Analytical formulae for computing dominance from species-abundance distributions

Author’s Accepted Manuscript Analytical formulae for computing dominance from species-abundance distributions Tak Fung, Laura Villain, Ryan A. Chisholm

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To appear in: Journal of Theoretical Biology Received date: 15 April 2015 Revised date: 6 August 2015 Accepted date: 12 September 2015 Cite this article as: Tak Fung, Laura Villain and Ryan A. Chisholm, Analytical formulae for computing dominance from species-abundance distributions, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2015.09.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Analytical formulae for computing dominance from speciesabundance distributions

Tak Funga,1, Laura Villainb,2, Ryan A. Chisholma,c,3

a

National University of Singapore, Department of Biological Sciences, 14 Science Drive 4, Singapore 117543

b

Institut National des Sciences Appliquées de Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne cedex, Lyon, France

c

Smithsonian Tropical Research Institute, Balboa, Ancón, Republic of Panamá

1

Corresponding author. E-mail: [email protected]

2

E-mail: [email protected]

3

E-mail: [email protected]

Type of article: Original Research Article

Keywords: Biodiversity, evenness, gamma, lognormal, log-series

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Abstract

The evenness of an ecological community affects ecosystem structure, functioning and stability, and has implications for biodiversity conservation. In uneven communities, most species are rare while a few dominant species drive ecosystem-level properties. In even communities, dominance is lower, with possibly many species playing key ecological roles. The dominance aspect of evenness can be measured as a decreasing function of the proportion of species required to make up a fixed fraction (e.g., half) of individuals in a community. Here we sought general rules about dominance in ecological communities by linking dominance mathematically to the parameters of common theoretical speciesabundance distributions (SADs). We found that if a community’s SAD was log-series or lognormal, then dominance was almost inevitably high, with fewer than 40% of species required to account for 90% of all individuals. Dominance for communities with an exponential SAD was lower but still typically high, with fewer than 40% of species required to account for 70% of all individuals. In contrast, communities with a gamma SAD only exhibited high dominance when the average species abundance was below a threshold of approximately 100. Furthermore, we showed that exact values of dominance were highly scale-dependent, exhibiting non-linear trends with changing average species abundance. We also applied our formulae to SADs derived from a mechanistic community model to demonstrate how dominance can increase with environmental variance. Overall, our study provides a rigorous basis for theoretical explorations of the dynamics of dominance in ecological communities, and how this affects ecosystem functioning and stability.

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1. Introduction

Human populations have caused massive and widespread changes to the structure of natural communities (Vitousek et al., 1997; Chapin et al., 2000; Steffen et al., 2007). This has resulted in sustained and ongoing loss of biodiversity (Chapin et al., 2000; Cardinale et al., 2012; Naeem et al., 2012). Since biodiversity is a fundamental determinant of ecosystem functioning, its loss affects functioning and associated ecosystem services (Cardinale et al., 2012; Naeem et al., 2012). Research over the last few decades has focused on the negative effects of species loss on functioning (Balvanera et al., 2006; Cardinale et al., 2006, 2012; Naeem et al., 2012), but the number of species is just one component of biodiversity. Another important component is species evenness, which describes the relative abundances of species in a community. Evenness is arguably more directly related than richness to the trait diversity on which the level of functioning depends (Reiss et al., 2009). Recent experiments have demonstrated substantial positive effects of evenness on biomass (Wilsey and Potvin, 2000; Mattingly et al., 2007; Zhang et al., 2012; Orwin et al., 2014), water retention (Orwin et al., 2014), resistance to invasion (Wilsey and Polley, 2002) and nutrient retention after leaching (Orwin et al., 2014). Moreover, evenness is expected to change quicker than richness in response to human stressors, since changes in abundance occur before species go extinct. Therefore, quantitative assessments of the causes and consequences of changing evenness are essential for a thorough understanding of ecosystem structure and function.

One metric of evenness is dominance, which has been used to quantify the degree to which the total abundance of a community is dominated by a subset of species (e.g., Grime, 1998; Barker et al., 2002; Binkley, 2004; Davidar et al., 2005; Hillebrand et al., 2008; Dornelas et

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al., 2011; ter Steege et al., 2013). Dominant species tend to have large effects on flows of energy and matter in an ecological network (Hillebrand et al., 2008; Berlow et al., 2009). In particular, empirical evidence suggests that in autotrophic and suspension-feeding invertebrate assemblages, the contribution of a species to functioning can be predicted largely by its biomass – the “mass ratio hypothesis” (Grime, 1998; Davies et al., 2011). Anthropogenic stressors such as targeted exploitation (e.g., stocks of Atlantic cod, Gadus morhua (Lotze and Worm, 2009)) may disproportionately affect dominant species, resulting in heavy loss of functioning (Gaston and Fuller, 2008). Ecosystems with high dominance also typically have many rare species, posing challenges for biodiversity conservation.

Studies have quantified dominance for particular communities (e.g., Grime, 1998; Barker et al., 2002; Binkley, 2004; Davidar et al., 2005; Hillebrand et al., 2008; Dornelas et al., 2011; ter Steege et al., 2013), but as yet there is no general theory for how dominance emerges from standard ecological models. In this study, we first derive mathematical formulae expressing dominance as a function of the parameters of four standard species-abundance distributions (SADs) – the log-series, exponential, gamma and lognormal. Specifically, for each SAD we derive formulae specifying dominance as a decreasing function of the number of species accounting for a proportion u of the total number of individuals (dominance is defined in this way because fewer dominant species equates to higher dominance). As u increases, more species are required for dominance of numerical abundance in the community considered. We examine four SAD forms that have been commonly studied and fitted to empirical data: the log-series (Fisher et al., 1943; Hubbell, 2001), exponential (Cohen, 1968), gamma (Brian, 1953; Plotkin and Muller-Landau, 2002; Forster and Warton, 2007) and lognormal (Preston, 1948; Engen et al., 2002; Volkov et al., 2003; Forster and Warton, 2007; Engen et al., 2011) forms. After deriving the dominance formulae, we

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proceed to apply them to (i) gauge the levels of dominance expected in natural communities under biologically plausible scenarios; (ii) assess dominance in the metacommunity of tree species found in Amazonia, using the data of ter Steege et al. (2013); and (iii) quantify how dominance is expected to change with increasing environmental variance, using the dynamic model of Engen and Lande (1996a). These three applications of our formulae highlight their three-fold utility: they allow dominance to be quantified quickly for discrete SADs, they permit dominance to be quantified for continuous SADs, and they can be mathematically analyzed further to derive general biological insights that cannot be obtained from simulations alone.

