Invariance in species-abundance distributions

Abstract

Many attempts to explain the species-abundance distribution (SAD) assume that it has a universal functional form which applies to most assemblages. However, if such a form does exist, then it has to be invariant under changes in the area of the study plot (the addition of neighboring areas or subdivision of the original area) and changes in taxonomic composition (the addition of sister taxa or subdivision to subtaxa). We developed a theory for such an area-and-taxon invariant SAD and derived a formula for such a distribution. Both the log-normal and our area-and-taxon invariant distribution fitted data well. However, the log-normal distributions of two adjoined sub-assemblages cannot be composed into a log-normal distribution for the resulting assemblage, and the SAD composed from two log-normal distributions fits the SAD for the assemblage poorly in comparison to the area-and-taxon invariant distribution. Observed abundance patterns therefore reveal area-and-taxon invariant properties absent in log-normal distributions, suggesting that multiplicative models generating log-normal-like SADs (including the power-fraction model) cannot be universally valid, as they necessarily apply only to particular scales and taxa.

Appendices

Appendix I: Composition of two SADs with correlated abundances

If abundances of the assemblages to be composed are correlated with each other, the SADs of the two assemblages interact in a particular way, which must be reflected in the definition of the mathematical operation that models a composition of the SADs. This operation was marked as *_c in Eq. 2, which refers to convolution (*) accounting for a correlation (index c).

If there is no correlation between abundances, the SADs f ₁(a) and f ₂(a) are convoluted as

$$f_1 \left( a \right) * f_2 \left( a \right) \approx \int\limits_0^a {f_1 \left( {a_1 } \right)} f_2 \left( {a - a_1 } \right)da_1 .$$

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This formula is based on the fact that the probability density of each abundance, a, is given by the sum (integral) of densities for all combinations of abundances, {a ₁;a ₂}, along a line of possible combinations of abundances, a ₂ = a−a ₁ (Fig. 1a). The reason is that (1) each point along the line represents one independent combination of abundances which gives the resulting abundance, a, and (2) the product of f ₁(a ₁) and f ₂(a−a ₁) is the probability (density) that the abundances a ₁ and a ₂ (=a−a ₁), which give the abundance a, occur simultaneously (i.e. each abundance applies to one assemblage). If abundances are not correlated, the line of the combinations is bounded by abundances of zero and a ($0 <a_1 <a$ and $0 <a_2 <a$; full line in Fig. 1a).

The simplest way to model correlation of abundances is by constraining the lines representing the possible combination of abundances which produce resulting abundance a by two increasing lines (a ₂ = σ _min a ₁, a ₂ = σ _max a ₁) intersecting the origin (Fig. 1b,c). If abundances are perfectly correlated, the abundances of the plots are proportional to each other — i.e. both lines approach each other.

In this case, we integrate probability densities (simply said, we sum the probabilities) not along the whole line between 0 and a (Fig. 1a), but only between two extreme abundances (${a \mathord{\left/ {\vphantom {a {\left( {\sigma _{\max } + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma _{\max } + 1} \right)}}$ and ${a \mathord{\left/ {\vphantom {a {\left( {\sigma _{\min } + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma _{\min } + 1} \right)}}$), which are imposed by the correlation (Fig. 1b). The composition of two SADs for species common to both subassemblages then obeys

$$f_1 \left( a \right) * _c f_2 \left( a \right) = \nu \int\limits_{\frac{a}{{\sigma _{\max } + 1}}}^{\frac{a}{{\sigma _{\min } + 1}}} {f_1 \left( {a_1 } \right)} f_2 \left( {a - a_1 } \right)da_1 ,$$

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where ν is a normalization constant.

Appendix II: A solution of the Eq. 2

Having the analytical form of the composition of two functional forms of SADs (Eq. 2 where *_c is defined by Eq. 5), we can check whether a multi-diffonential distribution (Eq. 3) is area-and-taxon invariant, i.e. whether the form composed from two multi-diffonentials is a multi-diffonential distribution. The evidence has been done by substituting the solution to Eq. 5. which gives

$$f_{1} * _{c} f_{2} \approx {\sum\limits_{i \ne j} {{\left( \begin{aligned} & \frac{{c_{i} c_{j} }}{{A_{i} - A_{j} }}{\left( {e^{{\frac{{A_{i} + A_{j} \sigma _{{\min }} }}{{\sigma _{{\min }} + 1}}a}} - e^{{\frac{{A_{i} + A_{j} \sigma _{{\max }} }}{{\sigma _{{\max }} + 1}}a}} } \right)} + \frac{{c_{i} c_{j} }}{{\alpha _{i} - \alpha _{j} }}{\left( {e^{{\frac{{\alpha _{i} + \alpha _{j} \sigma _{{\min }} }}{{\sigma _{{\min }} + 1}}a}} - e^{{\frac{{\alpha _{i} + \alpha _{j} \sigma _{{\max }} }}{{\sigma _{{\max }} + 1}}a}} } \right)} + \\ & + \frac{{c_{i} c_{j} }}{{A_{j} - \alpha _{i} }}{\left( {e^{{\frac{{\alpha _{i} + A_{j} \sigma _{{\min }} }}{{\sigma _{{\min }} + 1}}a}} - e^{{ - \frac{{\alpha _{i} + A_{j} \sigma _{{\max }} }}{{\sigma _{{\max }} + 1}}a}} } \right)} + \frac{{c_{i} c_{j} }}{{\alpha _{j} - A_{i} }}{\left( {e^{{\frac{{A_{i} + \alpha _{j} \sigma _{{\min }} }}{{\sigma _{{\min }} + 1}}a}} - e^{{\frac{{A_{i} + \alpha _{j} \sigma _{{\max }} }}{{\sigma _{{\max }} + 1}}a}} } \right)} \\ \end{aligned} \right)}} },$$

