Combining probabilities of occurrence with spatial reserve design


Mar Cabeza, Metapopulation Research Group, Department of Ecology and Systematics, University of Helsinki, FIN-00014 Helsinki, Finland (fax + 358 9 19157694; e-mail


  • 1There is a great concern about the loss of biodiversity that calls for more nature protection. Unfortunately, the funding available for conservation is limited, and often a compromise is needed between nature protection and economic interests. Reserve selection algorithms are optimization techniques that concentrate on identifying a set of reserves that represents biodiversity efficiently. Simple approaches to reserve selection use presence–absence data assuming that if a species occurs in a selected reserve, it will persist there indefinitely.
  • 2A refinement of this technique is the selection of reserves according to the local probabilities of occurrence of the given species. These can be estimated from habitat models fitted by empirical modelling techniques. However, local probabilities of occurrence are not static, and are influenced by the changing threats to biodiversity in and around the reserves.
  • 3The effects of landscape change can be minimized when compact reserves are favoured. Compact reserves that represent all species with a given target probability of occurrence can be achieved by combining the probability approach with spatial reserve design.
  • 4In this study we brought together two reserve selection approaches that implicitly deal with biodiversity persistence, by combining habitat models and spatial reserve design in a single algorithm. We applied the combined algorithm to a data set of 26 species of butterflies from north Wales.
  • 5For the particular case study addressed in this paper, clustered reserve networks could be identified at no or a small increase in cost.
  • 6A new, backwards, heuristic algorithm performed better and more consistently than the regular forwards heuristic approach.
  • 7Synthesis and applications. The reserve selection approach presented considers habitat quality (via probabilities of occurrence) and the spatial configuration of the reserves during the selection process. Emphasis on reserve networks that include high-quality sites in an aggregated manner increases the potential for long-term persistence of species in the reserve network.


The development of systematic approaches for selecting reserve networks is of increasing interest to researchers and conservation practitioners. These approaches often involve the use of quantitative reserve selection algorithms, which seek to optimize a conservation function (e.g. species representation) given a set of constraints (e.g. cost) (Pressey 1999; Williams 2001). Most methods have attempted to identify sets of reserves that represent biodiversity in the most efficient way (sensuPressey & Nicholls 1989). However, several studies have recently emphasized that this approach is insufficient to ensure long-term persistence (Margules, Nicholls & Usher 1994; Virolainen et al. 1999; Rodrigues, Gregory & Gaston 2000; Cabeza & Moilanen 2001; Araújo, Williams & Fuller 2002; Cabeza & Moilanen 2003).

Reserve selection approaches in common use seek to identify minimum sets of areas (minimum total area or minimum number of sites) that represent, say, at least one population of each target species (or a percentage of its original extent) in the region of interest (Csuti et al. 1997; Pressey et al. 1993; Margules, Nicholls & Usher 1994; Pressey, Possingham & Margules 1996). Alternative approaches seek to identify reserve networks that increase representation of biodiversity for a given cost or area (Church, Stoms & Davis 1996). All these approaches often use a single snapshot of presence–absence data and imply that species would persist within reserves if represented. Nonetheless, some studies have attempted to refine the assumptions of the most simplistic reserve selection approaches by incorporating criteria for persistence within reserve selection procedures. For instance, some studies have used thresholds to define the minimum size of selection units, or used abundance data instead of presence–absence data (reviewed by Cabeza & Moilanen 2001). More recent approaches have used the local probability of occurrence of target species as a surrogate for species local persistence (Araújo & Williams 2000; Williams & Araújo 2000; Araújo, Williams & Fuller 2002) while others have considered the spatial locations of the reserves, implicitly acknowledging the role of spatial population dynamics on species persistence (Nicholls & Margules 1993; Possingham et al. 1993; Possingham, Ball & Andelman 2000).

