## Recent Developments in Neutral Theory

Several advances have been made in neutral theory since publication of the UNT. Vallade and Houchmandzadeh, and Alonzo, McKane, and Sole were able to derive analytically the distribution of relative species abundance in a local community undergoing immigration and zero-sum dynamics, which was previously only available by simulation. One especially important advance has been made by Etienne, Alonso, and McKane, who separately and collectively produced a full sampling theory for the zero-sum, dispersal-limited version of the UNT, which was undeveloped in the original presentation of the theory. Next, Volkov, Banavar, Hubbell, and Maritan developed a more general and mathematically tractable formulation of the UNT. An important finding of their work was that Hubbell's zero-sum version of the UNT and Bell's noninteracting species version are asymptotically identical in the infinite metacommunity size limit, and that these versions quickly converge to the same dynamics as community size increases.

Volkov and colleagues also demonstrated a deep connection between neutral theory and Fisher's logseries. First, they reproved Watterson's result that Fisher's logseries was the expected distribution of relative species abundance in the metacommunity. More importantly, however, the new derivation by Volkov and colleagues led directly to an understanding of the population biological meaning of the two parameters of the logseries, Fisher's a and x. Fisher's a turns out to be identical to the fundamental biodiversity number 0, discovered by Hubbell. This result explains why Fisher's a is so stable in the face of increasing sample size. It means that Fisher's a is proportional to the product of two very stable, large-scale biogeographic numbers: metacommunity size JM and the average per capita speciation rate v over the entire metacommunity. Volkov and colleagues also proved that parameter x of the logseries is equal to the ratio of the metacommunity average per capita birth rate b to the average per capita death rate d. Values of x fitted to data are always very close to, but slightly less than, unity. The UNT says this must be so because at very large biogeographic scales, the total birth and death rates must be in near-perfect mass balance, that is, the metacommunity b/d ratio will be only infinitesimally less than unity. At the metacommunity diversity equilibrium, the mass imbalance caused by the very slight deficit in birth rates relative to death rates is made up by the very slow addition of new species. Volkov and colleagues also obtained another remarkable result. They proved that Fisher's log series necessarily implies that per capita birth and death rates are density independent at the spatial scale of the entire metacommunity. This is intuitively reasonable when it is realized that changes in population density of a species in one community can have little dynamic coupling with or constraint on changes in population density of the same species in a community at some arbitrarily large distance away.

That density independence rules in the metacommunity raises an apparent paradox, however, because much of theoretical population biology is built on the empirically well established and widespread occurrence of density-dependent population growth. To resolve this paradox, Volkov, Banavar, He, Hubbell, and Maritan introduced density dependence into the UNT. Density dependence would appear to violate the neutrality assumption because per capita vital rates are no longer the same. However, Volkov and colleagues introduced density dependence symmetrically, so that all species exhibit the same vital rates when they are at the same density. This generalization of the UNT is still neutral in the symmetric sense in that species still are not distinguishable, but their stochastic per capita vital rates are a function of their current abundance. This density dependence introduces a symmetric frequency-dependent advantage for rare species in the community. Whenever a species happens to be rare, its b/d ratio will be greater than unity (i.e., populations grow), whereas when it is common, its b/d is slightly less than unity (i.e., populations slowly decline).

With the incorporation of symmetric density dependence into the UNT, there are now two independent, symmetric mechanisms for explaining patterns of relative species abundance in local communities or on islands. One is the mechanism of dispersal limitation that is in the theory of island biogeography and the original version of the UNT. The second mechanism is density dependence, which generates frequency dependence (rare species advantage) at the community level. Because these mechanisms are independent, they are not mutually exclusive, and so both mechanisms can simultaneously operate on the assembly of local communities. The relative importance of these mechanisms can be distinguished empirically. If a trend is observed in the b/d ratio with species abundance in a community, such that rare species have b/d ratios greater than unity, whereas common species have ratios at or slightly below unity, then the density-dependence mechanism is indicated. However, if there is no trend in b/d ratios with species abundance, then dispersal limitation is likely to be more important. These two mechanisms lead to very different explanations of the extra rarefaction of rare species in local communities. Under the dispersal limitation mechanism, rare species are rarer in local communities than expected from their metacommunity abundance because they are more prone to local extinction, and when they go locally extinct, they take longer to re-immigrate. Conversely, under the density-dependence mechanism, the rarefaction or rare species is because they grow more abundant due to their rare species advantage, thereby depleting the frequency of rare species in local communities.

Zillio, Banavar, Hubbell, and Maritan have made advances in the UNT in regard to its predictions about species-area relationships and patterns of beta diversity (turnover ofspecies across landscapes). They demonstrate that if the axes of species-area curves are appropriately scaled by the speciation rate, then they obtain a scaling collapse. Under the UNT, the species-area curves differ depending on the speciation rate. By a scaling collapse, they mean that all of the species-area curves superimpose into a single curve, controlling for the speciation rate. They also show that observed patterns in beta diversity in several tropical tree communities can be explained by incorporating density dependence on local spatial scales, but not at large spatial scales. This finding is consistent with the generalized density-dependent UNT theory developed by Volkov and colleagues.

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