irreplaceability
2.18 + 20.86 max_REP + 7.52 avg_NREP 2.26 + 21.97 max_RARE + 7.37 avg_NREP + 0.81 max_IRRED 2.28 + 22.87 max_RARE + 8.06 avg_NREP 1.96 + 25.14 max_REP
0.954 0.958 0.956 0.939
1.47 1.59 1.43 1
2076 2077 2081 2101
The function g(X1, X2, X3) is substituted into the logistic equation, Eq. (4). All independent variables were normalized to a maximum value of 1. The variable abbreviations are max = maximum, avg = average, REP = representation, NREP = normalized representation, RARE = rarity, RICH = richness, and IRRED = irredundancy. The simple logistic regression model is shown for comparison only; it was not among the best models based on delta AIC less than 7.
B I O L O G I C A L C O N S E RVAT I O N
vulnerability graph. Whether irreplaceability or optimacity are used in the graph depends on whether efficiency is considered an important dimension for site prioritization. An index, such as average optimacity, can only guide conservation action. We will never be able to completely quantify the complex ecological, economic, social, and political factors that influence where biodiversity conservation can actually happen. Chance events and unforeseen opportunities will have a major influence on future conservation decisions, and local attitudes toward conservation can hinder or facilitate success. Biodiversity conservation requires difficult choices; we cannot protect every place with biodiversity value. Quantitative indices provide a more objective foundation upon which to establish priorities and weigh those choices.
779
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is determined by its penalty factor – the larger the fk, the firmer the constraint. Hard constraints can be established by setting an arbitrarily large fk. However, very large penalty factors can create ill-conditioned objective functions exhibiting sharp peaks or valleys, both of which make optimization more difficult (Gottfried and Weisman, 1973, pp. 253–254). To determine an acceptable value for each fk, we initially set all fk to 2 and then ran a simplistic iterative search which stopped when all elements met their targets in at least 38 consecutive Marxan runs. Thirty-eight is 1.5 times the number of replicates we used to generate optimacity. The resulting fk values ranged from 2 to 6 but at least 97% of the values equaled 2. We used the adaptive annealing with normal iterative improvement algorithm and each replicate consisted of 4 million iterations.
Acknowledgements Appendix B We thank E. Alverson, S. Cannings, J. Fleckenstein, J. Gamon, E. Scheuering, and M. Summers for assembling the species occurrence data. E. Alverson, C. Chappell, and A. Ceska helped to develop the ecological systems and rare communities data. W. Chang and K. Ryding provided advice on the statistical analyses. J. Pierce, M. Ingraham, P. Dunwiddie and four anonymous reviewers provided valuable critiques of the draft manuscript. This work was supported by a grant from the US Fish and Wildlife Service to the Washington Department of Fish and Wildlife.
S X
cj xj
ðA:1Þ
j¼1
subject to the constraints: ajk xj P tk
where ajk is the amount of element k in site j and Ak is the total amount of element k in the ecoregion. Maximum REPj and average REPj for site j were defined as maximum REPj ¼ maxðREPj1 ; REPj2 ; . . . ; REPjk ; . . . ; REPjB Þ B X
REPjk
ðB:2Þ ðB:3Þ
ðA:2Þ
j¼1
where xj is a binary variable that equals 1 when site j is included in the solution but 0 otherwise, cj is the cost of site j, ajk is the amount of biodiversity element k at site j, tk is the representation target for element k, S is the total number of sites, and B is the total number of biodiversity elements in the ecoregion (McDonnell et al., 2002). The ‘‘cost’’ of each site was area. Since all sites have equal area the algorithm minimized the number of sites. Simulated annealing, like other heuristic algorithms, is incompatible with constraints such as Eq. (A.2). Therefore, the constraints must be incorporated into the objective function as penalty terms, like so: S X j¼1
cj xj þ
where the max function returns the maximum value for site j among all biodiversity elements at site j and B is the total number of biodiversity elements in the ecoregion. The normalized representation of site j for element k was expressed as NREPjk ¼ ajk = maxða1k ; a2k ; . . . ; ajk ; . . . ; aSk Þ
for k ¼ 1; 2; 3; . . . ; B
minimize : FðxÞ ¼
ðB:1Þ
k¼1
Marxan minimizes an objective function using a simulated annealing algorithm. Our optimization problem was
S X
REPjk ¼ ajk =Ak
average REPj ¼ ð1=BÞ
Appendix A
minimize : FðxÞ ¼
Richness of a site equals the number of different biodiversity elements at the site. Representation (sensu Pressey and Nicholls, 1989) at site j for biodiversity element k was defined as
B X
fk PFðtk Þ
ðA:3Þ
k¼1
where PF is a function that determines the penalty when element k does not satisfy its target and fk is a penalty factor that determines the severity of the penalty for each target. The second term in the objective function imposes the constraints, however, they are soft constraints. ‘‘Soft’’ means that the constraints can be violated. Each constraint’s ‘‘hardness’’
ðB:4Þ
where the max function returns the maximum value for element k among all sites in the ecoregion and S is the total number of sites. If NREPjk equals 1, then element k has its highest representation at site j. For instance, if element y has 20 occurrences spread over 19 sites, then at one of those sites NREPjy = 1 and at the other 19 sites NREPjy = 0.5. In contrast, REPjy would equal 0.1 and 0.05, respectively. Maximum and average NREPj were calculated the same way as REPj. Rarity of element k is typically expressed as the inverse of the number of occurrences (Kershaw et al., 1995; Williams et al., 1996) RAREk ¼ 1=Ak
ðB:5Þ
where Ak is the number of occurrences of element k in the ecoregion. The definition works well for occurrence data that represent discrete locations but some ecological systems were represented as area. Area-based elements are incompatible with this rarity definition, and hence, a supplemental definition was needed. The rarity for area-based elements was defined as Pk 6 P0
RAREk ¼ 1
Pk > P0
RAREk ¼ ðP0 =Pk Þð1 Pk Þw
ðB:6Þ
780
B I O L O G I C A L C O N S E RVAT I O N
where Pk equals the proportion of ecoregion land area covered by element k, Po represents the proportion of ecoregion land area considered equivalent to an occurrence, and w determines how rapidly the function approaches zero. Pk and Po are not based on the size of a single patch; they are based on total land area in the ecoregion. Eq. (B.6) has three desirable properties: (1) elements with only one ‘‘occurrence’’ have RAREk equal to 1; (2) for w > 0, as Ak approaches the total land area of ecoregion, RAREk approaches zero; and (3) for w P 1, its concave shape resembles that of Eq. (B.5). The value assigned to Po was 0.00015 (0.015% of the ecoregion), which is 1 divided by the number of terrestrial sites, and the value assigned to w was 2. The average rarity for site j was defined as average RAREj ¼ ð1=bj Þ
bj X
RAREjk
ðB:7Þ
k¼1
where bj is the number of biodiversity elements at site j. We developed a novel measure of conservation value called irredundancy, which is the inverse of redundancy. We defined the redundancy of site j for element k as the number of sites that contain at least the same amount of element k: redundancyjk ¼
S X
LTEðajk ; aik Þ
ðB:8Þ
i¼1
where the function LTE equals 1 if ajk is less than or equal to aik, i.e., ajk is redundant relative to aik. LTE equals 0 if ajk is greater than aik. If site j has the most occurrences of element k, then LTE equals zero for all i except i = j, and therefore, redundancy equals 1. If the redundancy of site j for element k equals 1, then irredundancy also equals 1, which is the maximum possible value for irredundancy. If site j does not contain element k, then LTE equals 1 for all i and redundancy equals S. For very large S, irredundancy approaches 0, so for ease of interpretation irredundancy is set to zero when redundancy equals S. Average irredundancy for site j is defined similarly to Eq. (B.3).