probability of occurrence

Reserve selection methods have been adapted to deal with estimates of probability of occurrence (Williams & Araújo 2000; Williams & Araújo 2002 and references therein for an introduction to the use of probability of occurrence in reserve selection). Estimates of probability of persistence are obtained by modelling current species probability of occurrence from habitat quality (ability to survive) and dispersal (ability to colonize) factors. Probability of occurrence is generally expected to correlate to probability of persistence, at least for relatively short time periods, as shown for breeding birds in Great Britain (Araújo, Williams & Fuller 2002). However, the correspondence between probability of occurrence and probability of persistence within reserves depends on several additional factors. For example, the reported tendency of areas of high biodiversity to coincide with areas of high human pressure (Araújo 2003) may cause selected conservation areas to face greater challenges than expected (Araújo, Williams & Turner 2002). Furthermore, the factors used to calculate probability of occurrence are assumed to remain stable once the reserve system is established. However, in areas where competition for land is high, the pressure on non-selected sites will be greater than in designated conservation areas. Habitat fragmentation may follow and several studies have reported its negative consequences, from reduced migration success to several kinds of edge effects (Yahner 1988; Saunders, Hobbs & Margules 1991; Andrén 1994; Andrén 1997). Consequently, developments outside the reserves might have implications for the future probability of occurrence of the species in the reserve (Woodroffe & Ginsberg 1998). Hence, ideally, probability of occurrence should be estimated dynamically, in relation to the selected sites and to the consequent threats to biodiversity in and around the reserves.

spatial reserve design

Another way to address persistence within reserve selection procedures is to select reserves that are close together (Nicholls & Margules 1993; Possingham et al. 1993; Possingham, Ball & Andelman 2000; Leslie et al. 2003). Most minimum set approaches do not account explicitly for the spatial relationships among the sites selected and, therefore, the resulting reserve system is likely to be overdispersed. Reserves with high edge-to-area ratios will suffer from edge effects, including vulnerability to invasions, increased predation pressure and negative effects from abiotic factors like humidity and wind, etc. (Andrén & Angelstam 1988; Yahner 1988; Saunders, Hobbs & Margules 1991). From a metapopulation perspective, having a set of sites close together will enhance the regional persistence of a species by facilitating dispersal and allowing recolonization of unoccupied but suitable sites (Adler & Nürnberger 1994; Hanski 1999). From an economic perspective, boundaries need to be maintained, and longer boundaries often mean more neighbours and more cost. Nicholls & Margules (1993) and Possingham, Ball & Andelman (2000) introduced optimization methods that promote the clustering of reserves (i) by including a rule that chooses sites close together when breaking ties in iterative algorithms (Nicholls & Margules 1993) and (ii) by minimizing a linear combination of reserve network area and boundary length (Possingham, Ball & Andelman 2000). The level of aggregation of reserves achieved by the former method is low because compact spatial structure is not a primary aim of the algorithm.

integrating spatial constraints with high probabilities of occurrence

In this study we used models for the probability of occurrence coupled with spatial reserve design in order to select a compact network of reserves that represents all target species with a chosen target probability. By adding spatial constraints we can obtain more compact reserves. Consequently the potential negative effects of losing non-selected sites on the probabilities of occurrence are minimized.

Including additional constraints on the reserve selection process usually increases the cost of the solution in terms of the number of sites or the total area. For instance, Possingham, Ball & Andelman (2000) have shown how clustered reserve networks tend to need a larger number of reserves. When the largest weight for clustering was used, a completely clustered network of reserves was achieved but five times more sites were needed. The best balance between total area and clustering can be found by comparing the increase of area vs. decrease in boundary length when using different weights for the clustering. An intermediate value for the weighting can be chosen so that, even though the area is increasing, the boundary length is decreasing at a greater rate (Possingham, Ball & Andelman 2000).

We applied the new reserve selection approach to a data set of 26 species of butterflies from the 35-km2 Creuddyn Peninsula in north Wales. The distribution of these species has been mapped accurately on a 500-m grid throughout the landscape, and their densities have been measured on transect counts in 16 major habitat types within the peninsula. Therefore, we have detailed and accurate information on patterns of abundance, habitat association and distribution for each of the species, as well as some knowledge of the dispersal capacities of each species (Cowley et al. 2000, 2001a,b). Furthermore, logistic regression models applied to the distributions of these species have correctly classified between 58% and 98% (mean 83%) of presences and absences, based on the distribution of 16 habitat types (Cowley et al. 2000). The Creuddyn Peninsula is also an area of high conservation value, containing two endemic butterfly races restricted to the peninsula, as well as numerous other species that are more widespread or abundant in the study area than elsewhere in north Wales.

