# Unified neutral theory of biodiversity

The unified neutral theory of biodiversity and biogeography (here "Unified Theory" or "UNTB") is a theory and the title of a monograph[1] (http://www.pupress.princeton.edu/titles/7105.html) by ecologist Stephen Hubbell. The theory aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories of ecology, Hubbell's theory assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. Despite contradicting the principle of "survival of the fittest", the theory has been applied successfully to many groups of species, including forest tree species, bacterial populations, moths, British birds, and vascular plants.

 Contents

## Overview

Neutrality is defined as per capita ecological equivalence among all individuals of every species at a given trophic level in a food web; "per capita equivalence" means that all species are held to behave (ie reproduce and die) in the same way as one another; and individuals of a particular species reproduce and die (behave) in the same way. Early neutral theories include the broken stick hypothesis of R. H. MacArthur and the island biogeography theories of MacArthur and E. O. Wilson.

An ecological community is a group of trophically similar, sympatric species that actually or potentially compete in a local area for the same or similar resources (Hubbell 2001). Under the Unified Theory, complex ecological interactions are permitted among individuals of an ecological community (such as competition and cooperation), provided that all individuals obey the same rules. Phenomena such as parasitism and predation are ruled out by the terms of reference; but cooperative strategies such as swarming, and negative interaction such as competing for limited food or light are allowed (so long as all individuals behave in the same way).

The Unified Theory makes a large number of falsifiable hypotheses. Differences between predictions of the Unified Theory and observations are of very small magnitude. The Unified Theory also makes predictions that have profound implications for the management of biodiversity, especially the management of rare species.

Non-neutral theories of biodiversity would include niche assembly and dispersal assembly. These theories are non-neutral because they hold that different species behave in different ways from one another. Other examples of non-neutral explanations would be to hold that older organisms are fitter in the Darwinian sense.

Under Hubbell's theory, species drift is allowed to occur via speciation, which would occur with a specific probablity per birth. The neutrality of the Unified Theory implies that this probability would be independent of the parent's species (note that common species have a higher birth rate, and thus the UNTB predicts that speciation occurs more frequently for common species than rare species).

The theory predicts the existence of a fundamental biodiversity constant, conventionally written θ, that appears to govern species richness on a wide variety of spatial and temporal scales.

## The Unified Theory and saturation

Although not strictly necessary for a neutral theory, many stochastic models of biodiversity assume a fixed, finite community size. There are unavoidable physical constraints on the total number of individuals that can be packed into a given space (although space per se isn't necessarily a resource, it is often a useful surrogate variable for a limiting resource that is distributed over the landscape; examples would include sunlight or hosts, in the case of parasites).

If a wide range of species is considered (say, giant sequoia trees and duckweed, two species that have very different saturation densities), then the assumption of constant community size might not be very good, because density would be higher for if the smaller species were monodominant.

However, because the Unified Theory refers only to communities of trophically similar, competing species, it is unlikely that population density will vary too widely from one place to another.

Hubbell considers the fact that population densities are constant and interprets it as a general principle: large landscapes are always biotically saturated with individuals. Hubbell thus treats communities as being of a fixed number of individuals, usually denoted by J.

Exceptions to the saturation principle include disturbed ecosysems such as the Serengeti, where saplings are trampled by elephants; or gardens, where certain species are systematically removed.

### Species abundances

When abundance data on natural populations are collected, two observations are almost universal:

• The most common species accounts for a substantial fraction of the individuals sampled;
• A substantial fraction of the species sampled are very rare. Indeed, a substantial fraction of the species sampled are singletons, that is, species which are sufficiently rare for only a single individual to have been sampled.

Such observations typically generate a large number of questions. Why are the rare species rare? Why is the most abundant species so much more abundant than the median species abundance?

A non neutral explanation for the rarity of rare species might suggest that rarity is a result of poor adaptation to local conditions. The UNTB implies that such considerations may be neglected from the perspective of population biology (because the explanation cited implies that the rare species behaves differently from the abundant species).

Species composition in any community will change randomly with time. However, any particular abundance structure will have an associated probability. The UNTB predicts that the probability of a community of J individuals composed of S distinct species with abundances [itex]n_1[itex] for species 1, [itex]n_2[itex] for species 2, and so on up to [itex]n_S[itex] for species S is given by

[itex]

\operatorname{Pr}(S;n_1,n_2,\ldots,n_S)= \frac{J!\theta^S} {

 1^{\phi_1}2^{\phi_2}\cdots J^{\phi_J}
\phi_1!\phi_2!\cdots\phi_J!
\Pi_{k=1}^J(\theta+k-1)


} [itex]

where [itex]\theta=2J\nu[itex] is the fundamental biodiversity number ([itex]\nu[itex] is the speciation rate), and [itex]\phi_i[itex] is the number of species that have i individuals in the sample.

This equation shows that the UNTB implies a nontrivial dominance-diversity equilibrium between speciation and extinction.