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1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/biocon
Average optimacity: An index to guide site prioritization for biodiversity conservation George F. Wilherea,*, Mark Goeringb, Huilin Wanga a
Washington Department of Fish and Wildlife, 600 Capitol Way North, Olympia, Washington 98501-1091, USA The Nature Conservancy, 1917 1st Avenue, Seattle, Washington 98101, USA
b
A R T I C L E I N F O
A B S T R A C T
Article history:
We developed an index, called average optimacity, that can be used to guide site prioritiza-
Received 3 April 2007
tion for biodiversity conservation. Our index has three desirable properties: (1) it indicates a
Received in revised form
site’s potential contribution to an efficient representative reserve system; (2) it equals zero
28 December 2007
if and only if a site contains no elements of biodiversity; and (3) it accommodates multiple
Accepted 31 December 2007
sets of representation targets. The index is calculated with an optimization algorithm – simulated annealing as implemented in the computer program Marxan. We ran Marxan with 10 different representation target sets and averaged the results. The 10 target sets
Keywords:
approximated equal increments in conservation value. Data from the Willamette Valley-
Conservation planning
Puget Sound-Georgia Basin Ecoregion were used to evaluate the index. To help conserva-
Conservation assessment
tion planners understand average optimacity we: (1) compared it to an average of irreplace-
Irreplaceability
ability, a well-known index of conservation value, and (2) used logistic regression models to
Marxan
relate average optimacity to other measures of conservation value (e.g., rarity, richness). Average optimacity and average irreplaceability were highly correlated (Spearman rank correlation = 0.90), suggesting substantial agreement in site prioritization. However, there were clear differences in the rankings of highly valued sites, demonstrating that these two indices provide different perspectives on priorities. Simple regression models showed that average optimacity was strongly associated with maximum representation and maximum rarity. Two of the top multiple regression models for average optimacity exhibited a moderate increase in explanatory power relative to the best simple regression model and had three independent variables, which reflects the multi-dimensional nature of this index. The average optimacity index may be preferable to an irreplaceability index when efficiency is considered an important dimension for site prioritization. Ó 2008 Elsevier Ltd. All rights reserved.
1.
Introduction
Which places are the most important for biodiversity conservation? This fundamental question encapsulates two major challenges faced by organizations trying to conserve biodiversity – one scientific and one socio-economic. First, biodiversity, for our purposes defined as all native species, plant communities, and ecological systems within a geographic re-
gion, exhibits spatial patterns that confound simple formulations of importance. Species richness, species rarity, and other variables used to map ‘‘biodiversity’’ are often poorly correlated (Lennon et al., 2004). Peaks in richness may not coincide with peaks in rarity or endemism (Prendergast et al., 1993; Orme et al., 2005). Peaks in endemism for one taxon may not coincide with peaks in endemism for other taxa (Kerr, 1997). And, regional peaks in richness are unlikely to
* Corresponding author: Tel.: +1 360 902 2369. E-mail address: [email protected] (G.F. Wilhere). 0006-3207/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.biocon.2007.12.022
B I O L O G I C A L C O N S E RVAT I O N
include all species in that region (Williams et al., 1996). In short, any meaningful quantification of important places will most likely be a multi-dimensional relationship. Second, the funds that society has allocated toward conservation are insufficient to protect all places with biodiversity value. Consequently, efficiency is imperative in conservation planning (Pressey et al., 1993; Ando et al., 1998), and therefore, efficiency should be another dimension influencing importance. Furthermore, because funding is typically spread over time, conservation organizations must prioritize sites for protection. The problem of site prioritization has motivated extensive research in the development and application of various analytical techniques (for reviews see Justus and Sarkar, 2002; Williams et al., 2004). The first analytical approach is attributed to Ratcliffe (1971; cited in Justus and Sarkar, 2002). Ratcliffe (1971) proposed that sites be evaluated according to well-defined criteria such as community diversity, species rarity, and naturalness and that a multi-dimensional scoring system might be developed to determine the relative importance of sites. Scoring methods for site prioritization received considerable use through the 1970s (Margules and Usher, 1981) and similar prioritization schemes are still in use (e.g., Benayas and de la Montana, 2003). The main shortcoming of scoring methods is inefficiency (Kirkpatrick, 1983). High priority sites, i.e., those assigned high scores, may contain similar assemblages of species and communities resulting in unnecessary duplication (Pressey and Nicholls, 1989). Hence, this method of site selection could lead to inefficient nature reserve systems. In the pursuit of efficiency, scoring methods have largely been supplanted by approaches that reformulate the siteselection problem as an optimization problem. The solution identifies a set of sites which minimize an objective function while simultaneously satisfying various constraints. Typically, the objective function is a summation of the number, land area, or ‘‘cost’’ of selected sites, and the constraints are representation targets expressed as the number of occurrences per species or area per ecological system. Sites belonging to the solution constitute the most efficient nature reserve system needed to meet the representation targets. Kirkpatrick (1983) and Margules et al. (1988) were the first to use optimization algorithms for planning a reserve system. Their algorithms and many others since then (Csuti et al., 1997; Williams et al., 2004) have been heuristic, i.e., rule-based search procedures. Heuristic algorithms do not guarantee a truly optimal solution, only nearly-optimal solutions which are considered adequate for the purposes of conservation planning (Pressey et al., 1996; Csuti et al., 1997). Every optimal site-selection method exhibits at least one of three major shortcomings. First, many optimization algorithms generate a single indivisible solution. These optimization methods assume that the entire optimal set will be protected as a reserve system, but in nearly all real-world situations this is unrealistic (Meir et al., 2004). Rarely will sufficient funds become available within a reasonable time. Second, many optimization algorithms generate binary solutions; a site is either in or out of the reserve system. Such algorithms yield no information regarding the relative importance of sites, and therefore, cannot be used for site prioriti-
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
771
zation. Third, formulation of the optimization problem requires representation targets, which are effectively conservation goals. Conservation goals should be informed by science but are ultimately based on subjective values (Tear et al., 2005). Ideally, site prioritization would be the result of a totally objective analysis. The purpose of this study was to develop a site prioritization technique that might overcome the shortcomings of both scoring methods and optimization methods. We sought a numerical index for prioritizing site protection. The index would possess the following properties: (1) it is a good indicator of a site’s potential contribution to an efficient representative reserve system; (2) it equals zero if and only if a site contains no elements of biodiversity; and (3) it does not depend on a single set of representation targets. An index that comes close to possessing properties (1) and (2) is irreplaceability.