Materials and methods

study system and distribution data

As a result of its low elevation, limestone geology and habitat diversity, the Creuddyn Peninsula (53°18′N, 3°50′W) in north Wales is a regional hotspot for animal and plant diversity (Cowley et al. 2000). We used the distribution data for 26 species of butterflies and day-flying moths (hereafter ‘butterflies’) in the 35-km2 peninsula. The region has two endemic races, Plebejus argus caernensis (Thomson) and Hipparchia semele thyone (Thomson), localized species that are more abundant or widespread on the peninsula than for c. 100 km, and widespread species that are approximately equally common on and off the peninsula. All of these species are as abundant, and usually more abundant, on the peninsula than on the adjoining mainland. Therefore, it is reasonable to presume that the distribution and abundance of most species in the area reflect habitat and spatial population dynamics of the peninsula rather than immigration from elsewhere. The existence of endemic butterfly races is unusual in glaciated areas of northern Europe, illustrating the high conservation value of this landscape.

Between 1996 and 1998 a team of field workers made more than 14 000 site-specific records of butterflies (mapped to a 100-m resolution grid; Cowley et al. 2000). Analyses here and elsewhere have been carried out using a 500-m grid. At this resolution, the distribution records of each species had reached an asymptote: more recording effort would not have increased the known distribution by more than a few grid squares. (Gutiérrez et al. 2001). Indeed, further effort would have been more likely to reveal stray individuals, thereby obscuring genuine distributions, than to reveal new breeding colonies that had previously been overlooked. In addition to the landscape-scale censuses of all species, individual studies were made of the distribution of most of the more localized members of the butterfly fauna, such that their distribution and habitat requirements were known accurately (Gutiérrez et al. 2001; Menéndez, Gutiérrez & Thomas 2002; Thomas, Wilson & Lewis 2002; Wilson et al. 2003).

All butterfly species encountered in the field were recorded during the census. However, in the analyses we excluded Boloria selene (Denis & Schiffermuller), which existed as a single small population that became extinct during the study period. This species is more abundant outside the peninsula and may only exist on the peninsula as small transient populations. We also excluded two tree canopy species, Quercusia quercus (Linnaeus) and Srymonidia w-album (Knoch), that were under-recorded, and three ubiquitous species, Pieris rapae (Linnaeus), Cynthia cardui (Linnaeus), Aglais urticae (Linnaeus), that were predicted to be and actually were recorded in every 500-m grid cell: these provided no useful information with regard to site selection. We used the remaining 26 species for the study.

habitat-based statistical models

The reserve selection procedure requires estimates of the probabilities of occurrence of each species in each 500-m grid square. Typically, these probabilities are derived from relating presence–absence data directly to attributes of the landscapes. However, we adopted a different approach in three steps. First, 300-m long butterfly transect walks were established in each of the 16 habitat types (using the methods of Pollard & Yates 1993). Each transect site was monitored every other week from April–October 1997, with 10 separate sites monitored for most habitat types, giving relative densities of each species in each habitat type (Cowley et al. 2000). Secondly, the percentage cover of each of the 16 habitat types was calculated for each grid cell (Cowley et al. 2000). Thirdly, density and habitat cover data were combined for a given species for each grid cell to give a density score or ‘habitat value’ for each grid square. This ‘represents the predicted yearly count of individuals along a 1-km transect through a (given) grid square that sampled habitats in direct proportion to their extent’ (Cowley et al. 2000).

