Unified neutral theory of biodiversity
Encyclopedia
The unified neutral theory of biodiversity and biogeography (here "Unified Theory" or "UNTB") is a hypothesis and the title of a monograph by ecologist
Stephen Hubbell. The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories of ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. This implies that biodiversity arises at random, as each species follows a random walk
. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.
Neutrality means that at a given trophic level
in a food web
, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis. This can be considered a null hypothesis
to niche theory
. Hubbell built on earlier neutral concepts, including MacArthur
& Wilson's theory of island biogeography
and Gould's concepts of symmetry and null models.
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. 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. Asymmetric 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 also makes predictions that have profound implications for the management of biodiversity
, especially the management of 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.
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 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 ecosystems such as the Serengeti
, where saplings are trampled by elephant
s and Blue wildebeest
s; or garden
s, where certain species are systematically removed.
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
for species 1, 
for species 2, and so on up to 
for species S is given by
where
is the fundamental biodiversity number (
is the speciation rate), and 
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
, all other 
's being zero. The formula would give
which could be maximized to yield an estimate for θ (in practice, numerical methods are used). The maximum likelihood
estimate for θ is about 1.1478.
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 probability of observing a pattern of abundances will be the same observing any permutation
of those abundances. Here we would have
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
where 
is the abundance of the ith most abundant species: 
is the abundance of the most abundant, 
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:
where C is the total number of configurations,
is the abundance of the ith ranked species in the kth configuration, and 
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 http://www.nature.com/nature/journal/v424/n6952/abs/nature01883.html, it is shown that
, the expected number of species with abundance n, may be calculated by
where θ is the fundamental biodiversity number, J the community size,
is the gamma function
, and
. This formula is however an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne & Alonso 2005 http://www.blackwell-synergy.com/doi/abs/10.1111/j.1461-0248.2005.00817.x:
where
is a parameter that measures dispersal limitation.
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:
Here, the
for 
are coefficients fully determined by the data, being defined as
This seemingly complicated formula involves Stirling number
s and Pochhammer symbol
s, but can be easily calculated. See http://www.blackwellpublishing.com/products/journals/suppmat/ELE/ELE717/ELE717sm.htm
An example of a species abundance curve can be found here: http://www.sciam.com/article.cfm?articleID=000656C3-8F00-1CDA-B4A8809EC588EEDF.
and a so called metacommunity from which species can (re)immigrate and which acts as a heat bath to the local community. The distribution of species in the metacommunity is given by a dynamic equilibrium of speciation and extinction. Both community dynamics are modelled by appropriate urn processes
, where each individual is represented by a ball with a color corresponding to its species. With a certain rate
randomly chosen individuals reproduce, i.e. add another ball of their own color to the urn. Since one basic assumption is saturation, this reproduction has to happen at the cost of another random individual from the urn which is removed. At a different rate 
single individuals in the metacommunity are replaced by mutants of an entirely new species. Hubbell calls this simplified model for speciation
a point mutation
, using the terminology of the Neutral theory of molecular evolution
. The urn scheme for the metacommunity of
individuals is the following.
At each time step take one of the two possible actions :
Note that the size
of the metacommunity does not change. Note also that this is a point process
in time. The length of the time steps is distributed exponentially. For simplicity one can, however, assume that each time step is as long as the mean time between two changes which can be derived from the reproduction and mutation rates
and 
. The probability 
is given as 
.
The species abundance distribution for this urn process is given by Ewens's sampling formula
which was originally derived in 1972 for the distribution of allele
under neutral mutations. The expected number
of species in the metacommunity having exactly 
individuals is [Vallade, Houchmandzadeh 2003 http://prola.aps.org/abstract/PRE/v68/i6/e061902
where
is called the fundamental biodiversity number. For large
metacommunities and
one recovers the Fisher Log-Series as species distribution.
The urn scheme for the local community of fixed size
is very similar to the one for the metacommunity.