As an example, consider a community with 10 individuals and three species "a", "b", and "c" with abundances 3, 6 and 1 respectively. Then the formula above would allow us to assess the likelihood of different values of θ. There are thus S = 3 species and [itex]\phi_1=\phi_3=\phi_6=1[itex], all other [itex]\phi[itex]'s being zero. The formula would give

[itex]

\operatorname{Pr}(3;3,6,1)= \frac{10!\theta^3}{ 1^1\cdot 3^1\cdot 6^1 \cdot 1!1!1! \cdot \theta(\theta+1)(\theta+1)\cdots(\theta+9)} [itex]

which could be maximized to yield an estimate for θ (in practice, numerical methods are used). The R programming language can be used to show that the maximum likelihood estimate for θ is about 1.1478.

Note that we could have labelled the species another way and counted the abundances being 1,3,6 instead (or 3,1,6, etc etc). Logic tells us that the probablity of observing a pattern of abundances will be the same observing any permutation of those abundances. Here we would have [itex]Pr(3;3,6,1)=Pr(3;1,3,6)=Pr(3;3,1,6)[itex] and so on.

To account for this, it is helpful to consider only ranked abundances (that is, to sort the abundances before inserting into the formula). A ranked dominance-diversity configuration is usually written as [itex]Pr(S;r_1,r_2,\ldots,r_s,0,\ldots,0)[itex] where [itex]r_i[itex] is the abundance of the ith most abundant species: [itex]r_1[itex] is the abundance of the most abundant, [itex]r_2[itex] the abundance of the second most abundant species, and so on. For convenience, the expression is usually "padded" with enough zeros to ensure that there are J species (the zeros indicating that the extra species have zero abundance).

It is now possible to determine the expected abundance of the ith most abundant species:

[itex]

E(r_i)=\sum_{k=1}^C r_i(k)\cdot Pr(S;r_1,r_2,\ldots,r_s,0,\ldots,0) [itex] where C is the total number of configurations, [itex]r_i(k)[itex] is the abundance of the ith ranked species in the kth configuration, and [itex]Pr(\ldots)[itex] is the dominance-diversity probability. This formula is difficult to manipulate mathematically, but relatively simple to simulate computationally.

In a paper in Nature in 2003, it is shown that [itex] \langle \phi_n \rangle [itex], the expected abundance of the n-th most abundant species, may be calculated by

[itex]

\theta\frac{J!}{n!(J-n)!} \frac{\Gamma(\gamma)}{\Gamma(J+\gamma)} \int_{y=0}^\gamma \frac{\Gamma(n+y)}{\Gamma(1+y)} \frac{\Gamma(J-n+\gamma-y)}{\Gamma(\gamma-y)} \exp(-y\theta/\gamma)\,dy [itex]

where θ is the fundamental biodiversity number, J the community size, [itex]\Gamma[itex] is the gamma function, and [itex]\gamma=(J-1)/(1-m)[itex] (m is the immigration rate). This integral may be evaluated numerically, but no analytical solution is known.

This formula is important because it allows the Unified Theory to be tested using, for example, the chi square test.

Note that [itex]\langle \phi_n \rangle[itex] is zero for n > J, as there cannot be more species than individuals. An example of a species abundance curve can be found here: [2] (http://www.sciam.com/article.cfm?articleID=000656C3-8F00-1CDA-B4A8809EC588EEDF).

## Unified Theory and species-area relationships

The Unified Theory unifies biodiversity, as measured by species-abundance curves, with biogreography, as measured by species-area curves. Species-area relationships show the rate at which species diversity increases with area. The topic is of great interest to conservation biologists in the design of reserves, as it is often desired to harbour as many species as possible.

The most commonly encountered relationship is the power law given by

[itex]

S=cA^z[itex]

where S is the number of species found, A is the area sampled, and c and z are constants. This relationship, with different constants, has been found to fit a wide range of empirical data.

From the perspective of Unified Theory, it is convenient to consider S as a function of total community size J. Then [itex]S=kJ^z[itex] for some constant k, and if this relationship were exactly true, the species area line would be straight on log scales. It is typically found that the curve is not straight, but the slope changes from being steep at small areas, shallower at intermediate areas, and steep at the largest areas.

The formula for species composition derived above (not dont this bit yet) may be used to calculate the expected number of species present in a community under the assumptions of the Unified Theory. In symbols

[itex]

E\left\{S|\theta,J\right\}= \frac{\theta}{\theta }+ \frac{\theta}{\theta+1}+ \frac{\theta}{\theta+2}+ \cdots + \frac{\theta}{\theta+J-1} [itex]

where θ is the fundamental biodiversity number. This formula specifies the expected number of species sampled in a community of size J. The last term, [itex]\theta/(\theta+J-1)[itex], is the expected number of new species encountered when adding one new individual to the community. Note that this is an increasing function of θ and a decreasing function of J, as expected.

By making the substitution [itex]J=\rho A[itex] (see section on saturation above), then the expected number of species becomes [itex]\Sigma\theta/(\theta+\rho A-1)[itex].

The formula above may be approximated to an integral giving

[itex]

S(\theta)= 1+\theta\ln\left(1+\frac{J-1}{\theta}\right). [itex]

## Stochastic modelling of species abundances under the UNTB

The Unified Theory is perhaps best understood using stochastic process modelling. Consider a community, of fixed size, consisting of J individuals.

