# volkov

##### Expected frequency of species

Given a community size, biodiversity parameter \(\theta\), and an immigration rate \(m\), returns the expected frequency of species with \(n\) individuals, for \(0<n\leq J\).

- Keywords
- math

##### Usage

`volkov(J, params, bins = FALSE, give = FALSE)`

##### Arguments

- J
Size of community

- params
A two-element vector with first element interpreted as theta, the Fundamental biodiversity parameter and the second, m, interpreted as the probability of immigration. This argument will accept the output of

`optimal.params()`

- bins
Boolean, with default

`FALSE`

meaning to return the expected number of species with \(1,2,\ldots,J\) individuals, and`FALSE`

meaning to return the binned total, using a Preston-like binning system as used in`preston()`

- give
Boolean, with

`TRUE`

meaning to return*all*the output of`integrate()`

, and default`FALSE`

meaning to return just the value of the integral

##### Value

Returns an object of class “phi”.

##### Note

The method used is slightly inefficient: the terms to the left of the integral sign [in Volkov's equation 7] are integrated and this is, strictly, unnecessary as it is not a function of \(y\). However, taking advantage of this fact results in messy code.

##### References

I. Volkov and others 2003. “Neutral theory and relative species
abundance in ecology”. *Nature*, volume 424, number 28.

##### See Also

##### Examples

```
# NOT RUN {
volkov(J=21457,c(theta=47.226, m=0.1)) # Example in figure 1
# }
# NOT RUN {
volkov(J=20,params=c(theta=1,m=0.4))
data(butterflies)
r <- plot(preston(butterflies,n=9,orig=TRUE))
# }
# NOT RUN {
jj <- optimal.params(butterflies)
# }
# NOT RUN {
# needs PARI/GP
jj <- c(9.99980936124759, 0.991791987473506)
points(r,volkov(no.of.ind(butterflies), jj, bins=TRUE),type="b")
# }
```

*Documentation reproduced from package untb, version 1.7-4, License: GPL*