## Predicted number of species by Fisher's Logseries

### Description

Given a vector of species abundances, Fisher's alpha and total number of species and individuals in a sample, returns the number of species for each abundance value expected by the Fisher's logseries model

### Usage

```pred.logser(x, alpha, J, S)
```

### Arguments

 `x` Vector of (non-negative integer) abundances of species in a sample. `alpha` Fisher's alpha, the single parameter of log-series. `J` Total number of individuals in the sample. `S` Total number of species in the sample.

### Details

The Fisher logseries is a limiting case of the Negative Binomial where the dispersion parameter of the negative binomial tends to zero. It was originally proposed by Fisher (1943) to relate the expected number of species in a sample from a biological community to the sample size as:

S = alpha * log(1 + J/alpha)

Where alpha is the single parameter of the logseries distribution, often used as a diversity index. From this relation follows that the expected number of species with x individuals in the sample is

S(x) = alpha*X^x/x

Where X is a function of alpha and J, that tends to one as the sample size J increases:

X = J / (alpha + J)

Since the logseries model is a function that relates S to J using a single parameter (alpha), once two of these quantities are known the remaining is determined. So the function allows the input of any two among S, J and alpha. If the user does not provide at least two of these values, an error message is returned.

This function returns the expected number of species with abundance x, which is

E[S(x)] = x^(-1)*alpha*X^x

### Value

A (vector) of expected number of species to each abundance provided by argument `x`

### References

Pielou, E.C. 1977. Mathematical Ecology. New York: John Wiley and Sons.

Fisher, R.A, Corbert, A.S. and Williams, C.B. 1943. The Relation between the number of species and the number of individuals in a random sample of an animal population. The Journal of Animal Ecology, 12(1): 42–58.

`dls` for the log-series distribution; and `fitls`, `fishers.alpha` in package untb and `fisherfit` in package vegan for fitting the log-series to abundance data.
```data(moths) # Willians' moth data