At each time step take one of the two actions :
The metacommunity is changing on a much larger timescale and is assumed to be fixed during the evolution of the local community. The resulting distribution of species in the local community and expected values depend on four parameters,
, 
, 
and 
(or 
) and are derived in [Etienne, Alonso 2005 http://www.blackwell-synergy.com/doi/full/10.1111/j.1461-0248.2005.00817.x, including several simplifying limit cases like the one presented in the previous section (there called 
). The parameter 
is a dispersal parameter. If 
then the local community is just a sample from the metacommunity. For 
the local community is completely isolated from the metacommunity and all species will go extinct except one. This case has been analyzed by Hubbell himself [Hubbell 2001]. The case 
is characterized by a unimodal species distribution in a Preston Diagram and often fitted by a log-normal distribution. This is understood as an intermediate state between domination of the most common species and a sampling from the metacommunity, where singleton species are most abundant. UNTB thus predicts that in dispersal limited communities rare species become even rarer. The log-normal distribution describes the maximum and the abundance of common species very well but underestimates the number of very rare species considerably which becomes only apparent for very large sample sizes [Hubbell 2001].
The most commonly encountered relationship is the power law given by
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
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 may be used to calculate the expected number of species present in a community under the assumptions of the Unified Theory. In symbols
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,
, 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
(see section on saturation above), then the expected number of species becomes 
.
The formula above may be approximated to an integral
giving
This formulation is predicated on a random placement of 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 "e" to "i" 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 (i.e. 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. Preston
called this the veil line and noted that the cutoff point would move as more individuals are sampled.
A neutral model that can analytically predict both the relative species abundance (RSA) at steady-state and the STD at time t has been presented in http://www.nature.com/nature/journal/v444/n7121/abs/nature05320.html. Within this framework the population of any species is represented by a continuous (random) variable x, whose evolution is governed by the following Langevin equation:
where b is the immigration rate from a large regional community,
represents competition for finite resources and D is related to demographic stochasticity; 
is a Gaussian white noise. The model can also be derived as a continuous approximation of a master equation, where birth and death rates are independent of species, and predicts that at steady-state the RSA is simply a gamma distribution.
From the exact time-dependent solution of the previous equation, one can exactly calculate the STD at time t under stationary conditions:
This formula provides good fits of data collected in the Barro Colorado tropical forest from 1990 to 2000. From the best fit one can estimate
~ 3500 years with a broad uncertainty due to the relative short time interval of the sample. This parameter can be interpreted as the relaxation time of the system, i.e. the time the system needs to recover from a perturbation of species distribution. In the same framework, the estimated mean species lifetime is very close to the fitted temporal scale 
. This suggests that the neutral assumption could correspond to a scenario in which species originate and become extinct on the same timescales of fluctuations of the whole ecosystem.
Tests on bird and tree abundance data demonstrate that the theory usually is usually a poorer match to the data than alternative null hypotheses that use fewer parameters (a log-normal model with two tunable parameters, compared to the neutral theory's three), and are thus more parsimonious. The theory also fails to describe coral reef communities and is a poor fit to data in intertidal communities.
Ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
Stephen Hubbell. The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories of ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. This implies that biodiversity arises at random, as each species follows a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.
Neutrality means that at a given trophic level
Trophic level
The trophic level of an organism is the position it occupies in a food chain. The word trophic derives from the Greek τροφή referring to food or feeding. A food chain represents a succession of organisms that eat another organism and are, in turn, eaten themselves. The number of steps an organism...
in a food web
Food web
A food web depicts feeding connections in an ecological community. Ecologists can broadly lump all life forms into one of two categories called trophic levels: 1) the autotrophs, and 2) the heterotrophs...
, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis. This can be considered a null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
to niche theory
Ecological niche
In ecology, a niche is a term describing the relational position of a species or population in its ecosystem to each other; e.g. a dolphin could potentially be in another ecological niche from one that travels in a different pod if the members of these pods utilize significantly different food...
. Hubbell built on earlier neutral concepts, including MacArthur
Robert MacArthur
Robert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
& Wilson's theory of island biogeography
Island biogeography
Island biogeography is a field within biogeography that attempts to establish and explain the factors that affect the species richness of natural communities. The theory was developed to explain species richness of actual islands...
and Gould's concepts of symmetry and null models.
