Spatial Attributes and Reserve Design Models: a Review
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Environmental Modeling and Assessment (2005) 10: 163–181 # Springer 2005 DOI: 10.1007/s10666-005-9007-5 Spatial attributes and reserve design models: A review Justin C. Williamsa,*, Charles S. ReVellea and Simon A. Levinb a Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, MD 21218, USA E-mail: [email protected] b Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA A variety of decision models have been formulated for the optimal selection of nature reserve sites to represent a diversity of species or other conservation features. Unfortunately, many of these models tend to select scattered sites and do not take into account important spatial attributes such as reserve shape and connectivity. These attributes are likely to affect not only the persistence of species but also the general ecological functioning of reserves and the ability to effectively manage them. In response, researchers have begun formulating reserve design models that improve spatial coherence by controlling spatial attributes. We review the spatial attributes that are thought to be important in reserve design and also review reserve design models that incorporate one or more of these attributes. Spatial modeling issues, computational issues, and the trade-offs among competing optimization objectives are discussed. Directions for future research are identified. Ultimately, an argument is made for the development of models that capture the dynamic interdependencies among sites and species populations and thus incorporate the reasons why spatial attributes are important. Keywords: reserve design, biological conservation, spatial optimization, mathematical modeling 1. Introduction both the spatial attributes they address and the prospects for using them in tandem with optimization models. Methods for systematically selecting sites (land units) Finally, we conclude by discussing three other issues in for a nature reserve were first devised more than 20 years reserve design modeling: uncertainty, spatial scale, and ago, beginning with the pioneering work of Kirkpatrick [1] multiple objectives and trade-offs. Throughout the paper, (see also [2]). These methods sought to identify reserve our focus is on terrestrial reserves. Marine reserves, systems in which biodiversity, measured quantitatively, though, are now recognized as necessary components in a would be represented at desired levels. These initial global reserve network. Marine reserves are the topic of a methods were later criticized for emphasizing representa- recent special issue: Ecological Applications 13(1) Sup- tion over the long-term persistence of biodiversity (e.g., plement, 2003. [3]). Elsewhere in the conservation literature, however, guidelines had been suggested for the spatial design of reserves to promote the persistence of biodiversity (e.g., 2. Reserve site-selection models [4]), but these guidelines, too, were subject to criticisms for a variety of reasons. Quantitative methods for identifying an optimal or In this paper, we review how recent developments in efficient set of nature reserves originated in the 1980s in quantitative reserve design modeling have addressed, the field of conservation biology. The first such methods through spatial optimization, the important issues of repre- were iterative heuristic procedures for selecting sites of sentation and persistence of biodiversity within reserve land for a system of reserves. These methods sought to systems. We begin by presenting two basic mathematical achieve conservation objectives typically defined as some programming models for optimal reserve site selection and aspect of biological diversity: the representation of all discuss their limitations for achieving spatially coherent species from a list of target species (species richness) or the reserves. We next review important spatial attributes that representation of all features such as habitat types (e.g., are likely to be considered in the design of reserves. These [1,5,6]). Frequently, priority was given to rare or endan- attributes were initially proposed as design guidelines, and gered species or habitats. From this early work in heuris- we briefly review the arguments, both pro and con, sur- tic site-selection procedures, the problem of finding a rounding their use as guidelines. Third, we review the ways Bminimum reserve set^ emerged as the archetypal decision in which these attributes have been incorporated into recent problem. The minimum reserve is the smallest set of mathematical optimization models for reserve design. reserve sites (in number or in total area) needed to repre- Fourth, the rapid progress of optimization modeling in sent all species or other features. reserve design has been paralleled by the development of It was soon recognized that iterative heuristics could dynamic population models for evaluating reserve systems. guarantee only approximate minimum reserve sets, not the These dynamic models are briefly discussed in terms of true minimum (or global