Unified Neutral Theory of Biodiversity - Stochastic Modelling of Species Abundances Under The UNTB

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 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 :

1. With probability draw an individual at random and replace another random individual from the urn with a copy of the first one.
2. 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 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

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 :

1. With probability draw an individual at random and replace another random individual from the urn with a copy of the first one.
2. 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, 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 . 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 .