Ryan A. Chisholm and Mark A. Burgman. 2004. The unified neutral theory of biodiversity and biogeography: comment. Ecology 85:3172–3174.

Appendix A. Metacommunity dynamics and neutral ecology.

Neutral ecology was studied by both Hubbell (2001) and Volkov et al. (2003) in the context of metacommunity models undergoing zero-sum dynamics. However, whereas Hubbell’s metacommunity was a non-overlapping generations model, Volkov et al. used an overlapping generations model, and while Hubbell’s relative species abundance (RSA) distribution was Fisher’s logarithmic series (log-series) (Fisher et al. 1943) only in the limit as the community size J_M , the RSA distribution of Volkov et al. is Fisher’s log-series for all values of J_M.  Fisher’s log-series is defined as

,

(A.1)

where (originally referred to by Fisher as ) is a measure of diversity, 0 < x < 1, and  is the frequency of species containing n individuals in the metacommunity.  Using the basic properties of series one can derive the expressions

	(A.2)
and
,	(A.3)

which are very close approximations to the number of individuals and the number of species in the metacommunity.  The error in (A.2) is  while the error in (A.3) is readily shown to be less than , both of which are very small quantities.

In Hubbell’s Unified Neutral Theory of Biodiversity and Biogeography is defined as the “fundamental biodiversity number” (Hubbell 2001)

.

(A.4)

 By substituting Eq. A.4 into Eq. A.2, it is then apparent that

.

(A.5)

The number of species in the metacommunity is then given by substituting Eqs. A.4 and A.5 into Eq. A.3:

.

(A.6)

If we then set  = 3.8147 × 10^-4 and J_M = 65536, we obtain  = 50 and S_M = 358.95.  That is, there are approximately 359 species in Hubbell’s metacommunity at equilibrium.  This result is confirmed by simulations.

The metacommunity model of Volkov et al. (2003) is a time evolution problem with overlapping generations where p_n,k(t) denotes the probability that the kth species contains n individuals at time t. The steady-state solution to the problem is then

(A.7)

for n > 0, where b_n,k and d_n,k denote the probability of birth and death, respectively, in the kth species with n individuals, and b_-1,k = d_0,k = 0 (Volkov et al. 2003). In neutral ecology, where all species are competitively equal, Eq. A.7 reduces to

(A.8)

where b and d are the probabilities of a random individual in the metacommunity giving birth and dying respectively, is a non-zero probability of speciation, and P_n P_n,k. P₀ can then be determined by normalization condition (Volkov et al. 2003). A concise analytical expression for P₀ can be obtained as follows:

.

(A.9)

Of course, for a finite-size metacommunity the probability P_n that a species has n individuals at any given time, is only non-zero for n  J_M. As such, for a real metacommunity, Expression A.9 represents only a close approximation.

In the log-series distribution for metacommunity RSA that Volkov et al. derived, the parameters and x were shown to be

	(A.10)
	(A.11)

where S'_M is the number of species in the metacommunity (Volkov et al. 2003) including those with zero individuals.  The relationship between S_M and S'_M is simply

.

(A.12)

Under zero-sum dynamics in neutral ecology the probability d of a random individual dying is simply 1/J_M. Following a death, a new individual will be generated by a speciation event with probability v, or by a birth with probability 1 – . The unconditional probability b of a random individual in the metacommunity giving birth is then (1 – )/J_M. Substituting these expressions for b and d into Eqs. A.9, A.10 and A.11 yields

	(A.13)
	(A.14)
	(A.15)

By substituting Eq. A.14 into Eq. A.2 and rearranging, we obtain

.

(A.16)

We can now find the expected values of and S_M under the overlapping generations model of Volkov et al. for the same values  = 3.8147 – 10^-4 and J_M = 65536 as used in Hubbell’s model earlier. From Eq. A.14 we obtain x = 0.99962, and from Eq. A.16 we obtain  = 25.0095. From Eq. A.13 we obtain P₀ = 0.005054, and by substituting this into Eq. A.11 and rearranging, we then obtain S'_M = 197.86. Equation A.12 then gives S_M = 196.86. Alternatively S_M can be determined from Eq. A.3, which again gives S_M = 196.86. Thus there will be approximately 197 species in the metacommunity of Volkov et al. at equilibrium. This result is also confirmed by simulations.

From Eqs. A.4 and A.16 it can be seen that for similar values of and J_M, the fundamental biodiversity number of Volkov et al. will always be approximately half that of Hubbell’s providing that is small. The total number of species predicted by the two models also differs significantly. For most ecological applications an overlapping generations model is more realistic, and as such the model of Volkov et al. may be preferable. In certain circumstances, such as a community of annual plants, Hubbell’s non-overlapping generations model may be more appropriate.

LITERATURE CITED

Fisher, R. A., A. S. Corbet, and C. B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12:42–58.

Hubbell, S. P. 2001. The unified neutral theory of biodiversity and biogeography. Princeton University Press, Princeton, New Jersey, USA.

Volkov, I., J. R. Banavar, S. P. Hubbell, and A. Maritan. 2003. Neutral theory and relative species abundance in ecology. Nature 424:1035–1037.