An ecological community is a group of trophically similar, sympatric species
Species
In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...
that actually or potentially compete in a local area for the same or similar resources. 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. Asymmetric 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 also makes predictions that have profound implications for the management of biodiversity
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
, especially the management of 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 stochasticStochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
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
Sunlight
Sunlight, in the broad sense, is the total frequency spectrum of electromagnetic radiation given off by the Sun. On Earth, sunlight is filtered through the Earth's atmosphere, and solar radiation is obvious as daylight when the Sun is above the horizon.When the direct solar radiation is not blocked...
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 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 ecosystems such as the Serengeti
Serengeti
The Serengeti ecosystem is a geographical region in Africa. It is located in north Tanzania and extends to south-western Kenya between latitudes 1 and 3 S and longitudes 34 and 36 E. It spans some ....
, where saplings are trampled by elephant
Elephant
Elephants are large land mammals in two extant genera of the family Elephantidae: Elephas and Loxodonta, with the third genus Mammuthus extinct...
s and Blue wildebeest
Blue Wildebeest
The Blue Wildebeest , also called the Common Wildebeest, is a large antelope and one of two species of wildebeest. It grows to 115–145 cm shoulder height and attains a body mass of 168–274 kg. They range the open plains, bushveld and dry woodlands of Southern and East Africa, living for...
s; or garden
Garden
A garden is a planned space, usually outdoors, set aside for the display, cultivation, and enjoyment of plants and other forms of nature. The garden can incorporate both natural and man-made materials. The most common form today is known as a residential garden, but the term garden has...
s, 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
for species 1, for species 2, and so on up to for species S is given bywhere
is the fundamental biodiversity number ( is the speciation rate), and 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
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
of different values of θ. There are thus S = 3 species and
, all other 's being zero. The formula would givewhich could be maximized to yield an estimate for θ (in practice, numerical methods are used). The maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
estimate for θ is about 1.1478.
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 probability of observing a pattern of abundances will be the same observing any permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
of those abundances. Here we would have
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
where is the abundance of the ith most abundant species: is the abundance of the most abundant, 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
Expectation
In the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...
abundance of the ith most abundant species:
where C is the total number of configurations,
is the abundance of the ith ranked species in the kth configuration, and 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
Metacommunity
An ecological metacommunity can be described in a literal sense as a community of ecological communities. More formally, a metacommunity is defined as a set of local communities that are linked by dispersal of multiple, potentially interacting species...
. 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
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
in 2003 http://www.nature.com/nature/journal/v424/n6952/abs/nature01883.html, it is shown that
, the expected number of species with abundance n, may be calculated bywhere θ is the fundamental biodiversity number, J the community size,
is the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
, and
. This formula is however an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne & Alonso 2005 http://www.blackwell-synergy.com/doi/abs/10.1111/j.1461-0248.2005.00817.x:where
is a parameter that measures dispersal limitation. 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:
Here, the
for are coefficients fully determined by the data, being defined asThis seemingly complicated formula involves Stirling number
Stirling number
In mathematics, Stirling numbers arise in a variety of combinatorics problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second...
s and Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
s, but can be easily calculated. See http://www.blackwellpublishing.com/products/journals/suppmat/ELE/ELE717/ELE717sm.htm
An example of a species abundance curve can be found here: http://www.sciam.com/article.cfm?articleID=000656C3-8F00-1CDA-B4A8809EC588EEDF.
Stochastic modelling of species abundances under the UNTB
UNTB distinguishes between a dispersal-limited local community of size and a so called metacommunity from which species can (re)immigrate and which acts as a heat bath to the local community. The distribution of species in the metacommunity is given by a dynamic equilibrium of speciation and extinction. Both community dynamics are modelled by appropriate urn processesUrn problem
In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest are represented as colored balls in an urn or other container....
