Subjecting the theory of the small-island effect to Ockham’s razor

Authors


Abstract

Species–area curves from islands and other isolates often differ in shape from sample-area curves generated from mainlands or sections of isolates (or islands), especially at finer scales. We examine two explanations for this difference: (1) the small-island effect (SIE), which assumes the species–area curve is composed of two distinctly different curve patterns; and (2) a sigmoid or depressed isolate species–area curve with no break-points (in arithmetic space). We argue that the application of Ockham’s razor – the principle that the simplest, most economical explanation for a hypothesis should be accepted over less parsimonious alternatives – leads to the conclusion that the latter explanation is preferable. We hold that there is no reason to assume the ecological factors or patterns that affect the shapes of isolate (or island) curves cause two distinctly different patterns. This assumption is not required for the alternative, namely that these factors cause a single (though depressed) isolate species–area curve with no break-points. We conclude that the theory of the small-island effect, despite its present standing as an accepted general pattern in nature, should be abandoned.

Recently, there has been considerable debate on how a small-island effect (SIE) in species–area data should be identified. The SIE theory proposes that species–area relationships (SARs) for islands (or other types of isolated areas) have two distinctly different patterns that are dependent on the scale (or grain) (Lomolino & Weiser, 2001). The first pattern is for small islands (or fine scales), where the richness does not increase with island area, and, above a certain island size, there is a second pattern (for coarser scales) in which number of species increases with island area. The alternative to this theory is to assume that the isolate (or island) species–area relationship (as described by Preston, 1962a,b) has a single parametric form across the whole domain of the predictor; more precisely, it is sigmoid in arithmetic space, or at least depressed at the lower part, relative to a curve generated from sample areas from a mainland or from fractions of an island. Because the SIE introduces a more complex theory of the nature of these relationships, we discuss whether the prevalence of SIE theory is a case in need of Ockham’s razor – the principle that the simplest, most economical explanation for a hypothesis should be accepted over less parsimonious alternatives. We argue that we should indeed abandon this unnecessary and cumbersome explanation.

The SIE theory

Any search for the origin of the SIE theory must begin with Niering’s (1963) article on the plants of the Kapingamaringi Atoll. He notes a ‘two linear relationship’, and writes: ‘On those islands 3.5 acres or less there is little variation in number of species per island. However, on those larger there is a continuous increase in number of species with increase in island size’ (p. 137) (Fig. 1). A few years later, MacArthur & Wilson’s (1967) seminal work on the equilibrium theory of island biogeography states that: ‘If species turnover on very small islands is great enough, extinction rates can be essentially independent of area. Consequently, in the lower end of the range in island area, the equilibrial species number will not increase with area …’ (p. 31). In recent literature the most quoted definition of the SIE has been that of Lomolino & Weiser (2001), who argue that ‘Detection of the SIE, in essence, implies that the species–area relationship is actually comprised of at least two distinct patterns’ (p. 432).

Figure 1.

Niering’s (1963) data, from plants on islands of different sizes in the Kapingamaringi Atoll, plotted in log–linear space. The ‘... two linear relationships ...’ he suggested are drawn in, redrawn from the original paper. ‘S’ denotes the number of species, and ‘A’ denotes island area (in hectares).

Heatwole & Levins (1973), who introduced the term ‘SIE’, argue that ‘… very small islands seem to be special cases in that over a considerable range of small areas, there is little or no change in species number’ (p. 1054). This interpretation is still common. Triantis et al. (2006) have recently stated that: ‘The main feature of this SIE is that species number does not increase with increasing area in small islands’ (p. 915). Consequently, the reigning interpretation of Niering’s (1963) data set plotted in log–linear space (Fig. 1) is that the best curve-fit for small islands (with a SIE) is a horizontal line (see also Woodroffe, 1986).

