E.S: Functions for estimating Hubbell's theta from species - area...

In hubbell: Functions for Hubbell's Grand Unified Theory

Description

Functions for expected species number and its variance at certain sample size. These functions can be used in non-linear regression to estimate Hubbell's ultimate diversity number θ.

Usage

E.S(theta, J)
Var.S(theta, J)
E.Somega(theta, omega, J)
wtheta.fun(S, J, theta)

Arguments

theta	Hubbell's ultimate diversity number θ.
J	Community size J.
omega	A descriptor for community isolation ω.
S	Expected species richness S.

Details

Hubbell predicts a specific form for species area curve, so that the expected number of species in a metacommunity is dependent only on number of individuals J_M and Hubbell's ultimate diversity number θ. Thus the species–area curve can be used to estimate that ultimate θ. Further, Hubbell outlines analysis for local community.

The expected species richness in Hubbell's model is approximately S = 1 + θ ln(1 + (J-1)/theta). However, function E.S does not use this approximation but exact result from Hubbell's species generator (see rhubbell). Function Var.S similarly gives the exact estimator of variance of expected species number from Hubbell's species generator. Function E.Somega uses the species generator with isolation parameter omega >= 0 for a local community; no variance estimator is available.

Function wtheta.fun combines E.S and Var.S for variance weighted least squares fitting. It returns the standardized residuals, or prediction errors divided by the standard error.

These functions together can be used for non-linear least squares regression with nls.

Value

Functions return the specified value each.

Note

Hubbell gives only an approximate equation for expected species number. The exact form and its variance were derived by J. Oksanen and may be unpublished.

Hubbell's model for local community is based on wrong argumentation: Hubbell assumes falsely that his model is asymptotically similar to the classical Arrhenius model. Even with that model, the argumentation in deriving ω is invalid. Moreover, ω and θ are so strongly correlated (positively) that their estimates are very unreliable.

Author(s)

Jari Oksanen

References

Hubbell, S.P. (2001). The Unified Neutral Theory of Biodiversity and Biogeography. Princeton Univ. Press.

See Also

nls for non-linear regression.

Examples

## Use Carabid data. First get the total count per pitfall (J) and species
## richness (S). 
##
data(carabid)
J <- apply(carabid, 1, sum)
S <- apply(carabid>0, 1, sum)
plot(J, S)
##
## Non-linear least squares
##
sparea <- nls(S ~ E.S(theta,J), start=list(theta=10))
sparea
i <- order(J)  ## Order on x axis for nice lines
lines(J[i], fitted(sparea)[i], col="blue", lwd=2)
##
## Then variance weighted least squares.
##
sparea.w <- nls(~ wtheta.fun(S, J, theta),start=list(theta=coef(sparea)))
sparea.w
## We used one sided model formula, and so the fitted values would be
## actually the weighted residuals.  Therefore we need to use E.S to
## plot fitted values.
lines(J[i], E.S(coef(sparea.w), J[i]), col="red", lwd=2)
##
## Now with dispersal limitation omega.
## This takes time: so increase iteration limit and ask to
## trace iteration, so that you believe someting is happening
##
sparea.om <- nls(S ~ E.Somega(theta,omega,J), start=list(theta=4, omega=0),
   control=nls.control(maxiter=500), trace=1)
sparea.om
lines(J[i], fitted(sparea.om)[i], lwd=2)
##
## Some traditional alternatives
##
arrhenius <- lm(log(S) ~ log(J))
lines(J[i], exp(fitted(arrhenius))[i])
# Doubling J brings along constant number of new species:
logadd <- lm(S ~ log(J))
lines(J[i], fitted(logadd)[i], lty=2)

hubbell documentation built on May 2, 2019, 5:40 p.m.