For each species in turn, habitat values were converted into probabilities of occurrence using logistic regression, with the presence–absence of a species as the dependent variable and habitat value as the independent variable. The advantage of this approach is not only that habitat value and presence–absence data represent entirely independent data sets, but also that habitat value is based on predictions of abundance, which we might expect to be linked to likelihood of persistence. Cowley et al. (2000) tested the models statistically using a jack-knife procedure (leaving out each cell in turn): in the current study probability surfaces were based on all cells.

data sets

The selection of the reserve network was based on the original data set (predicted probability of occurrence as in Cowley et al. 2000). However, the species might have additional habitat requirements not necessarily described by the 16 habitat categories used in the analysis. In order to avoid the selection of sites with small species’ probabilities where species’ requirements might not be fulfilled, we produced a second data set that reflected the genuine variation in habitat quality. This was done by a filtering procedure in which species’ probabilities of occurrence were set to zero in all those sites that had a probability smaller than the minimum probability for that species in an occupied site (Fig. 1).

Figure 1.

Different representations of species richness (four classes). The darker the shade, the higher the richness. (a) Richness is the sum of probabilities across all species for each grid cell; (b) as (a) but using the filtered data; (c) richness is the sum of presences across all species for each grid cell, using the original data. Note that for (a) and (b) intermediate levels of richness might mean few species with high probabilities or many species with low probabilities.

the problem

Two algorithms were used to address a minimum set problem: what is the minimum combination of number of areas and boundary length required to represent all target species with at least a target probability Tj? Boundary length here measures the total number of cell edges neighbouring non-selected sites (including natural edges, e.g. the limits of the region).

The problem addressed by the algorithms can be formulated as a mathematical programming problem: let the total number of sites be N and the number of species SP. The information about the probability of a species being found at a site is contained in a site-by-species (N × SP) matrix, P, whose elements pij are the probability of finding species j in site i. Let S be the set of selected sites, | S | the number of selected sites and Ii an indicator variable determining whether or not site i is included in the reserve network:


The minimum set problem is:


image(eqn 1)

subject to:

pj  ≥ Tj for all species(eqn 2)

where ci is the cost of including site i (ci= 1 here, as all sites have the same area and are considered to be of the same cost), L′ is the ratio of boundary length of the selected reserve system to the total area, and b is the boundary length penalty, which determines the relative weights given to area and boundary length in optimization (note that when b= 0 the problem becomes a ‘classic’ minimum set problem, which only minimizes the number of sites). pj is the probability of having at least one occurrence of species j in any site [1 − (the product of probabilities of local non-occurrence); Williams & Araújo 2000]:

image(eqn 3)

Note that equation 3 assumes spatial independence between local probabilities. Tj (equation 2) is a chosen target probability for each species. To assess the performance of the algorithms and the difference between the spatial locations of the selected reserves, different targets Tj were used.

target probability

We used two different probability target schemes. (i) Assuming that there would not be species-specific information, a fixed probability target, equal for all species, was set. A range of fixed probability targets (0·8, 0·9, 0·95, 0·99, 0·999) was used. (ii) We used available species-specific information and experts’ opinions to choose species-specific probability targets. Species were classified into four categories according to their relative conservation importance in the Creuddyn Peninsula (see Fig. 6 for the classification and corresponding probability targets).

Figure 6.

Results obtained with the backwards algorithm at b= 0 and b=bmax, using different targets for different species, according to four categories: (v) endemic race of localized species; (x) local species, regionally significant or nationally declining; (y) more widespread species; (z) widespread species for which the local area is not important. (a) and (b) show different weighting strategies for the different categories. The probability target values used in (a) and (b) are, respectively, a= 0·9999, b= 0·99, c= 0·7, d= 0·5; a= 0·9999, b= 0·999, c= 0·95, d= 0·9.


We used two variants of a heuristic iterative algorithm for comparison: one works forwards, by adding the sites to become reserves one by one; and the other works backwards, starting with all sites selected and excluding lowest-quality sites one by one. Thus both algorithms choose one site at each iteration to be added (forwards) or removed (backwards) (equation 4) by examining how much each site contributes in reaching the target for all species (ΔKi in equation 4) in relation to how much boundary length relative to area is added or removed (Δinline image in equation 4), weighted by the boundary length penalty b. Let F be the value of the solution, and ΔFi the change in the solution caused by adding/removing site i:

image(eqn 4)

Then, the site with largest ΔFi value is chosen to be added or removed.