A renewed interest in SIEs began with Lomolino & Weiser’s (2001) meta-study, where they claim that they can identify SIEs by segmental (also named segmented, break-point or piecewise) regression. Segmental (or break-point) regression expects model parameters to be unlike within different domains of the predictor, and for the change to be abrupt. A significant number of papers that apply or discuss the use of segmental regression has since followed (see e.g. Dengler, 2010, for review). Typically, results from the fitted segmental regressions or comparisons between these and other fitted models are considered as proof of SIEs, and feature statements such as, ‘Evidence for a SIE was found’ (example from Gentile & Argano, 2005, p. 1715). Some authors have of late questioned these methods (Burns et al., 2009; Dengler, 2009, 2010). Burns et al. (2009) were the first to point out that the log-transformation of the dependent variable can cause spurious SIEs. Dengler (2010) also advocates the inclusion of islands with no species for the ‘unambiguous detection of SIEs’ (p. 256). Either way, the segmental regression tends to be a dubious method for the identification of curve patterns believed to indicate or even prove SIEs.

Two explanations, which are not mutually exclusive, have been used to validate the SIE: (1) habitat quality, and (2) differences in extinction rates. Factors that have been suggested to affect the curve at some scale, and thus explain change in habitat quality, include freshwater reservoirs (idea based on Wiens, 1962), protection from salt spray (Niering, 1963), inland habitat (Whitehead & Jones, 1969) (i.e. habitat diversity), and many others (see e.g. Triantis et al., 2006; for review). MacArthur & Wilson (1967) hypothesized avant la lettre that differences in extinction rates may cause a SIE. They argued that a rapid turnover of species on small islands ‘… seems the most economical explanation hypothesis to offer for the peculiarities of Niering’s data of the Kapingamarangi flora and those of Wilson & Taylor (1967) on the Polynesian ants’ (MacArthur & Wilson, 1967, p. 30).

The isolate curve

Species–area curves generated from islands or other types of isolated areas are also termed isolate curves (e.g. Tjørve & Turner, 2009). As early as almost four decades ago, Preston’s (1962a,b) magisterial work discussed differences in slope and curve shape between sample-area curves and isolate curves (in log–linear space). Islands or isolates typically have fewer species than sample areas of the same size generated from mainlands or sections of isolates (or islands). This difference between the isolate curve and the sample-area curve decreases with increasing area (i.e. as islands become larger). Thus, the isolate curve (relative to the sample curve in arithmetic space) becomes increasingly depressed in the direction of a sigmoid shape at finer scales. This shape is explained as a result of species’ minimum-area requirements. That there is a minimum (island or isolate) area for all species which can sustain a viable population or allow reproduction is recognized (Gurd et al., 2001). The size of this minimum area is affected by such factors as the presence of resources (e.g. breeding habitat, space, nutrients or food), point (or catastrophic) events, and inbreeding (Turner & Tjørve, 2005). Tjørve & Turner (2009, p. 391) state that ‘Minimum-area effects on actual islands and other isolates predictably cause species–area curves either to be sigmoid in arithmetic space or to be lowered for smaller areas’, and report sigmoid isolate curves in a number of data sets, which cover large-scale windows going from very fine to coarse scales. Sigmoid curves may not be obtainable for all isolate-area relationships because the size of islands with no species is so small that it will hardly affect the outcome of the regression. In a perfectly or considerably isolated island (or other type of isolate, where isolation is caused by distance, for example) the resulting minimum-area effects should increase gradually towards fine scales, thus depressing the curve increasingly (relative to corresponding sample-area curve). This is not to say that single factors might not cause unevenness or one or more ‘dents’ in the curve, but such irregularities do not justify a general SIE theory as described in the literature. The resulting pattern will be one where the lower end of the SAR curve becomes more or less evenly depressed relative to the sample-area SAR, and not a relationship where the lower part is raised (as is implied by the SIE theory). This difference between the isolate-SAR theory and a SIE theory becomes evident if we compare the two shapes between different spaces. Figure 2a–c illustrates the fundamental difference in shapes between log–log space, log–linear space and arithmetic space. The graphical SIE model assumes that the factors acting at fine scales keep the number of species from decreasing further, which results in a horizontal relationship (Fig. 2; unbroken line). In contrast, the sample-area SAR model assumes that the effect of these factors increases towards finer scales. The latter is realized in log–log space (Fig. 2a; dashed line) as a steepening of the curve towards finer scales. This curve shape tells us that the proportion of species lost increases for each halving of the area. In log–linear space this results in a convex downward shape (Fig. 2b; dashed line) [resembling in some ways the SIE model (Fig. 2b; unbroken line)], whereas in arithmetic space it results in a sigmoid shape (Fig. 2c; dashed line).