, where each individual is represented by a ball with a color corresponding to its species. With a certain rate
randomly chosen individuals reproduce, i.e. add another ball of their own color to the urn. Since one basic assumption is saturation, this reproduction has to happen at the cost of another random individual from the urn which is removed. At a different rate single individuals in the metacommunity are replaced by mutants of an entirely new species. Hubbell calls this simplified model for speciationSpeciation
Speciation is the evolutionary process by which new biological species arise. The biologist Orator F. Cook seems to have been the first to coin the term 'speciation' for the splitting of lineages or 'cladogenesis,' as opposed to 'anagenesis' or 'phyletic evolution' occurring within lineages...
a point mutation
Point mutation
A point mutation, or single base substitution, is a type of mutation that causes the replacement of a single base nucleotide with another nucleotide of the genetic material, DNA or RNA. Often the term point mutation also includes insertions or deletions of a single base pair...
, using the terminology of the Neutral theory of molecular evolution
Neutral theory of molecular evolution
The neutral theory of molecular evolution states that the vast majority of evolutionary changes at the molecular level are caused by random drift of selectively neutral mutants . The theory was introduced by Motoo Kimura in the late 1960s and early 1970s...
. The urn scheme for the metacommunity of
individuals is the following.At each time step take one of the two possible actions :
- With probability
draw an individual at random and replace another random individual from the urn with a copy of the first one. - With probability
draw an individual and replace it with an individual of a new species.
Note that the size
of the metacommunity does not change. Note also that this is a point processPoint process
In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...
in time. The length of the time steps is distributed exponentially. For simplicity one can, however, assume that each time step is as long as the mean time between two changes which can be derived from the reproduction and mutation rates
and . The probability is given as .The species abundance distribution for this urn process is given by Ewens's sampling formula
Ewens's sampling formula
In population genetics, Ewens' sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.-Definition:...
which was originally derived in 1972 for the distribution of allele
Allele
An allele is one of two or more forms of a gene or a genetic locus . "Allel" is an abbreviation of allelomorph. Sometimes, different alleles can result in different observable phenotypic traits, such as different pigmentation...
under neutral mutations. The expected number
of species in the metacommunity having exactly individuals is [Vallade, Houchmandzadeh 2003 http://prola.aps.org/abstract/PRE/v68/i6/e061902where
is called the fundamental biodiversity number. For largemetacommunities and
one recovers the Fisher Log-Series as species distribution.The urn scheme for the local community of fixed size
is very similar to the one for the metacommunity.At each time step take one of the two actions :
- With probability
draw an individual at random and replace another random individual from the urn with a copy of the first one. - With probability
replace a random individual with an immigrant drawn from the metacommunity.
The metacommunity is changing on a much larger timescale and is assumed to be fixed during the evolution of the local community. The resulting distribution of species in the local community and expected values depend on four parameters,
, , and (or ) and are derived in [Etienne, Alonso 2005 http://www.blackwell-synergy.com/doi/full/10.1111/j.1461-0248.2005.00817.x, including several simplifying limit cases like the one presented in the previous section (there called ). The parameter is a dispersal parameter. If then the local community is just a sample from the metacommunity. For the local community is completely isolated from the metacommunity and all species will go extinct except one. This case has been analyzed by Hubbell himself [Hubbell 2001]. The case is characterized by a unimodal species distribution in a Preston Diagram and often fitted by a log-normal distribution. This is understood as an intermediate state between domination of the most common species and a sampling from the metacommunity, where singleton species are most abundant. UNTB thus predicts that in dispersal limited communities rare species become even rarer. The log-normal distribution describes the maximum and the abundance of common species very well but underestimates the number of very rare species considerably which becomes only apparent for very large sample sizes [Hubbell 2001].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
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
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 may be used to calculate the expected number of species present in a community under the assumptions of the Unified Theory. In symbols
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,
, 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
(see section on saturation above), then the expected number of species becomes .The formula above may be approximated to an integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
giving
This formulation is predicated on a random placement of individuals.
