The unified neutral theory of biodiversity and biogeography (here "Unified Theory" or "UNTB") is a theory and the title of a monograph by ecologist Stephen P. Hubbell. It aims to explain the diversity and relative abundance of species in ecological communities. Like other neutral theories of ecology, Hubbell assumes that the differences between members of an ecological community of trophically similar species are "neutral", or irrelevant to their success. This implies that niche differences do not influence abundance and the abundance of each species follows a random walk. The theory 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 models, including Robert MacArthur and E.O. Wilson's theory of island biogeography and Stephen Jay 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 interactions such as competing for limited food or light are allowed (so long as all individuals behave alike).
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.
Saturation
Although not strictly necessary for a neutral theory, many stochastic models of biodiversity assume a fixed, finite community size (total number of individual organisms). 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 are 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. 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 community sizes 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.
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 suggests that it is not necessary to invoke adaptation or niche differences because neutral dynamics alone can generate such patterns.
Species composition in any community will change randomly with time. 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 random sample from the metacommunity and the formulas above apply. If m < 1, dispersal is limited and the local community is a dispersal-limited sample from the metacommunity for which different formulas apply.
It has been 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 an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne and Alonso in 2005:
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 appropriate 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 numbers and Pochhammer symbols, but can be very easily calculated.
An example of a species abundance curve can be found in Scientific American.
Stochastic modelling of species abundances
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 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 :
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.
The size of the metacommunity does not change. This is a point process in time. The length of the time steps is distributed exponentially. For simplicity one can 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 alleles under neutral mutations. The expected number of species in the metacommunity having exactly individuals is:
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 :
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.