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Encyclopedia > 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] 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. Ernst Haeckel coined the term oekologie in 1866. ... Stephen P. Hubbell, Ph. ... The neutral theory of molecular evolution (also, simply the neutral theory of evolution) is an influential theory that was introduced with provocative effect by Motoo Kimura in the late 1960s and early 1970s. ... Herbert Spencer coined the phrase survival of the fittest Survival of the fittest is a phrase which is a shorthand for a concept relating to competition for survival or predominance. ... A moth is an insect closely related to the butterfly. ... This article is about British Birds magazine. ... Vascular is an adjective for the word vessel and refers to tube-like structures. ...

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 Robert MacArthur and the island biogeography theories of MacArthur and E. O. Wilson. In ecology, the trophic level is the position that an organism occupies in a food chain - what it eats, and what eats it. ... Figure 1. ... Robert H. MacArthur (April 7, 1930-November 1, 1972) was an American ecologist who made a major impact on many areas of community and population ecology. ... The study of island biogeography is a field within biogeography that attempts to establish and explain the factors that affect the species diversity of a particular community. ... Edward O. Wilson Edward Osborne Wilson was born June 10, 1929 in Birmingham, Alabama (USA). ...

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). Sympatry is one of three theoretical models for the phenomenon of speciation. ... In biology, a species is one of the basic units of biodiversity. ... Ecology is the branch of science that studies the distribution and abundance of living organisms, and the interactions between organisms and their environment. ... A parasite is an organism that spends a significant portion of its life in or on the living tissue of a host organism and which causes harm to the host without immediately killing it. ... This snapping turtle is trying to make a meal of a Canada goose, but the goose is too wary. ...

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. This page discusses how a theory or assertion is falsifiable (disprovable opp: verifiable), rather than the non-philosophical use of falsification, meaning counterfeiting. ... A hypothesis is a suggested explanation of a phenomenon or reasoned proposal suggesting a possible correlation between multiple phenomena. ... Rainforests are among the most biodiverse ecosystems on earth Biodiversity or biological diversity is the diversity of life. ...

Non-neutral theories of biodiversity would include niche construction 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. Niche construction is the process in which an organism alters its own environment in order to increase its chance of survival. ...

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 (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). Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... Prism splitting light Sunlight in the broad sense is the total spectrum of electromagnetic radiation given off by the Sun. ...

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 if the smaller species were monodominant. Binomial name Sequoiadendron giganteum (Lindl. ... Genera Landoltia Lemna Spirodela Wolffia Wolffiella Lemnaceae, or the Duckweed family, is a family of monocot flowering plants containing the duckweeds (also known as water lentils). ...

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 ecosystems such as the Serengeti, where saplings are trampled by elephants and Blue wildebeests; or gardens, where certain species are systematically removed. It has been suggested that this article or section be merged with Serengeti National Park. ... Genera and Species Loxodonta Loxodonta cyclotis Loxodonta africana Elephas Elephas maximus Elephas antiquus â€  Elephas beyeri â€  Elephas celebensis â€  Elephas cypriotes â€  Elephas ekorensis â€  Elephas falconeri â€  Elephas iolensis â€  Elephas planifrons â€  Elephas platycephalus â€  Elephas recki â€  Stegodon â€  Mammuthus â€  Elephantidae (the elephants) is a family of pachyderm, and the only remaining family in the order Proboscidea... Binomial name Connochaetes taurinus (Burchell, 1823) The Blue Wildebeest, is a large ungulate mammal of the genus Connochaetes,. Which grows to 1. ... A garden is a planned space, usually outdoors, set aside for the display, cultivation, and enjoyment of plants and other forms of nature. ...

### 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 n1 for species 1, n2 for species 2, and so on up to nS for species S is given by

$operatorname{Pr}(n_1,n_2,ldots,n_S| theta, J)= frac{J!theta^S} { 1^{phi_1}2^{phi_2}cdots J^{phi_J} phi_1!phi_2!cdotsphi_J! Pi_{k=1}^J(theta+k-1) }$

where θ = 2Jν is the fundamental biodiversity number (ν is the speciation rate), and φi 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 φ1 = φ3 = φ6 = 1, all other φ's being zero. The formula would give In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ...

