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 n_1 for species 1, n_2 for species 2, and so on up to n_S for species
S is given by : \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!\cdots\phi_J! \Pi_{k=1}^J(\theta+k-1) } where \theta=2J\nu is the fundamental biodiversity number (\nu is the speciation rate), and \phi_i 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 \phi_1=\phi_3=\phi_6=1, all other \phi's being zero. The formula would give : \Pr(3,6,1| \theta,10)= \frac{10!\theta^3}{ 1^1\cdot 3^1\cdot 6^1 \cdot 1!1!1! \cdot \theta(\theta+1)(\theta+2)\cdots(\theta+9)} 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 : \Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6) 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 \Pr(S;r_1,r_2,\ldots,r_s,0,\ldots,0) where r_i is the abundance of the
ith most abundant species: r_1 is the abundance of the most abundant, r_2 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: : 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, r_i(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 random sample from the metacommunity and the formulas above apply. If m \langle \phi_n \rangle , the expected number of species with abundance n, may be calculated by : \theta\frac{J!}{n!(J-n)!} \frac{\Gamma(\gamma)}{\Gamma(J+\gamma)} \int_{y=0}^\gamma \frac{\Gamma(n+y)}{\Gamma(1+y)} \frac{\Gamma(J-n+\gamma-y)}{\Gamma(\gamma-y)} \exp(-y\theta/\gamma)\,dy where
θ is the fundamental biodiversity number,
J the community size, \Gamma is the
gamma function, and \gamma=(J-1)m/(1-m). This formula is an approximation. The correct formula is derived in a series of papers, reviewed and synthesized by Etienne and Alonso in 2005: : \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. \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 appropriate likelihood function should be used. For the metacommunity this was given above. For the local community with dispersal limitation it is given by: : \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},1\right) }{\overline{s}\left( n_{i},1\right) } 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==