2. Methods

2.1. Deriving analytical formulae for calculating dominance: Discrete SADs

To derive a formula specifying dominance given an SAD for a community, the minimum fraction of species (i.e., the most abundant ones) required to account for a fraction u of the individuals in the community, denoted by pu , must first be computed. We consider the case

u ³ 0.5 , for which the minimum set of species is called the set of dominant species, making up at least half of all individuals. As pu decreases, fewer species are required to account for a fraction u of all individuals, such that dominance increases. Thus, dominance is defined as

Du =1- pu .

To calculate pu , we introduce a quantity nu , which is defined as the abundance of the least abundant dominant species. Three simple examples are now given to illustrate clearly the

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meaning of pu and nu , together with some difficulties with interpretation in specific cases. First, consider a community with S = 8 species and abundances {1,1, 3, 4, 4, 7,8,12} . In this example, p0.5 = 0.25 because we can account for half of the individuals with just one quarter of all species (i.e., the two most abundant ones). We also have n0.5 = 8 , because species with abundance greater than or equal to 8 comprise half of the community. Second, consider another community with eight species but with abundances {1, 4, 4, 4, 5, 5, 5, 8} . Here we have

p0.5 = 0.375 because we can account for half of the individuals with the three most abundant species. However, we note that the value of n0.5 is biologically not well-defined here. It might seem logical to take n0.5 = 5 , but this is problematic because to get 50% of all individuals we only want two of three species with abundance 5. Thus, the quantity nu is generally not as useful as pu , although it is necessary to calculate nu in the derivation of pu . As a final simple example, consider yet another community with eight species, this time with abundances {1, 3, 4, 6, 6, 7, 8,15} . In this case, the top three species are required to reach at least 50% of individuals, but we actually overshoot and get 60% of individuals. Such overshoots arise from the fundamental discreteness of individuals and pose problems of interpretation. Thus, in our mathematical treatment, we will work in the high-diversity, large-community size limit in which overshoots are negligible, with the understanding that in practical applications, theoretical values for pu S , where S is the expected total number of species, will need to be rounded to the nearest integer to get a whole number of species.

We first detail a method for deriving exact formulae for pu and hence Du for discrete SADs, in the high-diversity, large-community size limit. Consider a discrete SAD defined by a

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probability distribution y (n) , specifying the probability that a randomly selected species will have abundance n. We have:

y (n) =

s(n) s(n) , = ¥ S å s(n)

(1)

n=1

where s(n) is the expected number of species with abundance n. To obtain nu , the equation

¥

¥

å ns(n) å ny (n)

F(n) = n=m ¥

= n=m ¥

n=1

n=1

å ns(n) å ny (n)

=u

(2)

can be solved analytically for m. Here, F(m) is the fraction of individuals belonging to species with abundance greater than or equal to m.

In general, the value of nu obtained would not be an integer, so it is rounded up to the nearest integer, éênu ùú. Species with an abundance equal to or greater than éênu ùú are insufficient to make up a fraction u of all individuals, so a proportion h of species with abundance éênu ùú -1 are required. The quantity h satisfies

¥

¥

å ny (n) + h (éên ùú -1) y (éên ùú -1) = uå ny (n) . u

n=éênu ùú

u

n=1

Rearranging gives

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(3)

¥

uå ny (n) -

h=

n=1

¥

å ny (n)

n=éênu ùú

(éênu ùú -1) y (éênu ùú -1)

. (4)

The quantity pu , which is the minimum fraction of species required to make up a fraction u of all individuals, can then be calculated as

¥

pu = hy (éênu ùú -1) +

¥

å y (n) =

n=éênu ùú

uå ny (n) n=1

¥

å ny (n) n=éênu ùú

éênu ùú -1

¥

+

å y (n) . (5)

n=éênu ùú

The exact formula for dominance is then Du =1- pu . Usually, the formula for pu cannot be simplified further because of the ceiling function. However, simpler analytical approximations for pu can be derived by retaining the generally non-integer but analytical form of nu and then using

¥

pu » pu* = å y (n) .

(6)

n=nu

We derive pu and pu* according to equations (5) and (6), respectively, when y (n) follows a discrete log-series distribution and when it follows a discrete exponential distribution. We then derive Du and Du* for these two distributions as 1- pu and 1- pu* , respectively.

2.2. Deriving analytical formulae for calculating dominance: Continuous SADs

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Dominance formulae for continuous SADs can be derived analogously to those for discrete SADs. For these distributions, nu can take non-integer positive values and can be found by solving

ò G(m) = ò

¥ n=m ¥ n=d

ny (n)dn ny (n)dn

=u

(7)

for n, where y (n) is the probability density corresponding to abundance n and d is the lower limit of the domain of the corresponding probability distribution. An analytical formula specifying pu can then be derived using

pu =

ò

¥ n=nu

y (n)dn ,

(8)

allowing the corresponding dominance formula to be derived using Du = 1- pu . We follow this method to derive dominance formulae for four continuous distributions that approximate discrete SADs: the continuous log-series, exponential, gamma and lognormal distributions. Values of Du for the continuous log-series and exponential distributions can be considered as approximations to the exact discrete cases because the distributions permit non-integer species abundances, which are not observed in reality. Because the abundance n is assumed to be continuous, there are conceptual issues with choosing d , the lowest value of n for a continuous distribution. With d = 1, values between 0.5 and 1 are omitted, but these values become 1 when rounded to the nearest integer, so could be considered biologically meaningful. With d = 0 , values between 0 and 0.5 are included, but these values are rounded

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to 0, which is not biologically meaningful since it represents species with no individuals. Thus, it may be better to use d = 0.5 instead. However, using d = 0.5 for the continuous log-series, gamma and lognormal distributions results in loss of analytical tractability, which is why we used 1, 0 and 0 for the three distributions, respectively. For the continuous exponential, all three values of d (0, 0.5 and 1) are examined since analytical formulae for

nu and pu can be derived for all three d values.

