ARTICLE IN PRESS Journal of Theoretical Biology 264 (2010) 604–612 Contents lists available at ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi How to resolve the SLOSS debate: Lessons from species-diversity models Even Tjørve Lillehammer University College, P.O. Box 952, N-2604 Lillehammer, Norway a r t i c l e in f o a b s t r a c t Article history: Received 7 October 2009 Received in revised form 19 January 2010 Accepted 6 February 2010 Available online 10 February 2010 The SLOSS debate – whether a single large reserve will conserve more species than several small – of the 1970s and 1980s never came to a resolution. The ﬁrst rule of reserve design states that one large reserve will conserve the most species, a rule which has been heavily contested. Empirical data seem to undermine the reliance on general rules, indicating that the best strategy varies from case to case. Modeling has also been deployed in this debate. We may divide the modeling approaches to the SLOSS enigma into dynamic and static approaches. Dynamic approaches, covered by the ﬁelds of island equilibrium theory of island biogeography and metapopulation theory, look at immigration, emigration, and extinction. Static approaches, such as the one in this paper, illustrate how several factors affect the number of reserves that will save the most species. This article approaches the effect of different factors by the application of species-diversity models. These models combine species–area curves for two or more reserves, correcting for the species overlap between them. Such models generate several predictions on how different factors affect the optimal number of reserves. The main predictions are: Fewer and larger reserves are favored by increased species overlap between reserves, by faster growth in number of species with reserve area increase, by higher minimum-area requirements, by spatial aggregation and by uneven species abundances. The effect of increased distance between smaller reserves depends on the two counteracting factors: decreased species density caused by isolation (which enhances minimum-area effect) and decreased overlap between isolates. The ﬁrst decreases the optimal number of reserves; the second increases the optimal number. The effect of total reserve-system area depends both on the shape of the species–area curve and on whether overlap between reserves changes with scale. The approach to modeling presented here has several implications for conservational strategies. It illustrates well how the SLOSS enigma can be reduced to a question of the shape of the species–area curve that is expected or generated from reserves of different sizes and a question of overlap between isolates (or reserves). & 2010 Elsevier Ltd. All rights reserved. Keywords: Species–area relationship Species extinction Species overlap Reserve design Minimum-area requirements 1. Introduction The greatest challenge in environmental management is to preserve enough habitats in order to save as many species as possible from extinction. In the mid-1970s researchers proposed six of rules for reserve design (Diamond, 1975; May, 1975; Terborgh, 1974; Wilson and Willis, 1975). The ﬁrst of these rules states that, for a ﬁxed total area, one large reserve will conserve more species than several small ones. Later, others began to question the validity of most of these original rules of thumb. The debate over whether a single large area or several small areas will hold the most species, denoted by the acronym SLOSS, became the most infamous of these (Abele and Connor, 1979; and more recently Etienne and Heesterbeek, 2000; Gilpin and Diamond, Tel.: + 47 61288219; fax: +47 61288170. E-mail address: [email protected] 0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.02.009 1980; Groenveld, 2005; Ovaskainen, 2002; Simberloff, 1988; Simberloff and Abele, 1982; Simberloff and Abele, 1976). A large number of empirical and theoretical studies have addressed the SLOSS question, but it is still considered to be unresolved. Despite this impasse, the ﬁrst rule of reserve design, which many authors considered an unproved over-simpliﬁcation, became the norm for reserve planning. According to Connor and McCoy (1979), it is unfortunate that species–area relationships (SARs) were used as justiﬁcation for conservation practices that led to the preference for large areas being preserved over small areas. Soule´ and Simberloff (1986) claimed that the SLOSS concept is not useful and should be abandoned. They suggested that the minimum area needed to sustain a viable population, habitat diversity, environmental quality, and distance between reserves should decide the size of reserve areas. Whether a single large area or several small areas will hold the most species is, therefore, expected to depend on the situation considered. Yet, far from ending the SLOSS debate, Soule´ ARTICLE IN PRESS E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 and Simberloff’s arguments, by highlighting the crucial factors, actually justify the furtherance of SLOSS modeling. Accordingly, this study aims to integrate into the modeling of species-diversity such important factors as species overlap, minimum-area requirements, habitat diversity, and the distance between reserves. There are numerous empirical studies showing whether a single large or several small reserves will hold the most species (see Ovaskainen, 2002 for review). Though much of the discussion has advocated either a single large or several small reserves as the universally best design, we should expect neither to be always the best strategy. The optimal solution depends on factors that vary from place to place. Also the choice of focal species will