$operatorname{Pr}(3,6,1| theta,10)= frac{10!theta^3}{ 1^1cdot 3^1cdot 6^1 cdot 1!1!1! cdot theta(theta+1)(theta+1)cdots(theta+9)}$

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. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and graphics. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ...

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 Pr(3;3,6,1) = Pr(3;1,3,6) = Pr(3;3,1,6) and so on. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

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 $Pr(S;r_1,r_2,ldots,r_s,0,ldots,0)$ where ri is the abundance of the ith most abundant species: r1 is the abundance of the most abundant, r2 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: expectation in the context of probability theory and statistics, see expected value. ...

$E(r_i)=sum_{k=1}^C r_i(k)cdot Pr(S;r_1,r_2,ldots,r_s,0,ldots,0)$

where C is the total number of configurations, ri(k) is the abundance of the ith ranked species in the kth configuration, and $Pr(ldots)$ is the dominance-diversity probability. This formula is difficult to manipulate mathematically, but relatively simple to simulate computationally.

The model discussed so far is a model of a regional community, which Hubbell calls the metacommunity. Hubbell also acknowledged that on a local scale, dispersal plays an important role. For example, seeds are more likely to come from nearby parents than from distant parents. Hubbell introduced the parameter m, which denotes the probability of immigration in the local community from the metacommunity. If m = 1, dispersal is unlimited; the local community is just a sample from the metacommunity and the formulas above apply. If m < 1, however, dispersal is limited and the local community is a dispersal-limited sample from the metacommunity for which different formulas apply.

In a paper in Nature in 2003, it is shown that $langle phi_n rangle$, the expected number of species with abundance n, may be calculated by First title page, November 4, 1869 Nature is one of the oldest and most reputable scientific journals, first published on 4 November 1869. ... 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ...

$thetafrac{J!}{n!(J-n)!} frac{Gamma(gamma)}{Gamma(J+gamma)} int_{y=0}^gamma frac{Gamma(n+y)}{Gamma(1+y)} frac{Gamma(J-n+gamma-y)}{Gamma(gamma-y)} exp(-ytheta/gamma),dy$

where θ is the fundamental biodiversity number, J the community size, Γ is the gamma function, and γ = (J − 1) / (1 − m). This formula is however an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne & Alonso 2005 [2]: The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

$frac{theta }{(I)_{J}} {J choose n} int_{0}^{1}(Ix)_{n}(I(1-x))_{J-n}frac{(1-x)^{theta -1}}{x}dx$

where I = (J − 1) * m / (1 − m) is a parameter that measures dispersal limitation (and resembles γ above).

$langle phi_n rangle$ is zero for n > J, as there cannot be more species than individuals.

This formula is important because it allows a quick evaluation of the Unified Theory. It is not suitable for testing the theory. For this purpose, the approptiate likelihood function should be used. For the metacommunity this was given above. For the local community with dispersal limitation it is given by:

$operatorname{Pr}(n_1,n_2,ldots,n_S| theta, m, J)= frac{J!}{prod_{i=1}^{S}n_{i} prod_{j=1}^{J}Phi_{j}!}frac{theta ^{S}}{(I)_{J}} sum_{A=S}^{J}K(overrightarrow{D},A)frac{I^{A}}{(theta) _{A}}$

Here, the $K(overrightarrow{D},A)$ for A = S,...,J are coefficients fully determined by the data, being defined as

$K(overrightarrow{D},A):=sum_{{a_{1},...,a_{S}|sum_{i=1}^{S}a_{i}=A}} prod_{i=1}^{S}frac{overline{s}left( n_{i},a_{i}right) overline{s} left( a_{i},1right) }{overline{s}left( n_{i},1right) }$

This seemingly complicated formula involves Stirling numbers and Pochhammer symbols, but can be easily calculated. See [3] In mathematics, Stirling numbers arise in a variety of combinatorics problems. ... In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...

An example of a species abundance curve can be found here: [4].