Although in reality individuals die and reproduce, it is often realistic to assume that the community changes at regular intervals, the timestep being J times an individual's lifespan. At each timestep, one individual dies and one is born (community size remaining constant at J); the dynamical process simulated is known as "zero-sum", by analogy with zero sum game theory.

Each individual occupies one space or unit of limiting resources. The individual dies with probability μ per timestep and is replaced by a new individual. Under the UNTB, the replacing species is drawn randomly from the community. It is possible to use this fact to calculate the probabilities of species' abundance changing with time:

Consider species i, which at time t has abundance [itex]N_i[itex]. For the species to increase abundance to [itex]n_i+1[itex] at time t+1, two separate events must happen: firstly, the individual that dies must be of species i; and secondly, the individual that is born must be of some other species.

For the species to decrease abundance to [itex]N_i-1[itex], then again two separate events must happen: the individual that dies must not be species i, and the individual that is born must be of species i.

For the species to remain at abundance [itex]N_i[itex], one of two things might happen:

• Either the individual that dies is of species i and the individual that is born is some other species; or
• The individual that dies is not of species i and the individual that is born is species i.

In symbols,

[itex]

{\rm Prob}({\rm Pop}_i(t+1)=N_i-1|{\rm Pop}_i(t)=N_i)= \mu\cdot\frac{N_i}{J}\cdot\frac{J-N_i}{J-1} [itex]

[itex]

{\rm Prob}({\rm Pop}_i(t+1)=N_i+1|{\rm Pop}_i(t)=N_i)= \mu\cdot\frac{J-N_i}{J}\cdot\frac{N_i}{J-1} [itex]

[itex]

{\rm Prob}({\rm Pop}_i(t+1)=N_i|{\rm Pop}_i(t)=N_i)= 1-2\mu\cdot\frac{N_i}{J}\cdot\frac{J-N_i}{J-1}. [itex]

It is always possible to choose the time increment so that [itex]\mu=1[itex]. Note that the probability of species i increasing is equal to the probability of it decreasing. The abundance of species i, if viewed as a discrete time sequence of random variables, is thus a martingale because the expectation of species i 's abundance at time t+1 is equal to its abundance at time t.

### Example

Consider the following (synthetic) dataset, of 23 individuals:

a,a,a,a,a,a,a,a,a,a,b,b,b,b,c,c,c,c,d,d,d,d,e,f,g,h,i

There are thus 27 individuals of 9 species ("a" to "i") in the sample. Tabulating this would give:

    a  b  c  d  e  f  g  h  i
10  4  4  4  1  1  1  1  1


indicating that species "a" is the most abundant with 10 individuals and species "d" to "h" are singletons. Tabulating the table gives:

   species abundance    1    2    3    4    5    6    7    8    9    10
number of species    5    0    0    3    0    0    0    0    0     1


On the second row, the 5 in the first column means that five species, species "e" through "i", have abundance one. The following two zeros in columns 2 and 3 mean that zero species have abundance 2 or 3. The 3 in column 4 means that three species, species "b", "c", and "d", have abundance four. The final 1 in column 10 means that one species, species "a", has abundance 10.

This type of dataset is typical in biodiversity studies. Observe how more than half the biodiversity (as measured by species count) is due to singletons.

For real datasets, the species abundances are binned into logarithmic categories, usually using base 2, which gives bins of abundance 0-1, abundance 1-2, abundance 2-4, abundance 4-8, etc. Such abundance classes are called octaves; early developers of this concept included F. W. Preston and histograms showing number of species as a function of abundance octave are known as Preston diagrams.

Note that these bins are not mutually exclusive: a species with abundance 4, for example, could be considered as lying in the 2-4 abundance class or the 4-8 abundance class. Species with an abundance of an exact power of 2 (ie 2,4,8,16, etc) are conventionally considered as having 50% membership in the lower abundance class 50% membership in the upper class. Such species are thus considered to be evenly split between the two adjacent classes (apart from singletons which are classified into the rarest category). Thus in the example above, the Preston abundances would be

abundance class 1    1-2   2-4   4-8  8-16
species         5     0    1.5   1.5   1


The three species of abundance four thus appear, 1.5 in abundance class 2-4, and 1.5 in 4-8.

Note that the above method of analysis cannot account for species that are unsampled: that is, species sufficiently rare to have been recorded zero times. Preston diagrams are thus truncated at zero abundance Fisher called this the veil line and noted that the cutoff point would move as more individuals are sampled.

### Proponents and critics of the UNTB

Notable proponents of the Unified Theory include Stephen Hubbell (University of Georgia); notable critics include Brian McGill (Michigan State University) and J. Timothy Wootton (University of Chicago).

## Reference

The Unified Neutral Theory of Biodiversity and Biogeography (MPB-32) Stephen P. Hubbell Paper | 2001 | \$45.00 / £29.95 | ISBN: 0-691-02128-7 448 pp. | 5 x 8 | 123 line illus.

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