2.3. Applying the formulae to quantify the typical level of dominance in ecological communities

The formulae derived using the methods described in Subsection 2.1 and 2.2, for two types of discrete SAD and four types of continuous SAD, are applied to quantify dominance across a broad range of biologically possible scenarios. Specifically, parameter values are specified for each type of SAD examined to give a broad range of average species abundances ( J S ), from small to very large, and then dominance values calculated for these parameter values using the formulae. To facilitate comparison of dominance values between different types of SAD, dominance for each type is plotted against J S , which is defined for all SADs. It is noted that J S for an SAD is equal to the mean of the underlying distribution; this can be seen for discrete SADs because

¥

J = S

¥

å ns(n) å ny (n) n=1 ¥

å s(n) n=1

=

n=1 ¥

åy (n)

¥

= å ny (n)

(9)

n=1

n=1

10

and it can be seen for continuous SADS using analogous calculations. In addition, we mathematically analyze the dominance formulae for the continuous log-series, continuous exponential and lognormal SADs to gain general insights into how they change with J S .

For the discrete log-series SAD with parameter x, we examine the parameter range

éë4.65´10-10 , 0.980ùû , which gives J S from 1.01 (corresponding to the maximum x) to 108 (corresponding to the minimum x). 1.01 is close to the theoretical minimum of 1 whereas 108 is on the order of magnitude of the median number of individuals found for 207 dominant tree species in Amazonia, which covers an area of 6.29 million km2 (ter Steege et al., 2013). Thus, the range of J S that we examine is a wide range encompassing most values likely to be encountered in reality. The range of l for the discrete exponential SAD is chosen in the same way as éë1.00 ´10 -8, 4.62ùû, with the maximum value corresponding to J S =1.01 and the minimum value corresponding to J S = 108 . For each of the two types of discrete SAD, Parameter values for the continuous SADs are chosen using the same rationale as for the discrete SADs. Thus, the range of x for the continuous log-series SAD is chosen to be

éë4.79 ´10 -10 ,1.00ùû , corresponding to a range of J S of éë1.01,108 ùû. Similarly, the ranges of

l for the continuous exponential SAD are chosen to be éë1.00 ´10-8, 0.990ùû, éë1.00 ´10-8,1.96ùû and éë1.00 ´10 -8,100ùû for lower abundance limits of d = 0 , 0.5 and 1 respectively, to correspond to the same range of J S . For the gamma SAD with shape and rate parameters a and

b , it was found that Du

is independent of

b

and hence scale

( q = 1 b ). This is expected because the scale parameter does not change the shape of the gamma distribution. However, J S depends on both a and

b . Therefore, we (i) calculate a

range of b values from previous studies on plant and insect communities (Brian, 1953;

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Plotkin and Muller-Landau, 2002; Forster and Warton, 2007); (ii) calculate the minimum and maximum values of b and for each, calculate the corresponding range of a that would give a range of J S of éë1.01,10 8 ùû; (iii) calculate values of Du corresponding to the two ranges of a found; and (iv) plot curves of Du against J S , one for each of the two ranges of a . The minimum and maximum b values found were 0.000321 and 0.00761 respectively, with the corresponding ranges of a being [ 0.000324,32100 ] and

[ 0.00769, 761000] respectively. Similarly, for the lognormal SAD with log-scale and shape parameters

m and s , it was found that Du is independent of m , but that J S depends on

both parameters. Therefore, we calculated the minimum and maximum values of

m from

previous studies on plant and bird communities (Preston, 1948; Volkov et al., 2003; Forster and Warton, 2007) as 1.39 and 3.47 respectively, and for each value calculated a corresponding range of s that would give a range of J S of éë32.2,108 ùû . Values of J S lower than 32.2 are not considered because they correspond to negative values of s when the maximum value of m = 3.47 is used. For each range of s , we calculate corresponding values of Du and plot them against J S . The values of u examined range from 0.5 to 0.9 in increments of 0.1, as for the discrete SADs. In addition, for each of the log-series and exponential SADs, the error in Du arising from using the continuous version is quantified as the difference between Du derived using the exact formula for the discrete version minus

Du derived using the formula for the continuous version, for each combination of u and J S.

2.4. Applying the formulae to assess dominance for tree species in Amazonia

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To estimate the total number of tree species in Amazonia (the Amazon and Guiana Shield), ter Steege et al. (2013) fitted a discrete log-series SAD to species population abundances estimated from their sample data and then extrapolated from this distribution. Using this method, they estimated that Amazonia has approximately 16,000 tree species. However, their low sampling intensity of 0.0002% (not 0.002% as ter Steege et al. state on p. 4 of their Supplementary Material) means that the distribution of rare species is essentially unknown. Thus, other theoretical SADs, not considered by ter Steege et al. (2013), provide equally good fits to their estimated population abundances and give virtually the same sample SAD (Electronic Supplementary Material 1). These other theoretical SADs lead to vastly different estimates of the total Amazon tree species richness S. The examples in Electronic Supplementary Material 1 show plausible Amazon-scale discrete log-series SADs truncated such that S = 12,000, 9,000 and 6,000, with removal of increasingly more of the least abundant species. To calculate corresponding dominance values for these alternative distributions, we adapt our dominance formulae for log-series SADs and apply them to the truncated log-series SADs. This provides an idea of the range within which the true dominance may vary. Dominance is expected to decrease with greater truncation because the dominant species would likely form a higher proportion of a smaller total number of species.

Given the fundamental uncertainty in the distribution of rare species in poorly sampled species-rich communities, a more informative approach could be to fit only the observed portion of the SAD, calculate the dominance of the fitted SAD and use it as an estimate of a lower dominance bound. We apply this approach to the Amazonian tree metacommunity by calculating dominance for a log-series SAD fitted to the sample data and truncated to give

S » 5, 000 , corresponding approximately to the number of observed species.

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2.5. Applying the formulae to quantify how dominance changes with environmental variance

In the dynamic community model of Engen and Lande (1996a), the abundance ( n ³ 1) of each species changes according to

dB(t) dn , = rn - ng(n) + s r (n) dt dt

(10)

where r is the per-capita growth rate, g(n) is the density-dependent per-capita mortality rate, and B(t) is a Wiener process representing stochastic variation in the growth rate, with

(

)

2 2 2 variance s r (n) = s e + s d n . This variance is the sum of two components, the first

representing environmental variance and the second demographic variance. Under a Gompertz growth curve g(n) = g ln(n + e ) with e = s d2 s e2 (there is a typo on p. 175 of Engen and Lande (1996a) – they mistakenly stated that e = s e2 s d2 ), the SAD at the steady state is given by

S(n) =

A (n + e )s e

é éln(n + e ) - r g ù2 ù ( )û ú 1ë exp ê, (11) 2 ú ê 2 s e 2g p g ë û

where A is a scaling constant (Engen and Lande, 1996a). Explicitly,

A=

2w p g

se

2 ìï g é r ù üï exp í 2 êln(1+ e ) - ú ý , g û þï îï s e ë

(12)

14

where w is the per-capita speciation rate. Using the transformation z = n + e , S(z) follows a truncated lognormal distribution with log-scale parameter m = r g and shape parameter

s =se

2g over the range [1+ e, ¥) . This distribution can be approximated by the full

lognormal distribution with range [ 0, ¥) if the density of the full distribution is approximately zero in the range [ 0, 1+ e ) , which is the case for large communities. In this case, we apply our dominance formula for the lognormal distribution to explicitly relate dominance to environmental variance. We mathematically analyze this relationship to show how dominance changes with increasing environmental variance, and illustrate the trend found with example parameter values.

