## Metacommunity zero-sum multinomial distribution

### Description

Density, distribution, quantile function and random generation for Alonso & McKane's mZSM distribution with parameter `theta`.

### Usage

```dmzsm(x, J, theta, log = FALSE)
pmzsm(q, J, theta, lower.tail=TRUE, log.p=FALSE)
qmzsm(p, J, theta, lower.tail = TRUE, log.p = FALSE)
rmzsm(n, J, theta)
```

### Arguments

 `x` vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundance of species in a sample. `q` vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundance of species in a sample. `n` number of random values to return. `p` vector of probabilities. `J` positive integer 0 < J < Inf, sample size. In the context of species abundance distributions, usually the number of individuals in a sample. `theta` positive real theta > 0; Hubbell's ‘fundamental biodiversity number’ `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

The metacommunity Zero-sum multinomial distribution (mZSM) describes the probabilities of abundances (population sizes) in a random sample of size J taken from a collection of populations (the metacommunity). The total number of individuals in the metacommunity is fixed (zero-sum assumption). The populations in the metacommunity undergo a stochastic birth-death-immigration process, with equal demographic rates (neutrality or ecological equivalence assumption, Hubbell 2001). Alonso and McKane (2004) proposed an approximation for the density function for a large Poisson sample (J>100):

p(x) = N(x)/(sum(from 1 to S) N(x))

where S is the number of populations in the sample, and N(x) is the expected number of sampled populations of size x :

N(x) = theta/x (1 - x/J)^(theta -1)

Therefore, the mZSM is a model for species abundances distributions (SAD) in a sample taken from a community under the assumptions that (a) species abundances in the community follows the stationary distribution of a neutral, zero-sum stochastic process of birth, death and speciation (or migration); (b) sampling is a Poisson process with expected value well approximated by N(x), (c) individuals are sampled with replacement, or the fraction of total individuals sampled is small enough to approximate a sample with replacement.

### Value

`dmzsm` gives the (log) density function, `pmzsm` gives the (log) distribution function, and `qmzsm` gives the quantile function.

Invalid values for parameters `J` or `theta` will result in return values `NaN`, with a warning.

### References

Alonso, D. and McKane, A.J. 2004. Sampling Hubbell's neutral model of biodiversity. Ecology Letters 7:901-910.

Hubbell, S.P. 2001. The Unified Neutral Theory of Biodiversity. Princeton University Press.

`fitmzsm` for maximum likelihood estimation; `alonso.eqn12` in package untb which is based on the exact formulation of mZSM.

### Examples

```## Alonso & McKanne (2004) figure 2
data(moths) #Fisher's moths data
m.tab <- hist(moths, breaks = 2^(0:12), plot = FALSE)
plot(m.tab\$density~m.tab\$mids, log="xy",
xlab = "Abundance", ylab = "Probability density",
ylim=c(1e-7,1))
X <- 1:max(moths)
Y <- dmzsm(X, J = sum(moths), theta = 39.8)
lines(Y ~ X)
```