Forwards algorithm

This implementation of a ‘best-in greedy algorithm’ (using terminology of Korte & Vygen 2000) is a modified version of the method described in Williams & Araújo (2000) that incorporates a penalty for the boundary length. For this algorithm, equation 4 becomes:

image(eqn 5)

where Ri is the contribution of site i to the achievement of the target probability for all species. Ri is computed as the goal–gap difference (difference between target and current representation of species j), summed over all species, that the addition of site i would cover (Williams & Araújo 2000). L is the boundary length of the current solution, and ΔL is the change in boundary length caused by the addition of site i.

The first term of the equation (ΔKi in equation 4) is a positive value; the smaller the value, the worse the site. The second term (bΔLi) is larger the more isolated is the site. Therefore, adding the site with the largest ΔFi adds good sites in terms of goal–gap contributions, or clustering, or both. The addition of sites to the solution is continued until the target probability (Tj) is achieved for all species.

Backwards algorithm

Unlike familiar heuristic algorithms used in the field of reserve selection, this is an implementation of a ‘worst-out greedy algorithm’ (Korte & Vygen 2000) that starts with all candidate sites selected as reserves, and at each iteration removes the site that would damage the solution least if not included (≈ the site that reduces fragmentation most while decreasing representation little). The removal of the sites is continued until no more sites can be removed without going below the probability target for any species.

For this algorithm, in order to avoid the strong non-linearities in the probability calculations, all quantities related to species representation are calculated in terms of the expected number of populations:

image(eqn 6)

where Rj is the current representation of species i in terms of number of populations (equation 7) and inline image is an approximation of the species target (Tj) in terms of the number of populations. Rj − inline image represents the extra number of populations still present in the set of selected sites:

image(eqn 7)

The symbols in the second term of equation 6 are as described for the forwards algorithm.

To keep probability targets comparable between the forwards and backwards algorithms, we first convert the probability target used by the forwards algorithm (Tj) to a ‘number of populations’ target used by the backwards algorithm. inline image is calculated by first ranking all sites in decreasing order of pij value, giving a set of values prj where pr+1,j ≤ prj, and then finding the smallest Nmin < N for which Aj ≥ Tj in the equation below:

image(eqn 8)

Then the corresponding probability target framed as the expected number of populations is:

image(eqn 9)

To compare with the forwards algorithm, and looking back at equation 4, the first term, ΔKi, is negative and smaller (larger in absolute value) the more a site contributes to the representation goal. The second term (−bΔLi) has a positive value; the removal of a site would cause more fragmentation the larger its value for this term. Therefore, removing the site with the largest ΔFi value removes sites with either small representation contributions or isolated sites or both.

Note that the two algorithm variants, although addressing the same problem, use different heuristic steps, not only differing in the direction (forwards/backwards) but also in the quantities that are maximized. Therefore the results obtained using the same b but different algorithms are not directly comparable; only the solutions at the extreme b values are comparable: b= 0 and b=bmax. The situation for which no weight is given to the value of the boundary length is represented by b= 0. At the other end, bmax, is a sufficiently large value of b (b→∞) at which the spatial configuration of the solution is such that increasing the value of b would not change it (i.e. in practice only the boundary length is optimized). For different values of b the results between algorithms are comparable when either the costs of the solutions (area) are equal or when the boundary length of the solution is the same.


Both algorithms selected clustered reserves when the boundary length penalty (b) was larger than zero, i.e. when the cost of the boundary length was considered. In general, the boundary length of the solutions decreases when increasing b until it converges to the minimum value found by the particular algorithm at the particular probability target (Fig. 2). As expected, the sizes of the reserve systems, in terms of number of sites and boundary length, were larger for larger probability targets. By increasing the probability target we can observe how the solutions might shift from needing only one cluster, to two clusters and again to one cluster (Fig. 3)

Figure 2.

Decrease in boundary length for different probability target levels, when increasing the value of the boundary length penalty (b). The four graphs show the results for both algorithms and for filtered and non-filtered data. (a) and (b) use non-filtered data, (c) and (d) use filtered data (diamonds, target = 0·999; triangles, target = 0·99; circles, target = 0·9).

Figure 3.