## Unified Theory and species-area relationships

The Unified Theory unifies biodiversity, as measured by species-abundance curves, with biogeography, 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

S = cAz

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 S = kJz 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 done 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

$Eleft{S|theta,Jright}= frac{theta}{theta }+ frac{theta}{theta+1}+ frac{theta}{theta+2}+ cdots + frac{theta}{theta+J-1}$

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, θ / (θ + J − 1), is the expected number of new species encountered when adding one new individual to the community. This is an increasing function of θ and a decreasing function of J, as expected.

By making the substitution J = ρA (see section on saturation above), then the expected number of species becomes Σθ / (θ + ρA − 1).

The formula above may be approximated to an integral giving In calculus, the integral of a function is a generalization of area, mass, volume and total. ...

$S(theta)= 1+thetalnleft(1+frac{J-1}{theta}right).$

## 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. In the mathematics of probability, a stochastic process is a random function. ...

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. Zero-sum describes a situation in which a participants gain or loss is exactly balanced by the losses or gains of the other participant(s). ... Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ...

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 Ni. For the species to increase abundance to ni + 1 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 Ni − 1, 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 Ni, 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,

${rm Prob}({rm Pop}_i(t+1)=N_i-1|{rm Pop}_i(t)=N_i)= mucdotfrac{N_i}{J}cdotfrac{J-N_i}{J-1}$
${rm Prob}({rm Pop}_i(t+1)=N_i+1|{rm Pop}_i(t)=N_i)= mucdotfrac{J-N_i}{J}cdotfrac{N_i}{J-1}$
${rm Prob}({rm Pop}_i(t+1)=N_i|{rm Pop}_i(t)=N_i)= 1-2mucdotfrac{N_i}{J}cdotfrac{J-N_i}{J-1}.$

It is always possible to choose the time increment so that μ = 1. 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. Discrete time is non-continuous time. ... A stopped Brownian motion as an example for a martingale In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i. ... expectation in the context of probability theory and statistics, see expected value. ...

### 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.

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.

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. Sir Ronald Aylmer Fisher, FRS (17 February 1890 â€“ 29 July 1962) was a English statistician, evolutionary biologist, and geneticist. ...

### Proponents and critics of the UNTB

Notable proponents of the Unified Theory include Stephen Hubbell (University of Georgia); notable critics include Brian McGill (McGill University) and J. Timothy Wootton (University of Chicago). Stephen P. Hubbell, Ph. ... The University of Georgia, is located approximately 70 miles north-east of Atlanta in Athens, Georgia and is the largest institution of higher learning and research in the State of Georgia. ... McGill University is a publicly funded, non-denominational, co-educational research university located in the city of Montreal, Quebec, Canada. ... The University of Chicago is a private university located principally in the Hyde Park neighborhood of Chicago. ...

## Reference

• Hubbell, S.P. (2001). The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press. ISBN 0-691-02128-7.
• Preston, F. W. (1962). "The Canonical Distribution of Commonness and Rarity: Part I". Ecology 43 (2): 185-215.

Stephen P. Hubbell, Ph. ... The Princeton University Press is a publishing house, a division of Princeton University, that is highly respected in academic publishing. ...

Diademed Sifaka, an endangered primate of Madagascar Biodiversity Action Plan (BAP) is a an internationally recognized programme addressing threatened species or habitats, which is designed to protect and restore biological systems. ... Warren Ewens is a professor of biology at the University of Pennsylvania. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... Metabolic rate may relate to organismal temperature and dimensions Researchers at the Santa Fe Institute, including ecologists James Brown, Brian Enquist, Jamie Gillooly and physicist Geoffrey West, helped to develop the metabolic theory of ecology. ... The neutral theory of molecular evolution (also, simply the neutral theory of evolution) is an influential theory that was introduced with provocative effect by Motoo Kimura in the late 1960s and early 1970s. ...

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 NodeWorks - Encyclopedia: Neutral theory of molecular evolution (823 words) The neutral theory of molecular evolution (also, simply the neutral theory of evolution) is an influential theory that was introduced with provocative effect by Motoo Kimura in the late 1960s and early 1970s. Thus, the neutral theory is the foundation of the molecular clock technique, which evolutionary molecular biologists use to measure how much time has passed since species diverged from a common ancestor. Tomoko Ohta extended the neutral theory to include the concept of "near-neutrality", that is, genes that are affected mostly by drift or mostly by selection depending on the effective size of a breeding population.
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