1.1.
Irreplaceability
Irreplaceability indicates a site’s potential to contribute to a representative reserve system. The original operational definition of irreplaceability was given by Pressey et al. (1993, 1994) – the proportion of representative reserve systems in which a site occurs. However, this definition has a problem with site redundancy. The inclusion of a site in a reserve system that is already fully representative increases that site’s irreplaceability even though the site is replaceable. Consequently, Pressey helped to develop a revised definition of irreplaceability that eliminated the redundancy problem – the proportion of representative reserve systems in which a site is necessary to be fully representative (Ferrier et al., 2000). The most direct way to calculate this revised irreplaceability is, at present, computationally impractical because the number of possible site combinations quickly approaches astronomical numbers. For this reason, Pressey et al. (1994) and Ferrier et al. (2000) developed indirect ways to estimate site irreplaceability. Irreplaceability has become a fundamental concept in conservation planning (Margules and Pressey, 2000), and the revised definition is quite elegant. Unfortunately, the appeal of ‘‘irreplaceability’’ has led to other definitions that alter its meaning (e.g., Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003). These other definitions arise from the use of optimization algorithms which offer a means to estimate a pseudo-irreplaceability. However, as it was originally defined, irreplaceability is not influenced by optimality in any way, and therefore, optimization algorithms are the wrong tool for estimating irreplaceability. This mistake is most prevalent in studies that have used the simulated annealing algorithm to create optimal reserve systems (Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003; Leslie et al., 2003). Simulated annealing performs a stochastic, heuristic search for the optimal solution (Rutenbar, 1989; Lockwood and Moore, 1993). Since it is stochastic, or random, simulated annealing can find more than one nearoptimal solution to a single optimization problem. Hence, a single site may belong to many different near-optimal solutions. The output of such algorithms enables the calculation of an index that is similar to but not the same as irreplaceabil-
772
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ity – the number of near-optimal solutions that include a site divided by the number of near-optimal solutions. This ratio has been misnamed ‘‘irreplaceability.’’
1.2.
Optimacity
The number of near-optimal solutions that include a site should be a good index of a site’s conservation value. This is the assumption made by Andelman and Willig (2002) and Leslie et al. (2003). Both used a simulated annealing algorithm to calculate an index which can be expressed as Oj ¼ ð1=RÞ
R X
xjr
ð1Þ
r¼1
where Oj is the conservation value of site j, R is the number of near-optimal solutions generated by R replicate runs of the optimization algorithm, and xjr is a binary variable that equals 1 when site j is included in solution r but 0 otherwise. The input data and parameter values, including representation targets, are identical for all replicates. Oj indicates a site’s potential contribution to an efficient representative reserve system. Specifically, it is the probability that site j will be included in an optimal or near-optimal reserve system. We call this index optimacity. Optimacity is different than optimality. Optimality is a quality of the entire reserve system, but optimacity is a quality of a site. Renaming ‘‘irreplaceability’’ (sensu Leslie et al., 2003) to optimacity was partly motivated by conservation planners. We found that some planners objected to efficiency influencing a site’s ‘‘irreplaceability.’’ They believed that ‘‘irreplaceable’’ was synonymous with ‘‘priceless’’, and therefore, efficiency was irrelevant to irreplaceability. The definitions of Pressey et al. (1993, 1994) and Ferrier et al. (2000) were much closer to the planners’ conception of irreplaceability. Because Pressey et al. (1993) was the first to operationally define irreplaceability, their definitions were adopted for our study. While new technical jargon is best avoided, we believe that the word ‘‘optimacity’’ effectively conveys the computational basis and meaning of the index. Optimacity is an index based on optimization; irreplaceability (sensu Pressey et al., 1993) is an index based on representation. Optimacity is partly a function of the representation targets. Changing the representation targets changes the number of sites included in each solution and it changes the number of solutions in which each site is included. As representation targets increase, optimacity steadily increases for most sites but for some sites optimacity may fluctuate as it increases (Stewart et al., 2007). In general, low representation targets typically yield a small number of sites with high optimacity, some sites with moderate optimacity, and many sites with zero optimacity (Leslie et al., 2003). The fact that some sites go from zero optimacity to positive optimacity demonstrates that this index is somewhat misleading – at low representation targets, some sites are shown to have no conservation value when they actually do. To address this shortcoming, we created an average optimacity index defined as j ¼ ½1=ðM RÞ O
M X R X m¼1 r¼1
xjrm
ð2Þ
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
where M is the number of different but closely related optimization problems. For instance, each of the M optimization problems might correspond to slight changes in a particular assumption or parameter value in the problem formulation. j , R replicates of the optimization algorithm To calculate O are run, a parameter value is changed, and R replicates are run again. This is repeated until M sets of replicate results are obtained.
2.
Study area
Our case study was associated with the Willamette ValleyPuget Trough-Georgia Basin Ecoregional Conservation Assessment (WPG ECA; Floberg et al., 2004). The WPG Ecoregion is a broad valley, approximately 810 km long, flanked by the Olympic Mountains and Coast Range to the west and the Cascade Mountains to the east. The ecoregion encompasses 3,977,000 ha of land; 82% in the United States and 22% in Canada. Its boundaries in the United States, specifically the states of Washington and Oregon, were delineated by Omernik and Gallant (1986) and the boundaries in Canada, specifically, British Columbia, followed those of Demarchi (1996). The ecoregion also encompasses Puget Sound and part of Georgia Strait, 1.5 million ha of marine waters. The effects of these marine waters, the adjacent mountain ranges, and pre-historic glaciation have created dramatic localized differences in climate, soils, and topography. From distinctive combinations of these factors spring an array of ecological systems and rare plant communities ranging from coniferous forests to herbaceous grasslands, oak savannahs, and rocky balds (Floberg et al., 2004). Historically, the predominant vegetation types were 39% conifer or conifer-deciduous forest, 25% oak or oak-conifer forest, 12% grasslands, and 23% wetlands (Johnson and O’Neil, 2001, pp. ix–xi).