3. Results

3.1. Analytical dominance formulae for SADs

For the six types of SAD that we considered, explicit formulae were derived specifying dominance, measured as 1 minus the minimum proportion of species required to account for a proportion u of all individuals. This measure of dominance is denoted by Du and varies between 0 and 1. In addition, for the two types of discrete SADs, we derived formulae specifying Du* , which is an approximation to the exact dominance measure Du . Table 1 lists the formulae obtained, together with formulae specifying the probability distributions underlying the SADs. Appendix A provides details of how the formulae were derived following the methods detailed in Subsections 2.1 and 2.2. It is seen that the formulae for the

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(continuous) gamma and lognormal SADs depend only on the shape parameter, that is, a for the gamma distribution and s for the lognormal distribution.

3.2. Quantifying dominance typically found in ecological communities

In general, Du values obtained from the formula for the discrete log-series were high, indicating that numerical abundance was dominated by relatively few species (Fig. 1). When u increased from 0.5 to 0.9 (in increments of 0.1), Du remained above 0.7 for average species abundances ( J S ) greater than 100. As J S decreased below 100, Du decreased non-linearly to low values, dropping below 0.2 when u = 0.9 (Fig. 1). Values of Du for the discrete exponential were smaller than for the discrete log-series (Fig. 1), indicating that numerical abundance was dominated by more species. Nonetheless, Du was generally quite high: when u increased from 0.5 to 0.7, it remained above 0.6 for J S > 100 (Fig. 1a-c), only dropping below 0.6 for J S > 100 when u increased further to 0.8 and 0.9 (Fig. 1d-e). As for the discrete log-series, Du for the discrete exponential decreased non-linearly when

J S decreased below 100, to below 0.2 when u = 0.9 (Fig. 1). The Du* values for both the discrete log-series and exponential were found to be very close approximations of the exact dominance values ( Du values) (Fig. 2). The absolute error between the approximate and exact values was almost zero except for very low values of J S below 10, when the absolute error increased but still remained below 0.1 (Fig. 3). Therefore, trends in Du* closely followed trends in Du .

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Values of Du for the continuous log-series are close approximations of the corresponding values for the discrete log-series (Fig. 2a), with absolute errors typically < 0.05 and always < 0.11 (Fig. 3a). Similarly, values of Du for the continuous exponential were typically close approximations of corresponding values for the discrete exponential, regardless of whether a lower limit of d = 0 , 0.5 or 1 was used (Fig. 2b and Fig. S2 in Electronic Supplementary Material 2); absolute errors were virtually zero for J S above 100 (Fig. 3b and Fig. S3 in Electronic Supplementary Material 2). The absolute errors were typically higher as J S decreased below 100, reaching values close to 0.4 with d = 0 (Figs. 3b, S2). However, with

d = 0.5 , the errors remained virtually zero down to J S = 10 and remained below 0.2 for lower J S values (Fig. 3b), and with d = 1, the errors remained below 0.1 as J S decreased below 100 (Fig. S3). Thus, dominance trends for continuous log-series and exponential SADs closely follow those for their discrete counterparts. In addition, we showed analytically that for the continuous log-series, dDu d ( J S ) > 0 for small and large J S (Appendix B). These analytical results hold for all u, thus generalizing the trends found from simulations for particular values of u (Fig. 1). Furthermore, we proved that for the continuous exponential with lower abundance limit d > 0 , dDu d ( J S ) > 0 for all u (Appendix B), again generalizing the simulated dominance trends (Fig. 1).

Values of Du for the lognormal distribution followed a similar trend to those for the logseries and exponential, exhibiting high values greater than 0.6 for J S above 200, regardless of the value of u tested, and decreasing non-linearly as J S decreased below 200 (Fig. 1). For large J S above 10,000, Du reached values close to 1 for all values of u tested, exceeding Du for both the log-series and exponential (Fig. 1). Values of Du for the gamma

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distribution showed the opposite trend to that of the other three distributions, increasing rather than decreasing with J S (Fig. 1). For small J S below 50, Du attained high values above 0.6, but as J S increased above 50, Du decreased according to a non-linear sigmoidal shape, to values below 0.6 (Fig. 1). We mathematical analyzed the form of Du for the lognormal and found that dDu d ( J S ) > 0 for all u (Appendix B).

3.3. Assessing dominance of tree species in Amazonia

In their study of tree species in Amazonia, ter Steege et al. (2013) fitted a discrete log-series SAD with a parameter value of x = 1.93´10 -9 to species abundances estimated from sampled data. This parameter value corresponds to J S = 2.58´10 7 and S » 16, 000 . Application of our exact dominance formula (Table 1) shows that for this SAD, 1.89% of species are required to account for 50% of all tree individuals, i.e. Du = 0.981 for u = 0.5 . However, a truncated discrete log-series fits the data just as well (Electronic Supplementary Material 1). Following the method used to derive the exact and approximate dominance formulae for the full log-series, we derived corresponding formulae for a truncated log-series as

Du = 1-

é u - (1- x)éênu ùú-nmin ù 1 én ù-n ê + (1- x)ê u ú min F (1- x, 1, éênu ùú)ú F (1- x, 1, nmin ) êë x (éênu ùú -1) úû

and

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(13)

æ log(u) ö uF ç1- x, 1, nmin + ÷ log(1- x) ø è * Du = 1F (1- x, 1, nmin )

(14)

respectively. In these formulae, nmin is the species abundance at which the SAD is truncated, such that species with a lower abundance are removed; also, in (13),

nu = nmin + ( log(u) log(1- x)) . The dominance formula for the continuous version of the truncated log-series can be derived analogously to that for the continuous log-series, as

Du = 1-

li ( u(1- x)nmin ) li ((1- x)nmin )

.

