Geometric series (Motomura 1932)

Figure 4. Generic Rank-abundance diagram of three common mathematical models used to fit species abundance distributions: Motomura's geometric series, Fisher's logseries, and Preston's log-normal series (modified from Magurran 1988)^[10]

Figure 5. Plant succession in abandoned fields within Brookhaven National Laboratory, NY. Species abundances conform to the geometric series during early succession but approach lognormal as the community ages. (modified from Whittaker 1972^[11])

I. Motomura developed the geometric series model based on benthic community data in a lake.^[12] Within the geometric series each species' level of abundance is a sequential, constant proportion (k) of the total number of individuals in the community. Thus if k is 0.5, the most common species would represent half of individuals in the community (50%), the second most common species would represent half of the remaining half (25%), the third, half of the remaining quarter (12.5%) and so forth.

Although Motomura originally developed the model as a statistical (descriptive) means to plot observed abundances, the "discovery" of his paper by Western researchers in 1965 led to the model being used as a niche apportionment model – the "niche-preemption model".^[8] In a mechanistic model k represents the proportion of the resource base acquired by a given species.

The geometric series rank-abundance diagram is linear with a slope of –k, and reflects a rapid decrease in species abundances by rank (Figure 4).^[12] The geometric series does not explicitly assume that species colonize an area sequentially, however, the model fits the concept of niche preemption, where species sequentially colonize a region and the first species to arrive receives the majority of resources.^[13] The geometric series model fits observed species abundances in highly uneven communities with low diversity.^[13] This is expected to occur in terrestrial plant communities (as these assemblages often show strong dominance) as well as communities at early successional stages and those in harsh or isolated environments (Figure 5).^[8]

Logseries (Fisher et al 1943)

${\displaystyle S=\alpha \ln {\left(1+{N \over {\alpha \,\!}}\right)}}$

where:

S = the number of species in the sampled community

N = the number of individuals sampled

${\displaystyle \alpha \,\!}$ = a constant derived from the sample data set

The logseries was developed by Ronald Fisher to fit two different abundance data sets: British moth species (collected by Carrington Williams) and Malaya butterflies (collected by Alexander Steven Corbet).^[14] The logic behind the derivation of the logseries is varied^[15] however Fisher proposed that sampled species abundances would follow a negative binomial from which the zero abundance class (species too rare to be sampled) was eliminated.^[1] He also assumed that the total number of species in a community was infinite. Together, this produced the logseries distribution (Figure 4). The logseries predicts the number of species at different levels of abundance (n individuals) with the formula:

${\displaystyle S_{n}={\alpha \,\!x^{n} \over n}}$

where:

S = the number of species with an abundance of n

x = a positive constant (0 < x < 1) which is derived from the sample data set and generally approaches 1 in value

The number of species with 1, 2, 3, ..., n individuals are therefore:

${\displaystyle \alpha \!x,{\alpha \,\!x^{2} \over {2}},{\alpha \,\!x^{3} \over {3}},\dots ,{\alpha \,\!x^{n} \over {n}}}$

Log normal (Preston 1948)

Figure 6. An example of Preston's veil. Fish species abundances sampled using repeated trawling over a one-month (blue bars), two month (gold bars) and one-year period (yellow). One year of sampling indicates that the fish community is log-normally distributed. (derived from Magurran 2004^[2])

Using several data sets (including breeding bird surveys from New York and Pennsylvania and moth collections from Maine, Alberta and Saskatchewan) Frank W. Preston (1948) argued that species abundances (when binned logarithmically in a Preston plot) follow a normal (Gaussian) distribution, partly as a result of the central limit theorem (Figure 4).^[7] This means that the abundance distribution is lognormal. According to his argument, the right-skew observed in species abundance frequency histograms (including those described by Fisher et al. (1943)^[14]) was, in fact, a sampling artifact. Given that species toward the left side of the x-axis are increasingly rare, they may be missed in a random species sample. As the sample size increases however, the likelihood of collecting rare species in a way that accurately represents their abundance also increases, and more of the normal distribution becomes visible.^[7] The point at which rare species cease to be sampled has been termed Preston's veil line. As the sample size increases Preston's veil is pushed farther to the left and more of the normal curve becomes visible^[2][10](Figure 6). Williams' moth data, originally used by Fisher to develop the logseries distribution, became increasingly lognormal as more years of sampling were completed.^[1][3]

Yule model (Nee 2003)

The Yule model is based on a much earlier, Galton–Watson model which was used to describe the distribution of species among genera.^[16] The Yule model assumes random branching of species trees, with each species (branch tip) having the equivalent probability of giving rise to new species or becoming extinct. As the number of species within a genus, within a clade, has a similar distribution to the number of individuals within a species, within a community (i.e. the "hollow curve"), Sean Nee (2003) used the model to describe relative species abundances.^[4][17] In many ways this model is similar to niche apportionment models, however, Nee intentionally did not propose a biological mechanism for the model behavior, arguing that any distribution can be produced by a variety of mechanisms.^[17]

Unified neutral theory (Hubbell 1979/2001)

The Unified Neutral Theory of Biodiversity and Biogeography (UNTB) is a special form of mechanistic model that takes an entirely different approach to community composition than the niche apportionment models.^[1] Instead of species populations reaching equilibrium within a community, the UNTB model is dynamic, allowing for continuing changes in relative species abundances through drift.

A community in the UNTB model can be best visualized as a grid with a certain number of spaces, each occupied with individuals of different species. The model is zero-sum as there are a limited number of spaces that can be occupied: an increase in the number of individuals of one species in the grid must result in corresponding decrease in the number of individuals of other species in the grid. The model then uses birth, death, immigration, extinction and speciation to modify community composition over time.

Hubbell's theta

The UNTB model produces a dimensionless "fundamental biodiversity" number, θ, which is derived using the formula:

θ = 2J_mv

where:

J_m is the size of the metacommunity (the outside source of immigrants to the local community)

v is the speciation rate in the model

Relative species abundances in the UNTB model follow a zero-sum multinomial distribution.^[19] The shape of this distribution is a function of the immigration rate, the size of the sampled community (grid), and θ.^[19] When the value of θ is small, the relative species abundance distribution is similar to the geometric series (high dominance). As θ gets larger, the distribution becomes increasingly s-shaped (log-normal) and, as it approaches infinity, the curve becomes flat (the community has infinite diversity and species abundances of one). Finally, when θ = 0 the community described consists of only one species (extreme dominance).^[1]