affect this issue: do we want to ‘save’ as many species as possible, or do we want to preserve speciﬁc species, e.g., rare or globally threatened ones? 1.1. Different approaches The ﬁrst rule of reserve design (that states that a single large supersedes several small) has its origins in MacArthur and Wilson’s (1963, 1967) equilibrium theory of island biogeography, which they based on immigration and extinction. The island equilibrium model does not, however, take into account the recolonization between the reserves in a network of reserves. Recolonization, as discussed by metapopulation dynamics, may serve as a buffer against species extinction (e.g. Etienne and Heesterbeek, 2000; Hanski and Ovaskainen, 2000; Ovaskainen, 2002). If a population at one small reserve (in a system of several reserves) becomes extinct, then individuals from other small reserves can recolonize it. Island equilibrium theory and metapopulation modeling can thus be applied to discuss reserve conﬁguration by considering the effects of reserve size, immigration, emigration, and extinction rates. Neither of these two types of dynamic approaches considers speciﬁcally the effect of habitat diversity, resource restrictions or minimum-area requirements. Resource restrictions and minimum-area requirements will affect population sizes and may determine whether the reserve (or area) can sustain viable populations, and therefore the probability of occurrence (see e.g. Gurd et al., 2001; Lomolino, 2000). Such minimum-area effects (MAEs) will restrict the number of species at smaller scales by decreasing the slope of the species area curve (Tjørve and Turner, 2009; Turner and Tjørve, 2005). Moreover, most approaches do not take into considerations spatial attributes as shape and connectivity (see Williams et al., 2005, for review). Indeed, no workable model can be expected to include every factor that may affect the system. We are thus left to restrict an approach as the one in this article, to a few key factors. Early on, certain theoreticians realized that species overlap between areas is a key to the SLOSS enigma (Higgs and Usher, 1980; Simberloff and Abele, 1976), though no attempt at furthering the modeling approach followed. Instead, others suggested that, since the SLOSS question is expected to vary from case to case, the answer lies in the study of empirical data (Quinn and Harrison, 1988; Rosenzweig, 2004). Several other authors (Bolger et al., 1991; Simberloff and Martin, 1991; Wright and Reeves, 1992) have mentioned overlap (or species nestedness) as an important consideration to the SLOSS-question, though no systematic approach can be found in the literature. Recently, some studies have applied overlap to species-diversity models, discussing optimal number of areas or optimal proportions of different area (or nature) types (Bascompte et al., 2007; Tjørve, 2002). Models that combine diversity for different reserves (or areas) have to correct for the species overlap between areas. What affects species overlap, and thereby combined species numbers 605 for several areas or types of areas (e.g. landscape mosaics or nature reserve systems) is therefore a key to understanding not only species-area modeling but also the SLOSS enigma itself. Different biotic and abiotic factors may affect species overlap between reserves or between subsections of reserves. Species nestedness, i.e. that several small reserves contain largely subsets of species found in a single large area, causes larger overlap. Several authors have argued that the level of nestedness affects the optimal reserve conﬁguration (Bolger et al., 1991; Simberloff and Martin, 1991; Wright and Reeves, 1992). Fukamachi et al. (1996), for example, found that even though more species were found in a number of small fragments than in a single large, rare species were found only in the largest patches. Therefore, if the aim is to preserve as many species as possible at a coarser scale than the total reserve area (e.g., globally), a single large reserve may be a better strategy even when several small will preserve the most species locally. Diamond and May (1976) noted that if smaller areas were spread out, they would be more likely to sample complementary distinct communities and therefore contain more species. Simberloff (1988) later pointed out that islands and other discontinuous areas of different sizes do not all have nested species composition, and that two smaller reserves could, therefore, have more species than one larger reserve. The SLOSS question then becomes a discussion of whether small reserves have unshared species or one about the number of overlapping species between smaller areas. When May et al. (1995) calculated the effect of extinction pattern on SLOSS with regard to random extinction, they did not take species minimum-area requirements into account (Lomolino, 2000; Turner and Tjørve, 2005), though others, like Diamond (1976), already commented that reserve size determines whether a species can maintain a viable population. Minimum-area requirements are the result of minimum viable populations or vice versa. We may, therefore, use the two terms interchangeably as an argument for a reserve species– area curve depressed at smaller scales (Turner and Tjørve, 2005). The species–area curve has for a long time played an important part in conservation biology, including the SLOSS debate (see e.g. Rosenzweig, 2004, for review). In this paper, the question of when a single large reserve is better and when several small are better is assessed through species-diversity models that combine species– area curves for several areas, or as in this case; reserves. The few papers that have discussed how to