Unified neutral theory of biodiversity

The unified neutral theory of biodiversity (here UNTB) is a\nscientific hypothesis that aims to explain the relative abundance of\nspecies in ecological communities. The theory is named in analogy to\nthe neutral theory of molecular evolution, to which it is closely\nrelated. The theory has been applied successfully to many diverse\necosystems including forest tree species, bacterial populations,\nmoths, British birds, and vascular plants. The UNTB makes a large number of falsifiable hypotheses.\nDifferences between predictions of the UNTB and observations are of\nvery small magnitude. The UNTB also makes predictions that have\nprofound implications for the management of biodiversity, especially\nthe management of rare species. Neutrality is defined as per capita ecological equivalence among all\nindividuals of every species in a given trophically defined community.\nIn other words, all species are held to behave (ie reproduce and die)\nin the same way as one another; and individuals of a particular\nspecies reproduce and die (behave) in the same way. Early neutral\ntheories include the broken stick hypothesis of\nR. H. MacArthur and the island biogeography theories of\nMacArthur and E. O. Wilson. In the UNTB, an ecological community is defined as a group of\ntrophically similar, sympatric species that actually or potentially\ncompete in a local area for the same or similar resources (Hubbell\n2001). Under the UNTB, complex ecological interactions are permitted\namong individuals of an ecological community (such as competition and\ncooperation), provided that all individuals obey the same rules.\nPhenomena such as parasitism and predation\nare ruled out by the terms of reference; but cooperative strategies\nsuch as swarming, and negative interaction such as competing for\nlimited food or light are allowed (so long as all individuals behave\nin the same way). Non-neutral theories of biodiversity would include niche assembly\nand dispersal assembly. These theories are non-neutral because\nthey hold that different species behave in different ways from one\nanother. Other examples of non-neutral explanations would be to hold\nthat older organisms are fitter in the Darwinian sense. Under the UNTB, species drift is allowed to occur via speciation, which would occur with a specific probablity per birth. The neutrality of the UNTB 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.

Table of contents

1 UNTB and saturation 1.1 Species abundances 2 UNTB and species-area relationships 3 Stochastic modelling of species abundances under the UNTB 3.1.1 Example

UNTB and saturation

Although not strictly necessary for a neutral theory, many stochastic models of \nbiodiversity assume a fixed, finite community size. There are unavoidable physical\nconstraints on the total number of individuals that can be packed into a given space\n(although space per se isn't necessarily a resource, it is often a useful surrogate\nvariable for a limiting resource that is distributed over the landscape; examples would \ninclude 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 UNTB refers only to communities of trophically similar, competing species, it is unlikely that population density will vary too widely \nfrom 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 \nthe 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\nobservations are almost universal:

• The most common species accounts for a substantial fraction of the individuals sampled;\n* 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\nare the rare species rare? Why is the most abundant species so much\nmore abundant than the median species abundance? A non neutral explanation for the rarity of rare species might suggest\nthat rarity is a result of poor adaptation to local conditions. The\nUNTB implies that such considerations may be neglected from the\nperspective of population biology (because the explanation cited\nimplies that the rare species behaves differently from the abundant\nspecies). 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

\nwhere is the fundamental biodiversity number \n( is the speciation rate), and is the \nnumber of species that have i individuals in the sample.

This equation shows that the UNTB implies a nontrivial dominance-diversity \nequilibrium 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\n, all other\n's being zero. The formula would give\n: which could be maximized to yield an estimate for θ (in practice,\nnumerical 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 and so on. To account for this, is is helpful to consider only ranked abundances (that is,\nto sort the abundances before inserting into the formula). A ranked dominance-diversity configuration is usually written as \nwhere is the abundance of the ith most abundant species: \nis the abundance of the most abundant, the abundance of the\nsecond 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:

\nwhere 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.

In a paper in Nature in 2003, it is shown that , the expected abundance of the n-th most abundant species, may be calculated by

where θ is the fundamental biodiversity number, J the community size, is the gamma function, and (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 UNTB to be tested using, for example, the chi square test. Note that is zero for n>J, as there cannot be more species than individuals.

UNTB and species-area relationships

The species-area relationship is the rate at which species accumulate as the area surveyed \nincreases. The topic is of great interest to conservation biologists in the design of reserves, \nas it is often desired to harbour as many species as possible. The most commonly encountered relationship is the power law given by\n:\nwhere S is the number of species found, A is the area sampled, and c and z are \nconstants. This relationship, with different constants, has been found to fit a wide range of \nempirical data. From the perspective of UNTB, it is convenient to consider S as a function of total community \nsize J. Then for some constant k, and if this relationship were \nexactly true, the species area line would be straight on log scales. It is typically found \nthat the curve is not straight, but the slope changes from being steep at small areas, shallower \nat intermediate areas, and steep at the largest areas. The formula for species composition derived above (not dont this bit yet) may be used to \ncalculate the expected number of species present in a community under the assumptions of the \nUNTB. In symbols\n:\nwhere &theta is the fundamental biodiversity number. This formula specifies the expected \nnumber of species sampled in a community of size J. The last term, \n, is the expected number of new species \nencountered when adding one new individual to the community. Note that this is an \nincreasing function of &theta and a decreasing function of J, as expected. By making the substitution (see section on saturation above), then \nthe expected number of species becomes . The formula above may be approximated to an integral giving\n:

Stochastic modelling of species abundances under the UNTB

The UNTB 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 . For the species to increase abundance to 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 , 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 , 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\n* The individual that dies is not of species i and the individual that is born is species i.

In symbols,

It is always possible to choose the time increment so that . 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.\nTabulating this would give:

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

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

   species abundance    1    2    3    4    5    6    7    8    9    10\n    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\n(viz "e" to "i") have abundance one. The following two zeros in\ncolumns 2 and 3 mean that zero species have abundance 2 or 3. The 3\nin column 4 means that three species have abundance four (viz "b", "c", and\n"d"). The final 1 in column 10 means that one species (viz "a") has\nabundance 10. This type of dataset is typical in biodiversity studies. Observe how\nmore than half the biodiversity (as measured by species count) is due\nto singletons. For real datasets, the species abundances are binned into logarithmic\ncategories, 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\n species         5     0    1.5   1.5   1

\nThe 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. Notable proponents of the UNTB include Stephen Hubbell.