(15)

Application of (13) with u = 0.5 and nmin = 84 , 3,646 and 157,027, corresponding to

S » 12, 000 , 9,000 and 6,000 respectively, gave Du = 0.975 , 0.966 and 0.949 respectively. Application of (14) and (15) gave dominance values that are the same to 3 decimal places; however, application of (15) is much faster than (13) and (14) because evaluation of the Lerch transcendent is slow with small x.

A lower bound for the dominance of tree species in Amazonia was estimated by applying dominance formulae (13)-(15) with nmin = 550, 755 , corresponding to S » 5, 000 , where 5,000 is the approximate number of observed species. All three formulae gave Du = 0.940 to 3 decimal places. By plotting Du against nmin from 1 to 550,755, we confirmed our expectation that Du decreases with nmin , such that Du = 0.940 can be taken as a lower bound.

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3.4. Quantifying how dominance changes with increasing environmental variance

We applied the dominance formula for the lognormal SAD (Table 1) to the lognormal SADs arising from the dynamic community model by Engen and Lande (1996a) (as described in Subsection 2.5). According to this formula, dominance increases with the shape parameter

s , and hence the average species abundance J S (Subsection 3.2). SADs from the dynamic model have a shape parameter that is proportional to s e , which is the square root of environmental variance. Thus, application of the formula shows that dominance increases with s e in the model.

The relationship between dominance and environmental variance is illustrated in Fig. 4 for example parameter sets that are the same except for s e increasing by a factor of 1.7 from -1/2 -1/2 0.05 = 0.224 yr to 0.380 yr . This gave a range of s e2 values that are in accordance

with empirical values collated by Lande et al. (2003; Table 1.2), typically on the order of 0.01–0.1 yr-1. Similarly, demographic variance s d2 was set at 0.2, giving a ratio e = s d2 s e2 from 1.38–4, in accordance with typical empirical ratios of 1 to 10 (Table 1.2 of Lande et al., 2003). The dynamic model explicitly models speciation, which operates over large spatial scales. Thus, values of the other parameters were chosen to give large values of J and J S , commensurate with communities at large spatial scales. These quantities together with S can be calculated using equations (11) and (12), after translation of the abundance variable and assuming that the transformed variable starts at zero (a reasonable assumption for large communities). For the example shown in Fig. 4, dominance increased with s e by 12–68% for values of u ranging from 0.5–0.9 in increments of 0.1. The sharpest increase was seen for

20

the highest u tested. In addition, as s e increased, S decreased from 139,000 to 7.52; J decreased from 7.76 ´108 to 136,000; and J S increased from 5,570 to 18,100.

4. Discussion

In this study, we examined four standard forms of SADs that together have commonly been fitted to empirical data (e.g, Fisher et al., 1943; Preston, 1948; Brian, 1953; Cohen, 1968; Hubbell, 2001; Engen et al., 2002; Plotkin and Muller-Landau, 2002; Volkov et al., 2003; Forster and Warton, 2007; Engen et al., 2011) and successfully derived analytical formulae quantifying dominance according to Du = 1- pu , where pu is the minimum proportion of species required to account for a proportion u of all individuals (Table 1). We also examined the discrete broken-stick (MacArthur, 1957) and Zipf-Mandelbrot (Frontier, 1985) SADs, but due to the complexity of the SAD formulae, we were unable to derive corresponding analytical dominance formulae. However, in the limit of large S, the broken-stick SAD converges to a gamma distribution (Longuet-Higgins, 1971), which we examined (Table 1). The dominance formulae derived facilitate quantitative investigation of dominance by speeding up calculation of dominance for discrete SADs (compared with simple summation, the formulae can be orders of magnitude quicker); allowing dominance to be calculated for continuous SADs; and permitting further mathematical analyses that yield general insights into how dominance changes with key ecological variables. We also derived analytical formulae to approximate Du for the discrete log-series and exponential SADs, and found that these approximations gave almost zero error when the average number of individuals per species was greater than about 10. In addition, under these circumstances, dominance formulae for the continuous versions of the SADs gave values that were very similar to those

21

from the exact formulae for the discrete versions. Thus, the formulae for the continuous versions can be used to give accurate estimates of dominance for any natural community of reasonable size that is distributed according to one of these two discrete SADs. The errors when using the approximate formulae for the discrete SADs were typically lower than those when formulae for the continuous SADs were used (Fig. 3). This result is intuitive because the continuous SADs assume that species abundance changes continuously, whereas in reality it is discrete. However, despite this continuity assumption, the resulting continuous SADs largely retain the dominance structure of their discrete counterparts (Fig. 2). Given lower errors, the discrete formulae would be preferred in practice, except for the discrete log-series SAD with low values of the parameter x. In this case, the approximate discrete formula is slow to compute because of the Lerch transcendent function, and so the continuous formula would be preferred.

A striking result from application of our formulae is that dominance was typically very high for reasonably large communities with individuals distributed according to a log-series, exponential or lognormal distribution. When the average abundance of a species exceeded 200 individuals, fewer than 40% of species were required to account for 90% of all individuals for the log-series or lognormal case ( Du > 0.6 for J S > 200 and u = 0.9 ), and 70% of all individuals for the exponential case ( Du > 0.6 for J S > 200 and u = 0.7 ). Given that many empirical SADs can be fitted by these three distributions (Fisher et al., 1943; Preston, 1948; Cohen 1968; Hubbell, 2001; Engen et al., 2002; Volkov et al., 2003; Forster and Warton, 2007; Engen et al., 2011; the lognormal distribution is typically mixed with a Poisson sampling distribution), the concentration of abundance in so few species that we typically found has important implications for community functioning and stability. Communities with high dominance attain a higher level of functioning if there is a positive

22

correlation between numerical dominance of a species and its level of functioning (Hillebrand et al., 2008) – this is the classic positive selection effect that has been demonstrated to be an important mechanism underlying positive relationships between richness and functioning for plant and invertebrate communities (Huston, 1997; Wilsey and Potvin, 2000; Hooper et al., 2005; Larsen et al., 2005; Orwin et al., 2014). However, another implication of high dominance is a decrease in the strength of multispecies processes, thus reducing the prevalence of niche complementarity and facilitation, both of which are important drivers of functioning (Loreau and Hector, 2001; Cardinale et al., 2002; Hooper et al., 2005; Carey and Wahl, 2010). In contrast to the selection effect, this acts to decrease community functioning. Therefore, the net effect of higher dominance depends on the balance of two countervailing forces – the selection effect and weaker multispecies processes. A meta-analysis of 54 studies suggests that the selection effect is typically stronger in forest communities, resulting in greater community functioning with higher dominance (Zhang et al., 2012). In addition to these effects on functioning, an increase in dominance results in an increase in intraspecific and corresponding decrease in interspecific interaction strengths, causing changes in community stability. Studies on model and empirical food webs suggest that weak interspecific interactions can increase stability (May, 1972; McCann et al., 1998; Neutel et al., 2002; Emmerson and Yearly, 2004), such that higher dominance is expected to result in more stable communities.