calculate the combined species diversities of several areas or types of nature (e.g., landscape mosaics of different habitat patches) either model the optimal number of reserves, the optimal size allocation between reserves, or both (Bascompte et al., 2007; Tjørve, 2002). Both my 2002 paper and the paper by Bascompte et al. (2007) started by discussing the optimal size allocation among two reserves. The latter then went on to discuss size allocation among several reserves. I shall not discuss here how much area should be allocated to each reserve in order to maximize the total number of species; rather, I shall restrict the approach to the basic question of the SLOSS debate of what number of reserves will hold the most species. I have attempted this through the type of speciesdiversity modeling whereby the total area (reserve system) is divided into a number of equally sized reserves. We may expect several different factors to affect such models and thereby the outcome of the SLOSS question. We know there must exist a trade-off in the SLOSS-debate, but we do not have a quantitative prediction for how SARs, compositional similarity (affected by isolation and distance), relative abundance, etc., shape this trade-off. The aim is to describe the effect of such factors on the optimal number of reserves in a reserve system, given that total reserve area is constant. I approach this through discussions of how maxima of species-diversity models vary with ARTICLE IN PRESS 606 E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 species–area curve shape and with species overlap between areas (or reserves). This approach is again based on knowledge about how factors as MAEs, species overlap between subdivisions of an area, spatial aggregation of individuals and species abundances affect the shape of species–area curves. 2. Modeling SLOSS Let us consider those reserves that display some degree of isolation, meaning that barriers exist so that species individuals cannot move or immigrate completely freely from the surroundings or similar areas. We may, therefore, perceive them as fragments, islands or what has come to be termed as isolates (Preston, 1962a, 1962b; Tjørve and Turner, 2009; Turner and Tjørve, 2005). When modeling SLOSS, therefore, we may expect the species–area curve generated from single areas (e.g. nature reserves) of increasing sizes to behave as isolate curves (Turner and Tjørve, 2005). (I shall use the terms isolates and reserves here synonymously.) Species–area curves from subsections of larger reserves (or areas) can be termed sample–area curves (e.g. Turner and Tjørve, 2005). 2.1. Candidate SAR models SARs are often presumed to be power relationships, S= cAz, where S is number of species, A is area, and c and z are constants (Arrhenius, 1921), though many other candidate models may often ﬁt the relationship better (see Tjørve, 2003, 2009, for review). A power relationship assumes scale-invariant overlap between bisections of a larger sample area (Harte et al., 1999). The exponential (logarithmic) model, ﬁrst proposed by Gleason (1922), assumes, on the other hand, a decreasing (proportional) overlap between bisections with increasing scale. (A logarithmic relationship results from the same number of species being added when the area is doubled, irrespective of scale.) The exponential model is usually given as S= C +b log A, where C and b are constants. One of the persistence models, S = cAz exp( b/A), applied by Ulrich and Buszko (2003, 2004), may be a useful alternative to the power model because it can display a j-shape at small scales (in arithmetic space). Here b and c are constants. This model is here referred to as the P2 function, for it is the second of two different persistence functions discussed by Plotkin et al. (2000a, 2000b) and Ulrich and Buszko (2003, 2004). Fig. 1a and b illustrated how the three above models behave in log–log space. Species–area curves are usually considered not to have an asymptote (e.g. Lomolino, 2000, 2002; Williamson et al., 2001, Fig. 1. (a) Compares the slope in log–log space of the power model (unbroken line) to the described persistence function (P2) (dashed line). The slope of the power model is scale constant, whereas the slope of the P2 function increases with decreasing scale. The slope of the two models converges as area becomes very large. In (b) the power model (unbroken line) is compared to the exponential model (dashed line) between two points. 2002), unless only a given predetermined number of species are considered. Then maximum species number is typically expected to be conﬁned to a ﬁnite species pool and the species–area curve expected to have an upper asymptote. One may object, however, that such homogenous systems, with no habitat diversity, are not likely to be found in nature, and that asymptotic relationships only result from sampling with replacement. Where a monotonically decelerating (convex upward) species–area curve is expected to have an upper asymptote (pool size), the negative exponential model, S = T(1 exp( bA)), where T is pool size, may be suitable. Where we might anticipate a sigmoid species–area curve (in arithmetic space) to have an upper asymptote (pool size), the cumulative Weibull distribution, S= T(1 exp( bAc)), may be a useful regression model. Here again T is pool size (and b and c are constants). Expected shapes of species–area curves and candidate models have recently been reviewed and discussed (e.g. Tjørve, 2003, 2009). We can expect isolate curves to be more or less depressed at ﬁner scales so as to become sigmoid in arithmetic space (i.e., space with untransformed axes). Sigmoid regression models, as the P2 model, should, therefore, ﬁt isolate curves. Convex models may, on the other hand, sufﬁce if the scale window discussed only comprises the upper decelerating (convex downward) part of the curve (Turner and Tjørve, 2005). As isolate data sets rarely include the very small scales, the power model and the exponential (logarithmic) model are usually satisfactory. This paper, then, employs both the sigmoid models and monotonically decelerating models to exemplify how species–area curves generated from reserves of different sizes behave as isolates. 