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which is, after substitution and reindexation, the multi-diffonential form (Eq. 3) as well (with different parameter N). The multi-diffonential distribution is obviously robust against proportional summation (linear combination), and thus follows the entire composition (Eq. 2) and is both taxon and area invariant. Note that in the cases in which the differences between respective As and/or αs approach zero the additive terms, whose denominators (see Eq. 6) are affected by these small values, turn into gamma distributions ($\gamma _i a^{\nu _i } e^{ - \beta _i a}$; $\nu _i \in {\mathbf{N}}$), and the area-and-taxon invariant distribution becomes more complicated. However, there is only a very low probability that this happens by chance, and thus here we present only the simpler solution. The cumulative distribution function of the multi-diffonential distribution obeys

$$P\left( {x <a} \right) = \sum\limits_{i = 1}^N {c_i \left( {\frac{{e^{ - \alpha _i a} - 1}}{{\alpha _i }} - \frac{{e^{ - A_i a} - 1}}{{A_i }}} \right)} ,$$

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and the expectation is

$$E\left( a \right) = \sum\limits_{i = 1}^N {c_i \left( {A_i^{ - 2} - \alpha _i^{ - 2} } \right)} .$$

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For the range of shapes see Fig. 2, for fitting procedure see Šizling and Storch (2007) or supplement SIII, and for fitting utility http://www.cts.cuni.cz/wiki/ecology:start.

Appendix III: Possible formal mechanism producing multi-diffonential SADs

First we show that a variety of distributions can be modelled as a sum of exponential distributions. According to the Taylor theorem almost any function f can be approached by $f\left( y \right) = \sum\limits_{k = 0}^N {c_i y^i }$. Denote abundance a as ln1/y ($y \in \left( {0;1} \right)$). Then a variety of shapes can be expressed as $f\left( {e^{ - a} } \right) = \sum\limits_{i = 0}^N {c_i e^{ - ia} }$. Because the result of the transformation is assumed to be a distribution (i.e. with finite integral between zero and infinity), we exclude all functions with i = 0. As a result, a variety of distributions can be approached by a sum of exponential forms

$$\phi \left( a \right) = \sum\limits_{j = 1}^N {c_j e^{ - \lambda _j a} } ,$$

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where $\lambda _{\min } \leqslant \lambda _j \left( {0 <\lambda _{\min } \leqslant 1} \right)$ for all j.

Now we show that the composition of two sums of exponential forms produces diffonential terms, which imposes additional constraints on a resulting distribution (e.g. an existence of indices m, n so that c _m  = −c _n , which cannot happen by chance). Assume two distributions expressed as $\phi _{10} \left( a \right) = \sum {c_j e^{ - \lambda _j a} }$ and $\phi _{01} \left( a \right) = \sum {c_l e^{ - \lambda _l a} }$. Their composition follows

$$\phi _{10} {\text{ composed with }}\phi _{01} = \pi _{10} \sum {c_j e^{ - \lambda _j a} } + \pi _{01} \sum {c_l e^{ - \lambda _l a} } + \pi _{11} \sum {\frac{{c_j c_l }}{{\lambda _l - \lambda _j }}} \left( {e^{ - \lambda _j a} - e^{ - \lambda _l a} } \right).$$

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(Eq. 2; c-convolution is defined as in Appendix I). If either of the component distributions comprises diffonential terms, the evidence does not change, because composition of two diffonentials remains diffonential (Appendix II). A gamma distribution results if both parameters λ happen to be equal to each other (Appendix II), which we consider unlikely.

If we express each additive term as a normalized distribution (terms in brackets ‘[]’ in Eq. 11) multiplied with its dominance δ, we get

$$\phi _{10} {\text{ cmp with }}\phi _{01} = \pi _{10} \sum {\delta _j \left[ {\lambda _j e^{ - \lambda _j a} } \right]} + \pi _{01} \sum {\delta _l \left[ {\lambda _l e^{ - \lambda _l a} } \right]} + \pi _{11} \sum {\delta _j } \delta _l \left[ {\frac{{\lambda _j \lambda _l }}{{\lambda _l - \lambda _j }}\left( {e^{ - \lambda _j a} - e^{ - \lambda _l a} } \right)} \right],$$

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where δ = c/λ. This suggests that a high Jaccard index (the proportion of species common to both plots π ₁₁; note that $\pi _{11} = 1 - \pi _{10} - \pi _{01}$ and all π ≥ 0) can make the diffonential terms dominant when composing SADs of several subplots.