In contrast to the log-series, exponential and lognormal distributions, dominance values for the gamma distribution were typically high only for communities with small average species abundances. When the average species abundance was below 50 individuals, fewer than 40% of species were required to account for 90% of all individuals ( Du > 0.6 for J S < 50 and u = 0.9 ). However, as the average species abundance increases beyond 50 individuals,

23

dominance decreases non-linearly, typically reaching values below 0.5. Thus, our formulae give opposite trends for how dominance in gamma- and lognormally-distributed communities scale with community size. This result is surprising given that these two distributions can have similar shapes. However, it can be explained by the tail of the lognormal distribution becoming heavier as the value of the shape parameter, which correlates with community size, increases, whereas the tail of the gamma distribution becomes less heavy. Therefore, for communities that are fit well by a gamma distribution (typically mixed with a Poisson sampling distribution; e.g., Brian, 1953), it is advisable to rigorously test whether a lognormal distribution can give a better fit or not, since this can have important implications for determining dominance and assessing its community effects. On the other hand, this is difficult given that statistical estimates of the rare species tail of an SAD are highly uncertain unless a community is intensively sampled. This is also evident from our analysis of the SAD underlying the tree metacommunity in Amazonia (discussed below), and together these results emphasize the sensitivity of dominance to sampling uncertainty in general.

A recent assessment of dominance among trees in Amazonia involved fitting a discrete logseries SAD to sample plot data and concluded that dominance is very high, in the sense that very few species are required to account for 50% of all individuals (ter Steege et al., 2013). We now see, from application of our analytical formula for discrete log-series SADs, that this result follows inevitably from fitting a log-series: the formula predicts high dominance for values of the parameter x corresponding to large communities with high average species abundances ( J S ). However, whether the actual distribution of Amazon trees is a log-series is highly uncertain, given a low sampling intensity of 0.0002% (ter Steege et al., 2013). Alternative types of SADs can be fitted to the Amazonian data equally well, and one of these

24

is the truncated discrete log-series (Electronic Supplementary Material 1). We adapted our calculations to derive dominance formulae for a truncated log-series and used it to show that dominance in the Amazonian tree flora can vary between 0.940–0.981, i.e., the percentage of species making up 50% of individuals could be anywhere between 1.9% and 6.0%. Thus, dominance is high, but the exact degree of dominance currently remains uncertain. The true dominance may even exceed the upper limit of 0.981 if there are more rare species than predicted by a log-series. Furthermore, the true level of dominance is probably fundamentally unknowable from small sample plots, because it is not possible to calculate an unbiased estimate of the total number of species from a small sample (Gotelli and Chao, 2013; Electronic Supplementary Material 1). A more justifiable approach in such situations may thus be to calculate dominance from a fitted distribution that covers only the empirical range of species abundances and use it as an estimate of a lower bound for dominance – for the Amazonian tree flora, such an estimate would be 0.940.

We also applied our dominance formula for lognormal SADs to the dynamic community model by Engen and Lande (1996a), which predicts lognormal SADs as a function of parameters representing the ecological mechanisms modeled. This allowed a quantitative, process-based assessment of how dominance is predicted to change with variations in the ecological mechanisms. In our application, we focused on how dominance is expected to change with environmental variance, which is modeled as stochastic fluctuations in the growth rate. We showed analytically that the model predicts increases in dominance with the degree of environmental variance, and illustrated this with an example whereby dominance increased by up to 68%. It is also possible to show analytically that the model predicts a decline in species richness with environmental variance, which was the case in the example shown. Therefore, the model predicts that greater environmental fluctuations homogenize

25

communities by causing extinctions and promoting dominance of abundance by proportionally fewer species, in effect pushing species abundances towards the extremes (Fig. 4a). This prediction is particularly relevant in the context of forecasted increases in environmental variance driven by climatic change, for example greater intra-annual variation in precipitation resulting in more severe floods and droughts (Knapp et al., 2008). Our predictions complement previous theoretical predictions on how environmental variance can increase extinction risk and decrease species richness (e.g., Drake and Adler, 2008; Ovaskainen and Meerson, 2010). Importantly, our predictions can be tested quantitatively using empirical data in future studies. In addition, the homogenizing effect of environmental variance has possible knock-on effects for functioning and stability, as discussed above. This can be further investigated in future studies by explicitly quantifying functioning and stability using model outputs and possibly other theory. Furthermore, future studies could use our dominance formulae in conjunction with other stochastic community models (e.g., Dennis and Costantino, 1988; Engen and Lande, 1996b; Hubbell, 2001) to further investigate the mechanistic links between evenness and ecosystem functioning and stability. Predictions from these studies would facilitate understanding of the ongoing effects of anthropogenically driven biodiversity loss (Cardinale et al., 2012), vital for conservation and sustainable management.

Acknowledgements

We would like to thank two anonymous reviewers, whose comments have helped to substantially improve the clarity and content of our work. TF and RAC are supported by the National University of Singapore start-up grant WBS R-154-000-551-133.

26

Appendix A. Details on derivation of dominance formulae

This Appendix contains details of how the dominance formulae in Table 1 were derived.