2.2. The case of two reserves As previously mentioned, the power relationship assumes scale-invariant overlap between bisections of an area, meaning that the resulting curve describes the case where the proportion of species that overlap between the two halves of the area is the same irrespective of the size of the area. Tjørve and Tjørve (2008) showed that the z-value of a continuous sample area following the power-law relationship translates directly to a proportional species overlap of bisections of a larger area, i.e., the proportion of the species found in the ﬁrst half that will also be fond in the second half. It was deduced that this proportional overlap, g, is described by z alone, given as g= 2–2z. Further, the proportional overlap between two isolates (or reserves) can be given as gðisoÞ ¼ 22zðisoÞ , where z(iso) represents the slope of the line between the data point for number of species in one isolate (or reserve) and the data point for total number of species in two isolates of the same size (given that they each contain the same number of species). Species-diversity models based on proportional overlaps between areas (as calculated from z-values) may be useful in addressing the SLOSS question. We should expect them to be able predict whether a single large reserve or several small reserves will hold most species will vary across scale. Let us start with the case of two reserves, each with the same number of species. Consider also the simple situation where the proportion of species that overlap, g(iso), between two equally sized reserves (or isolate areas) is scale-invariant; that is, the proportion of species in one reserve, which is also present in a second reserve of the same size, is constant across all scales. So if Si = 1 is the number of species in one reserve (or area), then the number of species in two reserves combined, Si = 2, is Si ¼ 2 ¼ 2Si ¼ 1 Si ¼ 1 gðisoÞ ¼ Si ¼ 1 ð2gðisoÞ Þ ð1Þ which can also be expressed as Si ¼ 2 ¼ 2Si ¼ 1 Si ¼ 1 ð22zðisoÞ Þ ¼ 2zðisoÞ Si ¼ 1 ð2Þ ARTICLE IN PRESS E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 where z(iso) is the slope of the SAR (in log–log space) between one reserve and the total number of species (combined) in two reserves each of this size. It then follows that one large reserve is better (holds more species than two small reserves) when Si ¼ 1ð2AÞ 4 Si ¼ 1ðAÞ ð2gðisoÞ Þ ð3Þ or Si ¼ 1ð2AÞ 4 2zðisoÞ Si ¼ 1ðAÞ ð4Þ where Si = 1(A) is the number of species in an single reserve of size A, and S i = 1(2A) is the number of species in a reserve of double this size, 2A. Let us now take the case where the proportion of new species added by the doubling of reserve size is also the same at all scales. Then inequality (3) becomes cð2AÞz 4 cAz ð2gðisoÞ Þ ð5Þ and it holds true as long as g(iso) 4g, because both g(iso) and g are constants. In this case the total size of the reserve system (i.e., how large an area can be preserved) will not affect the outcome of the SLOSS question. A constant number of reserves will always be best irrespective of the total reserve system area. Consider the cases where either the proportion of new species added by doubling the size of the reserve changes with scale or the proportional overlap between two equally sized reserves changes with scale. The ﬁrst is illustrated by substituting the exponential (logarithmic) model, S =C + b log A (where b and C are constants), for the power model in Eq. (3): C þ b logð2AÞ 4ðC þb log AÞð2gðisoÞ Þ: ð6Þ The resulting inequality holds true as long as g(iso) 41 (b log 2/(C + b log A)). Whether a single large reserve is better than two small will then vary with scale. Because the right-hand side of this inequality increases with scale (approaching one) as A becomes large, one single reserve is best only at small scales and two reserves at large scales. Other species–area models will produce different results, and so the shape of the species–area curve determines here whether a single large area or several small areas will hold the most species. The second case is illustrated when overlap between reserves, g(iso), changes with scale, but the proportion of new species added by doubling the reserve area is scale-invariant. If all species are drawn randomly from a common pool, then the proportional overlap between two equally sized areas becomes g(iso) = Si = 1/T (Bascompte et al., 2007; Tjørve, 2002), where T denotes pool size. The resulting inequality is thus: cð2AÞz 4 cAz ð2ðcAz =TÞÞ ð7Þ which holds true when cAz/T42 2z or cAz/T4g. In this case a single large reserve will hold the most species only when A is large. This situation may not be very realistic, since we can expect pool size to grow with increasing reserve size. Still, this may serve as a null hypothesis for the consideration of alternatives for further generalizations (e.g., where pool size becomes a function of area). The discussion above illustrates how the nature of the SAR (whether it be a power relationship or an exponential one) and species overlap between reserves both affect the outcome of the SLOSS question, or, as in this case, whether one single large reserve will hold more species than two small. Still, we should do well to expand this question to include the issue of how many reserves will hold the most species. One should expect that sometimes a system of more than two reserves is preferable. 