Selected grid cells at three values of b: when no clustering is imposed to the solution (b = 0), solution at an intermediate b value, and when the maximum clustering is achieved (b=bmax). The intermediate b value was selected so that the size of the solutions (in terms of area or boundary length) were similar. Solutions are shown for different probability targets (left column, T) and for both algorithms. Original probability data were used. Selected cells are marked in black, non-selected in grey.

The smallest solutions are generally found by the backwards algorithm but it varies for different scenarios. The backwards algorithm seems to be more efficient and consistent at intermediate b values (Figs 2–4) and there is a difference in the number of clusters in the final solution.

Figure 4.

As Fig. 3, but for the filtered data.

For this particular data set, clustering can be achieved at very little cost (in terms of number of sites). Across all solutions, the average increase in the number of sites was 8% while the average decrease of boundary length was 53% (Fig. 5). The backwards algorithm appears to achieve major reductions in boundary length with smaller increases in the number of sites (average increase in number of sites was 0·5%, average decrease in boundary length was 55%). This is in agreement with the fact that the backwards algorithm is not forced to select only one large cluster.

Figure 5.

The cost of decreasing the boundary length of the solution. Results show the boundary length (triangles) and the number of sites (circles) at different values of b, for both algorithms, and for two different targets, 0·9 (solid line) and 0·999 (dotted line). Results only shown for the filtered data.

These results are expected to be highly dependent on the data. Other data sets might require larger number of sites to achieve clustered solutions (Possingham, Ball & Andelman 2000). Nevertheless, comparing the rates of increase in the number of sites with the rates of decrease in boundary length, an intermediate point (corresponding to an intermediate value of b) can be chosen that satisfies both the cost and the clustering for the particular conservation problem. The choice of the level of cost (area) and clustering is likely to be a heuristic choice involving political and economical criteria.

For this particular case study the solutions for mid values of b seem to be the most pragmatic. The advantage of the algorithm is the ability to include both habitat quality (i.e. probability of occurrence) and spatial configuration. Intermediate b values do not give too much weight to either of the components. High values of b produce good solutions in terms of species persistence; however, many of the selected sites that link the clusters were already of low quality, and their inclusion in the final selection might not increase the probability of species persistence, especially for high probability targets.

Using filtered data reduces the flexibility in the choice of cells to be selected (sites that had low probabilities in the non-filtered data set do not contribute at all for particular species in the filtered data set). The final solutions, although clustered, have larger boundary lengths than the solutions for the unfiltered data set.

Also, it will be possible to achieve better levels of protection for critical or rare species at lower expenditure (i.e. relatively few cells) if information about the conservation priorities of species (Fig. 6) is available. For instance, a large probability target (0·9999 and 0·999) and a clustered solution can be achieved for particular species (endemic and regionally significant) without increasing the area needed to represent all species at a smaller probability target (0·95) (Fig. 4, T= 0·95; Fig. 6b).


In this study we bring together two different reserve selection approaches that implicitly deal with improving the chances of species persistence in reserve networks. (i) Using probabilities of occurrence, predicted from habitat models, to choose a set of areas that together represent each species with a targeted probability (probability of having at least one occurrence of species j in any site) (Araújo & Williams 2000; Williams & Araújo 2001; Williams & Araújo 2002). (ii) Clustering of the reserves by minimizing the boundary length of the reserve system in the optimization process (Possingham et al. 1993; Possingham, Ball & Andelman 2000).

Araújo, Williams & Fuller (2002) have shown how the probability of extinction for a set of species can be reduced if reserve areas are selected to maximize the probability of occurrence of those species at the present time. Using data on passerine birds for two time periods, 20 years apart, they showed that the networks selected with the probability method improved the persistence of the species in the reserve system (i.e. 100% of the species were still represented in the second time period) compared with the performance of areas selected with classical single representation algorithms. Unlike the present study, the estimates of occurrence in Araújo, Williams & Fuller (2002) were based on a contagion measure, i.e. for a species in a site the probability of occurrence is higher the larger the number of occupied neighbours. This means that, if the selected reserves include areas of high probability but do not include the neighbours, there is no guarantee that the probabilities of persistence in the reserves are still high following habitat loss in the non-selected neighbouring cells.