3.
Methods
The methods consisted of five major stages: (1) assembling and processing land cover and species occurrence data; (2) defining the representation target levels; (3) using an optimization algorithm to generate optimacity values; (4) evaluating the average optimacity index by comparing it to other measures of conservation value; (5) building a multivariate model that explained the average optimacity and index.
3.1.
Data
A coarse filter–fine filter approach (Hunter, 1990; p. 238) was employed to represent all biodiversity in the WPG Ecoregion. This approach hypothesizes that preservation of all ecological systems and plant communities, i.e., the coarse-filter elements of biodiversity, will also preserve the majority of species that occupy them. The fine-filter elements are rare or imperiled species thought to be inadequately preserved by the coarse-filter elements. Hence, fine-filter elements of biodiversity warrant special attention in a conservation assessment.
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3.1.3.
773
Sites
Two types of spatial data corresponded to coarse-filter elements – an ecological systems layer and points denoting the locations of rare plant communities. The spatial data for fine-filter elements were points denoting the locations of species. Point data are also known as occurrences. We assembled data for 84 coarse-filter elements and 325 fine-filter elements from various sources. Detailed information about the spatial data can be found in Floberg et al. (2004).
For the purposes of analysis, the locations of all biodiversity elements were referenced to a regular hexagonal grid. The grid cells were 750 ha, and were also known as ‘‘sites.’’ Site records consisted of a unique identification number and the amount of each biodiversity element at that site. The entire ecoregion had 8107 sites, but the analysis excluded sites that were in marine waters. Of the remaining 6772 sites, 6100 contained a biodiversity element of some sort.
3.1.1.
3.2.
Coarse-filter elements
Nineteen ecological systems (sensu Comer et al., 2003) were identified in the WPG Ecoregion. To develop an ecological systems layer for the entire ecoregion we combined the best available land cover and vegetation layers from the three geo-political jurisdictions. To combine layers within a jurisdiction and to create consistent land cover and vegetation categories across the entire ecoregion we developed a rulebased reclassification scheme. Places with no ecological systems (e.g., urban areas, farmland, etc.) were classified as null data in the final ecological systems layer. Four ecological systems could be accurately modeled and comprehensively mapped across the entire ecoregion. High quality occurrences of these four ecological systems were also mapped as points. The other 15 ecological systems, which typically occur in smaller, discrete patches, were more accurately represented as point occurrences. In total, 3840 points denoted the locations of ecological systems. Plant communities are described by the dominant plants within the community, e.g., tufted hairgrass-Henderson’s checker mallow or Douglas fir-common snowberryoceanspray. Rare plant community locations were obtained from the British Columbia Conservation Data Centre (CDC), Oregon Natural Heritage Information Center (ONHIC), and Washington Natural Heritage Program (WNHP). These locations were mapped as single points and had to meet minimum standards for ecological integrity to be included in the analysis. We identified 65 rare plant communities and their locations were represented by 439 occurrence records.
3.1.2.
Fine-filter elements
Most fine filter data were obtained from the CDC, ONHIC, WHNP, and the Washington Department of Fish and Wildlife. Occurrence data records gave spatial coordinates of actual observations of plants or animals. Occurrence records were eliminated if they were: (1) animal data collected before 1980; (2) plant data collected before 1975, (3) outside the species’ current geographic range, (4) not in the species’ habitat type or in a location where the species’ habitat was known to have been destroyed, (5) too imprecise spatially, or (6) spatially redundant. The analysis included 186 vascular plant species and 47 non-vascular plant species. The plant species had 2,055 occurrence records. We identified 127 animal species as potential fine-filter elements, but could only assemble adequate data for 92 elements: 10 mammals, 29 birds, 5 reptiles, 14 amphibians, 7 fish, 23 insects, 3 molluscs, and 1 annelid. In total, 1428 animal occurrence records were included in the analysis.
Representation targets
The average optimacity index was calculated by running the optimization algorithm with M different sets of representation targets that spanned a wide range. For this study M equaled 10. The set of largest representation targets were fixed at 100% of species and rare community occurrences and 100% of ecological systems area. Increments between successive sets of representation targets were designed to yield roughly equal increments of biodiversity conservation. These equal conservation interval (ECI) target sets produced an index called ECI average optimacity, henceforth known as average optimacity. Preserving fine-filter elements is accomplished by reducing extinction risk. Therefore, we wanted our ECI targets sets to provide roughly equal increments in risk reduction. Natural Heritage Programs (NHPs) in the United States use various factors to determine the conservation status of imperiled species (Regan et al., 2004). One such factor relates the number of occurrences to qualitative levels of extinction risk (Table 1). We assumed equal intervals of risk for this NHP factor, and equated codes A, B, C, D with levels 1, 2, 3, 4. Hence, the relationship was modeled as the function: upper threshold = 100.59d+0.11, where d is level of extinction risk (r2 = 0.99, p < 0.0001). We used this function to interpolate between the thresholds to yield 10 ECI targets for population-like occurrences: populations, subpopulations, or population segments (Table 2). For each species, if the target exceeded the total amount available in the ecoregion, then the target was set to the amount available. Some species occurrence data corresponded to the nest or territory of breeding adults. Occurrences of this type were dealt with somewhat differently. We followed the same approach as above but used a different NHP factor that relates the number of individuals in a population to extinction risk (Table 1). We converted the number of individuals to number of nests simply by dividing by 2. We assumed equal intervals of risk in this NHP factor, and equated codes A, B, C, with levels 1, 2, 3. This relationship was modeled as the function: upper threshold = 100.65d+0.76 (r2 = 0.99, p < 0.0001). Because the coarse-filter elements were intended to preserve the vast majority of species, ECI target sets for ecological systems and plant communities were based on the species–area curve. The curve follows the relationship species / areaz, where ‘‘species’’ refers to the number of species, ‘‘area’’ refers to the area of ecological systems, and z is a constant (Rosenzweig, 1995). An oft cited rule-of-thumb for z’s value is called Darlington’s Rule (MacArthur and Wilson, 1967, pp. 8–9). The rule states that a doubling of species occurs for every 10 fold increase in area, hence z = log10(2). We
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Table 1 – Thresholds for two factors used by Natural Heritage Programs (NHPs) to determine species status (Regan et al., 2004) Threshold code
Risk level
Number of occurrences Lower threshold
A B C D E
1 2 3 4 5
Number of mature individuals
Upper threshold
1 6 21 81 301
Lower threshold
Upper threshold
1 51 251 1001 2501
50 250 1,000 2,500 10,000
5 20 80 300 None
Threshold codes, defined by NHPs, were translated to risk levels. ‘‘Occurrences’’ refer to known locations of species. We derived the nest and den ECI representation targets by dividing the number of individuals by two.