607 2.3. The case of several reserves The modeling of the total number of species for more than two reserves requires more complex considerations of species overlap between areas. The simplest case would be that the overlapping species (between pairs of reserves) are always drawn randomly without any bias from being or not being part of the species found in both reserves of other pairs (of equally sized reserves). This assumption is not realistic, for we should be more likely to ﬁnd some species, especially very common ones, in several reserves than others. Still, this is again a useful starting point, which both Bascompte et al. (2007) and I (Tjørve, 2002) have employed. These two approaches are, as far as I know, the two ﬁrst attempts to model species diversity by combining species–area curves for several (more than two) areas. My study only considers equally sized reserves, whereas the other aspires to model asymmetric distributions of reserves. Unfortunately, they did not arrive at a simple way of ﬁnding the expected optimal area distribution. In the interest of simplicity, I shall restrict the present approach to the case where all reserves have the same size. The question will then be as follows: how many equally sized reserves will sustain the highest number of species in a reserve system of a given size? My original model proposes that the total number of species, Si = n, for n separate areas of the same size, and each with the same number of species, is given by Si ¼ n ¼ Si ¼ 1 ðð1ð1gðisoÞ Þn Þ=gðisoÞ Þ ð8Þ where g(iso) is the proportional species overlap between each any two reserves (or isolate areas). It was ﬁrst used to discuss species diversity in landscape mosaics (Tjørve, 2002), but is readily applicable for the assessment of the SLOSS question by combining species–area curves for reserve systems. We can apply an extension of the above model to identify the optimal number of reserves, that is, the number of reserves that hold the most species (given equally sized reserves and a ﬁnite total area for the reserve system). If g is a constant, thus making the overlap scale-invariant, and the total area available is k, then we can ﬁnd the optimal number of areas (the number that produces the maximal number of species) by the intersection between the model (9) curve and the model (8) curve closest to the maximum of the model (9) curve (from Tjørve, 2002). The equation is as follows Si ¼ k=A ¼ Si ¼ 1 ðð1ð1gðisoÞ Þk=A Þ=gðisoÞ Þ ð9Þ where Si = k/A is the number of species in a reserve system of total area k (and divided equally between k/A reserves), and where Si = 1 is the number of species in a single reserve. Fig. 2a and b shows model (8) curves for i= 1–4 and the resulting model (9) curve plotted in the same graph to illustrate graphically how many small reserves a total reserve system should be divided into in order to maximize the total number of species. The rationale behind the plots is explained in more detail in Tjørve (2002). The curves are plotted in both log–log space and arithmetic space, and the SAR (for reserves of different sizes) is assumed to be a power relationship. In this particular example (Fig. 2a and b) a reserve system divided into three reserves contains more species than systems of fewer or more reserves. Fig. 3a–d illustrates graphically the effects of changes in the SAR and in the overlap, g(iso), between equally sized reserves. Increased overlap, i.e., higher g-values, shifts the model maximum (the optimum) towards a smaller number of reserves (Fig. 3b rather than Fig. 3a). An increase in the proportion of new species added by doubling the reserve size shifts the model towards a ARTICLE IN PRESS 608 E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 Fig. 2. The model (9) curve is shown by the unbroken line, indicating the total number of species in a reserve system as the number of reserves changes but total reserve system area remains constant. The x-axis indicates the area of each of the reserves. Assuming power relationships, the four broken lines (model (8)) show, from lower right to upper left, the species diversity for one to four smaller areas (or reserves) of the same size. Given constant total reserve system area, the model (9) curve shows all possible outcomes represented by the intersections with the (model (8)) curves for the total number of species in one reserve, two reserves each of the same size, two reserves of the same size and three reserves of the same size. The optimal number of smaller areas is three, as indicated by where the dashed line intersect model (9) curve closest to its maximum. This is shown in both log–log space (a) and arithmetic space (b). smaller number of reserves (Fig. 3c and a). Increase in species density (or species richness), i.e., number of species per unit of area (represented by the c-value in the power model applied here), does not affect the outcome of the SLOSS question (i.e., it does not shift the maximum of the model (9) curve). This holds only as long as we do not anticipate average species density to vary with scale (c is a constant). In reality, we should expect both g(iso) and g to vary across scales. MAEs (caused by minimum-area requirements) should increase with decreasing scale but also with increasing isolation. This causes the lower end of the species–area curve (for number of species plotted against reserve size) to be depressed. If the curve becomes sigmoid by the lowering of the curvature at ﬁne scales, we may better represent it by the aforementioned P2 