Discrete log-series

In this case, F(m) = u is

(1- x)m-1 = u ,

(A.1)

which can be solved for m to give

nu = 1+

log(u) . log(1- x)

(A.2)

In addition,

¥

1- x

å ny (n) = x log 1 x

( )

n=1

¥

å

n=éênu ùú

,

(A.3)

én ù

ny (n) =

(1- x)ê u ú , x log (1 x )

(A.4)

and

27

¥

å y (n) =

n=éênu ùú

én ù

(1- x)ê u ú F (1- x, 1, éênu ùú) . (A.5) log (1 x )

Thus, using equation (5),

én ù

pu =

=

én ù

u(1- x) - (1- x)ê u ú (1- x)ê u ú + F (1- x, 1, éênu ùú) x log(1 x) (éênu ùú -1) log(1 x) én ù-1 ù (1- x) é u - (1- x)ê u ú én ù-1 ê + (1- x)ê u ú F (1- x, 1, éênu ùú)ú. log(1 x) êë x (éênu ùú -1) úû

(A.6)

Using Du =1- pu gives the formula for Du in Table 1. Furthermore,

¥

pu* = å y (n) = n=nu

(1- x)nu u(1- x) æ log(u) ö F (1- x, 1, nu ) = F ç1- x, 1, 1+ ÷. log (1 x ) è log(1- x) ø log (1 x )

Using Du* = 1- pu* gives the formula for Du* in Table 1.

Discrete exponential

F(m) = u is

(

)

e - lm 1+ ( e l -1) m = u ,

(A.8)

such that

28

(A.7)

W-1 ( uye- y ) 1 , nu = 1- e l l

(

(A.9)

)

l where y = l 1- e . In addition,

¥

å ny (n) = n=1

el , e l -1

¥

å ny (n) =

e

(A.10)

l -léênu ùú

(1- éên ùú + éên ùúe ) , l

u

u

(A.11)

e l -1

n=éênu ùú

and

¥

å y (n) = el l

- éênu ùú

.

(A.12)

n=éênu ùú

(A.10) and (A.11) can be derived using the identity

å

¥ k=a

2 kz k = éë z a ( a + (1- a)z )ùû (1- z ) ,

which holds for 0 < z < 1 . Thus,

pu =

(1- éên ùú + éên ùúe ) + e ( e -1) (éên ùú -1) éu - e (1- éên ùú + éên ùúe ) + e ê êë ( e -1) (éên ùú -1)

ue l - e

l - léênu ùú

l

u

u

l -léênu ùú

l

u

- léênu ùú

=e

l

l

u

l

u

u

- léênu ùú

ù ú. úû

(A.13)

Using Du =1- pu gives the formula for Du in Table 1. Also,

29

¥

pu* = å y (n) = e

= exp éël - y + W-1 ( uye- y )ùû .

l (1-nu )

(A.14)

n=nu

Using Du* = 1- pu* gives the formula in Table 1.

Continuous log-series

In this case, G(m) = u is

(1- x)m-1 = u ,

(A.15)

such that

nu = 1+

log(u) log(1- x)

(A.16)

(this is the same as in the discrete case (equation (A.2)). Therefore,

pu =

ò

¥ n=nu

y (n)dn =

li ((1- x)nu ) li(1- x)

=

li ( u(1- x)) . li (1- x )

Using Du = 1- pu gives the formula in Table 1.

Continuous exponential

30

(A.17)

G(m) = u becomes

e

l ( -m+d )

(1+ l m) = u .

(A.18)

1+ dl

Solving for m gives

nu =

-1- W-1 ( -e-1-dl u(1+ dl ))

l

,

(A.19)

such that

pu =

ò

¥ n=nu

é

æ u(1+ dl ) öù ÷ú . è e(1+dl ) øû

y (n)dn = el (-n +d ) = exp ê1+ dl + W-1 ç u

ë

Using Du = 1- pu gives the formula in Table 1.

Continuous gamma

Here, G(m) = u is equivalent to

G (1+ a, b m ) = u, G (1+ a )

(A.21)

which on rearranging gives

31

(A.20)

G (1+ a, b m ) = uG (1+ a ) ,

(A.22)

such that

nu =

Q -1 (1+ a, u)

b

.

(A.23)

Thus,

-1 G (a, b nu ) G (a, Q (1+ a, u) ) . pu = ò n=n y (n)dn = = u G(a ) G(a ) ¥

Using Du = 1- pu gives the formula in Table 1.

Continuous lognormal

In this case, G(m) = u is

æ m + s 2 - log ( m ) ö 1 1- erfc çç ÷÷ = u , 2 s 2 è ø

(A.25)

such that

nu = exp éëm + s 2 - s 2erfc-1 ( 2(1- u))ùû.

(A.26)

32

(A.24)

Therefore,

pu =

ò

¥ n=nu

1æ 2è

æ m - log ( nu ) öö ÷÷÷ è s 2 øø

y (n)dn = çç 2 - erfc ç

öö æs 1æ = ç erfc ç - erfc -1 ( 2(1- u)) ÷÷ . øø è 2 2è

(A.27)

Using Du = 1- pu gives the formula in Table 1.

Appendix B. Details on mathematical analyses of dominance formulae

Continuous log-series

Consider the dominance formula for the continuous log-series distribution:

Du = 1-

li ( u(1- x)) . (B.1) li (1- x )

Thus,

33

æ li ( u(1- x)) ö ÷÷ æ d çç öæ li ( u(1- x)) ö uli (1- x ) dDu 1 è li (1- x ) ø ç ÷ == ÷. (B.2) 2 ç ç éli (1- x )ù ÷è log(u) + log(1- x) log(1- x) ø dx dx û ø èë

For small x <<1,

æ öæ ö 1 dDu ç ÷ uli (1- x ) - li(u) ÷ . (B.3) » 2 ç dx ç éëli (1- x )ùû ÷è log(u) log(1- x) ø è ø

The two terms in the second brackets in (B.3) have derivative

-

u li(u) >> 0 , log(u)log(1- x) ( log(1- x)) 2 (1- x)

(B.4)

such that the sum of the two terms increases quickly with x and hence decreases quickly with x. Therefore, for small x, the sum is negative, such that dDu dx < 0 . Thus, Du decreases with x when it is small, i.e. increases with J S when it is large. For large x » 1,

æ öæ ö dDu 1 ç ÷ç li ( u(1- x)) ÷ < 0 . »ç éli (1- x )ù2 ÷è log(1- x) ø dx û ø èë

(B.5)

So Du decreases with x when it is large, i.e. increases with J S when it is small.

