## Introduction

The species–area relationship (SAR) is a fundamental empirical generalisation in ecology (Rosenzweig 1995; Lomolino 2000; Blackburn & Gaston 2003) and the prima facie reason for predicting declining biodiversity with on-going massive habitat conversion by humans (Millenium Ecosystem Assessment 2005). MacArthur & Wilson's (1963, 1967) theory of island biogeography was the first dynamic model to predict the SAR, followed by many other models (Ricklefs & Bermingham 2004; Kadmon & Allouche 2007; Rosindell & Cornell 2007, 2009; Whittaker *et al*. 2008). Given that very different models predict SARs, it is clear that just demonstrating a SAR for a particular system does not tell much about the mechanisms that have generated it. Nonetheless, it is widely agreed that the two main ecological processes that contribute to SARs are increasing habitat heterogeneity with increasing area, which allows a larger number of species with dissimilar ecological requirements to co-occur within larger areas (though see Allouche *et al*. 2012); and increasing probability of population survival with increasing population size and hence with increasing area, given that, other things being equal, larger areas tend to support larger populations (Hanski & Gyllenberg 1997).

SARs are most commonly described by the power-law *S* = *cA*^{z} (Arrhenius 1921; Rosenzweig 1995), where *S* is the number of species, *A* is the area of the habitat, and *c* and *z* are two parameters. In log–log space, the value of *z* gives the slope of the increasing number of species with increasing area. Starting with the pioneering study by Connor & McCoy (1979), researchers have attempted to relate the value of *z* to environmental and ecological variables, but with only limited success (Matter *et al*. 2002; Ovaskainen & Hanski 2003; Drakare *et al*. 2006; Tjorve & Tjorve 2008). One generalisation to emerge is that, across a large range of spatial scales, the SAR exhibits three phases: a relatively steep slope for very small areas, which harbour only small numbers of individuals per species; a shallower slope for intermediate (regional) scales; and again a steep slope for very large scales, in the extreme covering several continents, which tend to have different sets of species (Rosenzweig 1995; Hubbell 2001). Most analyses, including the present one, are concerned with the regional scale, typically ranging from roughly 1 to 10^{6} km^{2}.

Power-law SARs have been used to predict extinctions of species due to reduced area of habitat (Whitmore & Sayer 1992; May *et al*. 1995; Brooks & Balmford 1996; Cowlishaw 1999; Ney-Nifle & Mangel 2000; Pimm & Raven 2000; Brooks *et al*. 2002; Brook *et al*. 2003; Pereira & Daily 2006; Pimm *et al*. 2006; Hanski *et al*. 2007). If the original habitat area *A* is reduced to *A*_{new}, the fraction of species that is predicted to remain following habitat loss is given by

In other words, if *A*_{new} = *xA*, where *x* is the fraction of remaining habitat, fraction *x*^{z} of the original species will survive – the rest go extinct. Note that eqn 1 assumes that the power-law SAR applies both before and after habitat loss with the same value of *z*, which raises the question as to when, following habitat loss, the new species number *S*_{new} should be recorded, immediately after habitat loss or following any transient dynamics. We return to this question below.

Recently, He & Hubbell (2011) have argued that eqn 1 always overestimates extinctions, because the area required to remove the last individual of a species from a landscape, corresponding to extinction, is typically much larger than the area needed to encounter the first individual, on which basis SAR is constructed. To demonstrate their claim, He & Hubbell (2011) fitted to empirical data the power-law SAR as well as the following equation:

where *a* is the area destroyed and *S*_{loss} is the number of endemic species to area *a*, that is, species that only occurred within *a* and which were hence immediately lost when *a* was destroyed. He & Hubbell (2011) call eqn 2 the endemics–area relationship (EAR), which is however confusing, because eqn 2 does not describe the increase in the number of endemic species with area, given by *S*_{loss} = *ca*^{z} (Harte & Kinzig 1997; Kinzig & Harte 2000). Considering the slope values, *z* in eqn 2 is typically equal or less than *z* in the power-law SAR, whereas *z* in the power-law EAR, *S*_{loss} = *ca*^{z}, is greater than *z* in the corresponding SAR (Harte & Kinzig 1997; Kinzig & Harte 2000). Equation 2 may be rewritten as

which gives the fraction of species surviving habitat loss as a power of the fraction of remaining area. We hence call eqn 3 and the equivalent eqn 2 as the ‘remaining species–area relationship’, RAR, to avoid confusing it with EAR as defined by Harte & Kinzig (1997) (Table 1). Equation 3 describes the relationship between the number of remaining species and the remaining area immediately following habitat loss and the loss of endemic species. Noting that *S*_{new} = *S* *−* *S*_{loss}, eqns 1 and 3 are structurally equivalent, but because different data are used to estimate the value of *z*, the respective *z* values may be different. There is much confusion in the literature concerning the use of eqn 3; we describe our approach explicitly in the following section.