function. This results in fewer species at smaller scales, which shifts the maximum in the direction of fewer reserves (Fig. 3a–d). Isolation, therefore, moves the optimum further towards fewer reserves. An implicit assumption behind MAEs and the lowering of the curve at ﬁne scales is that some species that occur in a single large reserve will not occur in any of a series of several smaller reserves. Fig. 3. The panels illustrate the effects of changes in the SAR (changes in z, i.e. the proportion of new species added by increasing reserve size; or changes from the power model to the P2 model, thus accounting for larger minimum-area requirements at smaller scales) and changes in overlap between reserves (changing g(iso)). The effect of different factors affecting the optimal number of reserves, as demonstrated by model (8) and (9) applying the power model. In all panels the unbroken line represents model (9) and the broken lines (both dotted and dashed) represent model (8) curves. The curve with the optimal number of reserves is shown as dashed. Panels (b)–(d) shows the effect relative to panel (a). Panel (b) shows how the optimum shifts towards fewer reserves (relative to (a)) when the overlap between reserves increases (higher g). Panel (c) shows how the optimum moves towards fewer reserves (relative to (a)) when the proportion of new species added (by doubling reserve size) increases (higher z). Panel (d) shows how the optimum is shifted to fewer reserves (relative to (a)) when minimum-area requirements (MAEs) lower the reserve (isolate) curve (here represented by P2 curves instead of power curves). ARTICLE IN PRESS E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 609 Fig. 4. Model (8) and (9) demonstrates the effect of different spatial aggregation and uneven abundances. As aggregation and uneven abundances are known to lower the curvature of species–area curves, the exponential model (graph (a)) relative to the power model (graph (b)) (between the same two points) is applied to illustrate the increased aggregation or more uneven abundances. The decrease in curvature shows a decrease in the optimal number of reserves. 2.4. Scale-variant overlap between reserves We can study the effect of scale-variant overlap between reserves (of the same size) by substituting the constant g for an expression that is a function of area. One way of considering species overlap that varies with scale is again to let the overlap be determined by a ﬁnite species pools. Tjørve (2002) and Bascompte et al. (2007) recently attempted this approach. In my article (Tjørve, 2002), I presented a generalized model for the optimal size allocation among two reserves. This model assumes two areas with separate species pools and that species overlapping (between two reserves) are always drawn randomly. Bascompte et al. (2007) generalized the case of the area distribution of two reserves to a generic number n, but with a common species pool. I shall not elaborate on Bascompte et al.’s (2007) approach here, as it did not (as mentioned before) arrive at any easily applicable model. Instead, I shall present a model for several reserves (or areas) where all areas are the same size (and there is a single pool for all areas). In the above (inequality 7), we looked at two equally sized areas containing the same number of species and having all species drawn from the same pool. The proportion of species overlapping between two reserves was then g =Si = 1/T. If the problem is to be generalized from two reserves (inequality 7) to i= k/A reserves of equal size, we must add one extra assumption: overlapping species are always drawn randomly from the pool without any (prior) bias from being or not being part of the species overlapping between other pairs of reserves, then follows that Si ¼ k=A ¼ Si ¼ 1 ðð1ð1ðSi ¼ 1 =TÞÞk=A Þ=ðSi ¼ 1 =TÞÞ ¼ ð1ð1ðSi ¼ 1 =TÞÞk=A ÞT ð10Þ The above model is unrealistic for several reasons. Not only should the pool size grow with total reserve area, but the assumption that for each pair of reserves the drawing of overlapping species is random is also unrealistic. This is because we should expect to see some species having a higher probability of occurring in several reserves than others. When some species are more common than others (i.e., they occur in more of the reserves than expected with even abundances), the overlap will increase between smaller areas. Likewise, rare species will decrease the overlap between larger areas compared to even abundances. Still, this may be a useful exercise that demonstrates a case where overlap between reserves varies across scales. As a result of the model assumptions, overlap becomes very small as total area approaches zero. This is also improbable because it is likely that some species have smaller minimum-area requirements than others, which should (in addition to the effect of common species) increase overlap at ﬁne scales. Also, cooccurring species do not have equal abundances. If we were to model the SAR with a monotonically decelerating curves as the power or exponential model, then the maximum for model (9) would be at an inﬁnite number of reserves (i.e., when k/A-N). Sigmoid models (like the P2 model and the cumulative Weibull distribution) generate more meaningful results by displaying a model (9) curve with a maximum. This supports the expectation for the species–area curves for reserves or different types of isolates to be inherently sigmoid in arithmetic space (Tjørve, 2003; Tjørve and Turner, 2009; Turner and Tjørve, 2005). Aggregation of individuals (or occurrences) and uneven abundances between species (species-abundance distributions or occurrence distributions) should lower the curvature