Under the UNTB, the dynamics of an arbitrary species are governed by a generalized birth and death process (including speciation, emigration and immigration) (Volkov et al. 2003). The UNTB assumes that individuals die and reproduce continuously, but this is unrealistic for many species (such as most of the moths considered here) which have an annual life cycle. This suggests using a model with discrete generations. We build a discrete-time neutral model of (local) community dynamics which is identical to Hubbell's UNTB in all other aspects except in one detail: we relax the assumption of a constant community size.
The UNTB makes the underlying assumption that all individuals are equivalent, and hence each has the same probability of producing offspring. For a discrete-time model, the process of reproduction can be viewed as one of sampling individuals at random (with replacement) from the current population to provide offspring for the next generation. If we condition on the new population size, the number of individuals of a species is binomially distributed with probability of success equal to the proportion of individuals of the focal species in the current population (the whole community is then drawn from a multinomial distribution). Crucially for us here, the variance in the number of offspring is related to the total population size, so that the more variation we see in the numbers of a species in a neutral community over time, the smaller the community size. This is because, when population size increases, the variance of the binomial distribution increases but the coefficient of variation decreases. Hence, the variation in the proportion of the focal species in the community decreases. This is exactly analogous to genetic drift, where the variation in allele frequencies can be used to estimate the effective population size (e.g. Waples 1989). In the same way, the variation in species frequencies can be used to estimate the ‘effective’ community size. We develop this argument more formally below.
We wish to emphasize that because we are considering the dynamics over short time-scales (evolutionarily speaking), we can reasonably assume that speciation (which is a rare event) does not occur, and restrict our attention to the ecological dynamics of the species.
The process model
Throughout, Ni,t denotes the number of individuals of species i in the community at time t. So, Jt = ΣiNi,t and Ci,t = Ni,t × (Jt)−1 denote, respectively, the community size and the relative abundance of species i in the community at time t. We designate by Pi the relative abundance of species i in the metacommunity, which is considered constant over an ecological time-scale where the local community dynamics are examined, and mt denotes the immigration rate (i.e. the proportion of immigrants in the local community) at time t. For consistency with the data, we associate the value t = 1 with the first year of sampling.
As a consequence of the neutrality assumption, Ni,t is entirely determined by Ci,t–1 and Pi through the reproduction and immigration processes, respectively, allowing for the drift in species abundances. More specifically, we assume that the expected number of individuals of species i in the community at time t is
- (eqn 3)
This means, on average, a proportion mt of recruits to the community at time t are immigrants, and a fraction Pi of these are species i. The expression for λi,t can alternatively be written as
- (eqn 4)
where JPi,t = Jt–1 × Pi is real and non-negative.
The neutrality assumption entails no selective difference between species. So, the fraction (1 – mt) of individuals of species i derived locally will have the same expected relative frequency as in the previous generation (i.e. Ci,t–1). Eqn 3 is in essence identical to Hubbell's (UNTB) model for local community dynamics, except that the zero-sum assumption is relaxed here. Random drift is introduced into the model by thinking of λi,t as a birth rate, and then the actual numbers will follow a Poisson distribution. That is
- (eqn 5)
From the properties of the Poisson distribution, we know that E(Ni,t) = λi,t and Var(Ni,t) = λi,t. Further, if we condition on the total community size, Jt, then the vector of species abundances, will have a multinomial distribution with the expected proportion for species i at time t being E(Ci,t) = λi,t × (Σjλj,t)−1 (e.g. Agresti 1990, p. 38; Gelman et al. 2003, p. 431). The variance of the proportion of species i in the community is therefore (Ci,t × (1 − Ci,t))/Jt, so the fluctuations in Ci,t decrease with increasing community size Jt. This formalises the verbal argument given above.
The initial abundances of all species are also unobserved, and as such, need to be estimated from the data (e.g. Buckland et al. 2004; Clark & Bjørnstad 2004). In the Bayesian paradigm, this involves specifying priors on them. We assume that
- (eqn 6)
Immigration tends to stabilize communities around the metacommunity relative abundance. If we assume equilibrium, then we would expect to have Ci,t = Pi. It follows from eqn 3 that the expected abundance of species i in the community at equilibrium is Jt–1 × Pi as suggested by Hubbell (2001, p. 90) where Jt = J, ∀t.
It should be stressed that we are concerned here exclusively with the local community dynamics. We do not therefore make any assumption about the form of the distribution of the metacommunity. Instead, we allow the model to estimate the distribution. As with the total community size, restricting the model is not necessary for making our main point, so we choose to allow the model to be more flexible. This also means that our main results will not be due to these secondary assumptions.
The sampling model
The observed abundance of species i at time t (t ≥ 2), yi,t, (i.e. the number of individuals of species i trapped at time t) can be modelled assuming
- (eqn 7)
where qt > 0 is a parameter, henceforth referred to as ‘sampling rates’ at time t, and whose interpretation is as follows: at any time, q estimates the ratio of the observed community size to the (effective) size of a neutral community which corresponds to the observed level of variation. In case a community is completely observed, the expected sampling rate under neutrality is 1. If however the dynamics are not neutral, the excess of variation over the neutral expectations (i.e. a greater temporal fluctuation in numbers of each species than predicted from multinomial sampling) will tend to deflate the (effective) community size which would make the sampling rates to exceed 1, in virtue of the eminent inverse relationship between level of variation and effective population size. It is then clear that under neutrality, the sampling rates correspond to the observation probabilities (probability of capture in this case). We note that the sampling rates at time points t > 1 are intrinsically identifiable (e.g. Haslett et al. 2006), owing to the additional information coming from the previous state conveyed by the underlying Markov structure.
As can be seen from the model specification, Ni,t and qt are parameters, and mt and JPi,t are hyper-parameters, mt and JPi,t being at a higher level in the hierarchy than Ni,t and qt. A full Bayesian specification of the model requires priors to be explicitly specified on all independent parameters.
The model was fitted with relatively vague and independent priors on the fitted parameters (mt, qt, λi,1, JPi,t). We used Uni(0, 0·5) on mt which allows the abundance of immigrants up to half of the entire community size, and Exp(0·01)on λi,1 which is positive and typically ‘flat’ far from zero, to allow for large initial values. A non-informative approach to expressing ignorance about the relative species abundances of the regional species pool is to consider that all species are equally abundant in there. Under the equilibrium perspective, this would suggest setting all JPi,t at the average species abundance in the (local) community. We placed the non-informative Uni(5, 1000) on all JPi,t and assigned Gam(3, 6) to qt. Indeed the expected value and the variance of Gam(3, 6) match those of Uni(0, 1) whose support corresponds to the expected range of this parameter under neutrality.