Example
Consider the following (synthetic) dataset, of 27 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 "e" to "i" 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
Frank W. Preston
Frank W. Preston was an English-American engineer, ecologist, and conservationist. He helped found the Western Pennsylvania Conservancy and worked to reclaim the land that is now Moraine State Park in Butler County, Pennsylvania, in the United States. Preston was a leading expert in glass technology...
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 (i.e. 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. Preston
Frank W. Preston
Frank W. Preston was an English-American engineer, ecologist, and conservationist. He helped found the Western Pennsylvania Conservancy and worked to reclaim the land that is now Moraine State Park in Butler County, Pennsylvania, in the United States. Preston was a leading expert in glass technology...
called this the veil line and noted that the cutoff point would move as more individuals are sampled.
Dynamics under neutral hypothesis
All biodiversity patterns previously described are related to time-independent quantities. However, for biodiversity evolution and species preservation, it is crucial to compare the dynamics of ecosystems with models (Leigh, 2007). An easily accessible index of the underlying evolution is the so called species turnover distribution (STD), defined as the probability P(r,t) that the population of any species has varied by a fraction r after a given time t.A neutral model that can analytically predict both the relative species abundance (RSA) at steady-state and the STD at time t has been presented in http://www.nature.com/nature/journal/v444/n7121/abs/nature05320.html. Within this framework the population of any species is represented by a continuous (random) variable x, whose evolution is governed by the following Langevin equation:
where b is the immigration rate from a large regional community,
represents competition for finite resources and D is related to demographic stochasticity; is a Gaussian white noise. The model can also be derived as a continuous approximation of a master equation, where birth and death rates are independent of species, and predicts that at steady-state the RSA is simply a gamma distribution.From the exact time-dependent solution of the previous equation, one can exactly calculate the STD at time t under stationary conditions:
This formula provides good fits of data collected in the Barro Colorado tropical forest from 1990 to 2000. From the best fit one can estimate
~ 3500 years with a broad uncertainty due to the relative short time interval of the sample. This parameter can be interpreted as the relaxation time of the system, i.e. the time the system needs to recover from a perturbation of species distribution. In the same framework, the estimated mean species lifetime is very close to the fitted temporal scale . This suggests that the neutral assumption could correspond to a scenario in which species originate and become extinct on the same timescales of fluctuations of the whole ecosystem.Testing the theory
The theory has provoked much controversy as it "abandons" the role of ecology when modelling ecosystems. The theory has been criticized as it requires an equilibrium, yet climatic and geographical conditions are thought to change too frequently for this to be attained.Tests on bird and tree abundance data demonstrate that the theory usually is usually a poorer match to the data than alternative null hypotheses that use fewer parameters (a log-normal model with two tunable parameters, compared to the neutral theory's three), and are thus more parsimonious. The theory also fails to describe coral reef communities and is a poor fit to data in intertidal communities.
Application
Whilst the theory has been heralded as a valuable tool for palaeontologists, little work has so far been done to test the theory against the fossil record.See also
- Biodiversity Action PlanBiodiversity Action PlanA Biodiversity Action Plan is an internationally recognized program addressing threatened species and habitats and is designed to protect and restore biological systems. The original impetus for these plans derives from the 1992 Convention on Biological Diversity...
- Warren EwensWarren EwensWarren Ewens FRS, FAA is an Australian-born professor of biology at the University of Pennsylvania. He concentrates his research on the mathematical, statistical and theoretical aspects of population genetics. Ewens has worked in human population genetics, computational biology, and evolutionary...
- Ewens's sampling formulaEwens's sampling formulaIn population genetics, Ewens' sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.-Definition:...
- Metabolic theory of ecologyMetabolic theory of ecologyThe metabolic theory of ecology is an extension of Kleiber's law and posits that the metabolic rate of organisms is the fundamental biological rate that governs most observed patterns in ecology....
- Neutral theory of molecular evolutionNeutral theory of molecular evolutionThe neutral theory of molecular evolution states that the vast majority of evolutionary changes at the molecular level are caused by random drift of selectively neutral mutants . The theory was introduced by Motoo Kimura in the late 1960s and early 1970s...