of species–area curves, compared to random distribution and even abundances (Green and Ostling, 2003; Green and Plotkin, 2007; He and Legendre, 2002; Olszewski, 2004; Picard et al., 2004; Plotkin et al., 2000a; Solow and Smith, 1991). If we can use the power model (i.e. an assumption of scale-invariant overlap between bisections of a reserve) as a null model, we can also apply the exponential model to illustrate decreased aggregation or less uneven abundances, or both. Fig. 1b demonstrates the difference between the power model and the exponential model (between the same points) in log–log space. The power curve has a lower curvature than the exponential curve. Fig. 4 compares the power model to the exponential model between one unit of area and the total reserve system area (i.e., between A= 1 and k). Applied to model (8) and (9) system, this illustrates how increased aggregation and more uneven abundances shift the optimum towards fewer reserves. In general, factors that lower the curvatures will decrease the optimal number of reserves. 3. Implications for conservational strategies This approach to modeling based on combining species–area curves demonstrates how whether a single large area or several small areas will hold the most species will differ between cases. The discussion above has shown that whether a single large or several small reserves will maintain the most species is linked to the shape of species–area curves and to (the proportion of) ARTICLE IN PRESS 610 E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 species overlap between separate reserves, as they both affect the outcome. The shape of the species–area curve is a result of the proportion of new species added each time reserve area is doubled. From this understanding, we can generate predictions on how the optimal number of reserves (i.e., the SLOSS question) will be affected by such different factors as from minimum-area requirements, habitat diversity (or abundance and spatial pattern), environmental quality, and distance between reserves, as these all in turn affect curve shape and overlap between areas. 3.1. Predictions from the model I propose here rules of thumbs based on the predictions generated from the discussion of how different factors affect the optimal number of reserves (see Table 1). These factors are not only important for the question on the optimal number of reserves, but also whether or not we should minimize the distance between smaller reserves. First, (A) increased overlap between reserves and (B) increased proportion (or number) of new species when reserve area is enlarged will both decrease the optimal number of reserves. Moreover, it is evident that (C) higher minimum-area requirements lower the isolate curve at smaller scales, decreasing species diversity, though mostly at the lower end of the curve (producing a sigmoid species–area curve). This shifts the optimum in the direction of fewer reserves. In addition both (D) more uneven species abundances and (E) higher spatial aggregation will also decrease the optimal number of reserves. Both of these will depress the middle part of the reserve species–area curve (i.e. lower the curvature). If the aim is mostly to preserve the rare species, a single large reserve will more often be preferable, because many rare species lower the curvature the most at the small to medium scale end of the species–area curve (Tjørve and Turner, 2009; Turner and Tjørve, 2005). Greater distance between isolates (or islands and patches), according to both the equilibrium theory of island biogeography (MacArthur and Wilson, 1967, 1963) and metapopulation theory (Hanski, 1999; Hanski and Ovaskainen, 2000), decreases species density. Subsequently, Wilson and Willis’ (1975) third rule of reserve design stated that increasing distance decreases total species diversity. Decreased immigration, according to MacArthur and Wilson’s (1963, 1967) equilibrium theory, should result in fewer species. Further, more sessile species should reinforce this effect and should also strengthen the effect of minimum-area requirements because of added isolation. Table 1 Summary of predictions generated from the model approach, describing how different factors should affect the SLOSS question. Change towards (A) Larger species overlap between reserves (B) More new species with reserve area increase (C) Higher minimum-area requirements (D) More uneven species abundances (E) Higher spatial aggregation (F) Decreased species density with distance (G) Decreased overlap between areas with distance (H) Larger in total reserve system area Favors Fewer and larger reserves Fewer and larger reserves Fewer and larger reserves Fewer and larger reserves Fewer and larger reserves Fewer larger reserves More and smaller reserves Depends on how (A) and (B) vary with total reserve system area Fig. 5. The effect of increased distance between areas. The arrows indicate the pattern that will hold the most species: (a) less overlap between reserves is a result of large distances, as described by the distance-overlap decay curve. (b) Lower species density is a result of larger distances because of higher isolation, caused by minimum-area requirements. The ﬁrst also moves the optimum towards more reserves, whereas the second moves the optimum towards fewer reserves. Therefore, there are two conﬂicting mechanisms affecting whether the distance between reserves should be minimized or not. If (F) greater distance between reserves will decrease species density, it should move the optimum towards a smaller number of reserves. On the other hand, (G) increased distance between reserves should decrease species overlap and thus move the optimum towards a larger