optimum). However, researchers pointed out that by formulating the problem as a zeroYone * Corresponding author. integer program (IP) model, globally optimal (or Bexact^) 164 Williams et al. / Spatial attributes and reserve design models: A review solutions could be found by linear programming (LP) (2) and (4) can be adjusted so that each species is repre- methods [7,8]. We call this IP for finding the minimum sented not once but two or more times in the reserve sys- reserve set the Bspecies set covering problem^ (SSCP) tem. One important variation of the SSCP is the Bpercent because its formulation mirrors that of the earlier Blocation area^ problem, in which total area is minimized subject to set covering problem^ from facility location science [9]. a lower bound constraint placed on the percentage of area of each land or vegetation class that must be included in the reserve. 2.1. Formulation of the species set covering problem The appealing aspects of solutions Y whether exact or Y X approximate to the SSCP, MCSP, and their variations are efficiency and cost-effectiveness. The solutions identify, Min xj ð1Þ j2 J respectively, the minimum (or approximate minimum) monetary investment or amount of land resources needed X to achieve a specified level of biodiversity, or the s:t: xj 1 8i 2 I ð2Þ (approximate) largest amount of biodiversity achievable j 2 Mi within a particular budget or total land area. Efficient and where i and I are the index and set of species to be cost-effective solutions are understandably desirable, espe- represented in the reserve, respectively; j and J are the cially when conservation resources are severely limited. index and set of sites eligible for selection (candidate The benefits of efficiency and cost-effectiveness, however, sites), respectively; Mi is the set of candidate sites j that have tended to overshadow a major drawback of these re- contain species i; and xj = 0 or 1; it is 1 if site j is selected serve selection models, namely, that they do not adequately for the reserve system, and is 0 otherwise. take into account the reserve system’s spatial attributes. The objective (1)minimizesthenumberofsites Solutions to the SSCP and MCSP may well be selected, and the constraint set (2) requires the selection collections of scattered sites that lack spatial coherence of at least one site containing each species. Following the (e.g., figure 1). Under these problem statements, the con- formulation of the SSCP IP model, a second archetypal figuration of the reserve system depends entirely on the reserve selection problem was also structured as an IP: the geographic distributions of species or other features in Bmaximal covering species problem^ or MCSP [10,11]. In relation to site cost or site area. No consideration is given the MCSP, the total number of sites is fixed, and the to reserve shape, edge conditions, the number of reserves number of species that can be represented is maximized. created, or the connectivity and proximity of reserves to each other. Hence, although SSCP and MCSP solutions do a good job of representing species in the short term, it is 2.2. Formulation of the maximal covering species problem highly uncertain whether these solutions would effectively support the long-term survival of those species represented X at the time of site selection. Long-term persistence within Max y ð3Þ i scattered reserve sites is especially problematic when the i2 I surrounding matrix is inhospitable, as in the case of an X urbanizing area. Persistence requires evolutionary units s:t: y x 8i 2 I ð4Þ i j such as populations or metapopulation of species to be re- j2 M i silient to environmental fluctuations and to other intrinsic X xj ¼ P ð5Þ j2 J where yi = 0 or 1; it is 1 if species i is represented in the reserve system (i.e., is represented in at least one site se- lected from the set Mi), and is 0 otherwise. The objective (3) maximizes the number of species represented, whereas constraint set (4) enforces the con- dition that a species is represented only if a site containing that species is selected, and constraint (5) limits the num- ber of sites that may be selected. A number of variations of the SSCP and MCSP have appeared, which are reviewed in [12,13]. Variations of the SSCP include minimizing total area or minimizing the total cost of sites selected instead of simply minimizing the number of sites, whereas variations Figure 1. An optimal solution to a hypothetical minimum reserve set of the MCSP include placing an upper bound or budget problem (SSCP) is shown. A minimum of 20 sites is needed to represent constraint on total area or total cost. In addition, constraints all species. These sites are likely to be widely scattered. Williams et al. / Spatial attributes and reserve design models: A review 165 and extrinsic factors. Ultimately, persistence is promoted 3. Spatial attributes in reserve design by heterogeneity, redundancy, and modularity of habitats and of populations [14], and reserve systems should be Diamond [4] developed six geometric design guidelines designed in accordance with these principles.