Cabeza & Moilanen (2003) simulated species’ spatial population dynamics and showed that when reserves are selected far apart and manage to represent each species only one or a few times, species’ extinctions may be expected. The chances of species extinction in such a reserve system are even greater if the habitat surrounding the reserve is lost. When the group of reserves was selected as a cluster, the number of extinctions in the presence of habitat loss was much smaller (Cabeza & Moilanen 2003).

By incorporating spatial constraints in the selection process, the chances that the probabilities of occurrence remain more or less stable are larger, as effects of habitat fragmentation are reduced. Interestingly, as shown in this study, clustering of reserves may be achieved at a low cost (in terms of total area or number of sites). Therefore, we suggest that spatial constraints should be taken into account in reserve network design. The costs (in terms of increased total area or number of sites) of minimizing the boundary length of the reserve system should be evaluated and, whenever affordable, clustering of reserves should be favoured.

Nonetheless, there might also be reasons to avoid clustering. In situations where catastrophes can impact large areas and cause local extinctions, it may be less risky to choose reserves in at least two or three separate places, rather than having them all clustered together (Hess 1996; Lei & Hanski 1997). However, if reserves are widely separated from each other, they should be large enough to allow species to persist as independent populations.

The question of whether to use exact algorithms or heuristic algorithms in the field of reserve design has been a topic of major discussion (Cocks & Baird 1989; Church, Stoms & Davis 1996; Pressey, Possingham & Margules 1996; Pressey, Possingham & Day 1997; Cabeza & Moilanen 2001; Rodrigues & Gaston 2002). Exact algorithms may be theoretically preferred because they can be guaranteed to find an exact optimal solution (Cabeza & Moilanen 2001; Underhill 1994; Camm et al. 1996), but in practical applications heuristics are preferred for their clarity and speed, allowing interactive reserve selection actions. Also, exact algorithms cannot be easily adapted to solve all reserve selection problems. The problem addressed here is one of these examples: it is a highly non-linear problem because of the way combined probabilities are calculated and because the cost of adding a site to the reserve system depends on which other sites are already protected, and on the spatial relationships between candidate sites and those already selected.

We have introduced in this paper a new iterative heuristic algorithm that works backwards in respect to commonly used algorithms in reserve selection approaches. Common iterative algorithms select a set of sites, adding, at each iteration, the best site to the solution. The backwards algorithm, instead, starts by including all sites in the solution, and then removes one site at each iteration, until achieving the solution given the constraints. The forwards algorithm tends to select a single large cluster, whereas the backwards algorithm usually selects different clusters. This is because starting a new cluster is much more costly for the forwards algorithm than adding sites to the current selected cluster. Instead, the backwards algorithm starts by throwing away ‘bad’ patches, creating different clusters from the beginning of the iterative process, and then keeps on reducing the size of these clusters as long as the target is fulfilled for all species.

For the problem and data set used in this study, the backwards algorithm appears to be more consistent and yields, in general, better results than the forwards algorithm. However, the backwards algorithm does not guarantee to find the exact optimal solution. When the efficiency of the results is an important feature, numerical stochastic global search methods can be used (e.g. simulated annealing; Possingham, Ball & Andelman 2000; Leslie et al. 2003). In any case, when an exact algorithm cannot be used because of the characteristics of the problem, we recommend the use of more than one heuristic to increase confidence in the results.

A more complete consideration of the effects of spatial configuration on species persistence would require a more explicit consideration of spatial population dynamics. This can only be done if a habitat model (or a metapopulation model) with an explicit spatial component (e.g. species-specific dispersal ability) can be parameterized (Moilanen & Cabeza 2002; Cabeza 2003). In cases where such information is not available the reserve selection approach presented in this paper, which selects high-quality sites in an aggregated manner, increases the potential for long-term persistence of species in the reserve network.

Other issues that could be further considered in the context of the present approach to reserve selection include improved landscape modelling, irreplaceability of sites, uncertainty analysis and scheduling of conservation action.


Special thanks to Mark Burgman for useful comments on the manuscript. This work has been supported by a grant from the Academy of Finland (project no. 74125) to M.C.