number of reserves. Whether increased distance between reserves causes fewer or more species depends, therefore, on two opposing factors. Whether the third rule of reserve design, stating that distance between reserves should be minimized, holds true or not in each case must depend on which of these two mechanisms dominates. Fig. 5 illustrates the effect of distance on total number of species in several reserves. The arrow indicates the highest species diversity. Intuitively, one might have though that (H) larger total reserve area will increase the optimal number of reserves. In fact, the present model approach reveals that this is not necessary always true (Tjørve and Tjørve, 2008), or rather that it depends on the shape of the species–area curve and the overlap between equally sized areas. If the SAR is a power relationship (meaning that the proportion of new species added by doubling reserve area is scaleconstant) and the overlap between reserves is also scale invariant (inequality 5), the optimal number of reserves stays the same irrespective of total reserve system size (as long as on g(iso) and g are constants). This case can serve as a null model, illustrating how the size of the total reserve-system area in this case does not affect the optimal number of reserves, be it one large or several small. If on the other hand, the proportion of new species added by doubling reserve area decrease with scale, as will be the case if, e.g. the species–area curve (for a single reserve) is an exponential (logarithmic) relationship (inequality 6), then if one single reserve is best for small reserve systems, several small reserves will be best at large reserve-system scales. If the proportional overlap between reserves (g) increases with scale, as say the case where there is a ﬁnite total species pool as described for inequality 7 and Eq. (10), then the increase in total reserve-system area will move the optimum towards a single (or fewer) reserve(s). Some researchers have argued (Gilpin and Diamond, 1980; Higgs and Usher, 1980) that with a convex-upward (monotonically decelerating) species–area curve, several reserves cover more species than a single reserve (keeping the total area equal). ARTICLE IN PRESS E. Tjørve / Journal of Theoretical Biology 264 (2010) 604–612 The discussion above illustrates that this is not necessarily true. This approach shows how important the choice of species–area model is for the predictions generated by the species-diversity model (and the outcome of the SLOSS question). 611 Acknowledgements I am grateful to W.E. Kunin, K.M.C. Tjørve and K.I. Ugland for stimulating discussions and constructive comments, which greatly improved the article, and to S. Connolley for editing and correcting the language. 3.2. Can species-diversity models resolve the SLOSS debate? That most of the discussion aimed to establish one universal rule is one reason why the SLOSS debate from the 1970s and 1980s was never fully resolved. We can look back now and say that the question should have been when will one be better than the other. Soule and Simberloff (1986) had early on pointed out that several factors may affect which is the best strategy. There has been no shortage of suggestions and attempts to solve this issue, e.g., species overlap between areas (see Higgs and Usher, 1980; Simberloff and Abele, 1976), more recently metapopulation theory (Etienne and Heesterbeek, 2000; Hanski and Ovaskainen, 2000; Ovaskainen, 2002), and representative networks (Cabeza, 2003; Rodriguez and Gaston, 2001; Wiersma and Nudds, 2005). The present approach has sought to discuss the effect of species overlap between areas, minimum-area effects, and several other factors. This approach demonstrates that the answer SLOSS question, or rather how many reserves will hold the most species, varies and in each case depends on several factors. We have seen how we can reduce the study of how reserve systems behave (i.e., the calculation of their total number of species) to the consideration of only the species–area model (i.e., the shape of the species– area curve) and the knowledge about species overlap between two equally sized reserves (or areas) at different scales. This article hopefully will draw attention to several factors that affect the optimal number of reserves (SLOSS). It does not, on the other hand, take into account the possible relaxation of species numbers. A main problem with trying to determine the SLOSS question by empirical studies is that these do not provide a measure for the relaxation in species numbers caused by the loss of species (because of fragmentation habitat-loss) in areas not included in the reserve conﬁguration found to hold the most species (at the time of the count). Loss of species distribution outside the protected areas of the reserves may, therefore, induce extinction within the reserves, because reserves often bear more likeness to sample areas than real isolates. In reality, most areas lie on a continuum between sampleareas and complete isolates and are, therefore, susceptible to species loss outside the reserve. As a result, one could conclude erroneously several small reserves to be better than a single large. The discussion here has not only addressed Wilson and Willis’ (1975) ﬁrst rule of reserve design, but also the third (which states that distance between reserves should be minimized), since this is also closely linked to the SLOSS question. The species-diversity modeling in the present paper has considered several factors that should affect the outcome of the SLOSS question and the question of whether distance between reserves should be minimized or not. 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