Table 2 – Ten sets of representation targets for two types of species occurrences Occurrence type
‘‘Populations’’ Nests/dens
Target set 1
2
3
4
5
6
7
8
9
10
1 6
2 10
3 16
5 25
8 43
13 70
20 125
31 191
49 315
All All
The difference between each set represents an equal conservation interval. Values are number of occurrences, which are points in the spatial data. Population occurrences were actually populations, subpopulations, or population segments. For each element, if the target exceeded amount available, then the target was set to the amount available.
used this value to derive targets that would represent equal increments in the proportion of all existing species. That is 0.05%, 0.5%, 1.8%, 4.8%, 10%, 18%, 31%, 48%, 70%, and 100% of each ecological system should represent 10%, 20%, 30%, . . . , 100% of currently existing species. Representation targets for plant communities, which were point data, followed the species–area curve (i.e., 0.05, 0.5, 1.8, . . . percent of points) or the top row in Table 2, whichever was larger.
3.3.
The optimization algorithm
We generated optimacity indices using Marxan, a computer program developed to identify optimal reserve systems (Ball and Possingham, 2000; McDonnell et al., 2002). Marxan, or its closely related predecessor SPEXAN, have been applied to reserve design problems throughout the world (e.g., Andelman and Willig, 2002; Noss et al., 2002; Stewart et al., 2003; Cook and Auster, 2005). Appendix A explains the optimization problem and our use of Marxan. To generate the optimacity values, we did 25 replicates for each set of representation targets. Hence, the product MR equaled 250. Because the largest target set was 100% for all biodiversity elements, we did not use Marxan to calculate the optimacity values for that target set. The value for all sites containing a biological element must equal 1 according to the optimization constraints.
3.4.
Other measures of conservation value
We evaluated the average optimacity index by comparing it to other measures of conservation value. The measures used were richness, rarity, representation, normalized representation, and a novel measure called irredundancy (see Appendix
B). Maximum and average values were calculated for all measures except richness.
3.5.
Average irreplaceability
Because irreplaceability has been confused with optimacity, we also compared average optimacity to an average of irreplaceability values. Irreplaceability values are partly a function of the representation targets. For this reason, we calculated an average irreplaceability using the same ten sets of ECI representation targets: Ij ¼ ð1=MÞ
M X
Ijm
ð3Þ
m¼1
where Ijm is the irreplaceability value of site j for one set of targets. Irreplaceability was calculated using CPlan (Pressey et al., 2005). Because the largest target set was 100% for all biodiversity elements, we did not use CPlan to calculate the irreplaceability values for that target set. By definition, irreplaceability for all sites containing a biodiversity element equals 1 when the representation target is 100%.
3.6.
Statistical analysis
The usefulness of the average optimacity index for conservation planning can be judged, in part, by its relationship to other measures of conservation value. To compare average optimacity to other measures of conservation value we constructed simple regression models (i.e., models with one independent variable) for average optimacity versus each measure. We did the same for average irreplaceability. Optimacity and irreplaceability are proportions. Hence, we did a logistic regression based on a generalized linear model assuming a binomial error distribution and with the link
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function set to logit. The resulting regression equation is of the form: j ¼ egðX1 ;X2 ;X3 Þ =ð1 þ egðX1 ;X2 ;X3 Þ Þ O
ð4Þ
where g(X1, X2, X3) = b0 + b1X1 + b2X2 + b3X3. b2 and b3 equal zero in the simple regression model. The well-known coefficient of determination, r2, cannot be calculated for generalized linear models. Hence, to quantify the strength of the each regression relationship we calculated an analog to r2, r2L , based on the ratio of log-likelihoods (Menard, 2000). An analysis of deviance assuming a chi-squared distribution was done to determine whether each simple regression relationship was significantly different than the corresponding null model (Crawley, 2002, pp. 525–528). For all statistical tests we chose a = 0.05. We further explored the relationship between average optimacity and other measures of conservation value by building logistic multiple regression models. We did the same for average irreplaceability. All combinations of the nine conservation value measures resulted in 511 possible models for average optimacity and 511 possible models for average irreplaceability. We constructed and evaluated all 1022 models. Many candidate models were eliminated from further consideration because their r2L was less than the best simple regression model. More candidate models were eliminated because of high multi-collinearity amongst the independent variables. Multi-collinearity was dealt with in two ways. First, all of the simple regression models had positive slope coefficients, and therefore, a negative slope coefficient in a multiple regression model was a sign of high multi-collinearity. Second, we calculated the variance inflation factor (VIF; Kleinbaum et al., 1988, pp. 210–211) for each model and eliminated those that had a maximum VIF greater than 4. The ‘‘best’’ models were based on Akaike’s information criterion (AIC), which indicates the best compromise between goodness-of-fit and model parsimony (Burnham and Anderson, 2002). Models with delta AIC less than 7 were considered to be among the best possible models (Burnham and Anderson, 2002; p. 70). Using AIC for model selection can result in models with less than optimal parsimony (Crawley, 2002; p. 452). To determine whether terms in each of the best models made a significant contribution to the model, we dropped each term and compared the smaller model to the original model using an analysis of deviance assuming a chi-squared distribution. All statistical analyses were done with R statistical software (RDCT, 2005; Fox, 2006)
4.