Acronym | Formula | Description |
---|---|---|

SAR | Species–area relationship gives the number of species occurring in area A. The power-law SAR given here is the most commonly used functional form; SAR may be calculated for different spatial scales and for communities with different spatial population dynamics (Table 2) | |

EAR | Endemics–area relationship gives the number of species confined to area a that is a part of the total area A | |

RAR | Remaining species–area relationship gives the fraction of species that have populations outside the area a, that is, within A–a |

While constructing any SAR, two issues that deserve attention are the spatial configuration of the habitat and the way species are sampled. Concerning the spatial configuration, the primary distinction is between continuous vs. fragmented habitats. The latter are represented by true and habitat islands, which are also the natural units of sampling while constructing SARs. In the case of continuous habitat, researchers typically overlay a regular lattice over the landscape for sampling. For different sampling schemes, see Dengler (2009), Gotelli & Colwell (2001), Scheiner (2003) and Smith (2010). An important point to note is that all these sampling schemes involve subareas of the same landscape. In contrast, in conservation one typically asks about the capacity of different landscapes with more or less habitat to support viable populations and species (Simberloff & Abele 1976; Hanski & Ovaskainen 2000). To address this question, one may construct a SAR with individual data points representing different landscapes with dissimilar amounts of habitat rather than subareas of the same landscape. Table 2 summarises our view about the different types of situations for which a SAR may be constructed. We recognise three types of data sampled within a single landscape (region): nested or non-nested clusters of lattice cells from a continuous landscape, discrete habitat fragments from a mainland-island setting (as originally studied by MacArthur & Wilson 1963, 1967) and discrete habitat fragments in a landscape without a mainland (the metapopulation setting). On the other hand, when different landscapes are the units of sampling, we distinguish between cases where the habitat occurs as a single block, which is implicitly assumed by the conventional SAR (the ‘one-fragment’ (OF-)SAR, Table 2), and cases where the habitat occurs in many discrete fragments (the ‘fragmented landscape’ (FL-)SAR). The slope of the SAR depends on how it is defined (Rosenzweig 1995; Ulrich & Buszko 2004; Tjorve & Turner 2009), and hence, the definition affects any predictions concerning the survival of species in the face of habitat loss. In particular, applications of power-law SARs to predict extinctions are often based on the slope value 0.25, which stems from MacArthur & Wilson's (1967) original work and a large number of subsequent empirical studies analysing variation in the number of species on true islands (Connor & McCoy 1979; Rosenzweig 1995). This slope value does not necessarily apply to the other settings in Table 2, especially not to continental SARs as has been known for a long time (MacArthur & Wilson's 1967; Brown 1971).

Type of SAR | Spatial unit of sampling | Remarks |
---|---|---|

Continental | Subareas of different sizes | Subareas of a continuous landscape delimited in one way or another |

Mainland-island | Discrete habitat fragments | Habitat fragments receive migration from permanent mainland populations; this is the original mainland-island setting of the theory of island biogeography |

Metapopulation | Discrete habitat fragments | No mainland, species occur as metapopulations in a network of habitat fragments |

One-fragment (OF-SAR) | Landscape | Calculated for isolated habitat fragments (e.g. true and habitat islands) of different sizes with completely isolated populations |

Fragmented landscape (FL-SAR) | Landscape | Calculated for replicate fragmented landscapes with differences in the total amount of habitat and the degree of fragmentation |

To return to the task of predicting extinctions due to habitat loss, we reiterate that eqn 3 and EARs (Harte & Kinzig 2011; Kinzig & Harte 1997) by definition describe the number of species going extinct immediately following complete loss of habitat in the area occupied by the endemic species. In conservation, we are concerned not only with these immediate extinctions but also with subsequent extinctions due to the remaining habitat being insufficient to support, in the long-term, viable populations of other (non-endemic) species. Second, predictions about extinctions based on SAR, RAR and EAR are focused on the pooled amount of habitat that is lost across the region, while they ignore possible consequences of habitat fragmentation. Arguably, this is a major shortcoming when the total amount of remaining habitat is small or relatively small and fragmentation becomes severe (Hanski 2005). Third, most models that have been constructed to derive SAR, RAR and EAR ignore spatial correlation in habitat type and differences in species ecological requirements (e.g. Kinzig & Harte's 2000 derivation of EAR from SAR, Hubbell's 2011 neutral model, He & Hubbell's 2011 sampling models and Storch *et al*.'s 2012 model of range placement; but see Sizling & Storch 2004). Fourth, SAR, RAR and EAR are typically not derived from dynamic models, which is a shortcoming at regional scales, where the occurrence of most species is dynamic and affected by dispersal.

Here, we aim at providing a perspective on the use of SAR and RAR in predicting extinctions by taking a computational approach: we construct a dynamic simulation model of the spatial dynamics of a large number of species to analyse and clarify habitat area-dependent extinctions. We develop a simple yet robustly realistic model, with the following key features. We ignore interspecific interactions but assume differences in the ecological traits of the species. We model stochastic patch occupancy dynamics (Hanski 1994; Moilanen 1999) on a lattice, assuming an exponential dispersal kernel (Ovaskainen & Hanski 2004) with a parameter specifying the average range of dispersal. The lattice cells are characterised by spatially variable habitat type, which is spatially correlated at a scale that can be adjusted with a parameter. The performance of a local population in a lattice cell is determined by the match between the species phenotype and habitat type. The model includes regional stochasticity, that is, spatially correlated environmental stochasticity, which leads to spatially correlated extinctions. We construct continental SARs, OF-SARs and RARs for the simulation results, with the aim of examining differences in their slopes. We then analyse transient dynamics following habitat loss, to contrast extinctions that occur immediately following habitat loss vs. further extinctions in the course of transient dynamics towards the new quasi-stationary state. Finally, we analyse the influence of habitat fragmentation vs. the influence of habitat loss on the number of surviving species. We conclude that contrary to He & Hubbell (2011), RARs always underestimate even short-term extinctions, because following habitat loss many non-endemic species go quickly extinct in the remaining habitat, where they do not have viable populations. However, and what is very important, conventional SARs produce large or even very large underestimates of the number of species going extinct in highly fragmented landscapes with a small total amount of remaining habitat. We modify the power-law SAR to account for the effect of fragmentation, and we make a simple recommendation that helps reduce the fragmentation effect in practical conservation.