A novel method of partitioning beta diversity into species replacement and richness differences
Since we are interested in determining the relative roles of species replacement and richness differences in generating beta-diversity patterns we will use the term ‘beta diversity’ to refer all compositional changes between two sites irrespective of the process that originated it, in accordance with the original idea of Whittaker (1960). In the context of pairwise comparisons, a replacement occurs when one species is substituted by another species. For example, in Fig. 1, species 1 in site A is replaced by species 10 in site B. The expression ‘richness differences’ will be used to refer to the absolute difference between the number of species that each site contains, irrespective of being nested (site A and sites C, D and E, Fig. 1) or not (site A and site F, Fig. 1).
Figure 1. Hypothetical gradient of increasing dissimilarity. Sites A and B have the same richness but species 1 was replaced by species 10. Sites A, C, D, E differ in species richness and have at least one species in common (nested biotas). Sites A and F differ in species richness but do not share any species (non-nested biotas).
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Therefore, beta diversity can be defined, mathematically, in relation to its two possible components as:
In order to derive the measures that should represent the three terms of the equation, we use the following standard notation (Koleff et al., 2003): a is the number of species common to both sites, b is the number of species exclusive to the first site, and c is the number of species exclusive to the second site (Fig. 2). Therefore, the total compositional difference between two sites, independent of the process that originated it (replacement or richness differences), is given by the expression b+c. It is useful to think of compositional differences between two sites in the form of the proportion of these differences in relation to the total number of species recorded (a+b+c). Therefore, we obtain the Jaccard dissimilarity measure, also known as the βcc complementarity measure of Colwell & Coddington (1994):
Figure 2. Matching/mismatching components between two sites: a is the number of shared species, b and c are the species exclusive to each site. The total number of species in the system is given by the sum a+b+c. The number of compositional differences between the two sites (total beta diversity) is given by the sum b+c. Richness difference is given by the absolute value of b–c. Replacement is given by min(b,c) multiplied by 2 (one substitution involves two species).
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Conceptually, this approach corresponds to the variant of beta diversity defined as ‘proportional effective species turnover’ (Tuomisto, 2010a,b).
Replacement between two sites is the substitution of n species in a given site from n species in another site. Therefore, the number of substitutions between two sites is given by the minimum number of exclusive species, min(b,c), and the number of species involved in this process is 2 × min(b,c), since one substitution always involves two species. By expressing this quantity in the form of a proportion to the total number of species recorded (a+b+c), we obtain the measure β-3 of Williams (1996) as modified by Cardoso et al. (2009):
The species richness of each site is given by the expressions b+a and c+a. Therefore, the absolute difference in species richness between two sites, expressed as a proportion of the total number of species recorded (a+b+c), is given by a new measure, which we designate as βrich:
In conclusion, we postulate that beta diversity can be partitioned into two additive components, replacement and richness differences, according to the equation:
Related methods previously proposed
βrich is in some way similar to the previously proposed βgl (Lennon et al., 2001), the only measure of species richness difference to date:
Both measures differ in the fact that βgl double-weights the absolute richness difference (numerator) and the shared species component. This does not allow the richness differences value to be expressed as a simple proportion of the total number of species recorded (a+b+c), a very useful and intuitive property of βrich.
As we will extensively compare our novel method with the recent proposal of Baselga (2010), the only general framework proposed to date for partitioning beta diversity in its different components, we also represent this method. Baselga's framework can be represented by the equation:
βsor is deemed to represent all compositional differences, i.e. overall beta diversity. In the dissimilarity form the index can be formulated as (Sørensen, 1948):
βsor is conceptually similar to βcc by accounting for compositional differences (b+c). Both measures differ only because βsor double-weights the shared species component. This means that βsor expresses compositional differences in relation to the sum of species richness of both sites (usually larger than the total number of species recorded given that species may be shared).
βsim is deemed to reflect compositional differences attributable to replacement. The index has the form (Lennon et al., 2001):
It is worth noting that the numerator of βsim expresses the number of substitutions, min(b,c), but not the number of species involved in the replacement process, 2 × min(b,c), between two sites. Apart from that, βsim expresses the number of substitutions between two sites as a proportion to the number of species of the species-poorer site, unlike βsor which expresses compositional differences as a proportion of the sum of species richness of both sites. This is non-trivial, because in this way the replacement in βsor and the replacement in βsim is not mathematically equivalent. Actually, βsim is a measure of the proportion of the species-poorer site that is not nested in the species-richer site (Tuomisto, 2010b). Therefore, βsim overestimates replacement because it measures replacement relative to the species-poorer site and not as a proportion of all species.
βnes is deemed to account for dissimilarity due to nestedness (Baselga, 2010). The index is obtained by subtracting βsim from βsor:
Mathematically, βnes may be viewed as a richness difference measure (the first term of the product), multiplied by the proportion of the species-poorer site that is nested in the species-richer site (the second term of the product).
The hypothetical gradient of dissimilarity illustrated in Fig. 1 allows a first comparison of the performance of our novel partitioning method with Baselga's (Table 1). Apart from the similarity between βcc and βsor, the other measures have totally different behaviours. While β-3 remains constant, reflecting the constancy of the number of substitutions between site A and the other sites, βsim increases with increasing richness differences, which is a counter-intuitive behaviour of a measure that is deemed to reflect replacement. While βrich reflects all richness differences along the gradient, βnes is null when the last shared species disappears: βrich(A,F) = 0.8; βnes(A,F) = 0. This is a conceptual difference between the two measures, βnes only accounts for differences in richness when sites are nested, i.e. when sites have at least one common species (a > 0), while βrich reflects all richness differences independently of sites being nested or not. Apart from this difference, both measures try, in fact, to reflect richness differences when the sites shared at least one species (a > 0).
Table 1. Comparison of βcc, β-3, βrich,βsor, βsim and βnes values along a hypothetical gradient of increasing dissimilarity (see Fig. 1 for details).
It is worth noting that in the graphical example of Fig. 1 two paradoxical situations are apparent in the behaviour of βnes (Table 1): (1) βnes(A,D) > βnes(A,E), when in fact site E has one species less than site D; (2) βsim(A,D) > βnes(A,D), when clearly almost all species were lost and only one substitution has occurred. The behaviour of both partitions suggests that more formal testing is necessary to fully evaluate their usefulness.
Mathematical comparison of measures
The mathematical behaviour of the different measures was studied with ternary plots, as recommended by Koleff et al. (2003). Ternary plots allow visualization of the relationship between the matching/mismatching components (a, b and c), expressed as proportions, and the values of such measures, hence, a+b+c= 100% (Fig. 3). For that purpose, we used an artificial dataset with all possible combinations of the proportions of the components a, b and c, available in the simba package (Jurasinski, 2010) for the R statistical language (R Development Core Team, 2009).
Figure 3. Ternary plot of the components a, b and c in the form of proportions: a is the proportion of the shared species between two sites; b and c are the proportions of the species exclusive to each site. Grey lines represent 10% increments of the components. Dashed lines represent four hypothetical gradients of richness differences and replacement: A, pure gradient of richness differences; B, mixed gradient of richness differences with a fixed constant replacement (20% of the species in the system); C, pure gradient of replacement; D, mixed gradient of replacement with constant richness differences (20% of the species in the system). The arrows represent the direction of increasing gradients.
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