Results
Average optimacity is different than optimacity calculated with any of the 10 representation target sets (Table 3). For low targets (the first 5 sets), 75% of sites had optimacity equal to zero, but the third quartile for average optimacity was at 0.31. For moderate targets (sets 6 and 7), half of all sites had optimacity equal to zero, but the median value for average optimacity was 0.16. These comparisons show that optimacity can be a misleading index – for a wide range of representation targets, many sites are shown to have no conservation value when they actually do. For moderate and high targets (sets 6 through 9), 770–1621 sites had optimacity equal to 1,
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775
but only 90 sites had average optimacity equal to 1. This suggests that average optimacity is better for identifying the most important sites. Average optimacity and average irreplaceability showed some similarities in their distributions. There was little difference between their first quartiles, medians, and means (Table 3). The shapes of the two distributions began to diverge between the second and third quartiles – the third quartile equaled 0.31 for average optimacity but equaled 0.14 for average irreplaceability. Average irreplaceability had more high value sites than average optimacity. Two-hundred fifteen sites had average optimacity greater than 0.90 and 408 sites had average irreplaceability greater than 0.90. Only one site had average optimacity greater than 0.90 but average irreplaceability less than 0.90. Site prioritization implies ranking. The Spearman rank correlation between average optimacity and average irreplaceability was 0.90, suggesting little difference in how sites would be prioritized using either of these two indices. However, the rank correlation for high value sites is much less – 0.70 for sites with average optimacity greater than 0.90. In other words, prioritization of the most valuable sites was different. Potential differences in site prioritization become more apparent when plotting average optimacity versus average irreplaceability (Fig. 1). Simple logistic regression showed a strong association (r2L ¼ 0:797) and a fairly linear relationship between the two indices. However, when average irreplaceability is approximately 1, average optimacity is seen to range from 1 down to 0.7. For average irreplaceability in the range of 0.1–0.2, average optimacity ranges from 0.1 up to 0.4. Maps of average optimacity and average irreplaceability show different patterns of conservation value (Fig. 2). The maps generally agree on the location of high value sites, but keep in mind that the rankings within the high value categories are different. Many sites with intermediate value (0.2–0.4) on the average optimacity map have lower value (<0.2) on the average irreplaceability map. The analysis of deviance showed that all simple regression models for both average optimacity and average irreplaceability were significant. Average optimacity was most strongly associated with maximum representation and maximum rarity, r2L ¼ 0:744 and 0.738, respectively (Table 4, Fig. 3). In comparison, average irreplaceability had much stronger associations with these two measures, r2L ¼ 0:939 and 0.938, respectively. This makes sense because the definition of irreplaceability is based on representation and maximum rarity was highly correlated with maximum representation (Pearson’s correlation coefficient = 0.98). But for one exception, the associations of the all other conservation value measures with average irreplaceability were stronger than their associations with average optimacity. The one exception was maximum normalized representation. This makes sense because maximum normalized representation indicates which sites contain the largest amount of an element and one would expect such sites to be selected by an optimization algorithm. Based on AIC and r2L , multiple regression models were superior to simple models for both average optimacity and average irreplaceability (Table 5). The best simple regression model for average optimacity was based on maximum repre-
776
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1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
Table 3 – Summary statistics for optimacity calculated with each representation target set, average optimacity, and average irreplaceability Statistic
Representation target set 1
2
3
4
5
6
7
Average 8
9
10
Optimacity
Irreplaceability
Mean optimacity
0.03
0.04
0.06
0.08
0.11
0.15
0.23
0.33
0.488
0.90
0.24
0.21
Minimum 1st quartile Median 3rd quartile 90th percentile Maximum
0 0 0 0 0 1
0 0 0 0 0 1
0 0 0 0 0 1
0 0 0 0 0.08 1
0 0 0 0 0.52 1
0 0 0 0.12 1 1
0 0 0 0.32 1 1
0 0 0.08 0.68 1 1
0 0 0.48 0.96 1 1
0 1 1 1 1 1
0 0.10 0.16 0.31 0.57 1
0 0.11 0.12 0.14 0.64 1
Sites equal to 0 Sites equal to 1
6283 90
6249 192
6149 298
6038 419
5344 566
4441 770
3625 924
3003 1049
2384 1621
672 6100
672 90
672 102
N equals 6774, the number of sites.
1.0
but the best multiple regression model for average irreplaceability increased r2L by only 0.015.
average optimacity
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
average irreplaceability Fig. 1 – Average optimacity versus average irreplaceability, both calculated with ECI target sets. Points represent 6100 sites that had non-zero average optimacity and the solid line is the fitted logistic regression curve. Dashed line shows average optimacity equal to average irreplaceability and is shown for comparison only.
sentation, however, the dominant term in the best multiple regression model was maximum rarity. The best multiple regression model explained an appreciable additional amount of the variation in average optimacity relative to the best simple regression model; r2L increased from 0.744 to 0.884. The model for average optimacity with the highest r2L and nearly the lowest AIC value had three terms: maximum rarity, maximum normalized representation, and richness. The best models for average optimacity all included either maximum or average normalized representation. The best multiple regression model for average irreplaceability had maximum representation as the dominant term. Maximum rarity was the dominant term in the next best model. Irredundancy was a term in one of the best models for average irreplaceability but it had a small coefficient. The best average irreplaceability model had a greater r2L value than the best average optimacity model (0.954 versus 0.884),
5.
Discussion
5.1.
Average optimacity
We developed an index for prioritizing site protection. The index possesses the following properties: (1) it indicates a site’s potential contribution to a near-optimal representative reserve system; (2) it equals zero if and only if a site contains no biodiversity elements; and (3) it does not depend on a single set of representation targets. Optimacity possesses the first property because it was generated with an optimization algorithm. Irreplaceability does not possess this property. The index possesses the second and third properties because: (a) the optimization algorithm was run with a wide range of representation targets; (b) the largest representation targets were set to 100% of all biodiversity elements; and (c) the resulting optimacity values were averaged thus eliminating dependence on a single set of targets. When we introduced average optimacity to conservation planners they were curious about its interpretation. The planners wanted this index explained in terms they already understood, e.g., representation, rarity, and richness. To help planners understand average optimacity we (1) determined the statistical relationships between it and other well-known measures of conservation value, and (2) compared it with average irreplaceability. The regression models were not intended to be predictive. They were simply a means to gain insights regarding the new index. Using data from the WPG ECA and the ECI representation target sets, we found that average optimacity was strongly associated with maximum representation and maximum rarity and moderately associated with other measures of conservation value. The statistical modeling suggested that average optimacity is a somewhat more ‘‘multi-dimensional’’ index than average irreplaceability. The two and three variable models explained appreciably more variation in average optimacity than the best one variable model. The same could not be said for average irreplaceability.
B I O L O G I C A L C O N S E RVAT I O N
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
777
Fig. 2 – Maps of average irreplaceability (A) and average optimacity (B) for central portion of WPG Ecoregion. Six shades of gray represent six ranges of index values. Thick black line is ecoregion boundary and thin line is border between Washington and Oregon States. Each site is a 750 ha hexagon.