Figure 2.

 A comparison of the shape of species–area curves from a break-point regression model used to ‘identify’ small-island effects (SIEs) in data sets (unbroken line), and a sigmoid regression model used to accommodate the expectance of sigmoidality in sample-area SARs caused by minimum-area effects (MAEs). The former is represented by a power model that is replaced for finer scales with a horizontal line at a given isolate (island) size, as described by Lomolino & Weiser (2001), and the latter by the second persistence (P2) model as described by Ulrich & Buszko (2003). Panel (a) shows the two models in log–log space, panel (b) shows the models in log–linear space, and panel (c) shows the models in arithmetic space. ‘S’ denotes the number of species, and ‘A’ denotes island area.

A case fit for Ockham’s razor

The notion that evidence presents itself with only one possible interpretation is untenable. In reality any data set or results can be interpreted in several ways. The central issue, therefore is not the amount of evidence, but our interpretations of the evidence. Let us then examine more closely the interpretations according to the SIE and to the alternative theory. To explain the mechanisms behind the current theory of the SIE, we are required to accept the following propositions. (1) There is a monotonically increasing species–area curve at coarse scales. (2) An abrupt change occurs at a particular scale caused by one or more factors. Moreover, if there are several factors (of the many proposed), they have to take effect at the same point. (3) Below this (scale) point, the number of species does not decline further, which means that diversity is sustained even though area decreases. We must assume, then, that some factor, or factors, is preventing the expected fall in diversity towards finer scales. (4) Because the very smallest isolates (or islands) cannot be expected to hold any species (i.e. wave actions will cause tiny, wash-over skerries to have no terrestrial plants), there has to be a second point at which the curve starts to descend towards zero at the very finest scales. A hypothesis for the mechanism behind this change also has to be included. (5) Below this second point of change, the curve has to decrease monotonically towards the finest scales. Alternatively, more change points and patterns (with decreasing scale) may be hypothesized. This pattern, or more patterns, also requires justification. On the other hand, to explain the alternative theory, we need only to expect a single, sigmoid isolate curve (in arithmetic space with no break-point) in the presence of minimum-area effects (MAEs). We can also include various resource restrictions without the addition of further (or more specific) explanations.

The acceptance of the SIE as a general pattern in nature has entailed extensive theoretical ramifications owing to a need for a causative explanation. This a posteriori approach has given rise to a number of suggested explanations (or factors which may account for this effect), which include low habitat diversity, resource limitations or availabilities, species adapted to special conditions, isolation, exposure to stochastic events, natural and human-induced disturbances, small target area for immigrants, extreme isolation and recent geological age (e.g. MacArthur & Wilson, 1967; Whitehead & Jones, 1969; Heatwole & Levins, 1973; McGuinness, 1984; Losos, 1996; Anderson & Wait, 2001; Barrett et al., 2003; Triantis et al., 2003, 2006; Ackerman et al., 2007). There is no reason to assume that these ecological factors or patterns cause shapes of isolate (or island) curves to have two distinct patterns. They are more readily explained as artefacts of the transformation of axes or of the statistical method applied, or both. This leaves us with the more parsimonious explanation of a single-island SAR (in arithmetic space), which is depressed relative to a sample-area (or mainland) curve and which becomes sigmoid when the very smallest islands (or isolates) are included. We conclude, therefore, that the theory of the SIE, despite its present standing as a widely accepted general pattern in nature, should be abandoned. Although we reject the SIE theory, we acknowledge that this theory has made valuable contributions to ecological studies. Indeed, our present interest in and understanding of species diversity in general and SARs in particular would have been feebler without the development of the SIE theory.

Acknowledgements

We are grateful to Steven Connolley for editing and correcting the language.

Editor: K.C. Burns

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