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Journal of Theoretical Biology 264 (2010) 604–612
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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: even.tjorve@hil.no
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´
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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
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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
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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
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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).
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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)
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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).
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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. The species-diversity models presented generates predictions
about how these factors affect the optimal number of reserves.
Can we use the lessons from such models in order to resolve
the SLOSS debate? I hope that the species-diversity models
presented here will prove a useful tool in reserve design, or at
least contribute to the understanding of the SLOSS question. A
uniﬁed approach integrating dynamic models like the island
equilibrium model or metapopulation theory with static approaches, as I have discussed here, may bring us even closer to
being able to determine best strategies. Diamond (1976) suggested that the best compromise would be one large reserve plus
some smaller. Future model advances may enable us to discuss
such reserve systems, or even more complex reserve conﬁgurations, with greater precision.
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