Continuous exponential

34

The dominance formula for the continuous exponential distribution is:

Du = 1- exp éë1+ dl + W-1 ( -u(1+ dl )e-(1+dl ) )ùû . (B.6)

With d > 0 ,

(

d 1+ dl + W-1 ( -u(1+ dl )e -(1+dl ) )

)

dl ì ü -(1+dl ) ï ï æ dl ö W-1 ( -u(1+ dl )e ) = d í1- ç ý ÷ -(1+dl ) ù ïî è 1+ dl ø éë1+ W-1 ( -u(1+ dl )e )ûïþ -(1+dl ) ü ì ) ïý . d ï1+ dl + W-1 ( -u(1+ dl )e = í -(1+dl ) 1+ dl îï 1+ W-1 ( -u(1+ dl )e ) þï

(B.7)

The denominator of this derivative is thus negative, so that the derivative has the opposite

(

)

sign to 1+ dl + W-1 -u(1+ dl )e -(1+dl ) . Also, because d > 0 , the derivative is non-zero and therefore its sign does not change with l . For small positive l ,

1+ dl + W-1 ( -u(1+ dl )e-(1+dl ) ) » 1+ W-1 ( -ue-1 ) < 0 and the derivative is positive; hence the derivative is positive for all l > 0 . This implies that dDu d l < 0 , i.e. dDu d ( J S ) > 0 .

Continuous lognormal

Consider the dominance formula for the continuous lognormal distribution:

35

æs öö 1æ Du = 1- ç erfc ç - erfc-1 ( 2(1- u)) ÷÷ . è 2 øø 2è

(B.8)

The complementary error function erfc(z) is a decreasing function of z, so dDu ds > 0 , i.e.

dDu d ( J S ) > 0 .

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Table and figure captions

Table 1. Formulae specifying the six types of species-abundance distributions (SADs) examined and the corresponding dominance. Each SAD is specified by a probability mass function or probability density function (for discrete and continuous SADs, respectively),

y (n) , describing the probability or probability density of a species having abundance n, respectively. Dominance is measured as Du = 1- pu , where pu is the minimum proportion of species required to obtain a proportion u of all individuals. For each of the two discrete SADs, an approximation to Du is also given as Du* , which uses an approximation to pu that is specified by equation (6).

Fig. 1. Dominance values ( Du ) for the discrete log-series, discrete exponential, (continuous) gamma and (continuous) lognormal SADs, calculated using the formulae in Table 1. Panels (a)-(e) show how dominance changes across average species abundance (total number of individuals of all species, J, divided by total number of species, S) for five different values of u, which is the fraction of individuals accounted for by the set of dominant species.

Fig. 2. (a) Dominance values for the discrete log-series SAD using the exact and approximate formulae, compared with dominance values for the continuous log-series SAD (formulae shown in Table 1). The panel shows how dominance changes across average

44

species abundance (total number of individuals of all species, J, divided by total number of species, S) for different values of u, which is the fraction of individuals accounted for by the set of dominant species. (b) is the same as (a) but for the discrete exponential SAD and the continuous exponential SAD with a lower limit of d = 0.5 .

Fig. 3. Errors in approximating the exact dominance values for (a) the discrete log-series SAD and (b) the discrete exponential SAD when using the approximate formulae and the formulae for the continuous versions (formulae shown in Table 1). An error is calculated as the exact value minus the approximate value. Each panel shows how the errors change across average species abundance (total number of individuals of all species, J, divided by total number of species, S) for different values of u, which is the fraction of individuals accounted for by the set of dominant species.

Fig. 4. (a) Lognormal distributions underlying (stationary) SADs produced by the dynamic community model by Engen and Lande (1996a), for three different values of the square root of environmental variance (environmental standard deviation), s e (measured in units of yr1/2

, not shown on graphs to improve their clarity). Values of the other parameters are

r = 0.16 yr-1, g = 0.02 yr-1, s d = 0.447 yr-1/2 and w = 10 -4 yr-1 (for parameter definitions, see Subsection 2.5 in Methods). (b) Dominance values for the lognormal SADs, as calculated by applying the formula in Table 1 for five different values of u (fraction of individuals accounted for by the set of dominant species) and s e ranging from 0.224–0.380 yr-1/2. The log-scale parameter for the SADs is m = r g = 8 and the shape parameter varied from s = s e

2g =1.12 to 1.90.

45

Table

Table 1 Probability distribution underlying SADa Discrete log-

y (n) =

series

(1- x)n , æ1ö n log ç ÷ èxø

Dominance formula(e)

Du = 1-

én ù-1 (1- x) é u - (1- x)ê u ú ê + log(1 x) êë x (éênu ùú -1) én ù-1 (1- x)ê u ú F (1- x, 1, éênu ùú)ùû,

n ³1

nu = 1+

Du* = 1Discrete

y (n) = ( el -1) e- ln ,

exponential

n ³1

log(u) log(1- x)

u(1- x) æ log(u) ö F ç1- x, 1, 1+ ÷ log(1 x) è log(1- x) ø

é u - e- léênu ùú 1- éên ùú + éên ùú e l ( u u ) Du = 1- e ê l êë (e -1) (éênu ùú -1) l

+e nu =

W-1 ( uye 1 l 1- e l

y=

-y

- léênu ùú

ù, û

),

l

1- e l

Du* = 1- exp éël - y + W-1 ( uye- y )ùû , Continuous log-series

y (n) =

(1- x)n , nG ( 0, - log(1- x))

Du = 1-

li ( u(1- x)) li (1- x )

n ³1 Continuous

y (n) = l e l (-n+d ) ,

exponential

n ³d

é æ u(1+ dl ) öù Du = 1- exp ê1+ dl + W-1 ç - (1+dl ) ÷ú è e øû ë

46

Gamma

y (n) =

(continuous)

b a na -1e- b n , G(a )

Du = 1-

G (a, Q-1 (1+ a, u)) G(a )

n³0 æ ( m - log(n)) 2 ö ÷ exp çç ÷ 2s 2 è ø , y (n) = ns 2p

Lognormal (continuous)

öö æs 1æ Du = 1- ç erfc ç - erfc-1 ( 2(1- u)) ÷÷ øø è 2 2è

n³0 a

Special functions used in the SAD and dominance formulae are the Gamma function

gamma function

G(z) , the incomplete

G(a, z) , the Lerch transcendent F(z, s, a) , the Lambert W function W-1 (z) , the inverse

regularized gamma function

Q-1 (a, z) , the complementary error function erfc(z) , the inverse

complementary error function

erfc-1 (z) , and the logarithmic integral li(z) (Abramowitz and Stegun, 1972).

Highlights

·

Ecosystem stability and functioning critically dependent on species evenness

·

Evenness quantified as proportion of species that dominate total abundance

·

Formulae for dominance derived for different species-abundance distributions (SADs)

·

High dominance found for SADs with a log-series, lognormal or exponential form

·

Formulae used to show how dominance increases with environmental variance

47

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