Table 4 – Values of r2L from simple logistic regression of average optimacity and average irreplaceability versus various measures of conservation value Average optimacity Max. representation Max. rarity Avg. representation Avg. rarity Max. norm. representation Max. irredundancy Avg. norm. representation Avg. irredundancy Richness Average irreplaceability
0.744 0.738 0.682 0.652 0.624 0.576 0.556 0.543 0.324 0.797
Average irreplaceability 0.939 0.938 0.936 0.853 0.415 0.695 0.601 0.727 0.370 –
(1) (2) (3) (4) (8) (6) (7) (5) (9)
Regressions included only sites that had non-zero average optimacity and nonzero average irreplaceability (N = 6100). First column is ordered by rank of r2L values. The numbers in parenthesis equal rank of r2L within that column.
5.2.
Site prioritization and conservation policy
Why did we need a new index for prioritizing site protection? Site prioritization techniques have been the subject of extensive research (e.g., Margules and Usher, 1981; Kershaw et al., 1995; Freitag et al., 1997; Pressey and Taffs, 2001; Benayas and de la Montana, 2003), and consequently, many different techniques were available to us. We found that none of the currently available techniques met our needs. A new technique was needed because a government agency played a major role in the WPG ECA, and government agencies must make clear distinctions between science and policy. The agency desired a conservation assessment free from any subjective decisions that might be misconstrued as an expression of government policy.
The degree to which efficiency should influence site prioritization is a decision for policy makers, not scientists. Nevertheless, we believed the preferred policy was rather obvious, and consequently, we adopted an optimal site-selection algorithm for our conservation assessment. Such algorithms have three potential shortcomings that were unacceptable to the government agency. First, optimization techniques can be used to generate single indivisible solutions, which implies that the entire solution should be protected. Using optimization techniques, some conservation assessments have identified potential reserve systems covering 30–40% of an ecoregion (e.g., Klahr et al., 2000; Rumsey et al., 2003). A commitment of that magnitude would require a demanding public process, but we were not vested with the authority nor given the resources to engage in such a process. Second, most optimization algorithms generate binary solutions – a site is either in or out of a reserve system. This type of solution is too rigid. Most conservation organizations need flexibility to respond to changing circumstances and unforeseen opportunities. Furthermore, an ‘‘in or out’’ solution might be misinterpreted by the public as an authoritarian decree – all sites in the solution must be protected and all sites out of the solution need no protection. We wished to avoid the damaging loss of public confidence that such misunderstandings often engender. Third, formulation of the optimization problem requires representation targets, which are effectively conservation goals. Conservation goals are a policy decision (Tear et al., 2005), and therefore, should be established in collaboration with policy makers. We did not have access to the policy makers who possessed the requisite authority. Marxan generates an output, which we called optimacity, that partly circumvents the first two shortcomings, but optimacity did not meet our needs in two respects. First, even though optimacity is not binary, a single set of reasonable representation targets will result in zero optimacity for many sites. The government agency was concerned that zero
778
A
B I O L O G I C A L C O N S E RVAT I O N
0.8
optimacity would be misinterpreted to mean zero value for biodiversity conservation. In other words, optimacity could mislead policy and management decisions related to land use regulation and land acquisition. Second, optimacity is generated using one set of representation targets. Again, the government agency was concerned that one set of targets might be misconstrued as the government-sanctioned conservation goals. Average optimacity overcame the technical shortcomings of optimacity and effectively avoided major entanglements with conservation policy. We attempted to avoid the policy decisions implicit in a single set of representation targets. However, developing the representation target sets still entailed subjective judgments but of a different kind. The value for M, deciding to use the species–area curve, the value for z, and deciding to use the NHP risk levels were all subjective judgments. These judgments largely determined the representation targets, and therefore, strongly influenced average optimacity values. However, none of these subjective judgments could be described as policy decisions. These were technical decisions intended to yield roughly equal increments of biodiversity conservation.
0.6
5.3.
0.4
The purpose of our study was not to pit optimacity against irreplaceability (for an in-depth comparison see Carwardine et al., 2007). Clearly, the two indices are defined differently, convey different information, and will result in different site rankings. However, their applications in conserving planning can be quite similar. Both indicate conservation value, but according to Margules and Pressey (2000) and Pressey and Taffs (2001) conservation value alone is insufficient for prioritizing site protection. They suggest that site vulnerability is the other critical dimension to prioritization and that a graph of irreplaceability versus vulnerability should be used to prioritize sites. We have taken a similar approach in ecoregional conservation assessments by graphing average optimacity versus vulnerability (e.g., Floberg et al., 2004; Pryce et al., 2006). Conservation actions were guided by the relative positions of sites in the two-dimensional average optimacity–
1.0
average optimacity
0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1. 0
maximum representation
average irreplaceability
B
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
1.0
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1. 0
maximum representation Fig. 3 – Results of simple logistic regression with maximum representation as independent variable: (A) dependent variable is average optimacity, (B) dependent variable is average irreplaceability. Points represent 6100 sites that had non-zero average optimacity and solid line is the fitted logistic regression curve. Equations for curves given in Table 5.
Applying average optimacity
Table 5 – Best multiple logistic regression models Dependent variable
Regression equation g(X1, X2, X3) = b0 + b1X1 + b2X2 + b3X3
r2L
Max. VIF
AIC
Average optimacity
2.44 + 9.04 max_RARE + 1.95 max_NREP 2.54 + 8.29 max_RARE + 1.89 max_NREP + 1.72 RICH 2.50 + 7.60 max_REP + 1.85 max_NREP + 1.84 RICH 1.76 + 8.98 max_REP + 9.37 avg_NREP 1.46 + 12.61 max_REP
0.884 0.890 0.888 0.796 0.744
1.21 1.39 1.40 1.42 1
3033 3034 3038 3040 3067
Average irreplaceability
2.18 + 20.86 max_REP + 7.52 avg_NREP 2.26 + 21.97 max_RARE + 7.37 avg_NREP + 0.81 max_IRRED 2.28 + 22.87 max_RARE + 8.06 avg_NREP 1.96 + 25.14 max_REP
0.954 0.958 0.956 0.939
1.47 1.59 1.43 1
2076 2077 2081 2101
The function g(X1, X2, X3) is substituted into the logistic equation, Eq. (4). All independent variables were normalized to a maximum value of 1. The variable abbreviations are max = maximum, avg = average, REP = representation, NREP = normalized representation, RARE = rarity, RICH = richness, and IRRED = irredundancy. The simple logistic regression model is shown for comparison only; it was not among the best models based on delta AIC less than 7.
B I O L O G I C A L C O N S E RVAT I O N
vulnerability graph. Whether irreplaceability or optimacity are used in the graph depends on whether efficiency is considered an important dimension for site prioritization. An index, such as average optimacity, can only guide conservation action. We will never be able to completely quantify the complex ecological, economic, social, and political factors that influence where biodiversity conservation can actually happen. Chance events and unforeseen opportunities will have a major influence on future conservation decisions, and local attitudes toward conservation can hinder or facilitate success. Biodiversity conservation requires difficult choices; we cannot protect every place with biodiversity value. Quantitative indices provide a more objective foundation upon which to establish priorities and weigh those choices.
779
1 4 1 ( 2 0 0 8 ) 7 7 0 –7 8 1
is determined by its penalty factor – the larger the fk, the firmer the constraint. Hard constraints can be established by setting an arbitrarily large fk. However, very large penalty factors can create ill-conditioned objective functions exhibiting sharp peaks or valleys, both of which make optimization more difficult (Gottfried and Weisman, 1973, pp. 253–254). To determine an acceptable value for each fk, we initially set all fk to 2 and then ran a simplistic iterative search which stopped when all elements met their targets in at least 38 consecutive Marxan runs. Thirty-eight is 1.5 times the number of replicates we used to generate optimacity. The resulting fk values ranged from 2 to 6 but at least 97% of the values equaled 2. We used the adaptive annealing with normal iterative improvement algorithm and each replicate consisted of 4 million iterations.
Acknowledgements Appendix B We thank E. Alverson, S. Cannings, J. Fleckenstein, J. Gamon, E. Scheuering, and M. Summers for assembling the species occurrence data. E. Alverson, C. Chappell, and A. Ceska helped to develop the ecological systems and rare communities data. W. Chang and K. Ryding provided advice on the statistical analyses. J. Pierce, M. Ingraham, P. Dunwiddie and four anonymous reviewers provided valuable critiques of the draft manuscript. This work was supported by a grant from the US Fish and Wildlife Service to the Washington Department of Fish and Wildlife.
S X
cj xj
ðA:1Þ
j¼1
subject to the constraints: ajk xj P tk
where ajk is the amount of element k in site j and Ak is the total amount of element k in the ecoregion. Maximum REPj and average REPj for site j were defined as maximum REPj ¼ maxðREPj1 ; REPj2 ; . . . ; REPjk ; . . . ; REPjB Þ B X
REPjk
ðB:2Þ ðB:3Þ
ðA:2Þ
j¼1
where xj is a binary variable that equals 1 when site j is included in the solution but 0 otherwise, cj is the cost of site j, ajk is the amount of biodiversity element k at site j, tk is the representation target for element k, S is the total number of sites, and B is the total number of biodiversity elements in the ecoregion (McDonnell et al., 2002). The ‘‘cost’’ of each site was area. Since all sites have equal area the algorithm minimized the number of sites. Simulated annealing, like other heuristic algorithms, is incompatible with constraints such as Eq. (A.2). Therefore, the constraints must be incorporated into the objective function as penalty terms, like so: S X j¼1
cj xj þ
where the max function returns the maximum value for site j among all biodiversity elements at site j and B is the total number of biodiversity elements in the ecoregion. The normalized representation of site j for element k was expressed as NREPjk ¼ ajk = maxða1k ; a2k ; . . . ; ajk ; . . . ; aSk Þ
for k ¼ 1; 2; 3; . . . ; B
minimize : FðxÞ ¼
ðB:1Þ
k¼1
Marxan minimizes an objective function using a simulated annealing algorithm. Our optimization problem was
S X
REPjk ¼ ajk =Ak
average REPj ¼ ð1=BÞ
Appendix A
minimize : FðxÞ ¼
Richness of a site equals the number of different biodiversity elements at the site. Representation (sensu Pressey and Nicholls, 1989) at site j for biodiversity element k was defined as
B X
fk PFðtk Þ
ðA:3Þ
k¼1
where PF is a function that determines the penalty when element k does not satisfy its target and fk is a penalty factor that determines the severity of the penalty for each target. The second term in the objective function imposes the constraints, however, they are soft constraints. ‘‘Soft’’ means that the constraints can be violated. Each constraint’s ‘‘hardness’’
ðB:4Þ
where the max function returns the maximum value for element k among all sites in the ecoregion and S is the total number of sites. If NREPjk equals 1, then element k has its highest representation at site j. For instance, if element y has 20 occurrences spread over 19 sites, then at one of those sites NREPjy = 1 and at the other 19 sites NREPjy = 0.5. In contrast, REPjy would equal 0.1 and 0.05, respectively. Maximum and average NREPj were calculated the same way as REPj. Rarity of element k is typically expressed as the inverse of the number of occurrences (Kershaw et al., 1995; Williams et al., 1996) RAREk ¼ 1=Ak
ðB:5Þ
where Ak is the number of occurrences of element k in the ecoregion. The definition works well for occurrence data that represent discrete locations but some ecological systems were represented as area. Area-based elements are incompatible with this rarity definition, and hence, a supplemental definition was needed. The rarity for area-based elements was defined as Pk 6 P0
RAREk ¼ 1
Pk > P0
RAREk ¼ ðP0 =Pk Þð1 Pk Þw
ðB:6Þ
780
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where Pk equals the proportion of ecoregion land area covered by element k, Po represents the proportion of ecoregion land area considered equivalent to an occurrence, and w determines how rapidly the function approaches zero. Pk and Po are not based on the size of a single patch; they are based on total land area in the ecoregion. Eq. (B.6) has three desirable properties: (1) elements with only one ‘‘occurrence’’ have RAREk equal to 1; (2) for w > 0, as Ak approaches the total land area of ecoregion, RAREk approaches zero; and (3) for w P 1, its concave shape resembles that of Eq. (B.5). The value assigned to Po was 0.00015 (0.015% of the ecoregion), which is 1 divided by the number of terrestrial sites, and the value assigned to w was 2. The average rarity for site j was defined as average RAREj ¼ ð1=bj Þ
bj X
RAREjk
ðB:7Þ
k¼1
where bj is the number of biodiversity elements at site j. We developed a novel measure of conservation value called irredundancy, which is the inverse of redundancy. We defined the redundancy of site j for element k as the number of sites that contain at least the same amount of element k: redundancyjk ¼
S X
LTEðajk ; aik Þ
ðB:8Þ
i¼1
where the function LTE equals 1 if ajk is less than or equal to aik, i.e., ajk is redundant relative to aik. LTE equals 0 if ajk is greater than aik. If site j has the most occurrences of element k, then LTE equals zero for all i except i = j, and therefore, redundancy equals 1. If the redundancy of site j for element k equals 1, then irredundancy also equals 1, which is the maximum possible value for irredundancy. If site j does not contain element k, then LTE equals 1 for all i and redundancy equals S. For very large S, irredundancy approaches 0, so for ease of interpretation irredundancy is set to zero when redundancy equals S. Average irredundancy for site j is defined similarly to Eq. (B.3).
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